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$: Eindhoven University of Technology MASTER An … · EINDHOVEN UNIVERSITY OF TECHNOLOGY FACULTY OF ELECTRICAL ENGINEERING TELECOl\fMUNICATIONS DIVISION An optinlized radiOllleter antenna:$
Eindhoven University of Technology

MASTER

An optimized radiometer antenna : theory and design

Wittekamp, J.W.

Award date:1992

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

FACULTY OF ELECTRICAL ENGINEERING

TELECOl\fMUNICATIONS DIVISION

An optinlized radiOllleter antenna:theory and design

by J.W. Wittekamp

Report of graduation work,performed from June 1991 to June 1992.

ProfessorSupervisors

prof. dr. ir. G. Brussaardir. P.J.I. de Maagt, dr. ir. M.H.A.J. Herben

The faculty of Electrical Engineering of Eindhoven University of Technology disclaims all responsibility

for the contents of training and graduation reports.

Summary

A microwave radiometer is an instrument to measure noise power, and basically consistsof a very sensitive receiver and an antenna. A vast variety of applications of microwaveradiometers exists, ranging from gathering information on structure and behaviour of theatmosphere, to investigating the surface of the earth for geological purposes. The antennais an important system part of a microwave radiometer. The requirements to radiometer antennas differ profoundly from the requirements to communication antennas. For astandard ground-based communication antenna a high gain is the prime design objective,while the shape of the complete pattern is important to a radiometer antenna.

In the Telecommunications Division of the Eindhoven University of Technology a research project concerning a microwave radiometer system is executed. One of the objectivesis to determine an optimum antenna design.

By gathering information from radiometer system users and manufacturers, a representative set of requirements has been obtained, which leads to the selection of the mostimportant radiometer antenna system design parameters: beam efficiency, beamwidth andthe integrated pattern. The design parameters have been accurately related to the performance of the antenna system, which made it possible to determine target values on basisof the requirements gathered.

An existing antenna synthesis method has been extended, allowing simultaneous optimization of important antenna parameters. Furthermore, it is possible to impose constraints on the integrated pattern. By including the target values in the optimizationprocedure, an optimum theoretical antenna system has been determined. The optimumsystem can be used as a reference: an engineer designing an practical antenna system isable to see what could be achieved theoretically.

Two practical antenna systems have been modelled: a symmetrical front-fed paraboloidal reflector antenna and an offset paraboloidal reflector antenna. Important practicallimitations have been included in the models, in order to obtain a realistic impression ofantenna performance. The results are compared to the theoretical optimum, to see whetherfurther improvements should be made.

On basis of the selected radiometer antenna design parameters and their relation toantenna performance, a design procedure is presented, containing a step-by-step designmethod to determine an optimum antenna system for a given application.

1

Acknowledgements

The work described in this report could not have been performed without the help of manypeople. In the first place, I would like to thank my coaches Peter de Maagt and Mat Herbenfor their tremendous help and support, as well as for the regular PC-transports. Furthermore, I would like to thank Professor G. Brussaard for his interest and advices. Additionally, I would like to thank Drs. M. Mathiesen, W. Kummer, M. Pitsiladis, R.S. Orton,J.B. Snider, G. Doro, and D. Thompson for the correspondence on radiometer antennaspecifications. Among the many people in our division who offered help I would like tomention Gunnard Franssen for calculating the clear sky brightness temperature profiles,Rens Baggen for his help on Turbo Pascal, and Gerry van Dooren for his help on Matlaband Ib\TEX, In addition, I would like to thank my parents who made it possible for meto do this study and who have supported me continuously. Finally, I would like to thankMarjoke, who had to face the disappearance of yet another weekend.

2

Contents

List of symbols

1 General Introduction

5

8

2 Requirements to and performance of a radiometer antenna 102.1 Introduction............ 102.2 Radiometer applications . . . . . . . . . . . . . . . . . . 13

2.2.1 Prediction of attenuation. . . . . . . . . . . . . . 132.2.2 Water-vapour and liquid water contents retrieval. 15

2.3 Brightness temperature of the surroundings of the antenna 162.3.1 Brightness temperature of the sky. . . 162.3.2 Brightness temperature of the ground. 18

2.4 Pattern limitations . . . . . . . . . . . . . . . 212.4.1 Pattern model. . . . . . . . . . . . . . 222.4.2 Uncertainty in the brightness temperature observed in directions to

the sky " 222.4.3 Uncertainty in the brightness temperature observed in directions to

the ground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 252.4.4 Error due to finite beamwidth . . . . . . . . . . . . . . . . . . . .. 262.4.5 Calculations of the uncertainty in the brightness temperature observed 28

2.5 Reflector, feed and waveguide limitations . . . . . . . . . . . . . 392.5.1 Noise temperature at the input of the receiver . . . . . . 392.5.2 Uncertainty introduced by reflector, feed and waveguide. 41

3 Design of an optimum antenna3.1 Introduction .3.2 The antenna parameters . . . . . . . .3.3 Aperture illumination source function.3.4 The optimization procedure . . . . . .

3.4.1 Theorem .3.4.2 Optimizing a product of quadratic forms3.4.3 Optimizing a sum of quadratic forms

3.5 The optimizing procedure with constraints

3

434344485050515152

4

3.6 Integrated pattern and moments .3.7 Combinations of parameters .

3.7.1 Introduction... ....3.7.2 Weighted optimization .

3.8 The antenna synthesis program

Contents

5356565659

4 An optimum theoretical radiometer antenna 604.1 Introduction............................ 604.2 Optimum beam efficiency, beamwidth, and aperture efficiency 604.3 Optimum integrated pattern template. . . . . . . . . . . . . . 62

5 Practical radiometer antennas 66.5.1 Introduction........ 665.2 Configurations....... 665.3 Reduction of performance 68

5.3.1 Antenna efficiency 685.3.2 Beam efficiency . . 725.3.3 Other factors reducing antenna performance 72

5.4 Front-fed reflector. . . . . . . . . . . . . . . . . . . 765.4.1 Aperture, beam, and spillover efficiency and beamwidth . 765.4.2 Antenna efficiency, modified beam efficiency, and beamwidth 795.4.3 The complete pattern of the symmetrical paraboloidal reflector antenna 84

5.5 Offset reflector 845.5.1 Aperture, beam, and spillover efficiency and beamwidth. . . . . . 855.5.2 Antenna efficiency, modified beam efficiency and beamwidth . .. 905.5.3 The complete pattern of the offset paraboloidal reflector antenna. 90

5.6 A design procedure for radiometer antennas 90

6 Conclusions and recommendations 976.1 Conclusions . . . . 976.2 Recommendations.......... 98

A Calculation of antenna noise temperatureA.1 Coordinate transformation .A.2 Performing the integration of the brightness temperature

B Optimization of N parameters

C The antenna synthesis programC.1 Introduction .C.2 User Manual .C.3 Structure of the Antenna Synthesis Program

D Characteristics of Olympus Satellite and EDT ground station

103103105

108

110110110III

115

List of symbols

A Azimuth angle of antenna axis [degrees]; Matrixa Aperture radius [m]; Eigenvector [-]an Excitation coefficients [-]B MatrixC MatrixD Antenna diameter [m]; MatrixE Elevation angle of antenna axis [degrees]; Scalar far field pattern;

Matrixen Set of elementary real functionsF Focal distance of the antenna [m]; Matrixf Frequency [GHz]; Aperture illumination functionG Antenna pattern [-]; Matrix9 Antenna pattern [-]h Integrated pat tern function [-]; Height of a point in the atmosphere

[m]; Height of antenna above the ground [m]hn Normalized integrated pattern function [-]ho Ground level [m]L Loss (ohmic) [dB]mm The mth moment [-]N i Real part of compex refractivity at ground level ho [-]N j Real part of compex refractivity at height h [-]Pr Total power radiated by the aperture [W]Pt Total power radiated by the antenna [\V]PUmb Total power radiated in solid angle defined by Umb [W]P Antenna power patternPi Incident power of a wave [\V]pspec Specularly reflected power of a wave [\V]Pscat Scattered power of a wave [W]Qn Zernike polynomials [-]R Spherical coordinate, distance from antenna in far field [m]R~n Legendre polynomials [-]r Cylindrical coordinate, distance on antenna aperture [m]rE Standard earth's radius [m]s Pathlength between antenna and a point [m]

5

6 List of symbols

f3

€n

r1]

1]a

1]b

1]8

1]sur face

l]p

ofh F2

ToU

Umb

U pre

v pre

XyZ

Pathlength between antenna and a point in zenith direction [m]Brightness or physical temperature [K]Antenna noise temperature [K]Modified antenna noise temperature, entering the receiver [K]Atmospheric brightness temperature [K]Noise temperature due to correlated noise sources [K]Cosmic brightness temperature [K]Extraterrestrial brightness temperature [K]Galactic brightness temperature [K]Brightness temperature in directions to the ground [K]Physical temperature of the lowest layer of the. atmosphere [K]Maximum value of brightness temperature that could occur [K]Brightness temperature measured by the main beam [K]Receiver noise temperature [K]Brightness temperature of the atmosphere between the antenna anda reflection point [K]

Tsky Brightness temperature of the sky [K]T sur face Physical temperature of the ground [K]T true Actual brightness temperature along the main axis of the antenna

[K]Standard ambient temperature [K]Normalized angle, defining the antenna pattern [-]Normalized angle, defining the width of the main beam [-]Normalized angle, prescribing a point of the antenna pattern [-]Prescribed value of antenna pattern [-]Fraction of time in cumulative distribution [-J; Cartesian coordinateCartesian coordinateCartesian coordinateAttenuation coefficient [dB]; Angle limiting the area where a secantapproach is valid [degrees]Angle where the antenna is blocking radiation that would otherwisereflect and enter the antenna [degrees]Emissivity of the ground [-]Neumann factor [-]Reflection coefficient [-]Antenna efficiency [-]Aperture illumination efficiency [-]Beam efficiency [-]Spillover efficiency [-]Surface roughness efficiency [-]Polarization efficiency [-]Spherical coordinate defining the antenna pattern [degrees]Half-width of the first Fresnel zone [degreesJ

Sz

TTATa

Tatm

Tc

Tcos

Textra

Tga1

Tground

Tlayer

Tmax

Tmb

TR

Tr

List of symbols

8mb Half-width of the main beam [degrees]8pre Prescribes a point of antenna pattern [degrees]¢ Spherical coordinate defining the antenna pattern [degrees]¢' Cylindrical coordinate defining coordinates on the antenna aperture

[degrees]t/J Spherical coordinate, complementary elevation angle [degrees]n Solid angle [starradians]nmb Solid angle of the main beam [starradians]A Wavelength [m]; Eigenvalue [-](j Standard deviation of surface height [m]T Opacity [Np]TO Zenith opacity [Np]Tr Total attenuation between antenna and reiledion point [dB]p Reflection coefficient [-]

7

Chapter 1

General Introduction

Telecommunications via satellites is becoming more and more important in present-daysociety. As a result the need for communication capacity is growing, and higher frequencies have to be taken in use. At this moment the frequencies in use for civil purposes arebelow 20 GHz. Unfortunately frequencies in the range of 20 GHz and higher can be heavilyattenuated by atmospheric phenomena. Research is done to model the attenuation causedby the atmosphere at frequencies of 20 GHz and higher.

One way to measure attenuation is to make use of a stable satellite-based beacon. Thepower of the received signal can be measured at several ground-stations and under differentweather conditions. The information gathered can be used to obtain long-term statisticsof attenuation, which can serve as input for the design of new satellite communicationsystems. This method has one big disadvantage: one needs an expensive satellite.

A tool to research attenuation of propagating signals, without the help of a satellite, ispassive microwave radiometry. By means of a very sensitive receiver and an antenna,atmospheric noise is measured. By using the relation between atmospheric noise and attenuation, it is possible to gather information about the latter.

Predicting attenuation is just one application of a microwave radiometer. A huge range ofapplications is created by microwave remote sensing, which can be done actively and passively, satellite- and ground-based. Active systems are based on a radar approach: a strongsignal is transmitted and its reflections are received. As indicated above, passive systemsare based on measuring spontaneous emissions of noise. Applications of microwave remotesensing range from gathering information on structure and behaviour of the atmosphere,to investigating the surface of the earth for geological purposes.

An important part of a microwave radiometer system is the antenna. A standard groundbased telecommunication antenna, where a high gain is the prime design objective, requiresother design features than a radiometer antenna, where the shape of the complete antennapattern is important.

8

9

In the Telecommunications Division of the Eindhoven University of Technology a researchproject concerning a microwave radiometer antenna system is on-going. A synthesismethod has been developed to simultaneously optimize several antenna parameters [1].This method should be extended to be able to include parameters that more strictly prescribe the antenna pattern. By gathering information about the specific requirements fromradiometer system buyers and manufacturers, a representative set of requirements can beobtained. Combining the results of the evaluation of this set with the output of the synthesis procedure leads to an optimimum theoretical antenna design. Furthermore severalrealizable reflector antenna systems will be modelled for comparison with the theoreticalantenna. The models include a detailed study of the performance of the antenna systems,that will allow a reliable trade-off study leading to a proposal about the best antennasystem for radiometry purposes.

In this report the requirements that can be posed to the microwave radiometer antenna arediscussed in chapter 2. The optimization procedure is described in chapter 3. In chapter4 the requirements of chapter 2 and the optimization method of chapter 3 will be used todefine an optimum radiometer antenna system. In chapter 5 several practical antenna systems will be accurately modelled and compared to the ideal system, leading to a trade-off.Finally chapter 6 contains the conclusions and recommendations.

Chapter 2

Requirements to and performance ofa radiometer antenna

2.1 Introduction

The design of a microwave antenna requires a complex trade-off between many systemdesign parameters. Basic requirements, as the physical position and the application of theantenna, have a profound influence on the properties of the final design. The importance ofthe physical position of an antenna can easily be understood by considering both an antennain space and an antenna on earth. A spaceborn antenna requires a construction that cansuccessfully endure a rocket launch, but still has to be light and compact. Furthermore,large temperature gradients exist in space, mostly due to a partly sunlit structure, whichlead to extra design limitations. In contrast, the physical design of a ground-based antennais relatively simple: most constraints are set by the sensitivity to weather conditions astemperature and rain. In this report only ground-based systems will be dealt with.

Next to its physical position the application of an antenna stipulates many designfeatures as well. \Vhereas a communication antenna requires a high gain, the completepattern is of prime importance to a radiometer antenna. The latter constraint is oftenexpressed in terms of the integrated pattern or the fraction of the power that is receivedwithin a certain solid angle. As it is not always possible to make the communicationssystem approach compatible with the radiometer approach, antenna optimization fromthe radiometer point of view is necessary.

Since many desirable attributes of a system are mutually exclusive and can only beobtained at the expense of another, a trade-off is necessary. Since the number of relevantdesign parameters is large and the parameters are diverse, many different design approachesare feasible, which makes it difficult to find uniformity in design proposals for radiometer antennas. However, there is consensus about the fact that the most common designparameters are the beam efficiency 7]b and the integrated pattern function h. Althoughthe quantitative values of these parameters differ from proposal to proposal, their designgoals may be specified to be "as high as practically possible". In general, a trade-off might

10

2.1. Introduction 11

be influenced by for instance the knowledge that is already present within a company orinstitute, which leads to a design parameter that can be expressed as technical risk. Additionally, even the presence of specific equipment or tools may partly determine the resultsof a design study.

The basic objective of a radiometer antenna is to accurately measure a brightness temperature, as expressed by the equation

(2.1)

where TA is the antenna noise temperature, G(O, ¢J) is the antenna gain pattern, T(O, ¢J)is the brightness temperature observed and O,¢J are the polar coordinates of the far field,as defined in figure 2.1. It should be noted that a radiometer measures noise power,an incoherent signal with random polarization. Equation (2.1) clearly shows that thebrightness temperature contributes to the antenna noise temperature, from all directionsO,¢J. In the ideal case of a pencil beam antenna, the radiation pattern would consist of a

y

x

Figure 2.1: Antenna noise temperature and cOO1'dinate system

infinitesimal small main beam and no sidelobes; the antenna temperature, as far as patternlimitations are concerned, would correctly represent the brightness temperature observedalong the central axis of the antenna. Unfortunately the main beam of practical antennasystems is not infinitesimally small, and the sidelobes cannot be neglected. The latter canbe illustrated by the following academical example. The antenna in figure 2.2 observes aconstant brightness temperature of 3 K with the main beam, containing 99% of the total

12 Chapter 2. Requirements to and performance of a radiometer antenna

clear sky /

3K /

//

//

(

IJ----

Figure 2.2: Influence of antenna pattern on antenna noise temperature

received power, and observes a constant brightness temperature of 300 K with its sidelobes,containing 1% of the total received power. If no power is received from other directions,the resulting antenna noise temperature will be close to 6 K, leading to an error of 100%compared to the brightness temperature observed with the main lobe.

Since in most cases a radiometer antenna is applied to measure a brightness temperature in a certain direction, contributions from other directions are unwanted. Because thereal brightness temperature distribution of the sky is mostly unknown, the errors causedby the shape of the pattern are difficult to correct for and will result in a large uncertainty.Radiometer antenna performance is strongly determined by the uncertainty in the measured antenna noise temperature.

Next to the radiation pattern, system imperfections introduce errors in a measured brightness temperature. Possible sources are: reflection at feed/waveguide and waveguide/receiver transitions as well as ohmic losses in reflector, feed and waveguide. By careful measurement of losses, reflection coefficients and ambient temperatures, as well as accuratecalibration using a thermal load at known temperature, these errors can almost entirely becorrected for. Almost, because some uncertainty will remain due to the limited accuracyof the measurements and the calibration.

It should be kept in mind that in designing a radiometer, it is very important to matchthe performance of the various parts: a receiver with an accuracy of 0.01 K is useless, ifthe antenna noise temperature is determined with an accuracy of several K. Therefore,the critical system part dominating total system performance should be determined (ifpresent), and be used as a reference for other system pa.rts.

2.2. Radiometer applications 13

In this chapter radiometer antenna specifications will be discussed quantitatively, by considering the influence of two major error sources, illustrated schematically in figure 2.3:the radiation pattern (transition from Ts to TA ) and system imperfections (transition fromTA to Ta ). First of all different radiometer applications will be considered, leading to some

Figure 2.3: Antenna system configuration

basic requirements (section 2.2). In section 2.3 a distribution of the brightness temperature of the surroundings of the antenna is derived, which will be used in section 2.4 toestimate the uncertainty in TA due to the antenna pattern. The errors introduced by lossesin reflector, feed and waveguide are discussed in section 2.5.

2.2 Radiometer applications

Radiometer applications can be separated into three major sections: prediction of attenuation , retrieval of water-vapour and liquid water contents, and imaging. However, by usingsophisticated software-based reconstruction algorithms, which are based on the differencebetween the object observed and it's surroundings, satisfactory results in imaging can beobtained without severe constraints to the properties of the antenna [2]. Assuming environmental conditions vary at a slower rate than the objects observed, imaging does nothave to be considered in this report. The remaining two applications will be consideredseparately in sections 2.2.1 and 2.2.2.

2.2.1 Prediction of attenuation

Investigating the influence of atmospheric properties on electromagnetic waves propagating from and to satellites, is very important in present-day telecommunications. To satisfythe demand for more telecommunication-capacity via satellites, new frequency-bands have


to be considered. Since frequencies above 20 GHz can suffer from severe attenuation by(mainly) liquid water, water-vapour and oxygen, the influence of atmospheric constituentshas to be modelled. A project that is carried out in connection with this subject, isthe transmittance of beacon-signals from the Olympus satellite to earth, at accuratelyknown frequencies close to 12.5, 20 and 30 GHz. The European Olympus propagationexperiments are coordinated by the Olympus Propagation Experimenters Group (OPEX),with the assistance of the European Space Agency (ESA). In several places of Europe,ground-stations are continuously receiving the beacon-signals leading to data on e.g. attenuation and cross-polarization. One of these ground-stations is located at the EindhovenUniversity of Technology. Although much of the calculations and conclusions in this report are generally applicable to radiometer antennas, some expressions are evaluated usingthe characteristics of the Olympus satellite and the Eindhoven University of Technologyground-station (appendix D).

Next to direct measurements of beacon-signals, radiometer measurements are carriedout, which give the opportunity to relate the output of the radiometer receiver to the"beacon-data". Detailed knowledge on the accuracy and reliability of the radiometer, willmake it possible to do propagation experiments at any frequency without the need forsatellite beacon signals.

To be able to compare the radiometer-data to the beacon-data, two issues have to beconsidered: firstly, the radiometer antenna has to point exactly in the direction of thesatellite transmitting the beacon-signal and secondly, since the beacon receiver antennareceives power within half of the first Fresnel-zone, this should also be the volume "seen"by the radiometer antenna.

The first requirement must be met by the mechanical pointing mechanism: the smalleststep with which elevation and azimuth angles can be varied, as well as the accuracy of theread-out of these angles.

The half-width of the first Fresnel-zone, OlF has been calculated in [3] and is summarized in table 2.1. The relative size of the r~flector of the radiometer antenna has been

Table 2.1: Half-width of first H-esnel-zone and relative aperture size of radiometer antenna

f [GHz] O~F [degrees] D/>..12.5 0.14 50020.0 0.11 63630.0 0.09 778

given in this table too, calculated with the relation 03dB = 70· >../ D, which is actually validfor reflectors wi th an edge taper of 10 dB (a higher edge taper will result in even largerreflectors). It is clear that very large reflectors are required, to obtain the beamwidth thatcorresponds to half of the first Fresnel-zone of the beacon receiver antenna. A larger radiometer antenna beamwidth could result in errors, since atmospheric phenomena could be

2.2. Radiometer applications 15

included that do not influence the beacon-signal (for instance a heavy rain-cell, just outsidehalf of the first Fresnel-zone). In fact this assumption is not completely true, since even phenomena outside the first Fresnel zone might influence the reception of the beacon receiverantenna. However, in general the radiometer antenna beam spans several Fresnel-zones,and radiometer measurements will be more strongly correlated to atmospheric phenomenathan beacon measurements.

To limit the cost of the radiometer antenna, it is likely that a smaller antenna will beused, and the beamwidth will be larger than required. The resulting error is dependenton the scale on which atmospheric constituents vary. Even in case of a stratified atmosphere and with the antenna pointing at zenith an error will occur, which is explained insection 2.4.4.

