Noise and intercept point receiver planning

C.R.lversen and T.E.Kolding

calculation

Abstract: The paper presents analytical exmessions for determining

for modern radio

noise and intercept points for . _ - cascaded radio receiver stages. The-theory allows for active receiverstages with frequeiicyselectivity and flexible impedance levels. This makes the method highly usable for planning of modern receivers where baseband stages significantly influence the overall performance. A simple homodyne receiver example is used to demonstrate the scope of applicability and to exemplify the proposed theory.

1 Introduction

Traditional methods for calculation of cascaded noise and intercept points are not easily applied to modern receiver archtectures where a considerable portion of the signal processing is performed at baseband. Several issues make existing techniques for receiver planning less applicable. First, as opposed to power-matched receiver stages minded for the RF and microwave range, baseband and low- frequency stages are typically voltage-matched. This means that the input impedance of a stage is typically orders of magnitude lugher than the source impedance. Often, input and output impedances are not well defined, and circuit operation simply relies on a high ratio of input to output impedance in the stage interface. Consequently, traditional power-based approaches are less applicable to the analysis of modern receivers. Second, commonly used methods for calculation of noise in circuits with frequency translation [l] are unnecessarily complicated for receiver planning. Noise factors are not easily converted into noise voltage densitites typically applied when analysing baseband analogue circuits. For planning of modem receivers where baseband stages may have a large noise contribution, it is critical to have access to absolute RMS noise voltages at any inter- face in the receiver chain. When both absolute noise and interference values are known at some point in the receiver, it is possible to make an efficient trade-off between noise and linearity performance. Third, commonly used tech- niques for analysing intercept points do not consider general selectivity [2]. In Sagers' classic article from 1983 [3], selectivity is introduced by considering selectivity and nonlinearity in seperate stages and with selectivity limited to one interfering frequency. This is applicable to receivers where fdtering is primarily performed by passive linear devices and where linearity is dominated by the first stages in the receiver. In modern receiver architectures, active fdters play an important role, and the above h t a t ions are

0 IEE, 2001 . IEE Proceedings online no. 20010396 DOL 10.1049/ip-com:20010396 Paper received 29th March 2000 The authors are with the RISC Group Denmark, Aalborg University, Niels Jemes Vej 12, DK-9220 Aalborg, Denmark T.E. Kolding is also with Nokia Networks, Denmark

thus serious obstacles to accurately plan the receiver. The proposed theory facilitates noise and intercept point calcu- lations that are based on voltage definitions, incorporates mixed selectivity and gain for all stages, and facilitates arbi- trary impedance relations.

2 Noise calculation

The noise scenario is illustrated in Fig. 1. Each stage is characterised by an equivalent input noise voltage density vn,k in VIdHz , a noiseless input resistance Rin,k, a noiseless output resistance Rout,k, and a real loaded voltage gain G&,,), where f k denotes the input-referred pass-band frequency of stage k. The source generator has output impedance R, and equivalent noise temperature T,. Input and output impedances of all stages are considered real (resistive) which is usually sufficient for receiver planning, but may easily be generalised. It is assumed that noise generated in different stages is uncorrelated.

stage "2, out

Rin, I Fig. 1 Test scenario for cascdd mise amlysk

Generally, noise calculations require knowledge of equiv- alent input noise voltage and current as well as the correla- tion between them [5]. However, given the output impedance of the preceding stage, both noise sources can be represented by a single effective noise voltage at the input. The equivalent input mean noise voltage density is defined as [5]

(1)

255 IEE Pro<.-Commun., Vol. 148, No. 4, August 2001

where ( len,k21) is the equivalent input mean square noise voltage of stage k, and Af is the noise bandwidth. For RF and microwave circuits the noise factor, F k , or the effective input noise temperature, T,,k, are common noise quantities. The conversion to equivalent input noise voltage density is given by

Vn,k J 4 k ~ T e , k R o u t , k - 1 T e , k TO(Fk - 1) (2)

where is the output resistance of the preceding stage, kB is Boltzmann's constant, and To is the standard reference temperature (290 K). Note that for mixers the double side-band (DSB) noise factor must be used. Image frequency noise is considered separately in the following. Noise from the source, e.g. the antenna, is included in the first noise generator, Vn,,, by the following relation:

