plugin-11. pulse compression_2011

8/2/2019 Plugin-11. Pulse Compression_2011

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9. Pulse Compressionand Waveforms



7. Pulse compression and waveforms

7.1 Introduction!

7.2 The matched filter and pulse compression!

7.3 Linear FM and SAW devices!

7.4 Range side-lobes!7.5 The ambiguity function!

7.6 Phase and amplitude errors!



Introduction: The Waveform

• Waveform design has always taken into account:!

• Detection range!

• Ambiguity performance!

• Bandwidth!

• Range resolution!

• Doppler resolution!• Duty cycle!

!

A wide variety of waveform design have been used through thehistory of radar development.!

!The waveform determines detection performance and accuracy ofmeasurement.!!

However, the waveform component of radar is undergoing

something of a revolution.!



Introduction: The Waveform

• The waveform provides illumination for radar enabling it to view

the world within the specification defining the waveform.

• The waveform can be:

§ Continuous or pulsed

§ Modulated or un-modulated (in amplitude and/or phase and/

or frequency)

• The waveform is a key factor in determining radar capability and

performance.

• It is becoming more important as technology now allows adaptive

design of waveforms to maximize performance.

• This is an areas of vigorous research.



Waveforms: A Taxonomy



Pulse Compression

Start with the standard form of the radar equation :!

!!

!

!

!

For a simple pulsed radar, B , the receiver noise bandwidth, would be matched

to the pulse length t such that B ≈ 1/ t, so that :!!

!

!

!

!

The product P t t represents the energy of the transmitted pulse.!

!

!

High range resolution demands a small value of t , but good detection range

performance demands high transmit pulse energy, and hence (for a given

peak transmit power) a large value of t . How do we resolve this? !

( )

2 2

3 4

0

4

t r

n

PG L P

P r kT BF

λ σ

π

=

( )

2 2

3 4

0

4

t r

n

P G L P

P r kT F

τ λ σ

π

=



So instead of a short pulse, we transmit a long pulse (duration T ), modulated in

some way so as to spread the energy over a bandwidth B to give the requiredrange resolution (i.e. Dr = c /2B ).!!

The energy of the transmitted pulse is now P t T , so the radar signal-to-noise ratiofor a given target at a given range has been increased by the factor T / t , or BT .!

!

BT is known as the time-bandwidth product of the waveform, and is also theprocessing gain provided by the pulse compression processing.!

!

The maximum detection range for a given target is increased by a factor!!

For example, for BT = 100, the processing gain is 10 log10(100) = 20 dB and thesignal-to-noise ratio for a given target at a given range would be increased by this

factor.!

!

Alternatively, the maximum detection range for this target would be increased by a

factor .!

( )4 BT

( )4 100 3.3= ×

Pulse Compression



Let g (t 0) be the maximum value of g (t ). !

!The power spectrum of the noise at the output of the filter is :!!

!

!

!

where N 0/2 is the noise spectral density at the filter input. (The factor 1/2 occursbecause both positive and negative frequencies are used in this analysis, and

the usual definition of noise density uses only positive frequencies)!

!

The average noise output power is then :!

!

!

!

!

The energy of the input signal can be written :!

( ) ( )2

0

2

N G H ω ω =

( ) 20

2

N N H df ω

∞

−∞

= ∫

( ) ( )2

2 E u t dt F df ω

∞ ∞

−∞ −∞

= = ∫ ∫

The Matched Filter



( )( ) ( ) ( )

( )

2

2 0

0

20

exp

2

F H j t df g t

N N

H df

ω ω ω

ω

∞

−∞

∞

−∞

=

∫

∫

An optimum radar detector must maximize the ratio of peak signal power to

mean noise power at its output :!!!!!!!!!So we seek the receiver transfer function, H (w ), which maximizes this ratio.!!

This can be found using Schwarzs inequality :!

( ) ( ) ( ) ( )2

2 2

x y d x d y d ω ω ω ω ω ω ω

∞ ∞ ∞

−∞ −∞ −∞

≤ ∫ ∫ ∫

The Matched Filter



From this inequality, it follows immediately that :!

!!

!

!

!

The maximum output signal-to-noise ratio occurs when the two sides are

equal, which is true only if :!!