2.2.2 Water-vapour and liquid water contents retrieval

Although atmospheric attenuation is caused mainly by water-vapour and liquid water,retrieval of the latter two constituents themselves (e.g. for meteorological purposes), posesentirely different requirements to a radiometer antenna. Since as a function of frequencyclear peaks can be observed in the brightness temperature of the sky, which are due tooxygen and water-vapour, often two (or more) operating frequencies are used to be able tomeasure the brightness temperatures caused by each atmospheric constituent separately.The general problem in a multi-frequency system is, that the same sky volumes shouldbe observed at all frequencies. A multi-frequency system can be realized in two ways:using a single reflector/feed combination for each frequency, or using one reflector/feedcombination for all frequencies. The first solution will introduce difficulties in co-aligningthe beams of the antennas, as well as increased cost because of the presence of moreexpensive components (reflectors). With the second solution it must be ensured that thebeamwidth is equal for all frequencies. Furthermore the phase center of the feed is frequencydependent and a compromise position will have to be found.

Apart from the requirements associated with a multi-frequency system, some requirements are of a general nature. The width of the main beam of the radiometer antennadetermines the spatial resolution. Spatial and time resolution (the latter depends on theintegration time of the receiver) define the "radiometric resolution" [4]. The brightnesstemperature profile of the sky is averaged over the main beam. The degree up to whichaveraging is permitted, is dependent on the prescribed resolution of the radiometer, andon the scale on which atmospheric constituents vary [2]. Dynamic atmospheric processesrange from very small-scale and rapid physical processes up to very large-scale and slowlyvarying processes. A generally adopted subdivision has been made in [5], where it is foundthat to detect the smallest-scale processes (turbulence), the spatial resolution has to be ina range from several meters to 20 meters. In [6],[7, pp342] satisfying measurements wereperformed with a resolution of nearly 10 m. The characteristics of eddies in boundary layercumuli were investigated by another group of meteorologists [8], who obtained good resultswith a resolution of 5-10 m [7, pp426].

From these considerations it can be concluded that a spatial resolution of 5 to 20 m


would be required. Assuming clouds are at a minimum height of 500 m and a resolutionof 5 m leads to a maximum beamwidth of 1.1°. In [9] marine stratocumulus structure hasbeen investigated resulting in a histogram of fair weather cumulus cloud areas, where thesmallest is about 1.6· 102 m. This corresponds to a resolution of 46 m, assuming a circulararea.

2.3 Brightness temperature of the surroundings ofthe antenna

As indicated in section 2.1 it is very important to have knowledge of the spatial brightness temperature distribution of the sky and the ground in order to interpret the resultsof radiometer measurements correctly. In this section distributions of these brightnesstemperatures will be derived.

2.3.1 Brightness temperature of the sky

The contribution of the brightness temperature of the sky, Tsky('ljJ) (figure 2.1) to theantenna noise temperature depends on two sources: the extraterrestrial noise temperatureTextra and the atmospheric noise temperature Tatm [4]

T

- Tatm('ljJ) +Textrae-TTO

cos ('ljJ)

(2.2)

(2.3)

where TO (in [NpJ) is the total zenith opacity, which is dependent on water-vapour densityand frequency, and is a measure for the"optical thickness" of the atmosphere [4]. Thezenith opacity varies from 0.006 to 0.06 Np from frequencies from 15 up to 35 GHz andsurface water-vapour densities between 0 and 20 gjm3

• The extraterrestrial noise consistsof galactic and cosmic noise. The galactic noise, which originates from our own galaxy,can be neglected for frequencies above 10 GHz [4]. This is confirmed by the CCIR recommendations [10], where the "radio sky" is plotted for 10=408 MHz. The galactic noisetemperature Tga1 at any frequency Ii can be computed using

(2.4)

Since the highest galactic noise temperat ure (in directions of discrete sources as stars) isabout 500 K at 408 MHz and 5° angular resolution, and Ii will be exceeding 10 GHz theresulting brightness temperature will be smaller than 0.1 K. Since the antenna has to pointexactly in these directions and assuming the antenna beamwidth will not be much smallerthan 5°, this noise contribution can be neglected. The cosmic noise, due to the continuousexpansion of the universe, amounts to a frequency-independent value of 2.7 K [10].

2.3. Brightness temperature of the surroundings of the antenna 17

(2.5)

The atmospheric brightness temperature has been calculated as a function of the complementary elevation angle 1/J using the radiative transfer equation given by [11]:

Tatm = l°O T (s)a'(s)e-Jo'Ct'(B')dB'ds

with T(s) in [K] the physical temperature of a point at a distance s (in [km]) from theantenna. The attenuation coefficients d(s) (in [Np/km]) are computed up to a height of30 km using the 1989 version of the Millimeter-wave Propagation Model of Liebe [11]. Theatmosphere has been assumed to be stratified and homogeneous. Furthermore, the CCIRtemperature profile (which is identical to the one in the U.S Standard Atmosphere) andpressure profile have been used, as well as the following input data: a ground temperatureof 15°C, a ground level barometric pressure of 1013.25 hPa (both CCIR standard), anexponentional water-vapour profile with 2 km scale height and a water-vapour density atground level of 7.5 g/m3

.

Evaluation of (2.5) using the models mentioned above leads to a zenith brightnesstemperature. To calculate the brightness temperature in other directions a secant law isapplied to distance s:

szs = cos(1f;) (2.6)

where 1f; is the complementary elevation angle and Sz is the distance in the zenith direction.For low elevation angles (up to 10°) the secant law approach is not valid and the effects

of refraction and the curvature of the earth have to be included. This can be accomplishedby calculating s according to [12]:

(2.7)

where N j and N j are the real parts of the complex refractivity at ground level ho and atheight h respectively, and r e = 6357 km is a standard earth radius.

Using the input data above, the atmospheric noise temperatures have been calculatedfor the following six frequencies: 12.5, 20, 30, 10, 18, and 32 GHz, where the first threeare approximately the Olympus satellite beacon frequencies. The results are shown infigure 2.4. The brightness temperature curves are consistent with data presented in [13],which are used in the CCIR models [10]. However, a slight offset can be observed. This isprobably due to the differences in the models applied. In [13] millimeter-wave propagationhas been modeled according to Liebe's models in 1980, 1981 and 1982, while the morerecent Liebe-model in [11] has been applied in calculating figure 2.4.

For higher elevation angles the data has been fitted with the function Tatm = (a/ cos 1/J)+b, which is often used in approximations of atmospheric brightness temperature distributions [4, 14, 15]. The brightness temperature at low elevation angles (up to a = 4° or 2°,dependent on the best fit) has been fitted with a straight line Tatm = c1f; + d [15]. Thefit was accomplished by minimizing the mean square error. The resulting coefficients aregiven in table 2.2. As a result the brightness temperature of the sky is given by


9080706050403020

-+-.+- 1=1 B GHz. olpho=4 degrees

____ f=12.5 GHz. olpho=2 degrees

f= 10 GHz. olpho=2 degrees

..........,;.. ,

10

300,--------.----,------, ---r---,----,---.....-----,----l____ f=32 GHz. olpho=4 degrees _

f=30 GHz. olpho=4 degrees

f=20 GHz. olpho=4 degrees~ 250

II>L.

'"0L.

200II>a.E~.,.,OIl 150c:~

'"'c.J:l

U 100'COIl

.s::.a..,0

E500

00

complementary elevation angle [degrees]

Figure 2.4: Atmosphe7'ic brightness temperatures at 10, 12.5, 18, 20, 30 and 32 GHz

Table 2.2: Coefficients for brightness temperatw'e distributions at 10, 12.5, 18, 20, 30, and32 GHz

coeffi- f [GHz]cients 10 12.5 18 20 30 32a 2.438 2.796 6.838 11.298 10.238 10.468b 0.893 1.802 1.924 4.768 4.071 4.314c 27.761 26.048 32.268 26.734 28.227 27.350d -41.043 -38.367 -46.690 -37.217 -39.736 -38.357

T (7/J) - { a/ cos(¢) + b+ Textrae-T 7/J ~ 0'

sky - c¢ +d + Textrae-T 0' ~ 7/J ~ 11' /2 (2.8)

2.3.2 Brightness temperature of the ground

Besides radiation from the sky that enters the antenna directly, some radiation enters theantenna after reflection at the ground, or by emission from the ground, as depicted in figure2.5. Radiation incident on the surface of the earth (Pi) will be partly specularly reflected(Pspec), partly scattered (Pscat), while the remaining part will be absorbed, causing emissionof radiation [10]. Defining a reflection coefficient p, where

Pspec + pscatp=

Pi(2.9)

2.3. Brightness temperature of the surroundings of the antenna 19

h

IIIII

(1 -l)T(n-1/J) I--n -1/JIIII

Tlayer

Figure 2.5: Reflections and emissivity, increasing antenna noise temperature

the emissivity can be expressed asf=l-p (2.10)

The emitted and reflected radiation that enters the antenna will be attenuated while propagating through the atmosphere from the reflection point to the antenna. With a totalattenuation of T r the total attenuation will be e-Tr

• Furthermore, the atmosphere betweenthe reflection point and the antenna will add to the brightness temperature observed, acontribution Tr ( tP). By applying the radiative transfer approach of section 2.3.1, T r andTr ( tP) can be calculated:

r T - f/ Otds'dJo layero:e 0 S

- TlayerO: for e-OtSds

Tlayer (1 - e -Otr) [1<]T r - o:r [Np] (2.11)

where r is the distance between the antenna and the reflection point, 0: is the attenuationcoefficient in [Npjkm], s is the distance along the path from the antenna to the reflectionpoint, and where the physical temperature Tlayer of the lowest layer of the atmosphere is17°C. The attenuation coefficient 0: is listed in table 2.3 for six frequencies.

The brightness temperature of the sky is an incoherent, unpolarized signal. However, ingeneral reflections at the surface of the earth are dependent on polarization, as expressedby the Fresnel reflection coefficients. Since Fresnel reflection coefficients are only valid fora "smooth" surface, they should be corrected for a "slightly rough" surface. The roughnessof a surface is expressed in terms of the wavelength of the incident wave and the angle of


Table 2.3: Attenuation coefficients [Np/km] in the lowest air layer for 10, 12.5, 18, 20, 30,and 32 GHz

frequency Q

10 0.003232599512.5 0.004462605318 0.013288098020 0.024829330930 0.021496276332 0.0212940502

incidence. Often, the Rayleigh criterion or the more stringent Fraunhofer criterion [4] isused to determine whether a surface is rough. A surface is rough if:

>.(J' > --

a cos(~)(2.12)

with (J' the standard deviation of surface height and ~ the zenith angle of incidence onearth and where a = 8 for the Rayleigh and a = 32 for the Fraunhofer criterion. It isevident from (2.12) that for ~ close to 7r /2 (very small grazing angles) the fraction willbecome very large. Physically this means that the surface appears to be more smooth.However, as shown in [16] about 90% of radiation from the ground that enters the antennacomes from a circular area around the antenna with a radius of 20 m, assuming an antennaheight of 2 m. The grazing angle of incidence of radiation reflecting at the border of the20 m area is 5.7°. Therefore the Fraunhofer criterion has been calculated for ~ = 0° and~ = 85° at three different frequencies (Olympus) as displayed in table 2.4. In [17] three

Table 2.4: Fraunhofer c1'iterion for f=12.5, 20, 30 GHz and ~ = 0°,85°

f [GHz] ..\ [m] (J' [mm] ~ [degrees]12.5 0.024 0.75 0

8.61 8520 0.015 0.47 0

5.38 8530 0.010 0.31 0

3.59 8.5

examples of surface roughness are given: (J' is 0.88, 2.6 and 4.3 cm for smoothed, diskedand ploughed earth respectively, which can be considered good representatives of surfacesunder practical antenna sites. The standard deviation of surface height of the area aroundthe radiometer site at the Eindhoven University of Technology ground station is about

2.4. Pattern limitations 21

5 cm (short grass) [18]. It is clear that even for the lowest frequency and a small grazingangle the Fraunhofer criterion is satisfied for all (7, so that the surface can be consideredto be rough.

As mentioned above, the Fresnel reflection coefficients should be corrected for roughground. The effects of rough ground on the Fresnel reflection coefficients can be modeledaccording to [19]. However, as shown in [19, p 246] the corrected reflection coefficient dropsrapidly to zero for increasing roughness. So, for increasing roughness of the ground thereflection coefficient goes to zero and the emissivity is close to 1 and becomes polarizationindependent [4, pp.825-826]. It should be noted, however, that the reflection coefficientsincrease with growing moisture content of the ground [17]. But, since rough ground is lesssensitive to moisture content than smooth ground [17] an average emissivity f of 0.9 (andconsequently an average reflection coefficient for both polarizations of 0.1) will be assumed.In section 2.4 an additional uncertainty in the average emissivity and reflection coefficientswill be introduced to account for sensitivity to varying environmental conditions.

For 1/J close to 7r the antenna is blocking radiation that would otherwise reflect atthe ground and enter the antenna. In this case the brightness temperature becomes justETsurface. The angle where this effect starts to occur is denoted as f3 (figure 2.5).

Summarizing, the total brightness temperature that can be observed in directions tothe ground is given by

(2.13)

2.4 Pattern limitations

Due to the fact that an antenna receives power from every direction (see (2.1)) a largedifference between the antenna noise temperature and the brightness temperature observedalong the axis of the antenna can result, as has been illustrated in figure 2.2. In thissection the magnitude of the error due to the antenna pattern will be discussed. The mostimportant factors that influence the pattern error will be included: beamwidth, beamefficiency, integrated pattern and the (statistical) behaviour of the brightness temperatureof the surroundings of the antenna.

In section 2.4.1 an integrated pattern template is presented, which will function as amould for an ideal radiometer antenna integrated pattern. In the subsequent sections 2.4.2and 2.4.3 equations for the uncertainty in directions to the sky and directions to the groundrespectively, will be determined. In section 2.4.5 calculations will be performed on, andcurves will be presented for, the total cumulative uncertainty in the brightness temperatureobserved, as a function of the integrated pattern template, the elevation angle, and thecumulative uncertainty of the brightness temperature in directions to the ground and thesky. Finally, in section 2.4.4 the errors due to the finite size of the main beam will bediscussed.


2.4.1 Pattern model

Radiometer antenna performance is often described by the integrated pattern: the fractionof the total received power within a certain solid angle. When the antenna pattern is takenconstant in a number of regions, it will be possible to calculate the gain function G(O, ¢)in that region easily, using the prescribed fraction of total received power in the region:

WithG(O, ¢) = p(O, ¢)

Pt/41r

where Pt is the total radiated power, hi becomes:

which yields for a region i where G(O,¢) has the constant value Gi

1hi = 2"Gi( cos 01 - cos ( 2 )

It should be noted that the following relation can always be satisfied:

17< JG(O, ¢) sin OdOd¢ = 41r

(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

since the values specified by (2.17) are relative, and the gain can be varied. Furthermore,the integrated pattern can be normalized to one: Li hi = 1. In figure 2.6 the relationbetween hand G is illustrated. Here an integrated pattern template is presented. Thecorresponding gain pattern is an abstract form of a real antenna pattern, containing themain lobe (I), near-in sidelobes (II), spillover lobe (III), far-out sidelobes (IV) and backlobe(V).

2.4.2 Uncertainty in the brightness temperature observed indirections to the sky

The antenna noise temperature can be written as

(2.19)

assuming the pattern is divided into constant parts as introduced in section 2.4.1. Thebrightness temperature distribution has been determined in section 2.3 for a stratified

2.4. Pattern limitations

1- - - - - - - - ----- -- - -- - ----- - ---- -- - - --- -- --- ---- - -- - - - --- - -- - - - -- ----7'

_._--------23

iD~

I'"

oo .•

I II II I

v

IV

III

".-- -- -- ---------'---'

180

IIII

II

IIIIII•IIIIIIIII

r----

" III IV v

-60 -------- -~_.-,-------'-----'

Figure 2.6: Gain pattern and integrated pattern

'clear sky' atmosphere. In general terms TA represents the brightness temperature alongthe main axis of the antenna.

For a given brightness temperature profile, the antenna noise temperature TA can becalculated, while the true brightness temperature Ttrue along the main axis of the antennais given by eq. (2.8) or (2.13). The resulting error is the difference between these brightness temperatures. In this theoretical exercise the brightness temperature distribution isassumed to be known and the brightness temperature observed along the main axis ofthe antenna as well as the error caused by the brightness temperature observed in otherdirections are readily found. However, in practice the brightness temperature distributionis not known exactly and it should be considered what the brightness temperature mightbe. In other words, the uncertainty of the brightness temperature in a certain direction6T( ljJ) has to be calculated. In directions to the sky this can be achieved, by using cumulative distributions of the brightness temperature of the sky, which indicate the value ofTsky that is not exceeded for a certain fraction of time. The difference between the valuesgiven by the cumulative distributions Tcum (ljJ) and the clear sky brightness temperature


distributions Tsky( 'ljJ) derived in section 2.3.1 is the uncertainty:

(2.20)

The total uncertainty in the antenna noise temperature tlTA can be calculated by inserting(2.20) in (2.19). The physical importance of tlTA is, that it defines the limits of thedifference between the brightness temperature along the main axis of the antenna Ttrue

and it's representation TA • The total uncertainty in the antenna noise temperature isgiven by:

(2.21 )

It should be noted that even though rain has a profound influence on brightness temperature, it occurs during less than about 5 - 8% of the time, while clouds may be present fora far larger fraction of time. Therefore, this section will concentrate on the calculation ofthe uncertainty introduced by the presence of clouds. By considering basic meteorologicalinformation as the rain rate, in some cases a different brightness temperature distributionmodel can be used. For instance, in case of rain on site, which can easily be measured,this would result in assuming the presence of a homogeneous rain cell around the antenna.A second calculation of uncertainty can be performed to account for the presence of ahomogeneous rain cell on site.

The uncertainty in the atmospheric brightness temperature observed will be calculated using the results presented in [20], since similar data was not readily available at the E.U.T.groundstation. In [20] atmospheric noise temperature statistics are given for 15 climatologically distinct regions. Data is given in the form of cumulative distributions, which can beinterpreted as: for a fraction X of the time the brightness temperature at zenith is belowTmax • The zenith brightness temperature is extrapolated for use in other directions byapplying secant and linear relations as in section 2.3. The"average" year for each regionin [20] was chosen on basis of rainfall measurements. Using rainfall statistics to determinethe average cloud year has not been completely founded in [20] and may introduce errors.The curves presented for Saint Cloud, Minnesota have been chosen for use in this report,since the average rainfall per year is close to the Eindhoven thirty year average [21]. Someresults for f = 32 GHz from the Saint Cloud curves are presented in table 2.5. Thecumulative brightness temperature for 35 % of time corresponds to the clear sky value inzenith direction Tsky(O) in figure 2.4, since the latter is based on CCIR standard values forclear sky conditions, which occur about 35% of the time. Therefore, the difference betweenTcum and Tsky can be both positive (X > 0.35) and negative (X < 0.35). Before applyingthe cumulative distribution in (2.21) it will be transformed into a distribution giving totaluncertainty for values relative to Tsky '

It should be noted that the cumulative data in [20] do not include extra-terrestrialbrightness temperature, so that this contribution is also excluded from the clear sky brightness temperature distribution.


Table 2.5: Total year cumulative distributions of zenith atmospheric brightness temperaturefor Saint Cloud, ~Minnesota, f = 32 GH z

X T cum X% Tcum X% T cum

0.0 9.3 0.35 13.9 0.70 26.40.5 9.7 0.40 15.0 0.75 27.60.10 10.0 0.45 16.8 0.80 28.90.15 10.6 0.50 18.3 0.85 32.10.20 11.2 0.55 20.6 0.90 40.30.25 12.2 0.60 22.9 0.95 50.60.30 13.1 0.65 24.6

2.4.3 Uncertainty in the brightness temperature observed indirections to the ground

(2.22)

(2.23)

where b..Tsky is given by (2.20), which leads to

b..Tground( 1/J) (1 - t)e-Tr

b..Tsky ++ (Tsurface - T sky (1I" -1/J))e-

Trb..t) +

+ Ze -Trb..Tsur face ++ [-(1 - t)Tskll (1I" -1/J) - tTsurface + Tlayer] e-

Trb..Tr +

+ (1 - e-Tr )b..Tlaller

Next to the uncertainty in atmospheric noise temperature, additional uncertainty existsin radiation from the ground. In section 2.3 the emission and reflection coefficients havebeen estimated, but it has also been indicated that variations may occur. The uncertaintyof the brightness temperature observed in directions to the ground can be estimated fromthe first order derivative of (2.13)

{

OTground b..T (11" _ 01,) + OTground b..t + OTground b..T +OT.ky sky 'f/ Of oT.ur/ace sur face

b..T (01.) - +OTground b..-r + .5Tground b..Tground 'f/ - OTr r OT'ayer layer

OTground b..T + OTground b..-oT.ur/ace surface 6( t

for 11" /2 ~ 1/J ~ (3 and

b..Tground( 1/J) - (1 - e-Tr )b..Tlayer

[tTsur/ace - Tlayer] e-Tr

b..Tr

+ Tsur/acee-Trb..Z

+ te-Tr

b..Tsur face (2.24)

for (3 ~ 1/J ~ 11". It should be noted that although b..Tsky > 0 (since the cumulative values forT sky will always be larger than the clear sky value), b..Z, b..Tr , b..Tsur/ace, and b..Tlayer, can be


both positive and negative. Furthermore, since the uncertainties from all contributions areuncorrelated, they will be added 'root-sum-squared' (rss). All parameters and uncertaintiesof (2.23) and (2.24) are listed in table 2.6.

Table 2.6: Values of parameters and uncertainties that contribute to l::i.Tground

symbol value uncertainty unitTsky ( 'l/J) (2.8) (2.20) [K]Tsur/ace 290 10 [K]€ 0.9 0.1 [-]0' table 2.3 0.2·0' [Np/km]Tr 0' . r 0.20" r [Np]Trayer 290 10 [K]

2.4.4 Error due to finite beamwidth

Before the influence of a different allocation of power in the pattern template will bediscussed in the next sections, in this section the error due to the finite dimensions ofthe main beam will be calculated. The knowledge obtained will be used to define thebeamwidth in the pattern template applied in section 2.4.5 In literature the error dueto the finite dimensions of the mainbeam is sometimes described with the term elevationerror.