(3)

where Vn,, represents the noise from the first stage, R,y is the source impedance, and T, is the equivalent noise temperature of the source. To handle arbitrary impedance matching between stages, a parameter Kk is defined as

where Rin,k is the input resistance of stage k. In case of power-matched stages, Kk equals 1/4. For fully voltage- matched circuits, ?.e. Rin,k = w,

For frequency translating stages, e.g. mixers, a determi- nation of output spectral noise density calls for some addi- tional consideration. The problem is that noise from the image band of the mixer adds to the output signal band, thereby increasing the noise spectral density at the output. For most practical receivers, one of two cases can be assumed: (i) the image noise is sufficiently suppressed below the noise in the wanted band and therefore image noise does not contribute to the output noise, or (ii) image noise has the same level as the noise in the wanted band and they both contribute equally to the output noise. To consider the possible image noise, a factor a k is defined for each stage as

a k = output noise power from image band and pass-band

= 0, Kk equals 1.

output noise power from pass-band

For stages with no frequency translation, ffk = 1. For homodyne receiver mixers, equal noise from both pass- band and 'image band' (the lower signal side-band) trans- lates to baseband and, hence, ffk = 2. In superheterodyne receivers, common practice includes the use of an image suppression filter in front of the mixer. Such a filter effec- tively suppresses image noise and, hence, ak = 1.

With the above definitions, the actual noise voltage density at the output of stage k across the load terminals, vk,uuf , is determined as

(5)

When studying noise levels throughout the receiver chain, note that eqn. 6 does not include noise from stages k > K. Including all N receiver stages, the total input referred effective noise temperature of the receiver, T,, is calculated as

(7) Each of the Jerms in the sum of T, represents the erective noise temperature of each stage, and inspection of the terms provides useful information about the main noise contributors of the receiver. Having calculated the: total output noise voltage density, V,,,,, by eqn. 6, T, ma.y also be determined from

Note that T, by definition represents an infinite bandwidth noise source at the input. Using T,, the cascaded noise fac- tor F is calculated by the well-known relation

(9) Te TO

F = l + -

However, in the case of frequency translating cascades with contributing image noise, this definition of F is not appro- priate, and it is generally advisable to use T,.

3 Second-order input intercept point

For the analysis of intermodulation (IM) distortion, inter- cept points provide a convenient and measurable quantity. In the treatment %of IM distortion, three general issues must be defined: (i) The type and frequency of the interfering signals that cause the IM distortion. (ii) The frequency location of the IM distortion. For odd- order distortion, the products of concern most often arise on the fundamental frequency. For even-order IM distor: tion, the low-frequency products are often considered, .in particular for low-IF and homodyne receivers where most signal processing is conducted at baseband. (iii) The intercept point related to the above scenario. The IP measure must be measured with correct frequency and type of interferer, as well as the correct source and load impedances. Note that intercept points are small-signal nonlinearity measures, whch cannot be applied under large-signal conditions, i.e. near compression. Other key stage parameters are the pass-band gain, the stage (selectiv- ity at the interfering frequency, and the stage selectivity at the distortion frequency.

Ths Section describes how to calculate the second-order input intercept point, iIP2, for a cascade of receiver stages. The treatment assumes knowledge of iIP2, gain, and selec- tivity for the individual stages. The scenario is depicted in Fig. 2. The second-order input intercept point iIP;!, repre- sents a fictive input level at which the extrapolated desired response intersects with the extrapolated second-order IM product of interest [3] . Further, three different types of iIP2 are derived which can be used to analyse low-frequency IM products.

3. I Cascaded second-order input intercept point Adopting the notation of Fig. 2, the RMS amplitude Ao,k , of the second-order IM product at the output of receiver stage k, is given by

256 IEE Proc -Cummun, Vol 148, Nu 4 , Augu~ t 2001

where is the interferer input RMS amplitude, GkV;S) is the desired signal gain, and iIP2k is the second-order input interception point. Eqn. 10 applies generally to all types of iIP2. Second-order IM is created form either one or two interfering signals. If a two-signal i1P2 is used Ai,& is the RMS amplitude of each of the interfering signals.

