MS_7장 -Pulse Compression -Rev

Prof Kwag @ RSP Lab Korea Aerospace Univ.

Chapter 7.

Pulse Compression


7. Pulse Compression

- Using short pulse improved range resolution, decreased average power

- Desirable to increase pulse width while maintaining adequate range resolution

using pulse compression technique

: allows us to achieve the average transmitted power of a relatively long pulse,

while obtaining the range resolution corresponding to a short pulse

BN

N i2

2 0

< Input noise power >

Time-Bandwidth Product - Noise power available within the matched filter bandwidth

(7.1)

B


7.1 Time-Bandwidth Product

- average input signal power over a pulse duration

energy : signal E

ES

'i

'0 2 BSNR

tSNR

i

- matched filter input SNR

- output peak instantaneous SNR to the input SNR ratio

'

0 BN

E

N

SSNR

i

ii

Called the matched filter gain or compression gain

Unmodulated pulse approaches unity

Made much grater than unity by using frequency or phase modulation

(7.2)

(7.3)

(7.4)

'

0

0

2

N

EtSNR

'B - ( time-bandwidth product ) : corresponding matched filter

Perfectly matched compression gain 'B

Compression gain < 'B spectrum of the matched filter deviates form transfer

function of the input signal


7.2 Radar Equation with Pulse Compression

- radar equation for a pulsed radar

FLkTRGP

SNRe

t

43

22'

4

- pulse compression radar

: transmit relatively long pulses, process the radar echo into short pulses

: view the transmitted pulse to be composed of a series of short subpulses

(7.5)

FLkTR

GPSNR

e

ct

c 43

22

4

- for an individual subpulse Eq. (7.5)

(7.6)

widthpulsed :compressec


Radar Equation with Pulse Compression

RSP Lab

- SNR for the uncompressed pulse

FLkTR

GnPSNR

e

ct

c 43

22'

4

(7.7)

subpulsesofnumberthen :

using pulse compression

: maintained detection range (keeping the pulse width unchanged)

: improved range resolution (increasing the bandwidth)

BcR 2 (7.8)


7.3 Analog pulse compression

Correlation Processor

- Accomplishing by adding frequency modulation to a long pulse at transmission, using matched filter receiver in order to compress the received signal.

- By using long pulses and wideband LFM modulation,

we can achieve large compression ratios.

- In example, two targets with RCS=1, RCS=2 are detected. the two targets are not separated enough in time to be resolved.

- However, after pulse compression the two pulses are resolved as two target.

when using LFM, returns are resolved as long as they are separated by n1. ( where, n1 is the compressed pulse width )


< Composite echo signal for

two unresolved targets >

< Composite echo signal after

pulse compression >

7.3.1. Correlation Processor



- Returns from targets within the receive window are collected and passed

through a matched filter to perform pulse compression

- In the correlation processor digitally performed, digital implementation is

called Fast Convolution Processing (FCP) and can be implemented at base-

band.

< Computing the matched filter output using an FFT >



- Since the matched filter is a LTI system, its output can be described,

thtsty

where, s(t) is the input signal,

h(t) is the matched filter impulse response (replica)

- From FFT properties,

fHfSthtsFFT }{

- Both signals are sampled properly, the compressed signal y(t) can computed

from,

}{1 HSFFTy

where, FFT-1 is the inverse FFT

(7.9)

(7.11)

(7.10)


< Reducing the first side lobe to

-42dB doubles the main lobe width>


- When using pulse compression, it is desirable to use modulation schemes

to accomplish a maximum pulse compression ratio

to reduce the side lobe levels of the compressed waveform

- Weighting functions can be used on the compressed pulse spectrum in order

to reduce the side lobe levels.



- Weighting the time domain transmitted or received signal instead of the

compressed pulse spectrum will theoretically achieve the same goal,

but, since amplitude modulation burdens on the transmitter, rarely used.

- Consider a radar system that utilizes a correlation processor receiver

the receive window in meters is defined by,

minmax RRRrec

where, Rmax and Rmin define the maximum and minimum range

- Typically Rrec is limited to the extent of the target complex.

the normalized complex transmitted signal has the form,

'0)2

(2exp 20

tttfjts

- is the pulse width, = B/ , f0 is denotes the chirp start frequency.

(7.13)

(7.12)


Received Signal

- Consider a target at range R1, the echo received by the radar from this target is

cRdelaytime

a

ttfjatsr

11

1

2

1101

2

nattenuatiorangeandgain,antennaRCS,targettolpropotionaiswhere

)(2

)(2exp)(

- The first step of the processing consists of removing the frequency ,

accomplished by mixing with a reference signal whose phase is .

0f

)(tsr tf02

- Phase of the signal, after low pass filtering, is

20 )(

22)( ii tft

- Instantaneous frequency is

c

Rt

Btt

dt

dtf ii

12)()(2

1)(

(7.16)

(7.17)

(7.14)

(7.15)


Sampling the quadrature components - Quadrature components are

)18.7()(sin

)(cos

)(

)(

t

t

tx

tx

Q

I

- Sampling the quadrature components, the number of samples, N, must be chosen

so that foldover (ambiguity) in the spectrum is avoided.