As can be seen from (2.1) the antenna noise temperature is an integral operation,weighting the brightness temperature with the antenna pattern in all directions 0, <p. Upto now the cumulative uncertainty in the brightness temperature observed has been calculated, assuming the brightness temperature observed with the main beam represents thewanted signal. However, even in case all power would be allocated to the main beam, anerror will result. This is illustrated in figure 2.7 where an antenna beam is depicted, observing the brightness temperature of a stratified clear sky atmosphere. The -3 dB pointsin the antenna beam with an elevation angle slightly higher than that of the main axis,are observing brightness temperatures lower than the brightness temperature in the maindirection, while the corresponding -3 dB points with a lower elevation angle are observinghigher brightness temperatures. Therefore, averaging the brightness temperature distribution over the main beam, will result in brightness temperature higher than the temperaturethat would be observed with a infinitesimal small beam. It should be noted that the -3 dBpoints are only introduced, to illustrate the influence of the width of the main beam. Inthe remainder of this section the half-width to the first zero of the main beam is meantwhen the beamwidth is mentioned.

The contribution of the main beam to the antenna noise temperature is given by

TAmb = -'!"'J f T(O,<p)G(O,<p)sinOdOd<p (2.2.5)471" 10m b


-3dB

stro. tifledo.tMosphere

Figure 2.7: Main beam observing stratified atmosphere

27

where Omb is the solid angle of the main beam. The brightness temperature observed withthe main beam Tmb, can be reconstructed from TAmb by applying

TAmb = Tmb2-J [ G(B,¢»sinBdBd¢>471" 10mb

and Tmb can be calculated with

JJo G(O, ¢»T(O, ¢» sin OdOd¢>Tmb = _=m,,-:b----'--------'---

JJOmb G(0, ¢» sin OdOd¢>

(2.26)

(2.27)

The brightness temperature distribution T(B, ¢» used is based the 90% zenith-value ofthe cumulative distribution, which has been determined in section 2.4.2. The main beamof the antenna pattern can be modelled adequately [16] by

(2.28)

where the half-width off the main beam Omb can be specified by evaluating aBmb = 3.84,the position of the first zero in the function J1 (aO) / aO.

Using the definitions above for the antenna pattern and the brightness temperaturedistribution, Tmb can be calculated according to (2.27). The error due to finite beamwidth,relative to the brightness temperature observed along the main axis of the antenna (~Ttle)

can be expressed as:Tmb - TtTtle

TtTtle(2.29)


3 ~------r-----.---._,.-------r-----.----r----,---,-----,

9080

5

7

3

70

____ 9

60

beamwidth [degrees] :

50403020

,\,,

\ ,,,

10

-.-. _.--------_.......-.-._._._._.oL-----.2===~~~.::.--:..:..L-- -~--~--~--~--~--~--~-~--;;,:-- -=:--=-=.--=:---=-:=--~--d--o

2

0.5

1.5

2.5

InInI1lc::l:'"'C

.J:J

.S:

'"I1l

1:I1lII>

.J:Jo

elevation angle [degrees]

Figure 2.8: Error in brightness temperature due to the finite size of the main beam

which has been plotted as a function of E in figure 2.8 for half-beamwidths of 8mb is 10,

30, 50 and 70

• The effect of finite beamwidth decreases with increasing elevation angle, butis still noticeable at 900 elevation. To limit the error due to the dimensions of the mainbeam to 0.1 K, the beamwidth should be smaller than 30 for an elevation error of 300

• Itshould be noted that 'beamwidth' represents the half-width of the main beam anywherein this report, unless otherwise stated.

\Vhat remains is a discussion of how the error due to the finite size of the main beamshould interpreted. In essence the brightness temperature observed by the main beamdoes not reflect uncertainty, since that is the brightness temperature that the user wantsto measure. However, analogous to the uncertainty in the brightness temperature observedby the pattern outside the main beam, uncertainty exists about the spatial brightnesstemperature distribution in directions observed by the main beam. If a cloud remains inthe upper half of the main beam, the resulting average brightness temperature Tmb will betoo low, while if a cloud remains in the lower half of the main beam, the average brightnesstemperature will be too high. Therefore, the error due to the finite width of the main beamrepresents the absolute maximum value of possible errors, and can be seen as uncertainty.

2.4.5 Calculations of the uncertainty in the brightness temperature observed

The cumulative uncertainty in the brightness temperature observed will be calculated usingthe uncertainty in the brightness temperature profiles determined in section 2.3.1, expressed


in (2.20), (2.23) and (2.24), as well as the integrated pattern template of section 2.4.1. Thebrightness temperature observed in the main beam is assumed to be the wanted signal,while the difference between the actual brightness temperature and the clear-sky brightness temperature (see (2.20)) in any other direction, contributes to the total cumulativeuncertainty.

The cumulative brightness temperature distributions in [20] have not been given forthe Olympus frequencies 12.5, 20, and 30 GHz, but for 10, 18, 32, 44, and 90 GHz. Sinceclear-air and cloud attenuations behave differently with frequency, data cannot be simplyinter- or extrapolated. However, the brightness temperature at 32 GHz is very close to thebrightness temperature at 30 and 20 GHz [13], due to the dependence on frequency. To aless extent the same holds for the brightness temperature at 10 and 12.5 GHz. As a largenumber of curves have to be presented to show the effects of all parameters involved, onefrequency (32 GHz) has been chosen.

Care has to be exercised in evaluating (2.1), since the antenna pattern is normallydescribed in a coordinate system in which the z-axis is perpendicular to the aperture ofthe antenna, while the brightness temperature is expressed in a coordinate system with thez-axis in the direction of zenith (2.1). Therefore, in appendix A the functions describingthe brightness temperature profile, will be transformed into functions described in thecoordinates of the antenna pattern.

In figures 2.9 to 2.25 the influence of the 5 different parts of the (integrated) patterntemplate will be investigated subsequently, starting with the spillover lobe where normallya major part of the power outside the main beam is concentrated. In figure 2.9 threetemplates are depicted, with the 400 wide spillover lobe shifting from a position at 900

,

via 1100 to a position at 1300• The positions correspond to the spillover lobe in a single

reflector system with an increasing F ID ratio. In all cases it has been assumed the mainlobe contains 90 % percent of the power, while the remaining 10 % has been located in thespillover lobe. The main beamwidth is specified to be 50. It should be noted, that assuminga beam efficiency of 90% does not imply a premature choice. The uncertainty caused by thepattern outside the main beam is proportional to the fraction of power outside the mainbeam. So, if the beam efficiency is raised from 90 to 99%, total uncertainty is decreasedby a factor ten.

11'll SPI 10% ~P 2 10% SP 3 10%

r- r-~---[----,-~.,90% I·· !. [··1

l________ J__L _. __. -"-:_-----'1_o 5 90 1'0 130 150 170 180

Figure 2.9: Three pattern templates with spillover lobes at different post ions


5 ,------.----,-----.----.-------.-------,.-_-,-__,__-,-_-----,

,/,/

,,,,-'------_ ......

E = 30 degrees, 1/b = 0.9SP 1SP 2SP 3

------, "':::::-.--- ---=="

.---';::--'_.--.--.--.-:::::.:::-:.~

---

4.5

4

0.5

0.90.80.7o '---------'----------i__--'-----_-----'- ----'---__---'-__--'-----_---'--__'---_--l

o 0.1 0.2 0.3 0.4 0.5 0.6

Fraction of Time X

Figure 2.10: Cumulative uncertainty in brightness temperature observed, as a function ofthe position of the spillover lobe, E = 30°

E = 50 degrees, T/IJ = 0.9

SP 1----- SP 2_.- SP 3

5

L5

4

~3.5

.D 3E-<c: 2.5>,....,c: 2

'@....,.... 1.5Q)uc:~

0.5

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

--

0.9

Fraction of Time X

Figure 2.11: Cumulative uncertainty in brightness temperature observed, as a function ofthe position of the spillover lobe, E = 50°

2.4. Pattern limitatiol1s 31

5r----,---.,---.--.,---.--.,-----,----.------,-----,

4.5

4 E = 70 degrees, TJb = 0.9SP 1SP 2SP 3

0.5

0.9O.B0.70.60.50.40.30.20.1

OL-_---'-__L..-_----'-__"'---_----'-__L..-_--'-__"'---_----'-_-.J

oFraction of Time X

Figure 2.12: Cumulative uncertainty in brightness temperature observed, as a function ofthe position of the spillove1' lobe, E = 70°

:-:::,"::::::-=::::-::':=:':=:,':-::---:::==='::-:-:::-=::==:':=:":=:':::::-:'':-::

E = 90 degrees, TJb = 0.9

--------------------

SP 1SP 2SP 3

0.9O.B0.70.60.50.4

~.------.--------,I----.----,.---~-~

0.30.20.1

5

4.5

4

3.5

~

J:J 3E-<

c:: 2.5

».....,2c::

'@.....,I-< 1.5Q)uc::

:=>

0.5

00

Fraction of Time X

Figure 2.13: Cumulative uncertainty in brightness temperature observed, as a function ofthe position of the spillove1' lobe, E = 90°


It is evident that the uncertainty increases with increasing values of X. The differencesin uncertainty, and the sensitivity to changes in X, are mostly due to which part of thepattern is observing the ground, and which part of the pattern is observing the sky. Indetermining the best pattern, it should be considered which values of X are important.In most cases the uncertainty for X between 90 and 95 % should be considered, sinceradiometer performance should be guaranteed during all wheather conditions, except rain(the remaining fraction of time, performance should be evaluated using a different modelfor the uncertainty in the brightness temperature observed in all directions, as indicatedbefore in section 2.4.2). However, in some cases lower values of X could be relevant, forinstance when additional information about the brightness temperature distribution of thesky is present, or when measurements are performed during clear sky conditions. In theremainder of this section radiometer antenna performance will be evaluated for high valuesof X.

The influence of the elevation angle of the antenna can clearly be observed from figures 2.10to 2.13. The maximum uncertainty increases with decreasing elevation angle, and thepattern template providing the lowest uncertainty, changes from pattern SP 1 for E = 900

via pattern SP 2 for E = 700 to pattern SP 3 for E = 500 and E = 300• In the case

that E = 900 (figure 2.13), the spillover lobe is observing the ground for all three patterntemplates. The result is a high uncertainty for low values of X, but since the sky has onlya minor effect on the uncertainty in directions to the ground, the uncertainty will remainfairly constant for increasing values of X. In pattern SP 1 of figure 2.13 a small slope canbe observed. This is due to the fact that this template is observing the part of the groundwhere reflecting sky noise has the highest value (at small grazing angle) and therefore aslight increase does occur for increasing X. However, total uncertainty is still lower thanin case computations are performed with patterns SP 2 and SP 3.

For E = 300 (figure 2.10) pattern 3 should be preferred, although for higher values ofX the differences are small.

The influence of the width of the main beam on the cumulative uncertainty has beeninvestigated. It was found an increment in the width does not much influence the cumulative uncertainty, unless the beamwidth is very large (> 150

) and the elevation angle ofthe antenna is very low. However, the beamwidth does result in an additional error, whichwill be discussed in section 2.4.4.

At this point the influence of the spillover lobe has been determined, and the pattern template will be extended with the far-out sidelobes. In figure 2.14 two templates are shown,both with the spillover lobe positioned at 900

• In the first template the far-out sidelobesare concentrated next to the spillover lobe from 1300 to 1500

, in the second template thefar-out sidelobes are positioned in the backward looking region of the antenna pattern(from 1600 to 1800

)' overlapping the back lobe. Again the power in the main beam istaken equal to 90%, the spillover lobe contains 6.7% of the power and the remaining 3.3%is located in the far-out sidelobes.

In figures 2.15 to 2.18 the cumulative uncertainty in the brightness temperature ob-


II11 90%

Il __

SP/F 1

- ---,,IIII

3.3~ :III

SP/F2

r--tr----..,I ". II ". II . I

I II I

: 3.3%:I II II I

33

o 5 90 130 150 160 180

Figure 2.14: Two pattern templates with far-out sidelobes at different positions

served has been plotted, for the elevation angles 30°,50°, 70°, and 90°. As can be observed,differences are small. Only in the case of figure 2.15, where E = 30° , pattern SP IF 1 yieldsa higher uncertainty than pattern SP IF 2, since part of the far-out sidelobes in this patternare observing the sky. In all other cases pattern SP IF 2 should be preferred.

To complete the investigations into the pattern template the near-in sidelobes will be included. The resulting templates are shown in figure 2.19. The near-in sidelobes are shiftedfrom 10° via 30° and 50° to 70°. The width of the near-in sidelobes is 20°. The spilloverlobe is again positioned at 90°, and the far-out sidelobes at 160°. The main beam contains 90% of the total received power, the spillover lobe 5%, while the far-out and near-insidelobes both contain 2.5%.

The result of the calculations have been plotted in figures 2.20 to 2.23, for the sameelevation angles as before. From figure 2.23 it can be directly observed that the sky indirections parallel to the surface of the earth contributes the most to the uncertainty.Therefore, for each elevation angle the pattern template that directs (part of) its power inthe near-in sidelobes along the surface of the earth, will result in the highest cumulativeuncertainty. For E = 50°, 70° and 90° pattern SP IF IN 1 yields the lowest cumulativeuncertainty, for an elevation angle as low as 30° pattern SP IF IN 3 or SP IF IN 4 shouldbe preferred.


5 ,--------,---.-------,------,.-------,------,.--------.---.-------,-----.,

0.90.8

,,,,,,,,,........ ~-"",_ ...

0.7

------

0.60.50.40.30.2

E = 30 degrees, 7/b = 0.9- SP/F 1----- SP IF 2

----'-__-'__----'-__.L-_~I•• _ -----'----''--_-'--_---'__.....

4

4.5

'SZ3.5

.J:> 3~C 2.5>.

+-"c 2'@+-"I-<

1.5(1)uC~

o:L0 0.1

Fraction of Time X

Figure 2.15: Cumulative uncertainty in brightness temperature observed, as a function ofthe position of the far-out sidelobes, E = 30 0

,,

0.90.80.70.60.50.40.30.2

_ SP/F 1

----- SP IF 2

E = 50 degrees, 7/b = 0.9

0.1

,-------,---,----,-----,-----,---,-----,---r----,------,5

4.5

4

'SZ 3.5

.J:J 3E-<.S 2.5>.~

c 2'@-e(1) 1.5uC~

0.5

00

Fraction of Time X

Figure 2.16: Cumulative uncertainty in brightness temperature observed, as a function ofthe position of the far-out sidelobes, E = 500


5.....----.------,---.-------.---.------.---.-------.------,-------,

4.5

4

E = 70 degrees, 'lb = 0.9- SP/FI----- SP IF 2

g 3.5

b 3C

';:" 2,5

"'""C'@ 2"'""~~ 1.5C

:=>

0:[~_--'-----'-------'-o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0,8 0.9

Fraction of Time X

Figure 2.17: Cumulative uncertainty in brightness temperature observed, as a function ofthe position of the far-out sidelobes, E = 70 0

.---__--.--_------,-_-.--_----.-__-.--_---.-_---,~--.----_. -, -5

4.5 E = 90 degrees, TJb = 0.9

4SP/F 1

----- SP/F 2:z 3.5.r>

b 3c:::>.. 2.5

"'""c::'@ 2"'""....(l)u 1.5C

"""">-'

0.5

oo 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fraction of Time X

Figure 2.18: Cumulative uncertainty in b7'ightness temperature observed, as a function ofthe position of the far-out sidelobes, E = 90 0


fl'I'

90% SP/F/N 4

SP/F/N 3

SP/F/N 2

SP/F/N 1

Figure 2.19: Four pattern templates with near-in sidelobes at different positions

1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fraction of Time X

0.1oLI_----'- .L...-_-'-_----'-,-~----~

o

':f .~ ~ 30s~~~~~ ;b ~ 09

r SP /F/N 2~ -.>.51- SP/F/N 3

SP/F/N4~

E--< _\ .

.~ 2.5l

I':11

;:::, 1

0.5· ,.

Figure 2.20: Cumulative uncertainty in brightness temperature observed, as a Junction ojthe position oj the near-in sidelobes, E = 300


E = 50 degrees, T1b = 0.9

5

4.5

4

g 3.5

.03E-i

c::>. 2.5~c::'@ 21:Q) 1.5uc::

:::>

oj0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.B 0.9

Fraction of Time X

Figure 2.21: Cumulative uncertainty in brightness temperature observed, as a function ofthe position of the nea1'-in sidelobes, E = 500

5.----r----.--..,.-----r'--..-----,-------,,...--~-~-~

4.5

4

E = 70 degrees, J1b = 0.9SP/F/Nl ~~SP/F/N 2 ~/

__~.~::~>~:r~~::···········..... ,

0.5

0.90.4 0.5 0.6 0.7 0.8

Fraction of Time X0.2 0.30.1

o '-----_----'-_----J__--'--_--'--__-'-------_----'---__---'__--'---_--'-_----l

o

Figure 2.22: Cumulative uncertainty in brightness temperature observed, as a function ofthe position of the near-in sidelobes, E = 700


~-=-go'-d-eg-r-ee--rs-,-T/-b-="--O-.g-r---,-,-/-'-/="-

".;'

_ SP/F/N 1 /········SP/F/N2 '_.- SP/F/N 3----- SP IF /N _~_/", //

--,,--- / .....

----------- .'---'-- ------/- ..-.-.:~~.~:~ .. -

0.90.7 0.80.60.50.40.30.2o. ,o

~

2

3

5

~.5

2.5

t:S. 3.5

Fraction of Time X

Figure 2.23: Cumulative uncertainty in brightness temperature observed, as a function ofthe position of the near-in sidelobes, E = 90°

2.5. Reflector, feed and waveguide limitations 39

The strong influence of elevation angle on the optimum position of the near-in sidelobesis further illustrated in figures 2.24 and 2.25. Patterns SP IFIN 1 and SP IFIN 4, whichyield the highest differences in cumulative uncertainty, have been plotted as a function offour elevation angles. It is clear a different elevation angle also leads to a different optimumposition for the power in the near-in sidelobes.

5,....--..,...------r--,---..,...----r---,..-----.------r---,..-------.

SP /F/N 1, l1b = 0.9E=300E= 500E= 700E= 900

4.5

4

'5Z 3.5

~ 3c::

';:" 2.5~c::'@ 2~~ 1.5c::

:=>

..'......

.. '....

.,' .'........ /.~./.

"

.. ,;;...-:.:;;:.:.':;;:;:::-'-'-'-:..;.--- -------------

0.5

0.8 0.90.70.4 0.5 0.60.2 0.30.1OL....--...J..------'-----''---------'-------'-------'----'-------'------'-~

o

Fraction of Time X

Figure 2.24: Cumulative uncertainty in brightness temperature observed, as a function ofelevation angle, SPIFIN 1

2.5 Reflector, feed and waveguide limitations

Basically a microwave radiometer exists of two major elements: the antenna and the receiver. In section 2.4 the uncertainty in measured brightness temperature due to theantenna has been discussed. This section concentrates on the errors introduced by theohmic losses of reflector, feed and waveguide and the reflections that occur at transitionsbetween these components. Firstly, an expression for the noise temperature that enters thereceiver will be derived (section 2.5.1)' followed by an estimate of the contribution fromeach error source to the total uncertainty (section 2.5.2).

2.5.1 Noise temperature at the input of the receiver

A complete radiometer system is depicted in figure 2.26. Ohmic losses are introducedby the reflector (gr), the feed (9j) and the waveguide (gw), while reflections occur at thetransitions between reflector and feed (PI), feed and waveguide (PH) and waveguide and


5 ,--------.------.--.-------r--~-____._-~--__.__--__,_-_____.

4.5

4

g3.5b 3

~ ..- 2.5 r-~ !~

'@ 2...,......~ 1.5~

~

0.5

0.90.80.70.60.50.40.30.20.1

oL-_--'--_---l.__-'--_---'-__.l--_---'--_---' --'--_--'-_----J

o

Fraction of Time X

Figure 2.25: Cumulative uncertainty in brightness temperature observed, as a function ofelevation angle, SPIFIN 4

receiver (p II I). The ambient temperatures of the components are given by Tr , Tj, Tw forreflector, feed and waveguide respectively, while the noise temperature of the receiver isdenoted by TR .

The uncertainty introduced by the system described in figure 2.26 could be determinedequivalently to section 2.4 by taking the first order derivative of the equation that describesTA at reference plane 6. However, due to the large number of terms introduced by thevarious elements in the block diagram this would lead to a complicated equation, notgiving clear insight into the sensitivity to variation of a certain parameter. Therefore, thesimplified block diagram of figure 2.27 will be used.

The transition from TA to Tu can be described by the following equations:

(2.30)

where r is the assembled reflection coefficient, L is the total receiving system loss and Tois the ambient temperature. By applying an isolator at the transition between waveguideand receiver, the, in general very high, receiver noise temperature that otherwise could beobserved, becomes just the ambient temperature To.

In evaluating (2.30) it should be noted that the noise contribution from the system loss(1 -l/L)To and its reflected counterpart Tolrj2/L are correlated and cannot be added aspowers, but have to be added on an amplitude basis instead. This leads to an extra noisetemperature contribution, which is represented by T: in (2.30). However, as stated in [22,p.322-323] T: in general can be neglected and will be taken equal to zero.

2.5. Reflector, feed and waveguide limitations 41

TR+---

Pu H+.T. f-H P"

TA4 TAS

Figure 2.26: Complete block diagram of receiving system with ohmic losses and reflections

IIII L,ToTAI

TeAII.. J ..IIIJIII

Figure 2.27: Simplified block diagram with ohmic loss and reflection

2.5.2 Uncertainty introduced by reflector, feed and waveguide

In order to estimate the uncertainty introduced by reflector, feed and waveguide, the firstorder derivative of (2.30) is taken:

(2.31)


Table 2.7: Additional uncertainty introduced by reflector, feed and waveguide

contribution contributionsymbol unit value uncertainty for TA = 10 for TA = 100TA [K] - l:1TA 0.909· l:1TA 0.909· l:1TA

To [K] 290 10 0.91 0.91r [-] 0.02 0.002 0.001 0.005L [-] 1.1 0.002 0.46 0.30

(2.32)

Analogous to section 2.4.3 the contributions from the uncertainty in total loss ~L,

assembled reflection coefficient l:1r, ambient temperature l:1To in addition to the uncertainty in the input antenna noise temperature l:1TA will be added root sum squared. Theparameters, their values, as well as the uncertainty in their values, and the contributionsto the total uncertainty for two input noise temperatures TA are listed in table 2.7. Thevalues are believed to be a reasonal estimate of practical values [16].

