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1.0 STRETCH PROCESSING

1.1 INTRODUCTION AND BACKGROUND

Stretch processing is a way of processing large bandwidth waveforms using narrow band techniques. For our present purposes we want to look at stretch processing as applied to LFM waveforms. It turns out that the concepts of stretch processing appear in other applications such as FMCW radar and, as we will see later, SAR processing.

We consider a transmit waveform of the form

2

rectj t

T

tv t e

(1)

where

1 1 2

rect0 1 2

xx

x

, (2)

is the LFM slope and T is the pulse width. The instantaneous phase of v t

is

2t t (3)

and the instantaneous frequency is

12

f t d t dt t

. (4)

Over the duration of the pulse, f t varies from 2T to 2T . Thus, the

bandwidth of the LFM signal, v t , is

TB . (5)

We can also determine the bandwidth of v t by finding and plotting its

Fourier transform. Specifically,

2

2

1 2 22

2 2

j ft

j f

T T

V f v t e dt

e F f F f

(6)

where

F x C x jS x (7)

and C x and S x are the cosine and sine Fresnel integrals, respectively,

defined by

2

2

2

0

cos

x

C x t dt (8)

and

2

2

0

sin

x

S x t dt . (9)

A normalized plot of V f for an LFM bandwidth of 500 MHzB and a pulse

width of 100 sT is shown in Figure 1. It will be noted that, indeed, the

bandwidth is 500 MHz.

If we were to process the LFM pulse using a matched filter, the impulse response of the matched filter would be

2

rectj t

T

th t v t e

(10)

where we have made use of the fact that rect x is an even function.

The form of h t means that the signal processor (i.e. matched filter)

would need to have a bandwidth of TB . Herein lies the problem: large

bandwidth signal processors are still difficult and very costly to build. Two methods of building LFM matched filters (LFM pulse compressors, LFM signal processors) are SAW (surface acoustic wave) devices and digital signal

Figure 1 – Spectrum of an LFM pulse with B=500 MHz and PW=100 µs

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processors. According to the Skolnik Radar Handbook1 (page 10.11) one can build SAW compressors for bandwidths up to 1 GHz. However, I have not heard of hardware implementations with such devices. I would expect that the upper limit on bandwidth for practical SAW LFM compressors is in the 10s, or possibly low 100s, of MHz.

The bandwidth of digital signal processors is usually limited by the sample rate of the analog-to-digital converters (ADCs) needed to convert the analog signal to a digital signal. I think that the current limit on ADC rates is 300 MHz or so. If an upper limit on ADC sample rate is 300 MHz, then the maximum bandwidth of a LFM signal processor would also be 300 MHz (assuming complex signals and processors).

Stretch processing relieves the signal processor bandwidth problem by giving up all-range processing to obtain a narrow-band signal processor. If we

were to use a matched filter we could look for targets over the entire waveform pulse repetition interval (PRI). With stretch processing we are limited to a range extent that is usually smaller than an uncompressed pulse width. Thus, we couldn’t use stretch processing for search because search requires looking for targets over a large range extent, usually many pulse widths long. We could use stretch processing for track because we already know range fairly well but want a more accurate measurement of it. We must point out that, in general, wide bandwidth waveforms, and thus the need for stretch processing, is “overkill” for tracking. Generally speaking, bandwidths of 1s to 10s of MHz are sufficient for tracking

One of the most common uses of wide bandwidth waveforms, and stretch processing, is in discrimination, where we need to distinguish individual scatterers on a target. Another use we will look at is in SAR (synthetic aperture radar). Here we only try to map a small range extent of the ground but want very good range resolution to distinguish the individual scatterers that constitute the scene.

In the above discussion, we have focused on the signal processor and have argued, without proof at this point, that we can use stretch processing to ease the bandwidth requirements on a signal processor used to compress wide bandwidth waveforms. Stretch processing does not relieve the bandwidth requirements on the rest of the radar. Specifically, the transmitter must be capable of generating and amplifying the wide bandwidth signal, the antenna must be capable of radiating the transmit signal and capturing the return signal, and the receiver must be capable of heterodyning and amplifying the wide bandwidth signal. This places stringent requirements on the transmitter, antenna and receiver, but current technology has advanced to be point of being able to cope with the requirements.

