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
(1)
where
,(2)
is the LFM slope and is the pulse width. The instantaneous phase of is
(3)
and the instantaneous frequency is
.(4)
Over the duration of the pulse, varies from to . Thus, the bandwidth of the LFM signal, , is
.(5)
We can also determine the bandwidth of by finding and plotting its Fourier transform. Specifically,
(6)
where
(7)
and and are the cosine and sine Fresnel integrals, respectively, defined by
(8)
and
.(9)
A normalized plot of for an LFM bandwidth of and a pulsewidth of 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
(10)
where we have made use of the fact that is an even function.
The form of means that the signal processor (i.e. matched filter) would need to have a bandwidth of . 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 processors. According to the Skolnik Radar Handbook[1] (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 2contains a functional block diagram of a stretch signal processor. It consists of a mixer, a LFM generator, timing circuitry and a 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 is
(11)
where is a scaling factor that we will use when we address signal-to-noise ratio (SNR). 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
.(12)
In the above is the range delay to which the stretch processor is “matched” and is usually close to . Actually, usually is not a correct word. A more precise statement is that must be close to the of the scatterers that we wish to resolve. is the duration of the heterodyne signal and, as we will show, satisfies .
Notional sketches of and 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 and 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, , is greater than and the lower plot corresponds to the case where the range delay is less than . It will be noted that when the frequency of is greater that the frequency of . When the frequency of is less that the frequency of . Further, the size of the frequency difference between and depends upon the difference between and .
Figure 3 also tells us how to set the value of , the duration of the heterodyne signal. Specifically, we want to set so that is completely contained within for all expected values of relative to . From the bottom plot of Figure 3 we conclude that we want
.(13)
From the top plot we want
.(14)
This leads to the requirement on that it satisfy
(15)
where
(16)
is the range delay extent over which we want to use stretch processing. If satisfies the above constraint and
(17)
then will completely overlap and the stretch processor will offer almost the same SNR performance as a matched filter. If the various timing parameters are such that does not completely overlap , the stretch processor will experience a SNR loss proportional to the extent of that doesn’t lie within the extent of .
1.3 STRETCH PROCESSOR OPERATION
Given that and satisfy the above requirements, we can write the output of the mixer as
(18)
or
.(19)
The first exponential term of 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, , and the range delay to which the stretch processor is tuned, . Thus, if we can determine the frequency of the signal out of the mixer we can determine the target range. Specifically, since
(20)
we get that
.(21)
The spectrum analyzer of Figure 2 is used to determine . Ideally, the spectrum analyzer computes the Fourier transform of . Thus, we can write
(22)
or
(23)
where
.(24)
The information of concern is contained in , a normalized plot of which is contained in Figure 4. As we would expect, the sinc function is centered at and has a nominal width of . Thus we can measure 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 the nominal width of the main lobe is .
We would now like to examine the range resolution of the stretch processor. Since the nominal width of the sinc function is we normally say that the frequency resolution of at the output of the spectrum analyzer is also . Suppose we have a target that is at a range of and a second target at a range of . The mixer output frequency associated with the two targets will be
(25)
and
.(26)
Suppose further that and are such that
.(27)
That is, the frequencies are separated by a resolution cell of the stretch processor. With this we can write
(28)
or
(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 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
(30)
where and are the range delay and Doppler frequency mismatch, respectively, between the target return and matched filter. 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
.(31)
Substituting Equation (30) into Equation (31) yields
.(32)
The noise voltage at the output of the matched filter is given by
(33)
where is zero-mean, wide-sense stationary white noise with
.(34)
is the noise power spectral density and is the Dirac delta. Since is a random process so is . Thus, the average noise power out of the matched filter is given by
(35)
where we have made use of Equations (33) and (10). With this, we get the SNR at the matched filter output as
(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, . Thus, we are interested in the output of the spectrum analyzer at (This assumes that the stretch processor is matched to , i.e., ). With this we get, using Equation (23),
.(37)
If the noise into the mixer part of the stretch processor is the noise out of the mixer is
.(38)
Recall that the signal power was computed at the spectrum analyzer output where . The noise signal at this spectrum analyzer output is
(39)
and the average power at the output is
(40)
where we have made use of Equations (34) and (12).
The SNR at the output of the stretch processor is
.(41)
If we combine Equations (36) and (41) we get
.(42)
Thus, the stretch processor encounters a SNR loss of 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, . 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 mixer[2].
If and are the minimum and maximum range delays, relative to , over which stretch processing is performed then the corresponding minimum and maximum frequencies out of the mixer are
(43)
and
.(44)
Thus, the expected range of frequencies out of the mixer is
.(45)
Thus the ADC sample rate should be at least .
The FFT will need to operate on data samples taken between and or over a time window of at least
.(46)
The total number of data samples processed by the FFT will be
.(47)
This means that the FFT length will need to be some power of 2 that is greater than .
As an example of the above calculations we consider the following parameters.
-
-
-Stretch processing performed over 1500 m.
With this we get that
(48)
and
.(49)
To compute we first need to compute as
.(50)
With this we get
.(51)
Thus, the minimum required ADC sample rate is 50 MHz. The number of samples to be processed by the FFT is
.(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 and to increase.
If we continue the calculations we find that the time extent of the heterodyne window is
.(53)
The SNR loss associated with the use of stretch processing over a matched filter is 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 from Equation (1) to include a carrier term. We then specifically look at how range and rate affects the returned signal, . After this we examine the matched filter response to 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
(54)
where the first exponential is the carrier term and is the carrier frequency.
The signal returned from the target is
.(55)
It will be noted that the range delay, , 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
(56)
Where is the range at (the center of the transmit pulse in this case) and is the range-rate.
Substituting Equation (56) into Equation (55) results in
(57)
where is a phase term that we associate with Doppler effects due to the interaction of the target range-rate with the carrier and is a phase term that we associate with the interaction of range-rate with the LFM modulation. If we expand we get
(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 () is the same as we developed in EE619 when we discussed Doppler frequency.
The second phase term can be written as
.(59)
In this case we want to examine the frequency or
.(60)
It will be noted that the LFM slope of the received signal, , 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 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 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 filterto perform pulse compression. We further assume that the matched filter is matched to the transmit waveform plus some frequency offset, , 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 . Thus, we can write the normalized received signal as