It can be seen from this table the uncertainty due to the antenna pattern is slightlyreduced (factor 0.909) as it passes to the receiver. All contributions are relatively largefor a low input antenna noise temperature, but become relatively small for a high inputantenna noise temperature. In principal the contribution from ~To is the most important,but in practice it is possible to measure the ambient temperature quite accurately andthereby reduce the additional uncertainty to a lower level.

The uncertainty in the total loss L is smaller but more difficult to remove, since ingeneral it is hard to determine the loss accurately. The uncertainty l:1L = 0.002 correspondsto an accuracy of 0.01 dB, which is achievable in microwave measurements.

The influence of the uncertainty in the reflection coefficient is quite small, and hardlyaffects the performance.

In general it can be concluded the uncertainty introduced by reflector, feed and waveguideis much smaller than the uncertainty introduced by the shape of the antenna pattern.

Chapter 3

Design of an optimum antenna

3.1 Introduction

In chapter 2 it has been discussed which requirements should be imposed on a microwaveradiometer antenna, and which design parameters are important with regard to these requirements. In this chapter a synthesis technique will be presented, that gives the opportunity to optimize (combinations of) important design parameters, such as the apertureefficiency 1]a and the beam efficiency 1]b, with or without pattern structure constraints.Eventually this technique and the requirements of chapter 2 will be used to define an optimum antenna system.

The synthesis procedure described in [1] is based on an efficient technique to optimizeantenna parameters and differs from other synthesis procedures described in literature.Most of these procedures can deal with only one design objective or parameter. Sometimesconstraints in a specific form can be included. The more general synthesis technique ofde Maagt will be further extended in this chapter, so that it becomes possible to optimizethe integrated pattern function, by maximizing the fraction of power contained within acertain solid angle. Furthermore, restrictions can be made to the integrated pattern byprescribing the fraction of the power contained within a certain solid angle.

With the procedure described in [1] several antenna parameters can be optimized simultaneously with and without constraints. However, the antenna parameters have to be writtenin a form (section 3.2) suitable for evaluation with basic theorems from linear algebra (section 3.4), using the aperture illumination function described in section 3.3. In section 3.5it is explained how constraints can be included. In section 3.6 it will be explained in whichway the integrated pattern can be optimized with constraints. The differences between optimization of a sum and optimization of a product will be considered in section 3.7 Finally,in the last paragraph of this chapter the computer program containing the implementationof the synthesis procedure will be described.

43

44 Chapter 3. Design of an optimum antenna

3.2 The antenna parameters

Starting with a circular aperture in the x-y plane as shown in figure 3.1 the antennaparameters can be written in a form suitable for optimization. The aperture points are

z\

observationpoint

y

Figure 3.1: Coordinate systems

described with normalized polar coordinates (r, ¢/), while the far field observation point isindicated with the spherical coordinates (R, 0, </». The scalar far field pattern E(R, 0, </» isgiven by [23]

(3.1 )

where a small-angle approximation is used. The normalized integral part g(u, </» of the farfield pattern E(R,O,</» is related to the normalized aperture distribution f(r,</>') by:

r21r r1g( u, </» = a 2 J

oJo

fer, </>')ejurcos(t/J-t/J')rdrd</>' (3.2)

where:21ra . ° 1rD . ° ( )u=T sm =Tsm 3.3

When fer) is a </>'-independent uniform-phase aperture distribution, g(u) can be writtenas the first order Hankel transform:

(3.4)

with Jo is the Besselfunction of the first kind and zeroth order. An arbitrary phase aperture

3.2. The antenna parameters 45

(3.5)

distribution does not improve the optimization results in this report, as indicated in [24,p. 105]. If f(r) is written as:

f(r) = { EO;:'=0 anen(r) 0 ~ r ~ 1elsewhere

where an are the real excitation coefficients of the elementary real functions en(r), it ispossible to split the Hankel transform into:

N 1

g(u) = 21ra2Ean1en(r)Jo(ur)1'dr

N

- 21ra2L a.Jn( u)n=O

(3.6)

Now the equations for the power radiated by the aperture Pr , the power radiated withinthe main beam PumP the integrated pattern h, and the moments mm, are needed. Thepowers Pr and PUmb are given by:

21ra211 p(r )rdr

21r 1umb

p(u)udu

where:

(3.7)

(3.8)

p(u)

with Umb the first zero of g(u).Eq. (3.8) can be found by using

l(u)1rD . 0T sm mb

(3.9)

(3.10)

{BmbPBmb :::::: 21r J

op(O)OdO

which gives

(3.11)

(3.12)

which will yield the same result as (3.8). The presence of the constant expression (>"/1r D)2does not change the optimization result. In this report the power in the main beam willbe referred to as PUmb ' while the power in a larger part of the antenna pattern is given by


the integrated pattern function h(U pre ), with U pre denoting the normalized angle to whichthe power pattern is integrated:

(3.13)(Up••

h(upre ) = 211" io p(U )udu

Here the same small angle approximation is used as in (3.12).The normalized integrated pattern function hn has a maximum of 1 and can be found

by dividing the integrated pattern by the total radiated power Pr •

A problem might occur when u pre becomes that large that the assumption U ~ 11"D /,xis no longer valid. A second error could result from integrating the power pattern overthe entire space (411"), as reported by Zimmerman [25], because the surface currents aretruncated at the edge of the reflector. Numerical evaluation of the expressions (3.13) andt.he tota.l radiated power by the aperture as given in (3.7), which should lead to identicalresults, have shown that the discrepancies can be neglected. This is illustrated in figure3.2, where the error in (3.13) relative to (3.7) is plotted as a function of D /,x. Three caseshave been examined: optimization of TIn, Tlb and O.5T1a +O.5T1b. It is clear the relative error isvery small, even in case of low edge taper (optimization of etaa ), where the surface currentsare relatively large. Therefore, it is concluded that the error as reported in [25], is not ofsignificant magnitude concerning the calculations in this report.

100908070

-----

-- etaS

------- etoA

_.- 0.5etoA+0.5etoB

605040302010

.II,..

II,.

I\,

\\\\..

\,,,,,,,,, ,,,,

"'"

\\\\\\\

."".'--............---...- -----------------_.-._._._._._._.-OL----'_~===:;:==::::::i:::~=========do

0.02

0.005

~ 0.015t:I>

I>>:;::;ce 0.01

O/Iombda

Figure 3.2: Discrepancies between total radiated power as calculated with (3.7) and (3.13)

The mth moment of the far field radiated power with respect to the u=O axis is obtainedby integrating ump(u):

(3.14)

3.2. The antenna parameters 47

In section 3.6 it will be explained in which way the moments of the antenna power patterncan be used in the optimization process.

Now the following antenna parameters can be given:

the aperture efficiency

the beam efficiency

the normalized integrated pattern

2p(0)TJa =-

Pr

1 ( )_ h(upre )

~n U pre - Pr

(3.15)

(3.16)

(3.17)

The antenna parameters given above can be written in a form suitable for evaluation withbasic theorems from linear algebra by using equations (3.5) and (3.6). First f(r), g(u) andp(u) have to be written as given below:

f(r) = {!!OT. f. 0 ~ r ~ 1elsewhere

with:

(3.18)

(ao,al, ,aN)

(eo, el, , eN)

(3.19)

(3.20)

andg(u) = IT(e).!!

where IT(e) is an N + 1 element vector with elements:

(3.21 )

(i = 0, ..... , N) (3.22)

and:(3.23)

with Vij = I(ej)I(ej) the elements of an (N + 1) x (N + 1) element matrix V. The lineabove g(u) denotes the complex conjugate.

With help of these equations it is possible to write Pr ,PUmb , hupre and mm in a similar way:

(3.24)


where Aj are the elements of an (N + 1) x (N + 1) element matrix A.

(3.25)

where Xij are the elements of an (N + 1) x (N + 1) element matrix X.

where Yij are the elements of an (N + 1) x (N + 1) element matrix Y.

r D />.mm = Q.T];ymQ., ];Yij = io um+ll;ijdu

where W ij are the elements of an (N + 1) x (N + 1) element matrix W.

(3.26)

(3.27)

Now the basic antenna parameters can easily be written in a form suitable for the optimization procedure, namely as a ratio of two quadratic forms:

2Q.T V(0)g(3.28)TJa -

gTAQ.

Q.TXQ.(3.29)TJb -

Q.TAg

Q.T];ymQ.(3.30)n~m

Q.TAg

hngTyQ.

(3.31 )Q.T AQ.

For the theorem presented in the next paragraph it should be proven that the matrices arehermitian and positive definite. The matrices V,X,A,Y and Ware hermitian because [26]:

Zij = Zij with Z = V, A, X, ];y (3.32)

Furthermore the matrices A,X ,Y and Ware all positive definite because they representpowers.

3.3 Aperture illumination source function

In the preceding section antenna parameters have been written in a form suitable foroptimization by making use of a set of real functions en(r). Several source functions havebeen considered by de Maagt [1]. Among Bessel functions, Zernike polynomials and powerlaw functions, Zernike polynomials were found to give good results for any optimizationincluding TJb, which will cover most of the cases in this report.

3.3. Apel·ture illumination source function 49

The Zernike polynomials Qn can be reduced to Legendre polynomials ~2n by meansof the relation:

R~n(r) = Qn(2r2 - 1)

The elementary real functions en(r) are given by:

~T = (Rg(r), ~(r), ... , R~N(r))

Since 11R':(r)Jm(ur)rdr = (_I)t(n-m)Jn+~(U)

as defined in (3.22) the integrals In, as defined in (3.22), can be written as

I T ( , _ (J1(u) J3(u) J2N+l(U))_ e) - , , ... , ---'---'-u u u

The integral part of the far field pattern can be written as (using (3.21))

(3.33)

(3.34)

(3.35)

(3.36)

(3.37)

The elements of the matrices A,X and Y can be found by using (3.24),(3.25),(3.26) and(3.27). Since

11

R~i(r)R~j(r)dr = {

the elements of matrix A become

14i+2o

if i = jif i i- j

(3.38)

{

_1_ if i = jA .. - 4i+2

I) - 0 if i i- j

The elements of matrix X can be calculated by using

and

(3.39)

(3.40)

r l umb2 1 1 ~ 2Xij = J2i+1(U)-du = 2(,)' 1) L.J tnJ2i+l+n(Umb)

o u _Z + n=D

where t n is the Neumann factor defined as

t n = 1 for n = 0t n = 2 elsewhere

if i = j (3.41 )

(3.42)


(3.43)

The elements of matrix Yare calculated similarly to the elements of X, using u pre asdelimiter instead of Umbo

The pattern of a parabolic reflector antenna with uniform phase and amplitude distributionover the aperture is given by [23]:

()JI(U)

gUCX:--U

The first zero in this pattern is found at U = 3.84 [27]. Since the beamwidth is smallest incase of uniform aperture illumination, the delimiter of integral (3.40) is initially set to thisvalue: Umb = 3.84, which will be used throughout the report. However, after optimizationof an parameter the position of the first zero in that pattern will be determined and usedto calculate T]b correctly.

3.4 The optimization procedure

3.4.1 Theorem

The antenna parameters as given in (3.28), (3.29), (3.30) and (3.31) can be optimized bysolving an eigenvalue problem. Starting with the ratio:

(3.44)

with A and B hermitian and B positive definite, the maximum (or minimum) of the quantity is given by the largest (or smallest) eigenvalue of the generalized eigenvalue problemgiven by:

aTAaAg = >..BQ. with>. = -TB

Q. Q.(3.45)

The proof of this theorem can be found in [1, page 13,14]. The generalized eigenvalueproblem has some special properties that can be used in the optimization procedure. Someof them are mentioned below.

1. Because B is non-singular the equations can be reduced to the standard eigenvalueproblem:

(3.46)

2. Because A and B are both real and symmetric, and B is positive definite, the equations can be transformed into a suitable form using the Choleski decompositionB = LLT, with L a lower triangular matrix. Multiplying 3.45 with L -1 yields:

(3.47)

with>. the eigenvalues of the symmetric matrix L -1 AL-T, and LTQ. the correspondingeigenvectors.

3.4. The optimization procedure

3.4.2 Optimizing a product of quadratic forms

51

(3.49)

In case a product of antenna parameters has to be optimized, a special case occurs. Thefunction r(g.) in (3.44) can now be written as:

aTAa aTCarl(g.)r2(g) = -TB- -TD- (3.48)

g g,g g

The optimization of this function can be solved with the following expression,

[_1_ gTCg A+ _1_ gTAg C] a ==.A [_I_ B + _1_D] agT Bg gTDg gTDQ gTBg - gTBg gTDg -

which can be written as(3.50)

as proven in [1, p.19, p.72]. The latter expression is equivalent with the eigenvalue problemfound in 3.2.

The optimization is now done iteratively. Initially a suitable starting vector is calculated.For maximization the vector corresponding to the largest eigenvalue of the two ratiosof quadratic forms, appears to be a good choice. Then the matrices E and F can becalculated and the standard eigenvalue problem of (3.45) results. By solving this problema new eigenvector will be found which can be used in the next iteration, until the desiredaccuracy is reached.

3.4.3 Optimizing a sum of quadratic forms

A product is one way to combine two or more antenna parameters, another way is a sum.It appears that a sum of parameters can be written as a new quadratic form that can beevaluated in the same way as one parameter. The sum of two parameters, for example,leads to the following expression:

aTAa aTBarl (Q) + r2 (g) - =----= +=--=gTCg gTCg

gT(A + B)ggTCg

QTGg

gTCg

The last expression can be optimized using the theorem in 3.4.1, if G is hermitian.

(3.51 )

(3.52)

(3.53)

Since all matrices in the definition of the antenna parameters are real and since for allmatrices Zij = Zji, with i and j indicating row and column numbers of the matrix, thefollowing holds:

(3.54)

which makes the matrix hermitian.


3.5 The optimizing procedure with constraints

An important aspect of antenna synthesis is the possibility to add constraints to the optimization problem. vVith the algorithm as discussed in [1] it is possible to pose variousrequirements to the antenna pattern, and leaving a problem that can be solved. Addingconstraints to an optimization procedure usually results in a problem with more variables,but in this case a problem with less variables results. A constraint means that the antennapattern in one or more field-points Urn must satisfy prescribed values:

or, using (3.21)

IT (urn) . g = vpreIT(0) . g ::::}

(IT(u rn ) - vpreIT(O))' g = 0

(3.55)

(3.56)

where Vpre is the prescribed value in a point Urn relative to the value at U = O. In a shorterway this can be noted as:

aT. q = qT . a = 0- -m -m-

and the optimization problem with constraints becomes:

aTAaf(gJ = =-----= IQ.T. q = 0 (m = 1, ..... ,M with M < N + 1)

gTBQ. -m

(3.57)

(3.58)

Assuming V is the space spanned by q1 ...qM and V 1. (perpendicular V) is the space spannedby the vectors illM+1 ... WN+1, the vector Q. should be in V.L since Q.T ·!l.rn = O. This meansthat the number of variables is decreased by the number of constraints, leading to a reducedoptimization problem. Of course, this means that the optima that can be reached in aconstraint case will never be larger than in the same unconstrained case. Consequently thenumber of constraints should be taken as small as possible.

As mentioned above Q. should be in V.L, so:N+1

Q. = L 1lJjCj = lVf (3.59)M+1

where 1V is an (N + 1) x (N + 1 - AI) matrix, with columns formed by the vectors Wj'

and f is an N + 1 - Af vector. The original problem in (3.57) can now be written as:

f(c) = fT~VA1Vf = 0 (3.60)- f TWB1Vf

with WTAW and WTB1V (N + 1 - M) x (N + 1 - M) real square matrices.The problem that remains is finding a basis for V1.. This can be accomplished with help ofthe Householder transform [1]. The Householder transform reduces a (N + 1) x M matrixto an upper tridiagonal form with, as in figure 3.2, Q a (N + 1) x (N + 1) orthogonalmatrix and R a Af x Af upper tridiagonal matrix. SO Q2 forms a basis for V.L, becauseQ2' V = O. Matrix Q2 can be substituted for matrix W in (3.60).

3.6. Integrated pattern and moments

Q2 R AlN+ 1

x V -

Ql 0 N+1-M

N+1 Al M

Figure 3.3: The Householder transform

3.6 Integrated pattern and moments

53

As discussed in sections 3.2 it is possible to optimize the antenna parameters 7Ja, 7Jb andhn . Furthermore it is possible to impose requirements on the antenna gain pattern, whileoptimizing the parameters. Apart from constraining the antenna gain pattern by meansof a prescribed sidelobe level or a pattern zero, the fraction of the total power radiated bythe aperture, within a certain solid angle could be prescribed:

7f'Dhn(um) ~ vprehn(T) (3.61)

where Urn indicates a point of the integrated pattern and hn (7f'DjA) is the end value ofthe normalized integrated pattern, which is equal to one. Note that an under limit isgiven for the power in a fraction of the pattern, instead of a prescribed value that must beon the curve, as in (3.55). The reason to formulate constraints to the integrated patterndifferently, will be given in the remainder of this section.

The importance of the integrated pattern can be understood from figures 3.4a to d.Both patterns shown in 3.4a and b, have the same fraction of power in the main lobe,but the power in the sidelobes is distributed differently. Often the righthand figure ispreferred in radiometer applications, because the power is concentrated near the mainbeam, reducing the influence of interference by unwanted noise sources. In figure 3.4c andd the corresponding integrated patterns are plotted. The difference in the gain patternsof 3.4a and b is shown in the integrated patterns by a different inclination of the curve.A constraint to the integrated pattern corresponding to figure 3.4a, would force the gainpattern to keep the power closer to boresight.

A suitable expression for constrained optimization of the integrated pattern IS gIvenbelow:

or

hn(um) = V pre

V pre

(3.62)

54

o _~

Chapter 3. Design of an optimum antenna

o _

,,,

"" )0

I '"J '"')

..,o

(a)

;" J{l

I '"

."o

. y10 I~ ro

rtOf''''''''.Il .. ,,'lIh''

(b)

-; Ollllll

I"", u"I

I :::i

(! n",.

I "",,",

0

(c)

I Q 91

I 098

I 0"'! Olt4

10"co,

o

(d)

Figure 3.4: Gain patterns with equal Tjb, but different sidelobe distribution (a and b: farfield patterns; c and d cOr1'esponding integrated patterns)

aT . (yurn - V A)· a = 0_ pre_ (3.63)

The optimization problem can now be written as

aTYa=-----= laT. (yurn - V A)· a = 0 (3.64)g?Ag - pre_

It is not possible to solve this problem in a manner similar to §3.5. Instead it wouldbe necessary to turn to Lagrange multipliers, resulting in an increment of the number ofequations to solve. Because this is undesirable a different approach has been chosen.

Forcing the integrated pattern to reach a prescribed value at a given angle can be achievedby minimizing the power in the tail of the integrated pattern. This can be done by includingmoments in the optimization. It will be shown that the power in a region 8 = h to 8 = 7r

is limited by the mth moment:7TIm1- h (h) <-

n - hm(3.65)

3.6. Integrated pattern and moments 55

In probability theory the moments of a power density function (pdf) can completely describe the behaviour of the pdf. Since

Pn(f), ¢) sin(O) > 0 (0 ~ 0 ~ 71") (3.66)

121f l 1f

Pn(O, ¢) sin(O)dOd¢ 1 (3.67)

where Pn(O, ¢) = P((), ¢)/Pr , Pn((), ¢) sin(O) can be regarded as a probability density function and the integrated pattern as a cumulative probability density function [28]

hn(Opre) = 2121f l BprePn(O,¢)sin(O)dOd¢ (3.68)

assuming no power is radiated for 0 > 71"/2. When the gain pattern is taken to be¢-independent, hn becomes a marginal cumulative probability density function of themarginal pdf Pn ( 0) sin(0)

mm

[Bprehn(Opre) = 471" Jo Pn(0) sin(O)dO

The moments around the origin of the mpdf can now be defined as:

r/2mm = 471" Jo Ompn(O)sin(O)dO

Using the following derivation

r/2 [7r/2471" Jo om Pn(0) sin(O)dO ~ 471" Js om Pn(0) sin(O)dO

r/2

> 8m 471" Js Pn(O)sin(O)dO = 471"8mp(1 0 I~ 8) ~ 8mp(1 0 I~ 8)

it follows that

(3.69)

(3.70)

(3.71)

P(IOI~8)~1;: (3.72)

The probability P(I 0 I~ 8) represents the power in a region 0 = 8 to 0 = 71" and is equalto 1 - hn (8) which proves equation (3.65).

Equation (3.70) can be written in terms of u, analog to (3.8):

om = arcsinm (71"~/ )J ~ (71"~/ A) m

again assuming u ~ 7rf, the moments around the origin can be given as

(1 ) m+2 7rD/).

mm = 471" 7I"D/A 1 um+1pn(u)du

which can be normalized to (3.14).

(3.73)

(3.74)

The influence of the moments can be understood by considering the om term in the integrant. For greater 0 this term will become rapidly larger, forcing the optimization procedureto keep down Pn (()), as in figure 3.5. Constraints can be implemented by adding a momentto the optimization procedure, that can be increased until the constraints are satisfied.

109876

Cbapter 3. Design of an optimum antenna

5

m=l

\m=1.5

4

m=2

32

,-------,--------,---.------,--.,---.------.---..-----,-----,

56xl0-3

0.9

0.8

0.7EA

0.60.,2- 0.5;!...

0.,2- 0.4c

Cl.0.3

0.2

0.1

0 /0

theta

Figure 3.5: Effect of moments on antenna pattern

3.7 Combinations of parameters

3.7.1 Introduction

In the preceding sections it has been explained in which way antenna parameters can beoptimized. It is possible to optimize the aperture efficiency, the beam efficiency and theintegrated pattern at various angles (h-factor). When designing a system however it islikely that an engineer does not want to optimize just one parameter, but a combination ofparameters. Therefore in the antenna synthesis program the possibility has been includedto optimize a product or a sum of antenna parameters. In this chapter the significance ofsuch expressions will be discussed.

3.7.2 Weighted optimization

The existence of two methods to optimize a combination of parameters gives rise to a question: are the corresponding optima identical? This question can be answered intuitivelyand mathematically.