1.2 STRETCH PROCESSOR CONFIGURATION

Figure 2 contains a functional block diagram of a stretch signal processor. It consists of a mixer, a LFM generator, timing circuitry and a

1 M. I. Skolnik, Radar Handbook, Second Edition, McGraw-Hill, New York, 1990, ISBN 0-07-057913-X

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spectrum analyzer. If the transmit signal is as given in Equation (1) the

normalized signal returned from a point scatterer at a range delay of R is

2

rectRj t RS R S

T

tr t P v t P e

(11)

where SP is a scaling factor that we will use when we address signal-to-noise ratio

(SNR). SP is the peak signal power at the matched filter output and comes from the radar

range equation.

The normalized heterodyne signal generated by the LFM generator is

2

rectMj t Ms

h

th t e

. (12)

In the above M is the range delay to which the stretch processor is “matched”

and is usually close to R . Actually, usually is not a correct word. A more

precise statement is that M must be close to the R of the scatterers that we

wish to resolve. h is the duration of the heterodyne signal and, as we will

show, satisfies h T .

Notional sketches of r t and sh t are shown in Figure 3. The

horizontal axis is time and the vertical axis is frequency. The frequency of each

signal is shown only over the time that the signal itself is not zero. Since r t

and sh t are LFM signals, we note that their frequencies increase linearly over

their respective durations. Furthermore, by design, both frequency vs. time plots have the same slope of . The top plot corresponds to the case where the

target range delay, R , is greater than M and the lower plot corresponds to the

case where the range delay is less than M . It will be noted that when R M

Figure 2 – Stretch Processor

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the frequency of sh t is greater that the frequency of r t . When R M the

frequency of sh t is less that the frequency of r t . Further, the size of the

frequency difference between r t and sh t depends upon the difference

between R and M .

Figure 3 also tells us how to set the value of h , the duration of the

heterodyne signal. Specifically, we want to set h so that r t is completely

contained within sh t for all expected values of R relative to M . From the

bottom plot of Figure 3 we conclude that we want

2 2RMIN T M h . (13)

Figure 3 – Sketches of r t and sh t

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From the top plot we want

2 2RMAX T M h . (14)

This leads to the requirement on h that it satisfy

h R T (15)

where

R RMAX RMIN (16)

is the range delay extent over which we want to use stretch processing. If h

satisfies the above constraint and

M RMIN R RMAX M (17)

then sh t will completely overlap r t and the stretch processor will offer

almost the same SNR performance as a matched filter. If the various timing

parameters are such that sh t does not completely overlap r t , the stretch

processor will experience a SNR loss proportional to the extent of r t that

doesn’t lie within the extent of sh t .

1.3 STRETCH PROCESSOR OPERATION

Given that r t and sh t satisfy the above requirements, we can write

the output of the mixer as

2 2

rect rectM Rj t j tM Ro s S

h T

t tv t h t r t P e e

(18)

or

2 22

rectM R R M

j j t Ro S

T

tv t P e e

. (19)

The first exponential term of ov t is simply a phase term. However, the second

exponential term tells us that the output of the mixer is a constant frequency signal with a frequency that depends upon the difference between the target

range delay, R , and the range delay to which the stretch processor is tuned,

M . Thus, if we can determine the frequency of the signal out of the mixer we

can determine the target range. Specifically, since

m R Mf (20)

we get that

R m Mf . (21)

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The spectrum analyzer of Figure 2 is used to determine mf . Ideally, the

spectrum analyzer computes the Fourier transform of ov t . Thus, we can

write

22 2rectmj f tj ft j j ftRo o S

T

tV f v t e dt P e e e dt

(22)

or

2sincm Rj f fj

o T S m TV f P e e f f

(23)

where

2 2

M R . (24)

The information of concern is contained in oV f , a normalized plot of

which is contained in Figure 4. As we would expect, the sinc function is

centered at mf and has a nominal width of 1 T . Thus we can measure mf but

not with perfect accuracy. This is consistent with the result we would get with a matched filter. That is, the range measurement accuracy is related to the width of the main lobe of the output of a matched filter. For an LFM signal with

a bandwidth of B the nominal width of the main lobe is 1 B .