!

!

!

where K is a constant (gain) and t 0 is the time delay through the filter.!!

This says that the frequency response of the matched filter is equal (apart

from K and t 0) to the complex conjugate of the signal spectrum F (w ).!!

( )2

0

0

2

g t E

N N ≤

( ) ( ) ( )*

0exp H K F j t ω ω ω = −

The Matched Filter



In the time domain the impulse response of the matched filter is :

which is a time-delayed inverse of the input waveform, multiplied by a simple

gain constant.

Thus the time-domain output is the convolution of the input signal with the

impulse response, which is :

which is the autocorrelation function of the input signal (in the absence of noise)

( ) ( )*

2 0 h t K u t t = −

( ) ( ) ( )/ 2

0 0

/ 2

1

T

T

g t f f t t d

T

τ τ τ

−

= + −

∫

The Matched Filter



Summary

• The matched filter maximizes the ratio of the peak signal power to

the mean noise power.!

• Its frequency response is the complex conjugate of the spectrum ofthe input signal.!

• Its impulse response is the time-inverse of the input waveform.!

• The matched-filtered output waveform is the autocorrelationfunction of the input waveform.!

• The convolution or correlation is usually performed at baseband.!

• Note: We have assumed that the echo has not been modified fromthe transmitted signal. I.e. the target has the properties of a deltafunction. This suffices in almost all cases. If we want to know moreabout the target, it is the modulation which provides thisinformation.!



The Matched Filter Response foran Un-Modulated Pulse

Consider a simple pulse of unit amplitude with a square functionenvelope given by:!!

X(t) = A for –T/2 < t < T/2!

!

The match filtered response in a simple triangle given by:!!

Y(t) = A2 (t+T), -T < t < 0 or A2 (T-t) 0 < t < T!

Note: The length of the filter is twice that of the time delay



Practical Square Pulse Waveforms

Ideal square pulses are not realizable in practice and they will exhibitfinite rise and fall times (and often other modulations not desired).!

!

This leads to side-lobes which together with the main-lobe form the

total matched filter response, E.g.!



Practical Square Pulse Waveforms

Targets are not always centered within the illuminating pulse andthus there is a loss in SNR known as Straddling loss.!

!

The width of the pulse (usually determined at the half power or -3dB points is taken by rule of thumb to represent the range resolvingability of the waveform.!!

The more formal Rayleigh criterion states that two targets areresolved when they are separated in range such that the peak of theresponse of one target falls on the first null of the second target.This equates to the -6 DB position for a simple pulse and a point

target.!!

Side-lobes are undesirable as they “consume” energy that spreadsinto other ranges. A large target may conceal a small one due to theside-lobes of the large target being larger than the main-lobe

response of the small target.!!



The Linear FM Waveform

One of the simplest and most widely-used form of modulation used to spreadthe bandwidth of a long transmitted pulse is the linear frequency modulated

chirp.

time

frequency

T

B

( )2

0cos ,

2 2 2

t T T u t t t

µ ω

⎛ ⎞= + − ≤ ≤⎜ ⎟

⎝ ⎠

2

B

T

π

µ =



Spectrum of linear FM

The spectrum of a linear FM signal is given by its Fourier transform (see, forexample, Klauder et al, 1960) :!

!

!

!

!

!!

!

!

!

!

where!!

!

!

and!

( ) ( ) ( )exp 2U f u t j ft dt π

∞

−∞

= − ∫

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2

001

exp4 2 0

C X jS X C X jS X f f j

C X jS X C X jS X

µ

π µ µ µ

+ + − −

+ + − −

⎧⎡ ⎤ + + + >⎧ ⎫−⎪ ⎪ ⎪⎢ ⎥= − ⎨ ⎬ ⎨⎢ ⎥ − + − <⎪ ⎪ ⎪⎩ ⎭⎣ ⎦ ⎩

( ) ( )

2 2

0 0cos and sin2 2

X X x x

C X dx S X dx

π π ⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ∫ ∫

( )( )02 2

sgn2 2

f f X

π µ µ

π µ

±

± −⎡ ⎤⎣ ⎦=



Spectrum of linear FM

As the time-bandwidth product increases, the spectrum approaches arectangular shape of bandwidth B . The deviations are known as Fresnel

ripples .!