One of the properties of the optimization of a ratio of two quadratic forms using the eigenvalue method as discussed in section 3.4.1 is that the resulting optimum is absolute. Dueto the fact that the matrices A and B are hermitian and matrix B is positive definite thesolution of the eigenvalue problem is unique. Taking a closer look at the product and sumoptimization methods shows both techniques boil down to the same approach: productand sum are eventually written as a ratio of two quadratic forms. This means that theresults of those two optimizations should be identical. However when experimenting with

3.7. Combinations of parameters 57

the synthesis program the user will experience that for instance the expression 1Ja x 1Jbyields other optimal values for 1Ja and 1Jb as the optimization of 1Ja +1Jb. In the next sectionthis will be explained.

Before the question at the beginning of this paragraph can be answered mathematicallya new aspect has to be introduced. In optimization techniques weights are often used,because a designer might prefer one parameter above the other.

Now it is possible to prove that the results of a weighted sum and product optimizationare identical. The proof below is given for combinations of only two parameters for sakeof clarity, but as shown in appendix B this can easily be extended to N parameters.

A weighted sum B of two parameters a(x) and b(x) can be written as

with an extreme at

which yields

S = wa(x) +(1- w)b(x)

dB = 0dx

da dbw dx + (1 - w) dx = 0

da w - 1 db- ---dx w dx

(3.75)

(3.76)

(3.77)

A weighted product P of the same parameters a(x) and b(x) can be written as

with an extreme atdP-=0dx

which yields

These optima are identical if

(3.78)

(3.79)

(3.80)

A few possihIe cases are

w-1

w

a

a-1 a-- - ¢:}

a baw

b+ (a - b)w(3.81 )

58

(1) W = 0 =} Q = 0 =} { S = b( x )P = b(x)

{S = a(x)

(2) W = 1 =} Q = 1 =}P = a(x)

Chapter 3. Design of an optimum antenna

{S=~a(x)+~b(x)

(3) W = ~ ::} Q = asom/(asom + bsom )::::} P = aa/(a+b)b1-a/(a+b)

Obvious is the result for W = 0 and w = 1: sum and product are identical and bothhave the same optimum. When w = ~ a situation occurs which is representative for allw E (0,1). Starting with optimization of a sum a certain w yields two optimal values aand b. Using these a sum and bsum in a like in case 3 above yields a weight Q. Using thisweight when optimizing the weighted product will results in the same optima a and b. So,all optima found with sum optimization can also be found with product optimization andconversely.

In the antenna synthesis program it is easy to optimize a weighted sum as demonstratedin section 3.4.3. Weighted products are not implemented because of two reasons: firstlyfor each weight it should be proven that it is possible to write the resulting expression asa ratio of two quadratic forms and secondly this could only be achieved when the weightsare integers. Weighted sums can always be written as a ratio of two quadratic forms(section 3.4.3) on the condition that the denominators in the parameters to be combinedare identical. The weights can be any real number. However it is possible to check whetherthe results of sum and product optimization are identical for one simple case: Q = ~' Thiscase is identical to P = a x b, both parameters are taken to the first power. Calculatingthe optimum of P = aO' x lJ3 with Q = f3 = 1 yields:

P = aO'b{3 = a . b

using 3.80 yieldsda a db

----dx b dx

Since 3.83 should be equal to 3.77 the following holds

(3.82)

(3.83)

1- w

w

w

a

bb

a+b

(3.84)

(3.85)

Optimizing TJaTJb gives TJa = 0.9705 and TJb = 0.9121. Now w = Wt = 0.4849 and WITJa +(1 - Wt)TJb should be optimized as sum. The results of the latter optimization, TJa = 0.9705and TJb = 0.9122, are almost identical to the results of product optimization.

3.8. The antenna synthesis program

3.8 The antenna synthesis program

59

The algorithms discussed in this chapter have been implemented in a computer program.The program was written in Turbo Pascal, version 5.5, and is executable on any IBMcompatible personal computer. A user manual and a detailed description of the structureof the program have been included in appendix C.

Chapter 4

An optimum theoretical radiometerantenna

4.1 Introduction

As explained in chapter 2, a radiometer system must have special properties to meet theimposed requirements. This is certainly the case for the radiometer antenna. Unfortunately one cannot optimize a certain feature without deteriorating other features, as willbe illustrated in this chapter. The antenna synthesis method described in chapter 3 can beused to optimize an antenna for radiometry purposes, meeting the requirements of chapter 2. The result will be an aperture distribution with prescribed amplitude and uniformphase. This can be achieved theoretically, with a double shaped antenna system: onereflector to meet the uniform phase requirement and the other reflector to form the amplitude distribution. In practice it will be very difficult to manufacture the shaped reflectorswith sufficient accuracy. Therefore, the results of the optimization should be used as a reference: an engineer designing a certain antenna is able to see what could be achieved withan ideal antenna system. The performance of the (practical) system, which the engineer isdesigning, will of course be less than the performance of the optimum theoretical system.

The performance of an optimum antenna system will be described in this chapter.

4.2 Optimum beam efficiency, beamwidth, and apert ure efficiency

The requirements obtained in chapter 2 clearly indicate, that a high beam efficiency is veryimportant for a radiometer antenna design. In addition, the beamwidth should not be toolarge, to prevent a substantial error.

The aperture efficiency is a good indicator of changes in beamwidth and edge taper.A low aperture efficiency implies a high edge taper and a relatively wide main beam.Furthermore, including the aperture efficiency in the optimization procedure gives the

60

4.2. Optimum beam efficiency, beamwidth, and aperture efficiency 61

opportunity to investigate what could be achieved in case an antenna is applied in a dualpurpose system: communication and radiometry.

Before including other important design parameters and constraints, these parameterswill be discussed in order to get a first impression of the optimization process.

First of all, single parameter optimizations of 7]a and 7]b have been performed. The far fieldpatterns, resulting from optimizing 7]a and 7]b separately, are given in figure 4.1 (a) and (b).The corresponding aperture illumination functions are plotted in figure 4.1(c) and (d).

.""

yvyr10 I!)

(a)

'";: ",,§ ..,

..,o

·YYY~10 l~ '0

(b)

"o til 02 OJ 04 (l~ O. Ql ua U'oi

..a" I0 , ••• 1 ". ,

(c)

110 01 02 OJ 04 0," o. 01

(d)

o. 0 If

Figure 4.1: Single parameter optimization of 7]a and 7]b (a and b: Far field pattern afteroptimization of 7]a and 7]b; c and d: Aperture illumination functions after optimization of 7]a and 7]b)

Apart from single parameter optimization, also a combination of 7]a and 7]b can beoptimized. \\Then a weighted sum is used, viz. W7]a + (1 - W )7]b, it is possible to obtain aset of optimum combinations 7]a,7]b. For a certain 7]a no higher value for 7]b can be found.The meaning of a weighted sum has already been explained in section 3.7. The results of

62 Cllapter 4. An optimum theoretical radiometer antenna

optimizing the weighted sum of TJa and TJb are shown in figure 4.2. In figure 4.3 TJb is plotted

---~ -- ---.., ---.------,------.----...------------.-

0.98

0.96

0.94>-uc:.. 0.92'u-='Q;

E 0.90...0

0.88

0.86

0.84

0.820.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98

aperture efficiency

Figure 4.2: Combinations TJa,TJb after weighted optimization

as a function of the beamwidth. It is clear, that a higher beam efficiency is obtained atthe cost of a broader beam. By considering this curve, a radiometer antenna designer candetermine which combinations of TJb and beamwidth can be achieved.

4.3 Optimum integrated pattern template

In the previous section, it has been discussed in which part of the antenna pattern, powershould be located to minimize uncertainty. The most important conclusion that can bedrawn from chapter 2 is, that power outside the main lobe, in directions along the surfaceof the earth, contributes strongly to the total uncertainty in antenna noise temperature.Power outside the main lobe directed to the ground yields an uncertainty which is lower,but the lowest uncertainty is obtained if the remaining power is directed to the solid anglearound zenith. Initially, we will assume 90% of the power should be located in the mainbeam, but optimization will also be performed with the constraint that 99% of the poweris within the main beam. The latter case corresponds to a situation with an uncertaintywhich is ten times smaller than in the first case.

The beamwidth should be small, as indicated in section 2.4.4; a half-width of 2.50

guarantees an error smaller than 0.1 K for an elevation angle of 30 0•

Evaluating the restrictions to the antenna pattern for E = 30 0 yields, that spillover shouldbe in the region 1300 < 0 < 1700

• If no power is to be received in these regions, there theintegrated pattern template should not increase. The remaining power should be located

4.3. Optimum integrated pattern template----- -1-- -----0 .--, __-----,- .,.--_~_-__________,---

63

0.98

0.96

0.94>.uc:.,

0.92'u~

DE 0.90II.c

0.88

0.86

0.84

0.82 1

3.8 4 4.2 4.4 4.6 4.8 5 5.2

beomwidth in normolized ongle u

Figure 4.3: Combinations of beamwidth and T/b after weighted optimization

in the near-in sidelobes in the region 50° < () < 130°, or in the backward directions150° < () < 180°, as can be seen from figure 2.15 and 2.20. The differences in uncertaintybetween pattern 2 in figure 2.15 and patterns 3 and 4 in figure 2.20 are small, but since thepower in the near-in sidelobes can be located within a larger angle region (50° < () < 130°),this area is preferred to contain the majority of remaining power. The resulting optimumintegrated pattern template is depicted in figure 4.4. If optimization is performed with therequirement that 99% of the power should be located in the main beam, all constraints tothe remaining power in figure 4.4 should be changed accordingly.

A different template is obtained if an elevation angle of 90° is considered. Figure 2.23 clearlyillustrates the influence of power along the surface of the earth, since the uncertainty causedby pattern 4 is far greater than that of the other patterns. This is due to the fact that foran elevation angle of 90°, indeed all power in the near-in sidelobes of pattern 4 is receivedin the solid angle just above the surface of the earth, while for other elevation angles partof the power is received from other directions. In this case angles 0 in the antenna and t/Jin the zenithal coordinate systems are the same, so the antenna should be insensitive tonoise coming from the angle region 70° < () < 90°. Figure 2.18 shows that the position ofthe far out sidelobes has negligible effect on total uncertainty. As indicated by figure 2.13,the spillover lobe should be positioned from 900 to 130°. Comparison of the curve withlowest uncertainty due to spillover for X = 0.95 in figure 2.13, and the curve with lowestuncertainty due to the near-in sidelobes in figure 2.23, shows that differences are small,so it is chosen to locate the remaining power in the near-in sidelobes close to the mainbeam. The resulting optimum integrated pattern template is shown in figure 4.5. Thesame method as described in connection with figure 4.4 should be applied in figure 4.5 if

64 Chapter 4. An optimum theoretical radiometer antenna

0.9

o 2.5

0.9

50

0.99 0.99 1

130 150180

Figure 4.4: Optimum integrated pattern template, E = 30°, beam efficiency=90%

the main beam contains 99% of the power.

Using the integrated pattern templates derived above, the weighted sum of TJa and TJb hasbeen optimized. Optimization has been performed both for TJb = 90% and TJb = 99%,corresponding to maximum uncertainties of 4 and 0.4 K respectively for E = 30° , and 2.5and 0.25 K for E = 90°.

It appeared that the integrated pattern templates did not impose any restrictions on theoptimization for D / >. = 100, since the integrated pattern of the unconstrained optimizationof the weighted sum of TJa and TJb already satisfied the constraints. Therefore, the samecurve of combinations of TJa and TJb in figure 4.2 can be used to indicate the performance ofthe optimum, theoretical antenna. The integrated pattern of the weighted sum of TJa andTJb is plotted in figure 4.6 for the outermost values w = 0 and w = 1.

4.3. Optimum integrated pattern template 65

0.9

o 2.5

0.99

50

1

180

Figure 4.5: Optimum integrated pattern template, E = 900, beam efficiency=90%

------,---.-------.-------.------.-----,----.---------,

w=o

'00.99"2-

.r:.c: 0.980,.,uc:.2 0.97E

1?'0 0.96Q.

".,"§ 0.950'.,E" 0.94.,.~

0~ 0.9.3cc

0.920 0.5 1.5

w=l

2

theto [degrees]

2.5 .3 .3.5 4

Figure 4.6: Integrated pattern after (un)constrained optimization of the weighted sum of TJaand TJb, W = 0 and w = 1

Chapter 5

Practical radiometer antennas

5.1 Introduction

In chapter 2 of this report it has been investigated which requirements are important toa radiometer antenna. The performance of the antenna for various (integrated) patterntemplates has been studied, which lead to a template that guarantees the lowest uncertainty in measured brightness temperature for a high percentage of time. In chapter 3 anoptimization method has been discussed, which yields optimum values for 7]a, 7]b, and optimizes the integrated pattern. This method has been applied in chapter 4, where optimumcombinations of "lb and the beamwidth and of 7]a and 7]b have been determined, as well asthe optimized integrated pattern subject to the constraints determined in chapter 2.

In this chapter practical antenna systems will be compared to the optimum theoreticalradiometer antenna of chapter 4. Any substantial loss will be included in order to achievea realistic impression of the performance of the systems. Which configurations shouldbe dealt with, is discussed in section 5.2. In section 5.3 the efficiencies that reduce totalantenna performance will be considered. Models of a front-fed and an offset antenna systemas well as simulation results for both configurations are discussed in sections 5.4 and 5.5respectively. Finally, in section 5.6 a design procedure is presented, with which it is possibleto select an antenna system for a certain application.

5.2 Configurations

To limit the number of antenna systems that will be considered, some preliminary assumptions have to be made. Keeping in mind the requirements gathered in section 2.4, itis clear the uncertainty in the brightness temperature observed is the smallest when thepower outside the main beam is directed towards zenith or towards the ground. Therefore,in case of a low elevation angle, application of a Cassegrainian antenna system (where feedspillover is directed to the sky) would not improve performance. Furthermore, since another important aspect of an antenna design is cost, and the number (and size) of reflectors is a dominating factor in the prize of an antenna system, no double or more reflector

66

5.2. Configurations 67

(5.2)

(5.1)

antennas are taken into account.To be able to receive signals at different frequencies a narrow-band corrugated feed horn

can be used, which exhibits a beamwidth decreasing linearly with operating frequency. In[29] an example is given for an antenna operating at 20 and 30 GHz. The effective area ofillumination on the reflector at 30 GHz is about four ninths of the area at 20 GHz, therebyresulting in near-equal beamwidths for the overall antenna. The pattern of the feed isrepresented by a cosine-shaped illumination pattern, that is often applied in instructivecases [23]

GJ(t/J, f) = Go(n)cosn(t/J)GJ(t/J, f) = 0

where Go(n) = 2(n + 1) to satisfy the condition

Jr Go(n)cosn(t/J)dn= 47rIn=41r

The coordinates of the feed pattern are defined in figures 5.1 and 5.2 for a symmetrical

II

P II

"11" rR : \ ,I , rI \ I

0 I \lZI \I

I II II II II I,I~

II

II

Figure 5.1: Coordinates of feed and symmet1'ical reflector antenna

and an offset system respectively. An increase of the power n in the feed pattern will causea more directive feed pattern, increasing the beam efficiency and the beam width of thesecondary pattern, but decreasing the aperture efficiency.

Two antenna systems initially qualify because of a relatively cheap design: the symmetricalfront-fed paraboloidal reflector antenna and the offset paraboloidal reflector antenna. Thedirectivity of the feed can easily be changed, leading to a varying reflector edge taper.Furthermore, the size of the reflector in terms of the D/.x ratio, as well as the F / D-ratiowill be varied. In this way several sets of combinations of parameters like in figure 4.2 willbe obtained.

68 Chapter 5. Practical radiometer antennas

II

II

II

II

I

,l~

~F

/

Figure 5.2: Coordinates of feed and offset reflector antenna

5.3 Reduction of performance

In chapter 3 it has been shown that antenna performance can be expressed by a fewparameters: the beamwidth, the beam efficiency and the integrated pattern function h. Inchapter 4 performance of an optimum antenna system has been evaluated by consideringcombinations of beam efficiency and beamwidth, as well as aperture and beam efficiency.However, in practice more factors should be taken into account. The aperture efficiency TJawill be included in the total antenna efficiency 1] (section 5.3.1), while the beam efficiency1]b will turn into a modified beam efficiency TJb (section 5.3.2).

Next to 1] and TJb the performance of an antenna is also influenced by other factors.Due to the pointing error, caused by both the accuracy of the pointing mechanism andits read-out, an error will occur in the measured brightness temperature. Furthermore,reflections at various transitions in the antenna and receiver system have to be considered.Blockage by feed and feed support struts reduces total antenna efficiency, and, due to thestruts, also radiation cones will occur, of which direction and power have to be considered.These contributions will be discussed in section 5.3.3.

5.3.1 Antenna efficiency

The antenna efficiency is the ratio of the maximum theoretical gain and the actual gain ofthe antenna

G1]= -

Gmax

(5.3)

5.3. Reduction of performance 69

In case of a circular reflector antenna with diameter D, the maximum theoretical gain is[23]

(7r D)2

Gmax = T (5.4)

The antenna efficiency can be regarded as a combination of factors all bringing a source ofreduced efficiency into account [30,31,22,32].

'TI = 'TII'TI2 ••• 'TIn

The following factors which reduce antenna performance can be included:

• spillover efficiency

• illumination efficiency

• phase and surface efficiency

• polarization efficiency

• feed and feed support system blocking efficiency

• surface tolerance

(5.5)

In principle the aperture efficiency is a product of three factors: illumination efficiency,phase efficiency, and polarization efficiency. However, since the objective of a radiometer isto measure the noise power of a signal with random polarization, the latter efficiency canbe left out of consideration. It should be noted that since reflections at a smooth surfacesometimes strongly depend on polarization, the reflected signal in those cases can no longerbe regarded as randomly polarized. Due to the fact that for all operating frequencies theground under the radiometer site can be considered to be very rough, this problem doesnot occur (section 2.3.2).

In fact the surface efficiency is a special form of phase efficiency. Since the phaseefficiency of a conical feed horn can be very high, only surface efficiency will be considered.

In a simple case, the antenna efficiency can be taken equal to the product of apertureand spillover efficiency, as will be done initially in sections 5.4 and 5.5.

Spillover efficiency

A part of the power radiated by the feed will not be intercepted by the reflector and islooking past the reflector. Vvith the feed pattern described in (5.1) the spillover efficiencycan easily be determined:

'TIs27r Jail! Go cosn( 'l/J) sin( 'l/J )d'l/J27r J; Go cosn( 'l/J) sin( 'l/J )dt/J

_ 1 - cosn+1 (\II) (5.6)

where \II is the half-width of the subtending angle of a symmetrical reflector (figure 5.1).For an offset reflector 'l/Ja should be inserted instead of \II.

70

lIIumination or aperture efficiency

Cllapter 5. Practical radiometer antennas

The highest gain of a reflector antenna is achieved for uniform illumination. Reductionin gain by non-uniform illumination is described by the illumination efficiency. The illumination efficiency, which is in this case equal to the aperture efficiency, can be given by[23]

1 IfA f(r)dAI 2

1]a = AfA If(r)1 2 dA(5.7)

with A the area of the aperture and f(r) the amplitude of the e-independent illuminationfunction corresponding to the feed pattern defined in (5.1).

Blocking efficiency

In a symmetrical reflector antenna design blockage occurs, which will reduce the antennaefficiency by decreasing the gain and raising sidelobe levels. In offset reflector antennasystems blockage can be avoided. The effect of blockage will be an increment of sidelobelevels, giving rise to the question where the additional power is exactly located, since thatmight influence radiometer antenna performance strongly. In section 5.3.3 this effect willbe discussed.

Assuming an antenna design with one or several (straight) feed support struts, a circularfeed, and a uniform aperture illumination, the blocking efficiency 1]bl can be computed. Incase of a simple waveguide feed, the blocking efficiency can be calculated by consideringthe waveguide as one strut.

The influence of feed support struts on the antenna pattern has recently been investigatedin [33, 34]. In case of N feed support struts, the blocked fraction b of the aperture can begiven by

b = NIVL sin / = 2NIV1r(D/2)2 1rD

(5.8)

where N,l-V,L D and / are defined in figure 5.3 and assuming that the struts are attachedto the I'im of the reflector (Lsinb) = D/2). The blocked fraction of the aperture due toa circular feed is given by

1r(Df /2)2 = (D f )21r(D/2)2 D

(5.9)

with Df the diameter of the feed. Combining the aperture blockage by feed and feedsupport struts the total blocking efficiency 1]bl can be expressed as

(Df )2

1]bl = 1 - b - D (5.10)

5.3. Reduction of performance

0--

15:

71

Figure 5.3: Symmetrical paraboloidal reflector antenna with aperture blocked by three feedsupport struts

Surface tolerance

In the calculations above it has been assumed that the reflector is smooth. However, inpractice the reflector can only be produced with limited accuracy, which will lead to areduction of the antenna efficiency and of the beam efficiency as well. The latter will bediscussed in section 5.3.2.

The effect of irregularities of the reflector surface on the antenna efficiency has been determined by Ruze in the often cited [35]. Using a random phase error 8 and a correlationdistance c (the phase values are completely correlated within a diameter 2c and completelyuncorrelated for larger distances between individual aperture points) the gain function canbe given by

(5.11 )

The first term on the righthand side is the unperturbed pattern multiplied by a factorexp( -82) that reduces the gain. The second term represents a broad pattern, mainly affecting the sidelobe level. The importance of this term is determined by its value relative tothe desired sidelobe level, which is dependent on the effective surface error, the correlationdistance and the D /).. of the reflector [31].

In case of c ~ D, the second term in (5.11) can be neglected. The reduction of the totalefficiency by surface irregularities, or the surface efficiency 1]sur j, becomes the well-known

72

expressIOn:

Chapter 5. Practical radiometer antennas

TJsurJ exp( -82) = exp [_ (4~f)2]

exp [_ (4~ ~ ~) 2] (5.12)

where f is defined as the effective surface tolerance: the rms error which will produce thephase front vari ance 82• In the second equation the surface tolerance is expressed in termsrelative to the size of the reflector.

In fact the equations above only hold in case of shallow reflectors (large FID); for deepreflectors a correction should be made for the fact. that. t.he error norma.l to the reflectorsurface differs from the error in axial direction. In [36] curves are presented of the correctedeffective surface tolerance in a reflector antenna system with an edge taper of -10 dB. Incase of FID = 0.7, f should be corrected by a factor 0.96. Due to the fact that in general aradiometer antenna has a very high edge taper, surface errors at the centre of the reflectorwill have a greater effect on performance than surface errors at the edge of the reflector.Therefore, it can be assumed the correction factor will be close to one, and no correctionson f have to be made.