We would now like to examine the range resolution of the stretch

processor. Since the nominal width of the sinc function is 1 T we normally say

that the frequency resolution of at the output of the spectrum analyzer is also

1 T . Suppose we have a target that is at a range of 1R and a second target at

Figure 4 – Plot of o mV f f for B=500 MHz and PW=100 µs

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a range of 2 1R R . The mixer output frequency associated with the two targets

will be

1 1m R Mf (25)

and

2 2m R Mf . (26)

Suppose further that 1R and 2R are such that

2 1 1m m m Tf f f . (27)

That is, the frequencies are separated by a resolution cell of the stretch processor. With this we can write

2 1 2 11m m m T R M R Mf f f (28)

or

R 2 1 1 1res R R T B (29)

or that the stretch processor has the same range resolution as a matched filter.

Recall that with LFM we can use an amplitude taper, implemented by a filter at the input or output of the matched filter, to reduce the range sidelobes at the matched filter output. We can apply a similar taper to a stretch

processor by applying an amplitude taper to ov t before sending it to the

spectrum analyzer. We will look at this further when we discuss a specific implementation of the spectrum analyzer.

1.4 STRETCH PROCESSOR SNR

At this point we want to compare the SNR at the output of a matched filter to the SNR at the output of a stretch processor. Since neither processor includes nonlinearities we can invoke superposition and treat the signal and noise separately.

1.4.1 Matched Filter

For the matched filter case we can write signal voltage at the output of the matched filter as

2, mmMF j f tj

sm mm mm S mmv f P e v t v t e dt

(30)

where mm and mmf are the range delay and Doppler frequency mismatch,

respectively, between the target return and matched filter. v t is given by

Equation (1). We are interested in the power out of the matched filter at matched range and Doppler. That is, we want

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0,0 0,0sm sm smP v v . (31)

Substituting Equation (30) into Equation (31) yields

2

2MFj

sm S s TP P e v t v t dt P

. (32)

The noise voltage at the output of the matched filter is given by

nmv t h x n t x dx

(33)

where n t is zero-mean, wide-sense stationary white noise with

oE n t n N t . (34)

0o NN kT F is the noise power spectral density and x is the Dirac delta.

Since n t is a random process so is nmv t . Thus, the average noise power out

of the matched filter is given by

2

2

nm nm o TP E v t E h x n t x dx N

(35)

where we have made use of Equations (33) and (10). With this, we get the SNR at the matched filter output as

sm s Tm

nm o

P PSNR

P N

(36)

which we recognize from radar range equation theory.

1.4.2 Stretch Processor

For the stretch processor we are interested in the signal power at the

target range delay, R . Thus, we are interested in the output of the spectrum

analyzer at mf f (This assumes that the stretch processor is matched to R ,

i.e., M R ). With this we get, using Equation (23),

2 2

ss o m s TP V f P . (37)

If the noise into the mixer part of the stretch processor is n t the noise

out of the mixer is

nM sv t n t h t . (38)

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Recall that the signal power was computed at the spectrum analyzer output

where mf f . The noise signal at this spectrum analyzer output is

2 mj f t

ns m sV f n t h t e dt

(39)

and the average power at the output is

2

0ns ns m hP E V f N (40)

where we have made use of Equations (34) and (12).