For this purpose, sampling freq. based on the Nyquist sampling rate.

)19.7(2Bfs

the sampling interval is

)20.7(21 Bt

- Freq. resolution of the FFT is

)21.7(1 f

- The minimum required number of samples is

)22.7(1

ttfN


Sampling the quadrature components

- Eqs. (7.20) and (7.22) yields

)23.7(2 BN

A total of real samples, or complex samples, is sufficient to completely describe an LFM waveform of duration and bandwidth .

B2 B B

- For example, an LFM signal of duration and bandwidth requires

200 real samples to determine the input signal.

(100 samples for the I channel and 100 samples for the Q channel)

s 20 MHzB 5

- For better implementation of the FFT N is extended by zero padding.

thus, the total number of samples, for some positive integer m, is

)24.7(2 NN mFFT


FCP Processing

- The final steps of the FCP processing include :

1) Taking the FFT of the sampled sequence.

2) Multiplying the frequency domain sequence of the signal with the FFT of the

matched filter impulse response.

3) Performing the inverse FFT of the composite frequency domain sequence in order to

generate the time domain compressed pulse (HRR profile)

- Assume that I targets at ranges R1 , R2 , and so forth are receive window. the

phase of the down converted signal is

window.receivetheofstartthewithcoincideswhere

delays,timewaytwotherepresent,....,2,1);/2(timethe

)25.7()(2

2)(

1

1

2

0

IicR

tft

ii

I

i

ii


MATLAB Function matched_filter.m

* Input parameter values

* Input parameter


Echo Signal

< Uncompressed echo signal.

Scatterers are unresolved >

< Compressed echo signal.

Scatterers are resolved >

- Note that the compressed pulsed range resolution, without using window, is

R=9.3m


7.3.2 Stretch Processor

What is Stretch Processing - active correlation

- extremely high BW = very large BW LFM

Processing Technique

Radar returns are mixed with a Replica (Tx waveform)

Low pass filtering and detection A/D conversion

NBF extracts the tones, proportional to Target Range

-All returns from the same range bins produce

the same constant frequency


Stretch Processing - Concept

same LFM slope

Reference signal


MATLAB Function stretch.m

RSP Lab

- The function stretch.m presents digital implementation of stretch processing.


MATLAB Function stretch.m


7.3.3 Distortion Due to Target Velocity

- Assumed stationary targets

Target radial velocity, or equivalently Doppler shift degrades the quality of

the HRR profile generated by pulse compression

- When the target radial velocity

Doppler frequency is not zero

Effects are not compensated

The pulse compression processor output is distorted

< Compressed pulse output of a

pulse compression processor. No

distortion is present. >


7.3.3 Mismatched Compressed pulse

< Mismatched compressed pulse;

5% Doppler shift. >


10% time dilation. >


7.3.3 Distortion Due to Target Velocity


10% time dilation and 5% Doppler shift. >

Correction for the distortion caused by the target radial

velocity can be overcome by

using the following approach.

-First of all, over a period of few pulses,

the radar data processor estimates the

radial velocity of the target under track.

- Then, chirp slope and pulse width of the

next transmitted pulse are changed to

account for the estimated Doppler

frequency and time dilation.


- Expression for an LFM ambiguity function

7.3.4 Range Doppler Coupling

2

2

sin '( ) 1'

( ; ) 1'

'( ) 1'

d

d

d

f

f

f

- Distinctive property of range Doppler coupling associated with LFM was not presented

- Range Doppler coupling

phrase used to describe the shift in the delay/range response of an LFM ambiguity function due to the presence of a Doppler shift

- The nature of range Doppler coupling can be better understood by analyzing the LFM ambiguity function

' (7.44)



< Illustration of range Doppler coupling for an LFM pulse. >



- Ambiguity surface extends from to in range and from to in

Doppler.

- Maximum Response at the point

- Profiles parallel to the Doppler axis have maxima above the line

which passes through the origin.

- Presence of radial velocity forces the peak of the ambiguity surface to a point

that has a peak value smaller than the maximum that occurs at the origin.

- As long as the shift is less than the line , the ambiguity function

response exerts acceptable reduction in peak values

' '

( , ) (0,0)Df

Df

1/ 'Df


7.4.1. Frequency Coding (Costas Codes)

- Construction of Costas codes can be understood from the construction process

of Stepped Frequency Waveform

- long pulse of length :

- Frequency for the ith subpulse

- is a constant frequency and

- Time bandwidth product of this waveform

1 N

N1,i 0 fiffi

0f ff 0

2Nf


N X N Matrix row i=1,2,....,N and columns j=1,2,...,(N-1)

Possible Costas is less than N!