5.3.2 Beam efficiency

Due to surface irregularities part of the power that would otherwise have been locatedin the main beam will be radiated in other regions. Additionally, part of the power thatwould have been located in the sidelobes, will be radiated in the main beam angle region.The beam efficiency including surface tolerance effects will be denoted as TJb'

The modified beam efficiency TJb can be calculated by integrating (5.11) over the main beamand comparing the result with an integration over all angles, resulting in [31, 37]:

, -62 [ ~ 82( _("C6mb/~)2)]

TJb = e TJb + LJ I" 1 - e n

n=l n.(5.13)

with, analogous to (5.11), c the correlation distance, 8 the random phase error, and where()mb is the first zero in the unperturbed pattern.

5.3.3 Other factors reducing antenna performance

The factors that have been introduced in the preceding sections 5.3.1 and 5.3.2, could all beincluded in TJ and in TJb' However, more practical limitations exist, that cannot be includedin these two efficiencies. In this section these limitations will be discussed.


Pointing errors

Antenna pointing errors can be divided into two categories: the error caused by the fact thatthe antenna is not exactly pointing in the specified direction, due to the mechanical pointingmechanism, and the error due to the finite dimensions of the main beam of the antenna(elevation error). The latter error source has already been discussed in section 2.4.4. Theerror due to the mechanical pointing mechanism will be estimated in this section.

Assuming an atmospheric brightness temperature that varies as the cosecant of elevation,at low elevation angles a small pointing deviation can cause a considerable change in themeasured brightness temperature. For sake of simplicity (2.2) is rewritten as [14]

(5.14)

with Textra the extraterrestrial radiation incident on the atmosphere, T m the mean radiatingtemperature of the atmosphere, also known as the medium temperature, and T the opacity.

In case of a stratified atmosphere the opacity can be expressed as

TOT=---

cos(1/;)(5.15)

(analogous to section 2.3.1) with TO the total zenith opacity and 1/J the complementaryelevation angle.

The sensitivity of the observed brightness temperature to the elevation angle is givenby dTsky /d1/;, where

dTsky ( ) -7# = - Textra - T m e . T' tan(1/;) (5.16)

This function is plotted in figure 5.4 for various zenith opacities. In a worst case situation(low elevation angle and high opacity) a sensitivity to pointing of 5 K/degree elevation canbe found. The error due to pointing in brightness temperature measurements is dependenton the antenna pointing mechanism, e.g. 0.5 K in case of a pointing accuracy of 0.1 degree.

Ohmic losses and reflections

In the antenna and receiving system the apparent antenna temperature is higher than theantenna noise temperature, because of ohmic losses and reflections. Ohmic losses resultfrom reflector surface resistivity, waveguide attenuation, resistivity of the feed and, possibly,attenuation due to a window protecting the antenna from influence of the weather andlosses in a diplexer. Impedance mismatch between feed and reflector, feed and waveguide,and between waveguide and receiver (PI, PIl and PIlI in figure 2.26) will cause reflections.

In section 2.5 a simple model has been presented to estimate the influence of the assembled reflection coefficient and the total system loss on the uncertainty in the brightnesstemperature observed.

Most of the reflections and losses indicated above, will occur both in a symmetrical and in

74 Chapter 5. Practical radiometer antennas- 5 .-------.,~'_=__...,..---,---------,-----.----.____-_____,--...,.._-___,

- 4.5

-4~

OJ.,~ - 3.5.,"0;c- -3

.......

~ - 2.5'iic::: -20c:~ - 1.5'00..

-1

- 0.5

10

.0

20 30 40 50 60

.........

70 60 90

elevation angle [degrees]

Figure 5.4: Sensitivity to pointing error, for various zenith opacities

an offset reflector antenna system. However, two differences exist. The first is the length ofthe waveguide from feed to receiver. In an offset configuration the receiver can be mountedclose to the feed, which is more difficult in a symmetrical configuration. However, since theuncertainty in ohmic loss has a far greater influence than the actual loss itself, a waveguide with different length will only have minor influence. The second difference is thatreflections between feed and reflector do not occur in an offset system.

Blockage

Blockage of the aperture gives rise to reduction of antenna efficiency. This implies powerwill be scattered in other directions. In [34, 33] the contribution from these scatter conesto the antenna noise temperature has been calculated.

Due to the feed support struts, scatter cones will arise around the struts with an openingangle 2"( and a half-power width /3 given by

2,\/3=

D(5.17)

(5.18)

The half-power width has been determined by considering the projected length of a strut(D/2,\) as a radiating cylinder. The solid angle in the scatter cones can be determined(using an appropriately oriented coordination system) by calculating

fa21r 1"'1+(3/2

Os sin OdOd</>o "'1-(3/2

~ 471" sin "( . /3/2


Figure 5.5: Example of a scatter cone of a symmetrical antenna with an elevation angle of90°

which becomesOs = 4N1r>.sinh)

D(5.19)

for N cones and by inserting (5.17). The fraction of power in the scatter cones is simplyb, assuming radiation in the cone is constant and no power is radiated outside the cone.

In the preceding part of this section the fraction of power in the scatter cones has beenderived. The influence of the power in the scatter cones on radiometer antenna performance, is dependent on the direction of the scatter cones. The width of the scatter conesis dependent on the projected length of the struts, and is therefore determined by the sizeof the reflector, and the connection points of the struts to the reflector. The opening angleof the cones is defined by the angle / (figure 5.3), which is equal to \If in case the struts areconnected to the rim of the reflector. Therefore, besides the uncertainty curves given inchapter 2, the scatter cones should be considered too in determining the FI D ratio whichyields the lowest uncertainty.

Due to the large angle region that is covered by the scatter cones one can only indicatein general terms the influence of a certain angle /. For an elevation angle of 90° thelowest part of the cones will have an angle to zenith of 2\If (figure 5.5), assuming the strutsare connected to the rim. So, for an FID-ratio larger than 0.8, no scatter cone powerwill be radiated in directions 70° :::; t/J :::; 90°. If the struts are not connected to the rimthe angle will be even smaller, although in that case spherical blockage will have to beconsidered. For low elevation angles, a large fraction of scatter cone power will be radiatedin directions along the surface of the earth, due to the struts connected to the upper halfof the reflector. Therefore, it should be preferred to have the majority, or if possible all


of the struts connected to the lower half of the reflector. If one or more struts have to beconnected to the upper half of the reflector, the FID-ratio should be small, so that thelargest part of the cone will be 'looking' towards zenith or towards the ground. Furthercorrections are possible by considering different strut geometries (e.g. bent in stead ofstraight).

5.4 Front-fed reflector

In the preceding section, factors reducing antenna performance have been discussed in ageneral sense. In this section the knowledge obtained will be applied to a symmetricalfront-fed reflector antenna configuration. It will be possible to obtain a general impressionof the performance of the various front-fed systems. In section 5.6 the procedure which isto be followed in designing a radiometer antenna system on basis of the results of chapter 2,will be further explained.

Initially curves of T)a versus T)b for various FID-ratios will be shown (section 5.4.1).On basis of the requirements of chapter 2 an FID ratio will be chosen, and for thatconfiguration curves of T)a versus T)b and of beamwidth versus T)b will be presented, forvarious DI,X ratios. Finally, curves of beamwidth versus T)~ will be presented, which canbe used in the design procedure.

The complete pattern of the symmetrical reflector antenna will be investigated, whichwill be described in section 5.4.3.

5.4.1 Aperture, beam, and spillover efficiency and beamwidth

The main lobe and first few sidelobes of the secondary pattern of the front-fed paraboloidalreflector antenna have been determined using a computer program described in [24], whichis sufficient to calculate the aperture and beam efficiencies. The radiation pattern of themain beam of the front-fed paraboloidal reflector is almost completely symmetrical in ¢,certainly close to boresight, so only the O-integral has to be calculated to determine thebeam efficiency. The aperture efficiency has been calculated by applying:

T) = T)a • 7]s (5.20)

since, up to this point, no other factors reducing the antenna efficiency are present.The resulting combinations of T)a and T)b have been plotted in figure 5.6 for DI,X = 100,

where also the ideal curve of figure 4.2 is shown for comparison. The curves have beencalculated for four FID ratios. To investigate the influence of n on aperture, beam, andspillover efficiency, all three parameters have been plotted as a function of n in figures 5.7,5.8, and 5.9. Both aperture and beam efficiency are given for a system with FID = 1.42(which will proof useful later), while the spillover efficiency is given for various FI D-ratiosand DI,X = 100.

5.4. Front-fed reflector

ideal

77

0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98

"~':'.::.:""" .0.9 -----

0.8

0.7

~ 0.6c::II

1>;: 0.5II

E0 0.4II.a

0.3

0.2

0.1

a0.8

----

F/O-0.4F/O-O.SF/o-0.6F/O-0.7

aperture efficiency

Figure 5.6: Combinations of TJa,'rJb for the symmetrical paraboloidal reflector antenna,Dj>.. = 100

D/lambda=2SD/lambda=SOD/lambda=7SD/lambda= 100

0.9

0.8

0.7>-uc:: 0.6II

]'i O.Sf!::>

1:: 0.4IIa.0

0.3

0.2

0.1

0a 10 20 30 40 so 60 70 80 90 100

power n of feed pattern

Figure 5.7: Aperture efficiency TJa as a function of power n of feed pattern, symmetricalreflector, F j D = 1.42


0.9

O.B

Eo.,.0

O/lombdo=25D/lombdo=50D/lombdo=75D/lombdo= 100

100908070605040302010O'----------'-----'--------'------'-----'---------'-----'------'-----'-------.Jo


Figure 5.8: Beam efficiency 17b as a function of power n of feed pattern, symmetrical reflector, F / D = 1.42

F/O-0.7

F/O-0.6

F/O-O.S

F/O-O.4

F/O-0.3

0.9

0.3

------ --- --._.~~- _.- .

.' -' ~- ------./,,""./,/'

0.8 : :' ./

.... " /0.7 ! / /

j ! j': I: , .

0.6' ; /I .

0.5 ,I /! /

0.4 I

0.2 L-_---'-__--'-_----''--_--'-__.L.-_----L-__-'--_---'__--'-__

o 2 4 6 8 10 12 14 16 18 20


Figure 5.9: Spillover efficiency 178 as a function of power n of feed pattern, symmetricalreflector, D /). = 100

5.4. Front-fed reflector 79

Based on the uncertainty curves in chapter 2, a choice has been made for an F1D-ratio.Analogously to chapter 4 this choice will be based on the spillover lobe, which leads tothe selection of pattern 3 in figure 2.10 for minimum uncertainty. As can be seen fromfigure 5.10 the position of the spillover lobe of a symmetrical reflector antenna with asubtending angle of 200 is roughly between () = 1300 and () = 1700 for n = 10. Further,the F1D ratio can be calculated by applying

F 1D = 4 tan (\II 12)

(5.21)

which yields: F1D = 1.42 for a subtending angle of \II = 200• For this F1D-ratio, curves

of combinations of TJa and TJb are shown in figure 5.11, for several DIA ratios. In figure 5.12TJb is plotted against the beamwidth for the same four D1A-ratios.

5.4.2 Antenna efficiency, modified beam efficiency, and beamwidth

In this section the curves of combinations of 1Ja and 1Jb presented in the previous section,will be transformed into curves with combinations of "I and "lb' Furthermore, curves ofbeamwidth versus TJb will be presented. Since polarization and phase efficiency can beneglected in the antenna systems described in this report, these can be left out of consideration. The remaining factors that should be included in the antenna efficiency are theblocking efficiency and the surface efficiency. The latter is also included in the modifiedbeam efficiency.

Antenna efficiency

The blocking efficiency TJbl is dependent on size and number of feed support struts, as wellas the diameter of the feed itself. A simple waveguide feed will cause minor blockage effects,but it's pattern is not very directive and the beamwidth decreases with frequency. Theillumination pattern of a corrugated horn can be highly directive, but it is larger and itneeds a stronger support struct ure than a waveguide feed.

Assuming a scalar conical feed horn is used with three feed support struts, the blockingefficiency (5.10) can be calculated. The size of the feed horn is based on [38] where generalcurves of edge taper versus the required opening angle of the feed are presented for wideflare angle corrugated feed horns. The blocking efficiency has been calculated for strutwidths of 1 and 2 ,\ and for four D1A ratios. The curves are presented in figures 5.13and 5.14. It should be noted, that the size of the reflector, the size of the feed horn,and therefore the width of the feed support struts are all related linearly to frequency.Therefore the blocking efficiency remains essentially the same over a large bandwidth.

Finally, the surface efficiency has to be calculated, which is limited by material, structure,and size of the reflector as well as the manufacturing accuracy of the company involved.The antenna surface tolerance is often expressed in terms relative to the diameter of the


0

-10

'iIi' -20"C......c:0.. -30lJc:::I-E -40~cQ.

"C -50.,.!:!0E -600c:

-70

-800 20 40 60 80 100 120 140 160 180

theta [degrees]

Figure 5.10: Complete pattern of a symmetrical paraboloidal reflector antenna, W 20°,n = 10

0.9

0.8

0.7

>-0 0.6c.,·0;;::

0.5CiE0 0.4...

..Q

0.3

0.2

0.1

00.8

D/lambda=25

D/lambda=5D

D/lambda=75

D/lambda= 1DO

0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98

aperture efficiency

Figure 5.11: Comb7:nations of ""a,''7b for the symmetrical paraboloidal reflector antenna,F/D = 1.42


54.543.5

--<~- idealO/lambda=25O/lambda~50

O/lambda..75O/lambda=100

.3

0.9 flf/"0.8 I I

III

0.7 I IIII

~ I I

0.6 Ic ICI ,u I I

I;: I

G 0.5 •I

I,

E I

0 0.<4-,

CI I.Cl I I,

I

0.3 .I •I

II

0.2 IIIII

0.1 I,III

00 0.5 1.5 2 2.5

beamwidth [degrees]

Figure 5.12: Combinations of beamwidth and 7]b for the symmetrical paraboloidal reflectorantenna, F / D = 1.42

O/Iam - 100

......

0.98

0.96

>.c.>c.~c.> 0.9<4-I;:

G01C

:;;:c.>0 0.922:i

0.9

O/Iam - 75

F/O-0.7F/O-0.6F/O-0.5F/O-O.4

......

.....................................................::

. ··········· ..···~/;~~··~ .. 5-:g::·-:::·-:::::·-:::::·~·~::·~·~~~~~ ~~~.

..:::::::::::.::::::=:::-~O/Iam - 25 .

".'" ...........

20181612108620.88 '--_.........__-'-__.L..-_---L__....L..__-'--_----''--_--....__-'-_~

o


Figure 5.13: Blocking efficiency as a function of n with 3 struts of 1 A width

Chapter 5. Practical radiometer antennas

201816

O/Iam - 100

1412

O/Iom'" 50. - --------........................

1086

.......................:-.:-.-:------

4

F/O=0.7F/O-0.6F/O-0.5F/O-OA

2

O/Iom - 75

82

0.98

0.96

0.94

~c 0.92IIU

i 0.9co~ 0.88u0:is

0.86

0.84

0.82

0.8a


Figure 5.14: Blocking efficiency as a function of n with 3 struts of 2 A width

reflector: tiD. Assuming c « D (which implies the simplest version of surface toleranceefficiency (5.12) can be applied) TJsurface can be calculated. The results are presented infigure 5.15 as a function of the relative surface tolerance, for four DIA ratios.

Modified beam efficiency

The modified beam efficiency TJ~ can be calculated easily, using the surface efficienciespresented in figure 5.15. Analogous to the previous paragraph, the modified beam efficiencybecomes:

(5.22)

Combinations with TJ~

For the optimum FID-ratio derived in the preceding paragraphs, a new curve of combinations of TJ and TJ~ has been computed for four DIA-ratios, as shown in figure 5.16.Two values of the effective surface tolerance tiD have been selected (tiD = 10-4 andtlD = 5 . 10-4

) to indicate the effect of a whole range of surface tolerances. These valuesare believed to represent surface tolerances which can be achieved with commercial technology [32]. Furthermore, 71~ has been plotted against the beamwidth in figure 5.17, forthe same DIA-ratios.


0.9

0.8

li' 0.7c.!!u

!E0.6••uc

't: 0.5:3..0.4

0.3

0.210-~

D/lombdo-100

D/lombda-75

D/lombdo-50

D/lombda-25

10~ 10-3

relative effective surface tolerance epsilon/D

Figure 5.15: Surface efficiency as a function of relative surface tolerance f/D

ideal

D/lombda-25D/lombdo-50D/lombdo-75D/lombdo-100

0.92 0.94 0.96 0.980.9

............,,,,

, ,, ,, ,,I,,,

II,III,II

0.82 0.84 0.86 0.88

.---'--'--'---'---\

\\\\\\

0.8

li' 0.7c..u!E 0.6..E 0.5c..

J:J

"C 0.4..1;:

'is0

0.3E

0.2

0.1

00.8

ontenna efficiency

Figure 5.16: Combinations of TJ,TJ~ for the symmetrical paraboloidal reflector antenna,F/ D = 1.42, f/D = 2.5 . 10-4


0.9

0.8

~ 0.7c..u::: 0.6..~ 0.5..

.D

~ 0.4l;:

'6E 0.3

0.2

0.1

/7·:-------·• I

{,',' :" .• II : ,-- -- ~~'--"

,. I/ j//1'-'.::

i( ii jif ii t/'/' ~ f~ I

~ ~ f

(/1D/lombdo=25D/lombdo-50D/lombdo=75D/lombdo=100

54.543.532.521.50~----.L---.1.__...L..----L..__'--_--'---_-.1.__-'--_----l.._---l

o 0.5

beomwidth [degrees]

Figure 5.17: Combinations of beamwidth and "l~ for the symmetrical paraboloidal reflectorantenna, F / D = 1.42

5.4.3 The complete pattern of the symmetrical paraboloidalreflector antenna

It has been shown in section 2.4 that the integrated pattern, is very important in determining radiometer antenna performance. Before the integrated pattern can be determined theantenna pattern has to be known. This can be accomplished by using the work describedin [39), which accurately computes the far out sidelobes of a symmetrical paraboloidal reflector antenna, using uniform theory of diffraction (UTD) and the equivalent edge currentmethod (ECJ\l). The integration of the far field pattern has been performed for variousangles <P, and it appeared the pattern further away from boresight could be assumed <Pindependent, like the main beam.

In figure 5.18 integrated pattern curves are shown for n = 1,5,10,20,30,40,50 andF / D = 1.42. The optimum integrated pattern template is also shown in this figure forcomparison. The power in the spillover lobe decreases for increasing n.

5.5 Offset reflector

The calculations performed in section 5.4, will be repeated in this section for an offsetparaboloidal reflector antenna. In section 5.5.1 the curves of combinations of "la versus "lband of beamwidth versus "lb will be shown. In the subsequent section 5.5.2 the corresponding curves of "l versus "lb and beamwidth versus "lb are plotted. Finally, in section 5.5.3the integrated pattern of various offset paraboloidal reflector antenna configurations willbe discussed.

5.5. Offset reflector

n-50

0.9

EII~0Q.

"0.!Eco.!oS"0II

~0

~0c

0.2

0.1

00 20

n-'40 60 80 100 120 140 160 180

85

theta [degrees]

Figure 5.18: Integrated pattern of symmetrical paraboloidal reflector antenna for n1,5,10,20,30,40,50 and F / D = 1.42

It should be noted that an offset antenna configuration can be designed in two ways: thefeed may be positioned below the reflector (which corresponds to the position of the feed inthe parent symmetrical reflector), and the feed may be positioned above the reflector (whichis the same as the previous offset configuration turned upside down). Which configurationis preferred, depends on the elevation angle. For lower elevation angles, spillover powercan be directed more easily to the ground in an 'upside down' configuration.

5.5.1 Aperture, beam, and spillover efficiency and beamwidth

Aperture, spillover, and beam efficiency of an offset reflector antenna are calculated byusing a computer program described in [24], which computes the main lobe and first fewsidelobes of the far field pattern. A current integration method is used in [24], that approximates the current distribution over the reflector surface, which on its turn is basedon geometrical optics. This method yields good results only if the distance from illumination source to reflector is large in terms of wavelengths, which is the case for the offsetconfigurations used in this report.

To be able to compare the performance of the offset reflector configurations to the symmetrical reflector configurations, it has been decided that, initially, the (projected) aperturesof both systems should be equal. Furthermore, the offset and subtending angles of theoffset reflector are related to the subtending angle lIT of the symmetrical reflector, as willbe explained below. For a symmetrical antenna (5.21) can be used for the relation betweenthe F / D ratio and the subtending angle of the reflector lIT. The symmetrical reflector


(5.23)

functions as the parent of the offset reflector antenna; the reflector of the offset antennaconfiguration extends from 'l/Jc/ (feed clearance) to \l1, as illustrated in figure 5.19. After the

o

1_L--------------- ~f:;~--------------i r

,_LFigure 5.19: Relation between symmetrical and offset antenna configurations

offset angle t/Jo and the subtending angle 'l/Ja of the offset reflector have been determined,the system is scaled up to the point where the projected aperture of the offset antennaDoff is equal to the diameter D sym of the symmetrical reflector. The feed clearance 'l/Jc/is taken to be 5°, which in general is enough to assure an unblocked system. In table 5.1a summary is given of the F / D-ratios of the symmetrical reflector and the correspondingF / D-ratios of the offset antenna. The latter can be calculated using

F/D _ cos( 'l/Jo) + cos( 'l/Ja)off- 4 sin ('l/Ja)

Table 5.1: Corresponding F / D-ratios of symmetrical and offsetrefleetors

F/Dsym \II [0] 'l/Ja [0] 'l/Jo [0] F/Doff

0.7 39.31 17.15 22.15 1.590.6 45.24 20.12 25.12 1.340.5 53.11 24.07 29.07 1.100.4 64.01 29.51 34.51 0.860.3 79.61 37.31 42.31 0.63

Analogous to the symmetrical paraboloidal reflector, some results have been calculated

5.5. Offset reflector 87

for various offset configurations. Again, for the time being, neglecting other factors theaperture efficiency can be calculated using (5.20). The obtained combinations of "Ia andTJb are plotted in figure 5.20 for D/ A = 100, where again the ideal curve of figure 4.2 isshown. The offset configurations that correspond to larger F / D ratios of the symmetrical

1~-~__~__~_:::~~:::!::::=:::::==========~id;:;e~alll0.9

0.8

0.7

~ 0.6c::..U10:

0.5';

E0 0.4..