BT = 50 BT = 100 BT = 500



Autocorrelation function of linear FM

The output from the matched filter is by definition, the autocorrelationfunction, i.e. the convolution of the signal with the impulse response of the

matched filter.!!

The impulse response of the matched filter is :!!

!!

!

!

and the output of the matched filter is :!

( )

1 2 2

0

2cos

2

t h t t µ µ ω

π ⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

( ) ( )1 2

2 2

0 0

sin 2 Re exp2 2 2 4

Tt T t g t j t

Tt

µ µ µ π

ω

π µ

⎡ ⎤⎛ ⎞ ⎛ ⎞= + +⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎣ ⎦



Autocorrelation function of linear FM

The important part of this has the characteristic sinx / x form with apeak sidelobe level of -13.2 dB.!

( ) ( )

1 22 2

0 0sin 2 Re exp

2 2 2 4Tt T t g t j t

Tt µ µ µ π

ω

π µ ⎡ ⎤⎛ ⎞ ⎛ ⎞= + +⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎣ ⎦

-6 -4 -2 0 2 4 6

-120

-100

-80

-60

-40

-20

0

d e l a y

Ambiguity funct ion (dB)



Surface acoustic wave (SAW) dispersive delay lines

Surface Acoustic Wave technology has been developed to generateand compress waveforms. They consist of piezo-electrictransducers at input and output to convert the electrical signal to anacoustic signal (mechanical vibration), which propagates as asurface acoustic wave. Various geometries can be used; thereflective array compressor uses etched lines at varying spacing,which act as resonant reflectors when the spacing is equal thewavelength of the acoustic signal.!!

In this way the device acts as a dispersive delay line, since differentfrequency components of the input signal travel different pathlengths

through the device. Hence the frequency components of a shortpulse will be dispersed to generate a chirp waveform.!!

The inverse characteristic can be used to compress the echosignals in the receiver. Amplitude weighting can be included to lower

range sidelobes.!




λS

λC

λL

IMCON line

electro-acoustic

transducer

acoustic path(shear wave)

scribed line

thin steelstrip

selectivedamping

can beinserted

dispersion, µs

0.1 1 10 100 1000

1000

100

10

1

B a n d w i d t h , M H z

Surface Wave RAC

IMCON

steel strip

lumped constant

perpendicular diffraction

tapped delay

TB = 110

100

1000

10,000

Dispersive delay-line technology andlimitations

Reflective - Array!Compressor!




bandwidth 500 MHz

uncompressed pulse length 0.46 µs compressed pulse length 3 ns

centre frequency 1.3 GHz

peak range sidelobe level −24 dB peak insertion loss 40 dB

size 0.5 × 1.5 × 2.25 in

Typical device parameters (quoted by Skolnik) : !




frequency

time

power

time

power

time

transmitter

receiver H (f ) *

H (f )



Pulse compression - Summary

(i) More efficient use of the average power available at the radartransmitter and, in some cases, avoidance of peak power problems in the

high power sections of the transmitter.!!

(ii) Increased system resolving capability, both in range and velocity. In thecase of range resolution, the generation of extremely fast rise-time, high

peak power signals is bypassed when pulse compression techniques areused. !!(iii) Reduction of vulnerability to certain types of interfering signals that do

not have the same properties as the coded waveform.!!

But :!!

(iv) Increased minimum range!

!

(v) Range sidelobes!



The Ambiguity Function

The matched filter assumes that the received waveform is a scaledand time delayed replica of the transmitted signal.!!

When a radial velocity component exists between the radar and atarget, a Doppler shift is imparted on the received waveform.!!

This represents a potential miss-match in the filtering operation andmust be understood.!!

This is captured by Woodwardʼs “Ambiguity Function ” whichrepresents the range and Doppler resolution and ambiguity

properties of any given waveform or pulse train of waveforms.!!

The ambiguity function is often represented by a three-Dimensionalplot centered on zero Doppler and a desired delay range. Thesurface of this plot is the ambiguity surface.!



Philip Woodward

Courtesy of Lars Falk!




Mathematically, we have seen that the output of a matched filter for a givenwaveform as a function of time (the point target response) is given by the

autocorrelation function of that waveform.