Ao, k t l

I J I

GkU) Gk+l(f) ilP2k ilP2k+,

a Ai, k %, k+l

Gk(f).Gk+, (9 ilP2

Fig. 2 Signal scenarw for mbsing second-ordeu mjmc witercept point

At the output of two cascaded stages k and k + 1, there is a total second-order IM contribution given by

The first term in eqn. 11 is the IM product generated in k and transferred by stage k + 1. The second term is the IM product generated in stage k + 1 . Eqn. 11 assumes the worst-case scenario where IM products generated in stages k and k + 1 add in phase at the output of stage k + 1. As shown in Fig. 2, Gk+IV;m) is the gain of stage k + 1 with respect to the IM product. In eqn. 11 it is also assumed that the gain in stage k of the possibly two interfering sig- nals can be represented by a single number Gkvk + AA. This is a fair assumption in most practical cases where the two interfering signals are close in frequency. Collecting terms, the cascaded iIP2 of stages k and k + 1 is given by

- 1 - - Gk+l(f in) + G;( f k + af) ZIP2 Gk+i(fk+i) . iIp2k G k ( f k ) . i Ip2k+l

(12) Generally, the cascade of N stages gives a total order input intercept point of [4]

I k ' J k

k=l

second-

Eqn. 13 generally takes into account the selectivity of the individual stages.

IEE Proc.-Comriiun.. Vol. 148, No. 4, August 2001

3.2 Two-tone ilP2 A commonly employed iIP2 measure is the two-tone iIP2. In measurement and simulation, two closely spaced sinusoi- dal tones with equal levels are applied at the input. The considered second-order IM product at the output is a low- frequency signal with a frequency equal to the spacing between the two sinusoidal tones. A memoryless nonlinear system can be described by the following transfer function:

y ( t ) = ao. z"t) + a1 ( t ) +a2 ex2@) +as -x3 ( t ) + . . . (14)

where x(t) is the input voltage and y(t) is the output volt- age. Note that the factor al is the voltage gain. The two- tone input signal, Xtt(t), where both tones have RMS amplitude A,,, is given by

\/z * Att * (COS 27~fit + COS 27~f2t) (15) The output signal is a sum of the fundamental tones and IM products. The low-frequency second-order IM product at the output is

(16) The two-tone second-order input intercept point iIP2,, is defined as the input RMS amplitude that results in equal output amplitudes of a desired signal and the IM product JJ;,~,~~. Applying a desired input signal with an RMS ampli- tude of iIP2,, results in an output RMS amplitude of /all . iIP2,,. Hence,

yim,t t ( t ) = 2 . a2 . A& . ~ 0 ~ ( 2 7 ~ ( f i - f 2 ) t )

'(ii) 3.3 Single-tone ilP2 For the single-tone iIP2, a single sinusoidal, or potentially phase-modulated signal is applied at the input and the con- sidered second-order output IM product is located at DC. Assuming the transfer function given in eqn. 14 and an input signal of

(18) xst( t ) = JZ- A,t * cos(27~fit + p( t ) )

the resulting IM product at DC is given as

Y i m , s t ( t ) = a2 ' ASt (19) Using the same approach as above, the single-tone iIP2 becomes

relation between single-tone and two-tone intercept point is thus given as

Strictly speaking, eqn. 21 only applies to memoryless non- linear circuits. However, for many practical circuits with DC coupling, the conversion can be applied with good accuracy.

3.4 Time-varying-envelope signal ilP2 Modulation schemes that involve a time-varymg signal envelope are often applied in systems that require high spectral efficiency. For such signals, the two-tone and sin- gle-tone iIP2 measures cannot be applied directly to predict the resulting second-order IM product. A convenient and common way to represent modulated signals is on the vec- torial form:

~ r Q ( t ) = J Z . A I Q + (1(t) .cos27~fit+~(t) .sin27Tf~t) (22)

iIP2,t = Jz . ZIP2tt (21)

257

Assuming that I(t) and Q(t) are scaled such that

J(12(t)>t + (Q2( t ) ) t = 1 (23) the RMS voltage of xlQ(t) is simply AIQ. The symbol < . >, denotes the time averaged mean. By applying the signal xIQ(t) to the nonlinear transfer function of eqn. 14, the IOW- frequency IM product at the output is given by