The SNR at the output of the stretch processor is

2

0

ss s Ts

ns h

P PSNR

P N

. (41)

If we combine Equations (36) and (41) we get

s T

m h

SNR

SNR

. (42)

Thus, the stretch processor encounters a SNR loss of h T relative to the

matched filter. This means that we should be careful about using stretch processing for range extents that are significantly longer of the transmit pulse width. At first inspection it appears as if stretch processing could offer better SNR than a matched filter, which would contradict the fact that the matched filter maximizes SNR. This apparent contradiction is resolved by the stretch

processor constraint imposed by Eqution (15). Specifically, h R T . The

constraint if Equation (42) also demonstrates another reason why stretch processing should not be used in a search function: it would be too lossy.

1.5 STRETCH PROCESSOR IMPLEMENTATION

We next want to turn our attention to practical implementation issues. The mixer, timing and heterodyne generation are reasonably straight forward. We want to address how to implement the spectrum analyzer. The most obvious method of implementing the spectrum analyzer is to use an FFT. To do so, we need to determine the required ADC (analog-to-digital converter) sample rate and the number of points to use in the FFT. To determine the ADC rate we need to know the expected frequency limits of the signal out of the mixer2.

2 We will assume base-band processing in these discussions. In practice the mixer output will be at some intermediate frequency (IF). The signal could be brought to base-band using a synchronous detector or, as in some modern radars, by using IF sampling (i.e. a digital receiver). In either case, the effective ADC rate (the sample rate of the complex, digital base-band signal) will be as derived here.

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If RMIN M and RMAX M are the minimum and maximum range delays,

relative to M , over which stretch processing is performed then the

corresponding minimum and maximum frequencies out of the mixer are

mMIN RMIN Mf (43)

and

mMAX RMAX Mf . (44)

Thus, the expected range of frequencies out of the mixer is

mMAX mMAX mMIN RMAX RMIN Rf f f . (45)

Thus the ADC sample rate should be at least mMAXf .

The FFT will need to operate on data samples taken between 2RMIN T

and 2RMAX T or over a time window of at least

RMAX RMIN T R T h . (46)

The total number of data samples processed by the FFT will be

samp mMAX hN f . (47)

This means that the FFT length will need to be some power of 2 that is greater

than sampN .

As an example of the above calculations we consider the following parameters.

- 100 T s

- 500 MHzB

- Stretch processing performed over 1500 m.

With this we get that

10 sRMAX RMIN (48)

and

110 sh RMAX RMIN T . (49)

To compute mMAXf we first need to compute as

5 MHz/ sTB . (50)

With this we get

50 MHzmMAX RMAX RMINf . (51)

Thus, the minimum required ADC sample rate is 50 MHz. The number of samples to be processed by the FFT is

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5500samp mMAX hN f . (52)

This means that we would want to use a 8192 point FFT. The most logical method of getting to 8192 samples would be to increase the size of the range

window. This would cause both mMAXf and h to increase.

If we continue the calculations we find that the time extent of the heterodyne window is

110 sh R T . (53)

The SNR loss associated with the use of stretch processing over a matched filter

is 110 100h T or about 0.4 dB.

1.6 DOPPLER EFFECTS

We now want to examine the effects of Doppler frequency on the output of the stretch processor. Since we have established the equivalency between the stretch processor output and the output of a matched filter, we will approach the discussion from the perspective of matched filter theory. We start

by developing extending the definition of v t from Equation (1) to include a

carrier term. We then specifically look at how range and rate affects the

returned signal, r t . After this we examine the matched filter response to r t

from the specific perspectives of range resolution degradation due to Doppler frequency and range error due to Doppler frequency.

1.6.1 Expanded Transmit and Receive Signal Models

We extend the previous definition of the transmitted LFM pulse to include the carrier term. Thus we write

22rectcj f t j t

T

tv t e e

(54)

where the first exponential is the carrier term and cf is the carrier frequency.

The signal returned from the target is

22

rectc R Rj f t t j t t R

S R S

T

t tr t P v t t P e e

. (55)

It will be noted that the range delay, R t , is now shown as a function of time

to account for the fact that range changes with time because the range-rate is not zero. We will assume that the target range-rate is a constant. With this, we can write

0

2 2 2R

R t R Rt t

c c c (56)

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Where 0R is the range at 0t (the center of the transmit pulse in this case) and

R is the range-rate.