Woodward generalized this idea, to consider the response of a matched filter for a given waveform to an echo of that waveform with a Doppler shift ν :

The time-reversed matched-filter response is obtained by substituting -τ for τ

( ) ( ) ( )* g t u x u x t dx= − ∫

( ) ( ) ( ) ( )* exp 2 g t u x u x t j x dxπν = −

( ) ( ) ( ) ( )*, exp 2u x u x j x dx χ τ ν τ πν = +

*




The ambiguity function is defined as the square magnitude of this :!

!!

!

!

!

!

!!

!

If the signal amplitude is normalized so that!!

then !

!!

!

!

which is important, because it shows that energy suppressed somewhere inthe ambiguity function must reappear somewhere else !!

( ) ( ) ( ) ( )22 *, exp 2u x u x j x dx χ τ ν τ πν = − ∫

( )2

1u t dt = ∫

( )2

, 1d d χ τ ν τ ν = ∫∫



Range side-lobes (or time side-lobes)

The range side-lobes of echoes from strong targets can obscure weaker,close targets, in the same way as antenna side-lobes, but range side-lobes

do not have the two-way protection of antenna side-lobes.!!

Range side-lobes can be reduced in the same way as with antenna side-lobes, using an amplitude taper. The same taper functions can be used,

and they have the same effects of loss and broadening of main-loberesponse :!

Weighting function Peaksidelobe (dB)

Loss(dB)

Mainlobewidth(relative)

Sidelobedecayfunction

uniform −13.2 0 1.0 1/t

( )20.33 + 0.66cos f Bπ −25.7 0.55 1.23 1/t

( )2cos f Bπ −31.7 1.76 1.65 1/t

3

( )Taylor 8n = −40 1.14 1.41 1/t

Dolph-Chebyshev −40 …. 1.35 1

( )20.08 + 0.92 cos (Hamming) f Bπ −42.8 1.34 1.50 1/t



The Ambiguity Function Diagram

This is a powerful way of representing the performance of a given waveform.

A 3-D plot of the ambiguity function is referred to as an ambiguity function diagram :!

time!

(range)!

Doppler!(velocity)!



The Ambiguity Diagram

The ambiguity function and ambiguity diagram contain a large amount ofinformation about the performance of the radar waveform :!

!

• range resolution!

• range side-lobe level!

• range ambiguity spacing!

• Doppler resolution!

• Doppler side-lobe level!

• Doppler ambiguity spacing!

!

The ideal ambiguity function is the so-called thumbtack - i.e. an infinitely-narrow spike at the origin, giving infinitely-high resolution in range and

Doppler, no side-lobes, and no ambiguities.!



The Ideal Ambiguity Diagram!



The Monostatic Ambiguity Function

Ø Central peak gives an estimate of range and Doppler resolution.!

Ø Appearance of other peaks in the range – Doppler plane correspond topotential ambiguities.!

Ø Formulation:!

Ability to distinguish betweenmultiple targets

Clutter rejection

Ø Matched filtering:

)2exp()()( * fT j f kS f H π −=

),( D R

t S ω ),( D

t X ω

∫ ∞

∞−−= dt t jt T ut uT X D D )exp()()(),( *

ω ω



Ambiguity function examples – a single pulse

0 1 2 3 4 5 6 7 8 90

0.5

1

A m p l i t u d e

Signal used

0 1 2 3 4 5 6 7 8 90

0.5

1

p h a s e [ r a d ]

0 1 2 3 4 5 6 7 8 90

0.5

1

time (µsec)

f r e q u e n c y

-400 -300 -200 -100 0 100 200 300 4000

0.2

0.4

0.6

0.8

1

Doppler shift in KHz

a m b i g

u i t y f u n c t i o n X ( 0 , f ) cut along the doppler and range axis

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Range in Km

a m b i g u i t y f u n c t i o n X ( t , 0 )

Stationaryobject at

50 km

10μs pulseperiod

Range resolution!

Doppler resolution!

0

0

Range

D o p p l e r s h i f t

Ambiguity function contour plot



Ambiguity function examples – FM pulse

Stationary

object at100 Km

500μspulse

period

!