(24) ~ i m , l ~ ( t ) = a2 . AFQ . ( I 2 ( t ) -t Q 2 ( t ) ) The RMS amplitude of this IM product is given as

&,,IQ = la21 . A:Q * KIQ;

KIQ = d ( ( 1 2 ( t ) + Q2(t))')t (25) where the signal-constant KIQ describes the power variation of xIQ(t). It can be determined from proper knowledge of the statistics of Z(t) and Q(t). In practice, an estimate for KIQ can be calculated given a sufficient number of samples for I(t) and Q(t). Using the same approach as for the two- tone iIP2,

I ~ ~ I ' Z I P ~ I Q = I ~ ~ I . Z I P ~ ~ ~ Q ' E . ~ I Q ~ Z I P ~ I Q = la1 I K I Q . la21

(26) The relation to two-tone iIP2 is thus given as

Measurement of iIP2,Q is more difficult than of iIP2,,, since it requires a source with the correct modulation. Because of the modulated input signal, _simulation is also more cum- bersome since the harmonic balance simulation techniques cannot be applied. Instead, time-domain or circuit-envelope techtuques are needed. With proper knowledge of KTQ and iIP2,, from measurement or simulation, eqn. 27 provides a straightforward conversion to iIP21Q.

4 Third-order input intercept point

The third-order input intercept point, iIP3, represents a fictive input level at which the extrapolated desired response intersects with the extrapolated third-order response of interest [3]. Owing to third-order nonlinearities, two interferers at frequenciesfk + Af and& + 2.Af result in an TM product at the desired signal frequency j i Hence, this type of distortion is not separable from the desired signal. This Section describes how the iIP3 can be deter- mined for a cascade of receiver stages and how the commonly applied two-tone iIP3 can be determined for a memoryless nonlinearity.

4. I Cascaded third-order input intercept point (ilP3) Using the notation in Fig. 3, the output RMS amplitude of the thrd-order IM product generated in stage k is given by

Note that Al,k is the RMS amplitude of the interfering sig- nal located nearest to the pass-band frequency. Talung into consideration the selectivity of stage k, the third-order out- put contribution after stages k and k + 1 is given by

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assuming that generated IM products of stages k and k + 1 add in phase at the output of stage k + 1. Again, this is a worst-case scenario. Collecting terms, the cascaded iIP3 of the two stages k and k + 1 is given by

1 G;( fk -t af ) . G k ( f k + 2 n f ) + - -- 1 iIP32 dP32 . Gk ( f k ) . iIP3;+,

(30) Generally, the cascade of N stages gives a total thircl-order intercept point of [4]

k = l

which takes the relevant selectivity of all stages into account.

'1. k+ l

- - _./ stage bl stage k k + l

Fig.3 Test scenario for analysing third-order input intercept point

4.2 Two-tone ilP3 To achieve two-tone iIP3, the memoryless nonlinear system given by eqn. 10 is excited with an input signal

ztt(t) = Jz. A1 . cos((27rfO + 27rAf)t)

+h. Aa.cos((2nfo+2. 27rAf)t) (32)

where A , and A2 are the RMS amplitudes of the two-inter- feres. The resultant third-order IM product atfb is given as

The two-tone third-order input intercept point, iIP3, is defined as the input RMS amplitude that results in equal output amplitudes of the desired signal and the IM product Y i m p Hence,

(34) When testing the receiver, the interferer in offset 2 . Af is typically a modulated signal whereas the interferer in offset Afis a sinusoidal. For this setup, the iIP3 given by eqn. 34 still applies.

IEE Proc.-Commun., Vol. 148. No. 4 , .4ugust 2001

5 Demonstration example

The applicability of the derived theory can be demon- strated through a small example. The example covers a 2 GHz homodyne receiver consisting of three cascaded stages: a low noise amplifier (LNA), a direct down- converter (CNV), and a baseband filter (BBF). The receiver is illustrated in Fig. 4 and the associated stage data is listed in Table 1. The example includes the calculation of noise parameters, cascaded iIP2 for interferers offsets by lOMHz and 100MHz, and cascaded iIP3 for two-tone interference in offsets of 10MHz and 20MHz.