Substituting Equation (56) into Equation (55) results in

2

1 2

2 2 2 2 2 2 2rect

2 2rect

c o oj f t R c R c t j t R c R c t o

S

T

oj t j t

S

T

t R c R c tr t P e e

t R c R c tP e e

(57)

where 1 t is a phase term that we associate with Doppler effects due to the

interaction of the target range-rate with the carrier and 2 t is a phase term

that we associate with the interaction of range-rate with the LFM modulation. If

we expand 1 t we get

1 0 02 4 2 2 2 4 2c c dt f t R c R t f t R c f t (58)

where the first term on the right is the carrier term, the second is a phase shift associated with the initial target position and the third term is the Doppler

frequency term. This term ( 1 t ) is the same as we developed in EE619 when

we discussed Doppler frequency.

The second phase term can be written as

2

2 02 2t t R c R c t . (59)

In this case we want to examine the frequency or

2 0

2

00

21 2 21 1

2

22 21 1

FM

r

d t RR Rf t t

dt c c c

RR Rt t f

c c c

. (60)

It will be noted that the LFM slope of the received signal, r , is slightly different

from the LFM slope, , of the transmit signal. As we will see, this slight

difference can degrade the range resolution of the LFM waveforms in some

cases.

It turns out that the increase in slope of the received LFM signal is caused by a slight shortening of the pulse as it is “reflected” by the target. To see this we consider a specific example. Suppose we have a 1 ms pulse and a

target moving at 7500 m/s. Let 0t be the time that the leading edge of the

pulse reaches the target. During the time that the pulse is interacting with the target, the target moves about (7500 m/s)×(0.001 s) or 7.5 m. This translates to an effective round-trip time delay of 2×7.5/c or 50 ns. This means that the

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length of the pulse returned to the radar is shorter than the transmit pulse by 50 ns. Since the frequency still varies the same amount over the duration of the pulse, the LFM slope must increase.

1.6.2 Effect of Doppler Frequency on Range Resolution

To quantify the effect of the change in received waveform LFM slope on range resolution we will consider specific examples. We will assume that the radar uses a matched filter to perform pulse compression. We further assume that the matched filter is matched to the transmit waveform plus some

frequency offset, Mf , to account for the target Doppler frequency. As we did

earlier, we will shift our time reference so that the center of the received pulse is

at 0t . Thus, we can write the normalized received signal as

22rectd rj f t j t

PWr

tr t e e

. (61)

The first exponential term is the Doppler term discussed above (see the discussion related to Equation (58) – we have omitted the carrier frequency term since we assume that it has been removed by a heterodyning process in

the receiver). We have temporarily ignored the term 0f in Equation (60). We

will address this later.

We can write the matched filter impulse response as

22rectMj f t j t

T

th t e e

. (62)

The response of the matched filter to the received signal is the convolution of

r t and h t . That is,

ov t r t h t r h t d

. (63)

Substituting for r t and h t results in

22 22

rect rectMd rj f t j tj f j

o

PWr T

tv t e e e e d

. (64)

After considerable manipulation it can be shown that we can write ov t as

2 2

0o

K F U f F L f U Lv t

U L

(65)

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In Equation (65):

F is the Fresnel integral

r

d Mf f f

min 2, 2T PWrU t

max 2, 2T PWrL t

K is a complex constant that we will normalize away .

Equation (65) applies only to the case where the received LFM slope and rect() function widths are different (the mismatched case). If the LFM slope and rect()

function widths are the same, ov t can be derived from the ambiguity

function of v t and is

1, sinc rect2

o T T

T

tv t t f K t f t t

. (66)

In Equation (66), 1K is a complex constant that we will normalize away.