0 50 100 150 200 250 300 350 400 450 5000

0.5

1

A m p

l i t u d e

Signal used

0 50 100 150 200 250 300 350 400 450 500-4

-3

-2

-1

0

p h a s e [ r a d ]

0 50 100 150 200 250 300 350 400 450 500

-0.04

-0.02

0

0.02

0.04

time (µsec)

f r e q u e n c y

0 20 40 60 80 100 120 140 160 180-20

-15

-10

-5

0

5

10

15

20

Range in Km

D o p p l e r s h i f t i n K H z


-20 -15 -10 -5 0 5 10 15 20

0

0.2

0.4

0.6

0.8

1


a m b i g u i t y f u n c t i o n X ( 0 , f ) Cut along the doppler and range axis

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

Range in Km


A bi it f ti l



Ambiguity function examples –Stepped frequency pulse train

Stationaryobject at

50 km

10μs pulse

period

!

300μs

signalduration

0 50 100 150 200 2500

0.5

1

A m p l i t u d

e

Signal used

0 50 100 150 200 250

-2

-1

0x 10

-4

p h a s e [ r a d ]

0 50 100 150 200 250

-1

0

1

x 10-5

time (µsec)

f r e q

u e n c y

0 10 20 30 40 50 60 70 80 90

-80

-60

-40

-20

0

20

40

60

80

Range in Km

D o p p l e r s h i f t i n K H z


-100 -80 -60 -40 -20 0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1


a m b i g u i t y f u n c t i o n X ( 0 , f ) Cut along the doppler and range axis

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

Range in Km


ambiguity!

ambiguity!



Ambiguity diagrams

Rectangular Pulse!Linear Frequency

Modulation

(Chirped)!

Coherent Pulse

Train !

bi i di



Ambiguity diagrams

Barker

Code!

Pseudo Random Binary

Code!

Costas

Code!

Eff f h d li d



Effect of phase and amplitude errors

• Any practical radar will introduce phase and amplitude errors in thetransmitter and receiver stages. These errors can degrade therange side-lobe level.!

• The effect was studied by Klauder, Price, Darlington andAlbersheim (1960), who treated the phase and amplitude error

functions by expanding them each as a Fourier series, thenevaluating the effect of each term of the series. By analogy withamplitude modulation theory, it is easy to see that a givenamplitude error term will result in a pair of sidebands, either side ofthe main response. By analogy with narrowband FM the same is

true of phase errors. These are known as “paired echoes” .!

• Klauder et. al. evaluated the permissible level of amplitude andphase error for a given range side-lobe level. !

Ph d lit d i l i



Phase and amplitude errors in pulse compression

* phase and amplitude errors will degrade ! point target response ! * treat errors as periodic (Fourier series !

representation) ! * results in !paired echo ! sidelobes !

* can use !actual ! transmitted pulse as ! range reference function!

0.01 0.02 0.06 0.1 0.2 0.6 1.0 2 4

102030405060 L

E V E L O F F I R S T E C H O B

E L O W

M A I N S I G N A L ( d B )

AMPLITUDE DEVIATION, , (dB)(1+aoa1)

102030405060 L

E V E L

O F F I R S T E C H O B E L O W

M A I N S I G N A L ( d B )

0.1 0.2 0.6 1.0 2 4 6 10 20 40PHASE DEVIATION, b (degrees)

1

Note: Combinations of amplitude andphase errors are likely and will have a

larger overall effect.!



Further reading

Woodward, P.M., Probability and Information Theory, with Applications to Radar ,

Pergamon Press, 1953; reprinted by Artech House, 1980.!!

Arthur, J.W., ʻModern SAW-based pulse compression systems for radar applications ̓ ; part 1: SAW matched filters Electronics & Communication Engineering Journal,

December 1995, pp236-246; part 2: Practical systems, ibid April 1996, pp57-78.!!

Klauder, J.R., Price, A.C., Darlington, S. and Albersheim, W.J., 'The theory and design of chirp radars ', Bell System Tech. J., Vol.XXXIX, 1960, No.4, pp745-809;reprinted in Radars - Vol.3, Pulse Compression, D.K. Barton ed., Artech House,1975.!

!

And!

!

I have an electronic copy of Woodwardʼs original memo on the ambiguity function ifyouʼd like one let me know.!

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