Page 281 - Fundamentals of Radar Signal Processing

frequencies so as not to distort the scatterer amplitudes, but also a group delay 5
of –δt seconds at the frequency –β · δt /τ. It can be shown that the required
b
b
2
frequency response in analog radian frequency units is H(Ω) = exp(–jΩ τ/2β ),
Ω
which has unit magnitude for all frequencies and a quadratic phase in the
frequency domain (see Prob. 15). All of the beat frequencies will be aligned in
time at the output of this filter. As an extra benefit, this filter also corrects RVP
(Carrara et al., 1995).
The bandwidth of the stretch receiver output can be obtained by
considering the difference in beat frequencies for scatterers at the near and far
edges of the range window. This gives

(4.111)

If T < τ, the bandwidth at the receiver output is less than the original signal
w
bandwidth β. The mixer output can then be sampled with slower A/D converters
and the number of range samples needed to represent the range window data is

reduced. Thus, the stretch technique is most effective for systems performing
fine range resolution analysis over limited range windows. Also note that while
the digital processing rates have been reduced, the analog receiver hardware up
through the LFM mixer must still be capable of handling the full instantaneous
signal bandwidth.
As an example, consider a 100 μs pulse with a swept bandwidth of 750
MHz, giving a BT product of 75,000. Suppose the desired range window is R w

= 1.5 km, corresponding to a sampling window of T = 10 μs. In a conventional
w
receiver the sampling rate will be 750 megasamples per second. Data from
scatterers over the extent of the range window will extend over T + τ seconds,
w
requiring (750 MHz)(10 μs + 100 μs) = 82,500 samples to represent the range
window. In contrast, the bandwidth at the output of the stretch receiver will be

(T /τ)β = 75 MHz. The sampled time interval remains the same, so only 8250
w
samples are required. Restricting the analysis to a delay window one-tenth the
length of the pulse and using the stretch technique has resulted in a factor of 10
reduction in both the sampling rate required and the number of samples to be
digitally processed.
Stretch processing of linear FM waveforms preserves both the resolution
and the range-Doppler coupling properties of conventionally processed LFM.
Consider the output of the stretch mixer for a scatterer at differential range δt b

from the central reference point. This signal will be a complex sinusoid at a
frequency F = –β · δt /τ Hz observed for a duration of τ seconds. In the absence
b
b
of windowing, the Fourier transform of this signal will be a sinc function with