To see the effect of range-rate on the matched filter response we consider two examples. In both cases we consider the LFM waveform of previous examples. Specifically, we consider a waveform with a bandwidth of 500 MHz and a pulse width of 100 µs. We assume further that the matched filter is

matched to the target Doppler. That is, d Mf f . For the first case we consider

a typical aircraft range-rate of -150 m/s and for the second case we consider a ballistic missile with a (extreme) range-rate of -7500 m/s. Plots of the matched filter outputs for the two cases are shown in Figure 5 and Figure 6. Each plot contains a curve where we ignore the effects of range-rate on the LFM slope (the red curve) and another curve where the range-rate effects are included (the blue curve).

For the aircraft velocity case, it will be noted that the difference in LFM slope caused by the range-rate does not have a significant effect on the output of the matched filter. However, for the ballistic missile case, the difference in LFM slopes causes a significant degradation of range resolution. This, in turn, could cause problems in isolating closely spaced scatterers. To eliminate this effect, the LFM slope of the matched filter should be changed to match the expected LFM slope of the received waveform. Since changing the LFM slope of the matched filter could be difficult, the alternative would be to change the LFM slope of the transmit signal so that the LFM of the received signal matches that of the matched filter.

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Since we expect that the stretch processor behavior will be similar to the

matched filter behavior, LFM slope mismatch should have the same effect on the output of the stretch processor. That is, for large range-rates, and long, large bandwidth waveforms, LFM slope mismatch will cause a degradation of range resolution.

Figure 5 – Matched Filter Response – Target Range-Rate = -150 m/s

Figure 6 – Matched Filter Response – Target Range-Rate = -7500 m/s

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1.6.3 Effect of Doppler Frequency Mismatch on Range Error

We next want to examine the effect of Doppler mismatch between the received signal and the matched filter. We consider the aircraft target example from above. Figure 7 contains a plot of the matched filter output for the case where the matched filter is matched to the target Doppler (the red curve) and the case where the matched filter is matched to zero Doppler (the blue curve). It will be noted that the mismatched Doppler case has a range error of 0.3 m, which is the range resolution of the waveform. To understand the cause of this range error it will be helpful to look at the matched filter response in terms of the ambiguity function of the transmit signal. This ambiguity function is given by

, sinc rect2

LFM T T

T

tt f t f t t

(71)

In Equation (71) we note that if df f , the peak of ,LFM t f occurs at

1d d d Tt f f B . (72)

From Equation (72) we note that a Doppler shift of 1d Tf relative to matched

Doppler will cause a range error equal to the range resolution of the LFM waveform. Said another way, a Doppler mismatch equal to the reciprocal of the uncompressed pulse width will cause a range error of one range resolution cell. In the specific example above, if we assume the radar is operating at X-band

with 0.03 then a range-rate of -150 m/s causes a Doppler frequency of 10

KHz. It turns out that the uncompressed pulse width of our example was 100

µs so that 1 10 T KHz . Thus, by the above, we expect that the peak of the

Figure 7 – Effects of Doppler Mismatch on Matched Filter Response

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ambiguity function (matched filter response) will be at one range-resolution cell instead of zero. Given that our waveform bandwidth was 500 MHz, the range resolution of the waveform is 2 ns, or 0.3 m, which is where the peak is located!

As with the LFM slope, we expect that the response of the stretch processor to Doppler mismatch will be the same as the matched filter. Suppose

the return signal, r t , into the stretch processor has a Doppler frequency of

df . This will mean that the return signal into the mixer (see Figure 1) will be

2 dj f t

dr t e r t

. (73)

If we repeat the math of Section 1.3 using dr t in place of r t , the signal out

of the mixer will be

2 dj f t

od ov t e v t

. (74)

In other words, the frequency out of the mixer will be shifted by the Doppler frequency (albeit with a negative sign). This, in turn, will cause the peak of the

spectrum analyzer output shift by df . That is, the output of the spectrum

analyzer will be od o dV f V f f . If 1d Tf then the peak will shift by

1 T . From the discussions of Section 1.3, a frequency deviation in the

spectrum analyzer output of 1 T corresponds to a range shift of 1 B , or one

range resolution cell. From this we observe that the response of the stretch processor to Doppler is the same as for a matched filter. In other words, a

Doppler shift of 1 T cause the range to be in error by one range resolution cell.