.J:l

0.3

0.2

0.1

00.8

F/O=0.63F/O=0.86F/O-1.10F/0=1.34F/0=1.59

0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98

aperture efficiencyFigure 5.20: Combinations of "la, TJb for the offset paraboloidal antenna, D/A = 100

reflector are very shallow. For these configurations increment of n only slightly reduces theaperture and slightly increases the beam efficiency. Again, to investigate the influence ofn on aperture, beam, and spillover efficiency, all three parameters have been plotted as afunction of 11, in figures 5.21, 5.22, and 5.23, for F / D = 1.10 and four D / A-ratios.

The optimum F / D-ratio can be chosen on basis of the same arguments as in section 5.4.It should be noted that due to the asymmetry of the offset reflector, the subtending angleof the offset reflector cannot be related easily to the prescribed region. However, theuncertainty curves of chapter 2 have indicated that power in directions to the ground lessinfluences total uncertainty than power in directions to the sky. Therefore, the selectionof'l/Jo and 'l/Ja will concentrate on spillover around the upper part of the reflector.

For an elevation angle of 30°, spillover power should be radiated in the region 130° :::;() :::; 170° (pattern 3 of figure 2.10). This can partly be achieved in case of an upside downoffset configuration, if 'l/Jo + 'l/Ja :::::; 50° .

Calculations will be performed for tPa = 24.07 and 'l/Jo = 29.07, or F/Doff = 1.10. Infigure 5.24 curves are presented for several D / A-ratios and F / D = 1.10.


-- D/lambda=25............ D/lambda-50

------ D/lambda-75_.- D/lambda-100

0.9

o.e

0.7

~c 0.6~5:.. 0.5!:::J

1:: 0.4"Q.

" 0.3

0.2

O'J0 10 20 30

power n ot teed pattern

40 50 60

Figure 5.21: Aperture ejficlency 77a as a function of power n of feed pattern, offset reflector,FjD = 1.10

-- D/lambda-25............ D/lambda-50

-- - - -- D/lambda-75-.- D/lambda-100

0.9

0.8

0.7

~ 0.6c~:: 0.5..E" 0.4...D

0.3

0.2

O. ,

00 10 20 30


50 60

Figure 5.22: Beam efficiency 77b as a function of power n of feed pattern, offset reflector,FjD = 1.10

5.5. Offset reflector 89

60so

-- F/0-1.59

-.- F10-1.34

------ F/0-l.l0

............ F/0-0.B6

------ F/0-0.63

..............

302010

---- - ---------....... _--- --._0---- _.--..---,.. ' ",,~- -'

.' " .--'

///',::~;:///. ,t' I'! " j'//;

.! /j': I

!/IU j': I

: I: 1 .: I: I

0.9

O.B

>.<J 0.7cOJ·u

l;::

Oi 0.6L..OJ>,g

0.5Q..,

O.'!-

0.3

0.20


Figure 5.23: Spillover efficiency TIs as a function of power n offeed pattern, offset reflector,FID = 1.10


Furthermore, curves of the beamwidth versus "1b have been calculated and are plottedin figure 5.25.

5.5.2 Antenna efficiency, modified beam efficiency and beamwidth

Analogous to the symmetrical antenna, the curves of combinations of "1a and "1b will betransformed into curves of combinations of "1 and "1~. In addition "1~ will be plotted asa function of the beamwidth. In this case only the surface efficiency is important, sinceblockage can be avoided in the offset configurations considered in this report.

The surface efficiency of the offset reflector will be almost identical to that of the symmetrical reflector. For very large systems a difference may occur due to the fact that an offsetsystem is not symmetrical, and is therefore less rigid. However, to limit cost, reflectors willprobably be relatively small and differences in surface efficiency can be neglected. Therefore figure 5.15 can be used to determine the antenna efficiency and the modified beamefficiency. A value of €/D = 2.5 . 10-4 has been selected as a representative value for thecombinations of "1 and "1~ For combinations of "1~ and beamwidth two values of €/D havebeen selected (€/D = 10-4 and €/D = 5 . 10-4

) to indicate the effect of a whole range ofsurface tolerances.

The resulting curve is shown in figure 5.26. In figure 5.27 "1~ is plotted as a function ofthe beamwidth.

5.5.3 The complete pattern of the offset paraboloidal reflectorantenna

To calculate the integrated pattern of the offset antenna with the F / D and D/ A-ratiosselected up to this point, it is necessary to calculate the complete far field pattern. Suchcalculations have been performed in [39] using UTD and ECM for offset configurations,besides the calculations for symmetrical configurations, which have been used in section 5.4.Integration has been performed for various 4>-cuts and n = 1,5,10,20,30,40,50. For thecurves in 5.28 O-integration has already been performed. Analogous to the integratedpatterns of the symmetrical antenna, the influence of the spillover lobe can clearly beobserved.

5.6 A design procedure for radiometer antennas

In the previous section, performance of symmetrical and offset antenna configurations havebeen described in general terms, although some selections have already been made on basisof the results gathered in chapter 2. In this section the knowledge obtained in this reportso far, will be applied to the design of a practical radiometer antenna.

5.6. A design procedure for radiometer antennas 91

0.9

0.8

0.7

~ 0.6c.,UlO" 0.5'iEc 0.4.,.c

0.3

0.2

0.1

00.8

D/lambda=25D/lambda=50D/lambda=75D/lambda=100

0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98

aperture eHiciency

Figure 5.24: Combinations of 7]a, 7]b for the offset paraboloidal antenna, F/ D = 1.10

----<'..........1- ideal--- D/lombda=25

D/lambda=50D/lambda=75D/lambda=100I

(1fT./?! f ..:! / !! /! I! Ii /! /! f

1 I

0.3

0.1

0.9

0.7

0.5

0.2

0.4

0.6

0.8

ECl.,.c

0L-_--1-_---JL-_--1-_---JL-_--1-_-.JL-_--1-_-.JL-_--1-_-----:'a 0.5 1.5 2 2.5 3 3.5 4 4.5 5

beomwidth

Figure 5.25: Combinations of beamwidth and 7]b for the offset paraboloidal antenna, F / D =1.10


................

....

D!lombdo-25D!lombdo-50D!lombda=75o!Iombda=100

..................-''-'-'.-.----."'-."-,,

."".""."\\\\

---------------- ---""-

0.8

>- 0.7ucCl

U0.6::

Cl

E 0.50Cl~

"i 0.4l;:

'60

0.3E

0.2

0.1

0.92 0.94 0.96 0.980.90.82 0.84 0.86 0.88OL------L__--'-__"'"---_--'__----'-__-'--__'--_........__--'-_-----'

0.8

antenna efficiency

Figure 5.26: Combinations of7],7]~ for the offset paraboloidal reflector antenna, F/D = 1.10

,.---;=-._.-.. .I' j' .

,: (r·-L·~: I' :i! 1/--- f/:,,' I .:, ,

if IIif II:: I"•

!I,

D!lombdo=25D!lombdo=50D!lombda=75D!lombdo=100

0.9

0.8

~ 0.7cCl

U0.6::

Cl

E 0.5i..0

"l:I 0.4Cll;:

'60

0.3E

0.2

0.1

00 0.5 1.5 2 2.5 3 3.5 4 4.5 5

beomwidth

Figure 5.27: Combinations of beamwidth and 7J~ for the offset paraboloidal reflector antenna,F/D = 1.10


----n-50

-------- n-4O

n-20

n-5

0.9

0.2

c=:==:===:------===:-----"===:------===::----"==-.:----"=------

E~&.~..Er:E~....!:!"0E 0.3oc

0.1

18016014012010080604020OL...-_---'-__-.L-__-'--_---'-__---'----__-'---_----'-__-'-_-----'

o

theta

Figure 5.28: Integrated pattern of offset paraboloidal reflector antenna for n1,5,10,20,30,40,50 and F / D = 1.10

As indicated before, radiometer antenna performance can be described by three parameters:beam efficiency, beamwidth, and the integrated pattern. How to determine target valuesfor all these parameters, which will essentially define the configuration and cost of theantenna systems applied, will be explained in this section. An important aspect of thedesign procedure in general is that the complete radiometer system should be considered,in determining the target values for each system part. It is useless to connect a receiverwith an accuracy of 0.1 K to an antenna with a pattern that creates an uncertainty ofseveral K. The same holds for the accuracy of the mechanical pointing system and itsread-out, as well as for uncertainty introduced by the reflections and ohmic losses thatoccur in the antenna system.

Integrated pattern

A large fraction of the power received outside the main beam is located in the spilloverlobe. Therefore, by choosing the position of the spillover lobe on basis of the patternthat yields minimum uncertainty, the influence of the integrated pattern can be largelycontrolled. The solid angle in which spillover power is received is determined mainly bythe subtending angle of the reflector, in case of a symmetrical configuration. If an offsetreflector is applied, the relation between the subtending angle of the reflector and theregion of spillover power is less clear. However, by concentrating on the spillover power indirections to the sky, it is still possible to select the offset and subtending angles of the offsetreflector, for a given feed clearance. The corresponding F / D-ratios of the symmetrical andoffset configurations can be calculated easily by using (5.21) and (5.23).


Beam efficiency

The curves of total cumulative uncertainty in chapter 2, have shown which values of uncertainty are obtained for a given antenna pattern. The antenna pattern which yieldsminimum uncertainty has been approximated in the previous paragraph on basis of theposition of the spillover lobe. The absolute value of uncertainty for 90% of the time can befurther reduced by increasing the beam efficiency, as already indicated in chapter 2. Fora beam efficiency of 90% and an elevation angle of 300, a total uncertainty of about 4 Kis obtained if the best pattern is selected. A total uncertainty of 1 K, which is a reasonable estimate for the calibration-uncertainty of radiometer receivers, can be achieved byincreasing the beam efficiency from 90% to 97.5%.

Beamwidth

Finally, the beamwidth can be selected on basis of the curves depicting the error in thebrightness temperature observed as a function of elevation angle and the half-width of themain beam (figure 2.8). For an elevation angle of 300 a half-beamwidth of 50 limits theerror to 0.3 K.

Size and effective sUljace tolerance of the 1'eflector

By considering the curves for beamwidth versus 7Jb for the offset and symmetrical reflectorconfigurations, it is possible to determine the required size of the reflector, in terms of D / A,and the required effective surface tolerance of the reflector, in terms of E/ D. In figures 5.29and 5.30 again the curves of beamwidth versus 7Jb are depicted, which have already beenpresented in sections 5.4 and 5.5. In both figures lines have been drawn at the targetvalues of 7Jb and the beamwidth. The crossing of the lines indicate which D / A and effectivesurface tolerance E/D are required to achieve these target values. Of course only a limitednumber of D / A-ratios is depicted, and the required value of the surface tolerance has tobe estimated, but a general impression can be obtained.

The values of D / A and E/ D essentially determine the cost of the reflector. If the designeris not pleased with the results of the design procedure, he can choose out of two options.The first option is to accept higher cost or lower performance. The second option is toinvestigate whether a higher performance can be achieved at the same financial effort, byconsidering a different allocation of cost to the various antenna system components.

In table 5.2 some examples of the required D/ A and E/ D-ratios to obtain beam efficienciesof 90% and 97.5% and beamwidths of 30 and 50 degrees are presented for both the offsetand the symmetrical configuration. The power n of the cosine shaped feed pattern, whichis required to obtain 17b (figures 5.8 and 5.22), is given in this table too, as well as theresulting edge taper.

It can be concluded from this table that an upside down offset antenna configurationshould be preferred for radiometry measurements at 30 GHz and 300 elevation. It is


D/lombdo=25D/lcmbdo=50D/lcmbdo=75D/lombdo= 100

,,,. :' ,.-.-;7.--.

I' ..

'

0 :,/ i./;-f;"-"::

I( Ii if

/" i,i ::

J" ::.:' .

~ i I1/

0.8

0.1

0.2

0.91----------,f--/-----"--~~--__,

li' 0.7l:

"'0E 0.6

"~ 0.5".0il 0.4I::osE 0.3

54.543.532.521.50.5

oL-_-'-__'---_--'--__'---_-'-__l-_-'-__L--_-'-_-----'

o

beomwidth [degrees]

Figure 5.29: Beamwidth versus modified beam efficiency 77/', symmetrical reflector, F / D1.42

D/lombdo=25D/lcmbdo=50D/lombdo=75D/lombdc= 100

/

0.5

,·---r -0.9 t------.,':...-+--...,.:.---'"':':":':.,.,.,...,----~;...::::=----------I

,; { ~! I r: .L-.I/l[i;' IIif II:; f'""

f II

0.8

0.4

0.3

0.6

0.7

0.2

0.1

Eo

".0'0

~osoE

54.543.52.521.50.5oL----'----''-----'------'----'-----'----'---_----'-__--'---_--.Jo

becmwidth

Figure 5.30: Beamwidth versus modified beam efficiency 77/" offset reflector, F / D = 1.10


Table 5.2: Design procedure 1'esults for the symmet1'ical and offset antenna configuration,

F/ Dsym = 1.42, F/Doff = 1.10

antenna beamw. [0] "lb [%] D/).. f/D n edge tap. [dB]sym. 3° 90 50 5.10 4 80 -21.9sym. 3° 97.5 50 10-4 66 -18.1sym. 5° 90 25 5.10-4 47 -13.0sym. 5° 97.5 25 3.10-4 75 -20.5off. 3° 90 50 10-4 29 -11.8off. 3° 97.5 50 10-4 46 -18.5off. 5° 90 25 5.10-4 31 -12.6off. 5° 97.5 25 3.10-4 47 -18.9

possible to achieve the same performance with the front-fed symmetrical reflector antenna,but in that case a large and commercially difficult obtainable F / D-ratio is required. Inaddition the direction and power density of scatter cones in the symmetrical system haveto be considered, since they may worsen performance. For both systems the mechanicalpointing mechanism and its read-out should be chosen accurate enough, in order to preventdeterioration of performance.

Chapter 6

Conclusions and recommendations

6.1 Conclusions

It has been clearly indicated in this report that the requirements that can be posed aradiometer antenna differ profoundly from those that can be posed to a communicationantenna. On basis of several applications, the beam efficiency 7Jb, the beamwidth, andthe integrated pattern have been selected as the important factors in radiometer antennadesign.

It has been found that the antenna pattern is the most important source of uncertaintyin radiometer measurements. To limit total uncertainty the beam efficiency should be high,and no power should be received from directions along the surface of the earth. Ohmiclosses and reflections in the antenna and receiving system have a relatively small influenceon total uncertainty.

An existing synthesis and optimization method has been extended, so that weighted optimization of combinations of parameters is possible. Furthermore, it has been shown that aweighted product and a weighted sum yield identical optimization results. The integratedpattern can be optimized at various points, and by including moments in the optimizationprocedure, it is also possible to pose requirements to the fraction of power received withina certain solid angle.

By applying the optimization method to an integrated pattern template, determined onbasis of the requirements gathered, a set of optimum combinations of aperture and beamefficiency has been obtained, which can be used as a reference for practical systems.

On basis of the requirements gathered, a symmetrical reflector antenna system and anoffset reflector antenna system have been chosen as the most promising candidates forlow-cost radiometer antennas. By including substantial practical limitations an accurateimpression of antenna performance has been obtained.

It appeared to be possible to determine an optimum radiometer antenna design, onbasis of the target values of the selected design parameters. Results of the design procedure indicate that an 'upside-down' offset antenna configuration should be preferred for

97

98 Chapter 6. Conclusions and recommendations

radiometry applications at 30 GHz and 30° elevation.

6.2 Recommendations

The method applied in this report to calculate the uncertainty in antenna noise temperature, or in other words the accuracy with which the antenna noise temperature representsthe brightness temperature observed along the axis of the antenna, implies that differentmodels are used in case of rain on site and no rain on site. In that way, an accurate andfair image can be given of the performance of a certain antenna design. However, the samemethod should be applied in processing data from radiometer measurements. By usingone model for all weather conditions large errors can result. This might be the reason thatvalues for the mean radiating temperature of the atmosphere presented in literature, showrelatively large differences.

Although size and surface tolerance of the reflector largely determine the cost of the wholeantenna system, the trade-off concerning a practical antenna could be influenced by forinstance the cost of the feed in relation to the required n of the feed pattern. Therefore, to perform an accurate trade-off, a detailed study of the cost of all antenna systemcomponents should be carried out.

References

[1] de Maagt P.J.1., A SYNTHESIS METHOD FOR COMBINED OPTIMIZATION OF MULTIPLE ANTENNA PARAMETERS AND ANTENNA PATTERN STRUCTURE, Technical Report 90-E-246-X, Telecommunications Division, Eindhoven University of Technology,November 1990.

[2] de Maagt P.J.1., H.G. tel' Morsche and J.L.M. van den Broek, A SPATIAL RECONSTRUCTION TECHNIQUE APPLICABLE TO MICROWAVE RADIOMETRY, Technical Report 92-E-257, Telecommunications Division, Eindhoven University of Technology,March 1992.

[3] Gelissen P.J.W., COMPARISON OF BEACON-MEASUREMENTS OF THE OLYMPUSSATELLITE TO RADIOMETER DATA - DATA PROCESSING (dutch), Master's thesis,Telecommunications Division, Eindhoven University of Technology, 1989, Originaltitle: Vergelijking van bakenmetingen van de Olympus satelliet met radiometer data- dataverwerking.

[4] Ulaby F.T., R.K. Moore and A.I<. Fung, MICROWAVE REMOTE SENSING, volume 1,Addison-Wesley, 1981.

[5] Orlanski 1., A RATIONAL SUBDIVISION OF SCALE FOR ATMOSPHERIC PROCESSES,Bull. Am. Aleteorol. Soc., 56:527-530, 1975.

[6] Kuo H.C., DYNAMIC MODELING OF MARINE BOUNDARY LAYER CONVECTION, PhDthesis, Department of Atmospheric Sciences, Colorado State University, 1987.

[7] Cotton R.C. and R.A. Anthes, STORM AND CLOUD DYNAMICS, volume 44, AcademicPress Inc., San Diego, 1989, International Geophysics Series.

[8] Kitchen M. and S.J. Caughey, TETHERED-BALLOON OBSERVATIONS OF THE STRUCTURE OF SMALL CUMULUS CLOUDS, Q.J.R. Aleteorol. Soc., 107:853-874, 1981.

[9] Cahalan R.F. and J.B. Snider, MARINE STRATOCUMULUS STRUCTURE, Remote Sens.Environ., 28:95-107, 1989.

[10] International Telecommunication Union, International Radio Consultative Committee, RECOMMENDATIONS AND REPORTS OF THE CCIR, 1990, XVIIth plenaryassembly, volume V: Propagation in Non-Ionized Media.

99

100 References

[11] Liebe H.J., MPM - AN ATMOSPHERIC MILLIMETER-WAVE PROPAGATION MODEL,International journal of infrared and millimeter waves, 10(6):631-650, 1989.

{12] Blake L.V., RAY HEIGHT COMPUTATION FOR A CONTINUOUS NONLINEAR ATMOSPHERIC REFRACTIVE-INDEX PROFILE, Radio Science, 3:85-92, 1968.

{13] Smith E.K., CENTIMETER AND MILLIMETER WAVE ATTENUATION AND BRIGHTNESSTEMPERATURE DUE TO ATMOSPHERIC OXYGEN AND WATER VAPOR, Radio Science,12(6):1455-1464, 1982.

[14] Resch G.M., M.C. Chavez, N.!. Yamane, K.M. Barbier and R.C. Chandlee, WATER VAPOR RADIOMETRY RESEARCH AND DEVELOPMENT PHASE FINAL REPORT,Technical report, National Aeronautics and Space Administration, Jet PropulsionLaboratory, 1985, JPL publication 85-14.

{15] Dijk J., M. Jeuken and E.J. Maanders, ANTENNA NOISE TEMPERATURE, Proceedingsof the lEE, 115(10):1403-1410, October 1986.

{16] Brussaard G., RADIOMETRY A USEFUL PREDICTION TOOL?, PhD thesis, UniversiteCatholique de Louvain, 1985, European Space Agency Report sp-l071.

[17] Newton R.W. and J.\V. Rouse Jr., MICROWAVE RADIOMETER MEASUREMENTS OFSOIL MOISTURE CONTENT, IEEE Trans. Ant. Prop, 28:680-686, 1980.

[18] Gassel van P., DATA PROCESSING IN OLYMPUS PROPAGATION INVESTIGATIONS(DUTCH), Master's thesis, Eindhoven University of Technology, 1991, original title:Data processing binnen het Olympus Propagatie Onderzoek.

{19] Beckmann P. and A. Spizzichino, THE SCATTERING OF ELECTROMAGNETIC WAVESFROM ROUGH SURFACES, Artech House, Norwood, USA, 1987.

[20] Slobin S.D., MICROWAVE NOISE TEMPERATURE AND ATTENUATION OF CLOUDS:STATISTICS OF THESE EFFECTS AT VARIOUS SITES IN TIlE UNITED STATES, ALASKAAND HAWAII, Radio Science, 17(6):1443-1454, 1982.

[21] KNMI (Royal Netherlands :Meteorological Institute, CLIMATE ATLAS OF THENETHERLANDS, Staatsuitgevery, 's-Gravenhage, 1972.

[22] Collin R.E., ANTENNAS AND RADIOWAVE PROPAGATION, McGraw-Hill, New York,1985.

[23] Silver S., editor, MICROWAVE ANTENNAS THEORY AND DESIGN, Dover, New York,1965.

[24] Engelhart N.J.M., FROM COM1lUNICATION ANTENNA TO RADIOMETRY ANTENNA,Master's thesis, Telecommunications Division, Eindhoven University of Technology,October 1990.

References 101

[25] Zimmerman M.L., COMPUTATION AND OPTIMIZATION OF BEAM EFFICIENCY FOR

REFLECTOR ANTENNAS, Eleetromagnetics, 11:235-254, 1991.

[26] Gantmacher F.R., THE THEORY OF MATRICES, Chelsea, New York, 1959, transl.

from Russian.

[27] Jahnke E. and F. Emde, TABLES OF FUNCTIONS, Dover, New York, 1945.

[28] Peebles P.Z., PROBABILITY, RANDOM VARIABLES AND RANDOM SIGNAL PRICIPLES,

McGraw-Hill, New York, 1980.

[29]~ D.C., F.O. Guiraud, J. Howard, A.C. Newell, D.P. Kremer, and A.G. Repjar ,

AN ANTENNA FOR DUAL-WAVELENGTH RADIOMETRY AT 21 AND 32 GHz, IEEETrans. Ant. Prop., 27(6), 1979.