Figure 7.12 Frequency assignment for a burst of N subpulses .

(a) SFW (stepped LFM); (b) Costas code of length Nc=10

Frequency assignment for a burst of N subpulses


7.4.2 Binary Phase Codes

signal.referanceCWsometorelativeradiansoreither

aschosenrandomlyispulsesubeachofphasethen,widthofiseach;

pulsessmallerintodividediswidthofpulselongrelativelya,casethisIn

0

/N.

N

"""0")1(1

"""1")1(0

orVoltofamplitudephase

orVoltofamplitudephase

pulse.longtheofthatthanlargertimesisvaluepeaktheand

,toequaliscodedphasebinarywithassociatedrationcompressioThe

N

/

pulses.subindividualtheforphasetheofsequencerandom

theonheavilydependeswaveformcodephasebinarycompressedaofgoodnessThe


Barker codes

- One family of binary phase codes that produce compressed waveforms with

constant side lobe levels equal to unity is the barker code.

< Binary phase code of length 7 >

- A barker code of length n is denoted as Bn . There are only seven known

barker code that share this unique property.

- Note that B2 and B4 have complementary forms that have the same

characteristics.

since there are only seven Barker codes, they are not used when radar security

is an issue.


Barker Code Chart

< Barker Codes >


Barker Codes

- The autocorrelation function for a BN Barker codes will be 2N wide. The

main lobe is 2N wide, the peak value is equal to N.

- There are (N-1)/2 side lobes on either side of the main lobe.

< Barker code of length 13, and its corresponding

autocorrelation function >

* Note that the main lobe is equal to 13, while all side lobes are unity.


Combined Barker Codes

- The most side lobe reduction offered by a Barker code is -22.3dB, which may

not be sufficient for the desired radar application.

- Barker codes can be combined to generate much longer codes. In this case, a

Bm code can be used within a Bn code (m within n) to generate a code of length mn.

- The compression ratio for the combined Bmn code is equal to mn.

Ex.) a combined B54 is given by

B54={11101, 11101, 00010, 11101}

Unfortunately, the side lobes of a combined Barker code autocorrelation

function are no longer equal to unity.

< Ex. A combined B54 Barker code >


Barker Code Sidelobes

- some side lobes of Barker code autocorrelation function => zero

if matched filter is followed by,

N

Nk

k ktth 2

where, N is filters order, k(k= -k) are to be determined,

() is the delta function and is the Barker code sub-pulse width.

- A filter of order N produces N zero side lobes, the main lobe amplitude and

width do not change.

(7.58)


- Consider the case where the input to the matched filter is B11 and N=4

autocorrelation for a B11 code is

}1,0,1,0,1,0,1,0,1,0,11,0,1,0,1,0,1,0,1,0,1{11

- The output of the transversal filter is the discrete convolution between its

impulse response and the sequence 11 .

At this point we need to compute the k that guarantee the desired filter output.

Performing the discrete convolution as (7.58) and collecting equal terms(k= -k),

0

0

0

0

11

11 1- 1- 1 - 1-

1- 11 1- 2- 1-

1- 2- 10 2- 1-

1- 2- 2- 10 1-

2- 2- 2- 2- 11

4

3

2

1

0

Barker Code sequence

(7.59)

(7.60)

11


- Note that by setting the first equation equal to 11 and all other equations to 0

and solving for k guarantees that the main peak remains unchanged,

next four side lobes are zeros.

- So far we have assumed that coded pulses have rectangular shapes.

using other pulses of other shapes, such as Gaussian, may produce better

side lobe reduction and a larger compression ratio.

Barker Code sequence 11


Matlab Code Barker Code sequence

11

11

1/11

7

1/7


- a single pulse of width : divided into N equal group

7.4.3 Frank Codes

- each group is divided into other N sub-pulses each of width

- ploy-phase code

: use any harmonically related phases based on a certain fundamental phase increment

'

- total number of sub-pulses within each pulses : 2N

- compression ratio : 2N

- N-phase Frank code : Frank code of 2N sub-pulses

- fundamental phase increment

N

o360 (7.61)

decreases as the number of groups is increases

degrade very long Frank codes (phase stability)


Computing Frank Codes

- For N-phase Frank code the phase of each sub-pulse

21...131210

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

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

12...6420

1...3210

0...0000

NNNN

N

N

jj

jj

ooo

oo

ooo

11

1111

11

1111

901802700

18001800

270180900

0000

(7.62)

each row : a group, each column : sub-pulses for that group

- For example, 4-phase Frank code, and fundamental phase increment o90

(7.63)


Frank Code of 16 Elements

- Frank code of 16 elements

jjjjF 11111111111116 (7.64)

- phase increments within each row :

stepwise approximation of an up-chirp LFM waveform

- phase increments for subsequent rows : increase linearly versus time

< Stepwise approximation of

an up-chirp waveform, using

a Frank code of 16 elements >

Documents

MS_7장 -Pulse Compression -Rev