LNA CNV BBF

2 GHz

llLl - Fig. 4 Homodyne receiver in &nonstrution e m p l e

Table 1: Stage data for the homodyne receiver example

Parameter unit LNA CNV BBF

K 1 2 3

Gk,low frequency 99.0 10-3 63.2 10-3 1.00 103

Gk( fk) 10.0 3.16 1.00 io3 Gk(fk+ Af ) , A f = 10MHZ 10.0 1.62 15.7

G&fk+Af ) ,A f= 15MHz 10.0 1.17 4.64

Gk( fk+ Af ) , A f = 20MHz 10.0 906 x 10” 1.96

Gk( fk+Af ) ,A f= 100MHz 6.94 189 10-3 15.7 10-3

Rin, k 52 50 200 10000

Rout,k P 50 10 n.a.

Eq. input noise voltage nV/\/Hz 0.8 1.7 8.0

ilP3 (IOMHz, 20MHz) VRMS 0.200 1.00 0.500

ilP2 ( A f = 15MHz) VRMS 404 250 150

ilP2 ( A f = 100MHz) VRMS 404 250 150

To verify the methods presented in this paper, harmonic balance and nonlinear noise simulations have been conducted in Advanced Design System vl.3 (Agilent Tech- nologies, Palo Alto, CA, USA). Recall that the theory presented for intercept points assumes in-phase voltage addition of all components, which constitutes the worst- case scenario. Hence, iIP simulation results obtained by harmonic balance simulation can in fact be lower than the worst-case scenario, since IM products from different stages can be out of phase. To get consistent results, the verification example has been created such that all voltages

add in phase. As seen in Table 2, the resdts obtained using harmonic balance simulation are in total agreement with the analytical results.

Table 2: Comparison between calculated results and simula- tions

Test unit calculation ADS simulation

output noise pVfdHz 28.6a 28.6

ilP2 @ 15MHz VRMS ’ 14.5’ 14.5

ilP2 @ 100MHz VRMS 50.9‘ 50.9

48.5 ilP3 @ 10/20MHz mVRMs 48.5d

a ~ ~ = 0 . 2 5 , ~ ~ = 0 . 6 4 , ~ ~ ~ 1 , a ~ = 1 , a ~ = 2 , a n d n 3 = 1 . Vn,1=1.20nV/ d/(Hz) b 11 = 1, 12 = /I x (10.02/10.0) = 10.0, 5 = 12 x (1.172/3.16~ = 4.33, J~ = (63.2 x 10-3/3.16) (1.00 103/1.00 103) = 20.0 x io9, J2 = (1.00 x 103/l.00 x I O 3 ) = 1.00, and J3 = 1. See eqn. 13 c / l = 1, /2 = I1 x (6.942/10.0) = 4.82, 5 = 12 x (0. 189‘/3.16) = 54.4 x IO3. Since Jk is independent of A t values under bare valid dH2 = (10.02 x 10.0)/10.0 = 100, and H3 = H2 x (1.62’ x 0.906)/3.16 = 75.2

6 Conclusions

This paper has presented theory for noise and intercept point calculation for cascaded stages. The theory is based in the voltage-domain facilitating general impedance match between stages. Further, the theory incorporates the selec- tivity at important frequencies, which improves the accu- racy of intercept point considerations. Since only a small amount of information is required for each stage, the method is generally applicable to the planning of modem radio receivers where baseband circuits are important for the overall performance.

7 Acknowledgment

The authors would like to acknowledge the useful discus- sions on noise theory with Dr. Torben Larsen, Aalborg University, Denmark.

References

LARSEN, T.: ‘Noise temperature of multiresponse systems’, IEE Proc. H, Microw., Antennus. Propug ., 1992, 139, (4), pp. 309-313 ROHDE, U.L., WHITAKER, J, and BUCHER, T.T.N.: ‘Communi- cations receivers; principles and design’ (McGraw-Hill, 1997) SAGERS, R.C.: ‘Intercept point and undesired responses’, IEEE Trans. Veh. Technol., 1983, VT-32, pp. 121-133 KOLDING, T.E.: ‘Simple analysis of cascaded noise factor and inter- cept points for voltage-defined stages’. RISC Group Denmark, Aal- borg University, Denmark, technical report R99-1007, November 1999 ENGBERG, J., and LARSEN, T.: ‘Noise theory of linear and nonlin- ear circuits’ (Wiley, 1995)

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