[30] Dijk J. and E.J. Maanders, A COMPARISON OF EIGHT ANTENNA TYPES FOR SATEL

LITE COMMUNICATION GROUNDSTATIONS, In International Symposium on Space andTe1Testriai Microwave Propagation, Graz, Austria, 7-9 April 1975.

[31] Kummer W.H. , A.T. Villeneuve and A.F. Seaton, ADVANCED MICROWAVE RA

DIOMETER, ANTENNA SYSTEM STUDY, Technical report, Hughes Aircraft Company,

Antenna Department, 1976.

[32] Ha T.T., DIGITAL SATELLITE COMMUNICATIONS, Macmillan Publishing Company,

New York, 1986.

[33] Landecker T.L., M.D. Anderson, D. Routledge, R.J. Smegal, P. Trikha and J.F.

Vaneldik, GROUND RADIATION SCATTERED FROM FEED SUPPORT STRUTS: A

SIGNIFICANCE SOURCE OF NOISE IN PARABOLOIDAL ANTENNAS, Radio Science,26(2):363-373, 1991.

[34] Anderson M.D., T.L. Landecker, D. Routledge, J.F. Vaneldik, THE FAR SIDELOBES

AND NOISE TEMPERATURE OF A SMALL PARABOLOIDAL ANTENNA USED FOR RADIO

ASTRONOMY, Radio Science, 26(2):353-361, 1991.

[35] Ruze J., ANTENNA TOLERANCE THEORY - A REVIEW, Proceedings of the IEEE,54(4):633-640, April 1966.

[36] Lo Y.T. and S.W. Lee, ANTENNA HANDBOOK: THEORY, APPLICATIONS AND DE

SIGN, Van Nostrand Reinhold, New York, 1988.

[37] Johnson R.C. and Jasik H., ANTENNA ENGINEERING HANDDOOK, McGraw-Hill, New

York, 1984.

[38] Thomas B.M., DESIGN OF CORRUGATED CONICAL HORNS, IEEE Trans. Antennasand Prop., 26(2), 1978.

102 References

[39] Chen J., P.J.I. de Maagt and M.H.A.J. Herben, WIDE-ANGLE RADIATION PATTERN

CALCULATION OF PARABOLOIDAL REFLECTOR ANTENNAS: A COMPARATIVE STUDY,

Technical Report 91-E-252, Telecommunications Division, Eindhoven University ofTechnology, June 1991.

[40] Rahmat-Samii Y., USEFUL COORDINATE TRANSFORMATIONS FOR ANTENNA APPLI

CATIONS, IEEE Trans. Antennas Propagat., 27(4):571-574, 1979.

[41] Computer Department, Eindhoven University of Technology, DOCUMENTATION FOR

THE NUMERICAL TURBO PASCAL LIBRARY, 1 edition, 1991.

Appendix A

Calculation of antennatemperature

•nOIse

A.l Coordinate transformation

In order to calculate the uncertainty introduced by the shape of the pattern (2.1) has to beevaluated. Normally the brightness temperature T is expressed in terms of the sphericalcoordinates t/J, ~, while the antenna pattern is expressed in the spherical coordinates e, 1>(these angles are defined in figure 2.1). Therefore, a coordinate transformation has to beperformed. Since the integrated pattern is expressed in antenna coordinates too, and itwill be necessary to relate the antenna noise temperature TA to the fraction of the powerthat is received within a certain solid angle, T will be written in terms of the antennacoordinates.

The transformation from V', ~ to e, 1> and viceversa will be done in three steps. Firstly, thespherical coordinates of the brightness temperature t/J, ~ will be expressed in the cartesiancoordinates of system {e} = {x, y, z}. Secondly, the antenna coordinates e, 1> will bewritten in the cartesian coordinate system {e'} = {e', y', z'}. Finally, system {e'} will betransformed to {e} [40]. The coordinate systems are defined in figure A.I. The positionof the antenna axis in system {e} is defined by elevation angle E and azimuth angle A.The z'-axis is lying along the antenna axis. The x'-axis is lying in the x, y plane and isperpendicular to the plane spanned by z and z'. System {e} can be transformed to {e'} byrotating x to x' clockwise about the z-axis over the complementary azimuth angle A' andby rotating z to z' clockwise about the x'-axis over the complementary elevation angle E'.

Systems {e} and {e'} are given by:

x - r sin t/J cos ~

y = r sin t/J sin ~

z - r cos t/J

(A.l)

(A.2)

103

104 Appendix A. Calculation of antenna noise temperature

"z,z

,y

,Z

"y

y

Figure A.l: Spherical and cartesian coordinates of brightness temperature

,r sin 0 cos </>x -

y' - r sin 0 sin </>,r cos 0z

(A.3)

Transformation from {c} to {c'} with the rotations mentioned above can be implementedin a matrix, which is denoted as c' AC:

c' AC = (001

co~E' - si~ E' ) ( ~~:1: ~:~nA~' ~) (A A)sin E' cos E' 0 0 1

- (001

Si~E - c~s E ) ( :~~ ~ -s~:~A ~1) (A.5)cos E sinE 0 0

(sin~nc~s A si~~os~:A - c~s E ) (A.6)cos E cos A cos Esin A sin E

Transformation from {c'} to {c} can be achieved by multiplying with (C' Acf. Since

(A.7)

A.2. Performing the integration of the brightness temperature

the following relation between t/J, ~ and 0,4> can be given:

105

(

rsint/Jcos~ ) (SinA sinEcosA COSECOSA)r sin t/J sin ~ = - cos A sin E sin A cos E sin A

r cos t/J 0 - cos E sin E (

r sin 0 cos 4> )r sin 0 sin 4>

r cos 0(A.S)

The last equation shows that the brightness temperature of the sky can be written as

( Tz () TzT t/J =-- => T 0,4> = . . .

) cos t/J sm E cos 0 - cos E sm 0 sm 4>(A.9)

A.2 Performing the integration of the brightness temperature

Since the brightness temperature of the sky has been modeled using a secant law forhigher elevation angles, a linear relation for lower elevation angles and a combination ofemitted and reflected brightness temperature from directions to the ground, the averagebrightness temperature of the sky has to be divided into three contributions. The secantlaw T = a cos( t/J) + b is valid for elevation angles above a (or t/J ::; a'), where a' isthe complementary angle of a (figure A.2). For directions where a' ::; t/J ~ 90° thelinear relation T = ct/J + d should be applied. Finally, for directions to the ground, thetotal brightness temperature observed consists of reflected sky noise, emitted radiation(which are both attenuated along the path from reflection point to the antenna), and thecontribution from the atmosphere between the reflection point and the antenna.

Geometrically the constraint t/J ::; a' defines a cone around symmetry-axis z with anopening angle a' (cone 1). The antenna coordinates 0,4> define a cone around symmetryaxis z' (cone 2). \Vhile integrating the brightness temperature expressed in antenna coordinates (using section A.l), for each 0 possible intersections between cone 1 and cone 2, aswell as between cone 2 and plane z = 0 (the ground), have to be determined.

The cones in figure A.2 can be described by the equations

z'2 tan2()

_ z2 tan?a'(A.I0)

(A.ll)

By using the transformation in section A.l, (A.ll) can be expressed in the coordinatesx',y',z':

x - x' sin A + y' sin E cos A + z' cos EcosAy -x' cos A + y' sin E sin A + z' cos E sin Az -x' cos E + z' sin E

Inserting (A.13) in (A.ll) the following equation can be obtained

X,2 + (y' sin E + z' cos E)2 = (-y' cos E + z' sin E)2 tan2a'

(A.12)

(A.13)

106 Appendix A. Calculation of antenna noise temperature

cone 1

z cone 2z'

z=O

%:Figure A.2: The area where the secant law is valid} delimited by cone 1} and the directions

around the antenna axis for one () (cone 2)

Clearly (A.13) does not depend on the azimuth angle A. The conditions for (), </> are foundby inserting (A.3), which corresponds to (A.I0), in (A.13):

7.2 sin2

() cos2 </> + r 2 sin2 0 sin2 </> sin2 E + r 2 cos2() cos2 E

+ 2r2 sin () sin </> cos () sin E cos E = (A.14)

(r 2 sin2() sin2 </> cos2 E + r 2 cos () sin2 E - 2r2 sin 0 sin </> cos 0 cos E sin E) tan 2 c/

which can be reduced to

sin2() sin2 </>(sin2 E - cos2 Etan2 a') + cos2 O(cos2 E - sin2 E tan 2 a')

+2sinOcosOsin</>cosEsinE(1 + tan2 a')+sin2 0cos2 </>=O

orsin2 0 sin2 </> . A + cos2 0 . B + 2 sin 0 cos 0 sin </> . C + sin2 0 cos2 </> = 0

with

A sin2 E - cos2 E tan 2 a'

B = cos2 E - sin2 E tan2 a'

C cosEsinE(I+tan2 a').

(A.I5)

(A.16)

(A.I7)

For each () two intersections between the cones can be determined, if () > E - a ando< 7r - E - a. By defining

A' A· sin2 0 (A. IS)

A.2. Performing the integration of the brightness temperature

B' B· cos2()

0' - o· 2 sin () cos ()

D' sin2()

the following relation can be obtained

A' sin2 <p + B' + C' sin <p + D' cos2 <p = 0

which leads to

107

(A.19)

(A' - D') sin2 <p + 0' sin <p + B' + D' = 0

::::}

sin <p --0' ± Jon - 4(A' - D')(B' + D')

2(A' - D')

(A.20)

(A.21)

(A.22)

The last equation above yields zero, one, two, three, or four solutions for phi, dependent onthe value of (). Since equations (A.I0) and (A.H) define cones that can be mirrored in theplanes perpendicular to the z-axis and to the z'-axis respectively, care has to be exercisedto find the intersections for z > 0 only.

The intersections of cone 2 with the z = 0 plane are given by (see (A.13))

- y' cos E + z' sin E = 0

or

- cos E . r sin 0 sin <p + sin E . r cos () = 0

::::}

(A.23)

(A.24)

sin <p = tanEtan 0

(A.25)

which yields two solutions <Pl,z::O and <P2,z::O. In case of () < 1r/2 the correct solutionsare found for <p < <Pl,z=O and <p > <P2,z::O' If () > 1r /2 the correct solutions are found for<Pl,z::O < <P < <P2,z=O'

With the intersections determined above for each 0 a complete integration over <P can beperformed, using the correct brightness temperature function for each interval.

Appendix B

Optimization of N parameters

A sum of N parameters can be written as

N

S = l::>.vnan(x)n=1

with maximum

(B.l)

dS

dx

~ dan(x)L..J W nn=1 dx

dal da2 daNWI-+ W 2-+···+ W N-

dx dx dxdaldx

o=>

o

oW2 da2 W3 da3 WN daN- --------- ...---WI dx WI dx WI dx

(B.2)

(B.3)

(B.4)

(B.5)

Analogous a product of parameters can be written as

N

P = II a~n(x)n=1

with maximum

(B.6)

dP

dx

d N- 0 => dx II a~n(x) = 0

n=1(B.7)

~x(a~la~2 ···a~) - 0

O'laCl'l-1 daI (aCl'2 ... a~) +I dx 2

108

(B.8)

(B.9)

(B.lO)

a2a~2-1~(afla33 ... ac;.r) _ .. , _ aNae;.t-1~(afl ... a'!:--na1-1( 0'2 aN) a1-1( 0'2 aN)alaI a2 · " aN alaI a2" . aN

a2-1 d al 0'3-1 d al aN-1 d 0'1a2 a2 a2 a1 a3 a3 a3 a1 aN aN aNal- a1 afl-l dx a~2 - a1 afl-1 dx a33 - ••• - a1 afl-1 dx ae;.t

a2 al da2 a3 a1 da3 aN a1 daN-- - -- - - - -- - ... - - - --a1 a2 dx a1 a3 dx a1 aN dx

These optima are identical if

,-=--,"'-=--,a1 a2 WI a1 a3 WI a1 aN

109

(B.ll)

(B.12)

(B.13)

(B.14)

For every aI, a2, , aN the product optimization yields a set of parameters aI, a2, ... , aN,

with which WI, W2, , WN can be computed, giving the same aI, a2, ... , aN after optimizationof the sum.

The denominators WI en 0'1 indicate only the ratio between the weights is of importance,not their absolute values.

Appendix C

The antenna synthesis program

C.l Introduction

The Antenna Synthesis Program (ASP) has been written in TURBOPASCAL 5.5 and canbe executed on any IBM-compatible personal computer. The program has been given amodular structure; most functions and procedures performing similar tasks are groupedin UNITS. In the next section a user manual is presented discussing the requested inputand resulting output data. In section C.3 the complete structure of the program will bediscussed. The program listing is not included in this report, since it will never be themost recent version.

C.2 User Manual

Before executing the ASP, it should be checked whether the following files are present:

antsyn2.exeubegin.tpuucontrol.tpuuconstr.tpuueigenvl.tpu

uscreens.tpuuprodsum.tpuuplotscr.tpuufacejvv. tpuva1616.txt

xa1616. txtva1616. txtmatAenWs .txtmatAenYs. txt

To start the ASP type antsyn2. First a start-up screen will appear. By pressing Escexecution will stop (this is also the case for any other input screen), any other key willbring up the first menu. It is now possible to choose which optimization one wants toperform out of seven possibilities: aperture efficiency 7Ja, beam efficiency 7Jb, the normalizedsecond moment (72, the product 7Ja7Jb, the fraction 7Ja7Jb/(72 the sum W7Ja +(1- W)7Jb' and theintegrated pattern at one or more points. Pressing Control-Enter forces the program toproceed with the default value shown. All menus are very robust: it is not possible to entera variable of an incorrect type that would bring the program to a stop. Furthermore inputvalues are often limited to practicable values that will not cause any unwanted effects. The

110

C.3. Structure of the Antenna Synthesis Program 111

result of incorrect input will be a soft beep, and the cursor will remain in the current" cell" .In the next menu it is asked how many terms have to be included in the Zernike polyno

mials. The maximum number is 15 as indicated, usually 10 will be sufficient. Furthermore,it is asked whether and how many constraints have to be included. It is possible to giveconstraints with respect to the antenna pattern, as well as to the integrated pattern. Incase of the first, constraints can be 'fixed' (the point will be on the pattern) and 'nonfixed' (the maximum value of a sidelobe is prescribed). In case weighted optimization hasbeen chosen, the weights will be asked for as a number between 0 and 1. Furthermore, ifoptimization of the integrated pattern has been chosen, the point(s) where the integratedpattern should be optimized will be asked for.

In case the number of constraints is larger than zero, a new menu will appear for eachfixed pattern constraint, non-fixed pattern constraint and integrated pattern constraint. Incase of the first the coordinates of a point have to be specified, the value of the normalizedangle u and the gain value in [dB]. In case of the second a sidelobe number and a maximumlevel in [dB] have to be specified. Finally, in case of the last possibility, again the angle u

as well as the fraction of total received power have to be prescribed.At this time the input is finished and the program will proceed by extracting the cor

rectly sized matrices from one of the formerly mentioned . txt files, this is accompaniedby the message" searching matrices ... ". Next the eigenvalue problem will be evaluated once or several times, dependent on the chosen parameter, which is indicated by"calculating eigenvalues ... ". In case a constraint has been given with respect to theintegrated pattern, moments will be added to the parameter to be optimized, resulting ina message "moment .. , ", accompanied by the current order of the moment.

When the program has finished the calculations, a screen will appear showing the results ofthe optimization, together with a menu. It is possible to print the results, or to write themto a file (options 4 and 5 respectively). Furthermore, the antenna pattern, the illuminationpattern, and the integrated pattern can be shown on screen. If one of these options (1,2,4)is chosen, a message will appear showing the current point in the calculation of the chosenpattern. Option 3 will show both the gain and illumination pattern in one screen. If theintegrated pattern is chosen, a lower boundary for the presentation on screen can be chosen(when shown from 0 to 100% sidelobe effects cannot be observed) as well as the angle uup to which integration must be performed (integrating up to 7rD / >. is time consuming).Default values are set to 90% and u = 12. Again, pressing Esc will force the program tostop, while pressing Control-Enter will bring up the first menu, offering the possibilityto perform a new optimization.

C.3 Structure of the Antenna Synthesis Program

In the following paragraphs each unit will be treated separately, briefly discussing thefunctions and procedures in it. In some cases standard functions and procedures havebeen taken from software written by the EDT computer department, which is extensively

112

documented [41].

Appendix C. The antenna synthesis program

antsyn2

This is the start-up file containing the calls to the begin screen (initializegraphicsand mainwindowgraphics both from unit beginscr) as well as the calls to the procedurescontrolling the menu screens (parameter_input, optimization_configuration, get_fixed_constraints, get~oose_constraints,get...hfac_constraints, and get...hfactors, which are all defined in unit uscreens). After these calls all variables from themenus which are of the REAL type are converted to variables with slightly different namesof the type EXTENDED. This is due to the fact that the procedures in uscreens are notable to handle EXTENDED-type variables.

After the menu procedures, procedure start of unit ucontrol will be executed, controlling the input to the eigenvalue procedure, followed by plot_and-results of unituplotscr, which brings up the final menu showing the results of the optimization.

ucontrol

Optimization of a parameter is controlled in ucontrol. After one of the parameters hasbeen chosen in antsyn this parameter will be transported to procedure start in ucontrol.First of all the correctly sized matrices will be read from VA16i6. txt, WAi6i6. txt andXAi6i6.txt, which are written to the files VA.txt, WA.txt and XA.txt respectively. Nextone of a set of procedures in a case structure is called to perform optimization of oneparameter constrained and unconstrained (simple, consimple), of a product of two parameters constrained and unconstrained (simpleproduct, consimpleproduct), of a product of three parameters constrained and unconstrained (cornplexproduct, concomplexproduct), a sum of two parameters constrained and unconstrained (weighted-simplesum,conweightedsirnplesurn) and the optimization of the integrated pattern (hfactor). Theprocedures that execute constrained optimization will use the constraints procedurefrom unit uconstr. After executing one of these procedures the maximum eigenvector andeigenvalue will be known. In results it will be checked whether the optimum eigenvectorshould correspond to the minimum eigenvalue (in case (72 has been optimized) or to themaximum eigenvalue (in all other cases). Furhtermore, the resulting parameters will becalculated according to equations (3.28), (3.29), (3.30) and (3.31).

The remaining procedures in ucontrol are the following: initialize and initialize_cornplexproduct, which compute the starting eigenvalue in case of optimization ofa product of two and three parameters respectively (corresponding to section 3.4.2), andfindmax, which computes the sidelobe levels after an optimization with non-fixed constraints, to check whether the constraints have been satisfied.

ueigenvl

The heart of the ASP is procedure eigenjww in unit ueigenvl, where the eigenvalue problem is evaluated. The Choleski decomposition (section 3.4.1) is performed by procedure

C.3. Structure of the Antenna Synthesis Program 113

reduce, the eigenvalue problem is solved by allsyrnve and the resulting vectors are retransformed by reback successively. Procedure allsyrnve uses tridiag and alltrive.All three are properly described in [41]

uconstr

This unit contains the functions that are related to rewriting the matrices W,V,A and Xin case of constrained optimization. The Householder transform is performed in procedureconstraints. Many other functions and procedures in this unit are used by constraintsand are all taken from [41]: japlusn, jO, j1 for calculating Bessel functions, sign,maxi, maxr, faculty, rpower for determining the sign of a number, the maximum oftwo integer numbers, the maximum of two real numbers, the faculty of a number, and anumber to a certain power respectively, arnmag, arnrnagnl, lngarnma and gamma to calculateGamma functions and finally qadrat to perform an integration. Procedure backtransformcalculates the eigenvector with the correct length, since the result of constrained optimization is a vector of reduced length (section 3.5).

The other functions and procedures in uconstr are placed in this unit because theymake use of some of the standard functions. The antenna gain and illumination patternare calculated by gu and fr (section 3.3), while the integrated pattern is calculated inhu. Function integ-gu is the function that is integrated in hu, using qadrat, in order tocalculate the integrated pattern. Procedure makexa calculates the matrices X and A usingj aplusn. This procedure is used to calculate a new matrix X after an optimization hasbeen performed, in order to be able to calculate the beam efficiency up to the first zero ofthe resulting pattern (section 3.3).

uprodsum

In this unit products and summations of parameters are calculated. Procedure productevaluates (3.49), which leads to the new matrices E and F of (3.50). In etasigma TJa,TJb or (72 is calculated, depending on the input matrices. A weighted sum is calculated inweighted_sum and the total power radiated by the aperture is computed in P-radiated.Again amaalb, amaalv and sinprod are taken from [41] and calculate a product of twomatrices, a product of a matrix and a vector, and the dot product respectively.

ubegin

This unit contains two procedures that will bring up the begin screen with the name of theprogram and the EUT logo. Procedure initializegraphics investigates which graphicscard is present and calculates the resulting maximum number of colours and positionson the screen. The second procedure mainwindowgraphics will draw the screen with theinput title.

114

uscreens

Appendix C. Tbe antenna syntbesis program

In unit uscreens the procedures controlling the user input menus are assembled. Themenu asking which parameter has to be optimized is produced by parameter-input, themenu where the number of terms in the Zernike polynomial has to be inserted, as well as thenumbers of constraints of various types are controlled by optimization_configuration,while the different constraints are asked by get...iixed_constraints. get~oose_con

straints, and get..l1fac_constraints respectively. The points at which the integratedpattern is to be optimized can be entered in the menu, which is shown on screen byget..l1factors. All menus make use of procedures in ufacejww.

ufacejww

This unit contains the basic functions and procedures that control the menu operations.

uplotscr

The results after an optimization are presented in a screen, which is arranged by theprocedures and functions in uplotscr. By a simple menu it is possible to indicate whichfunction or combination of functions should be plotted on the graphics screen and whetherresults should be written to file or printer. This menu is controlled in uplotscr too.

Appendix D

Characteristics of Olympus Satelliteand EDT ground station

Beacon frequencies and wavelengths

12501.866 MHz 0.0240 m19770.393 MHz 0.0152 m29655.589 MHz 0.0101 m

Ground Station

Distance Olympus-EindhovenPosition Eindhoven

Height of station above sea levelAntenna diameterA Ground around radiometer antenna site

38908 [km]51.50 N latitude5.50 E longitude17 [m]5.5 [m]grass

11.5

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