Page 288 - Fundamentals of Radar Signal Processing

(4.118)

where K is the window length. The loss in SNR at the matched filter output is
called the processing loss (PL) and is

(4.119)

With these definitions LPG and PL are both greater than one so that the losses in
decibels are positive numbers. For a relatively long Hamming window, the LPG
is approximately 5.4 dB while the PL is approximately 1.4 dB. Both are weak
functions of K and are slightly larger for small K. These formulas are
approximate when applied to windowing of the LFM spectrum due to the finite-
width transition of the LFM spectrum and the designer’s discretion in choosing

the cutoff of the window in frequency. In the example above, the LPG is 5.35
dB. Derivation of these formulas is deferred to Chap. 5, where they will arise
again in the context of Doppler processing and where the results will be exact.

4.7.2 Matched Filter Impulse Response Shaping
The impulse response of the filter just obtained and illustrated in Fig. 4.34b
suggests that similar results could have been obtained by windowing the LFM
waveform in the time domain. Consider again the signal in Eq. (4.89) having an

arbitrary amplitude function A(t) and a quadratic phase function. The PSP
approximation to its spectrum was given in Eq. (4.93). The magnitude of the
spectrum is proportional to the time-domain amplitude:

(4.120)

If A(t) has finite support on –τ/2 ≤ t ≤ τ/2, it follows that X(Ω) will have finite
support on –β/2 ≤ F ≤ β/2 and in that interval|X(Ω)|has the same shape as the

window magnitude | A(t)|. Thus, a Hamming-shaped (for example) spectrum can
be obtained by applying a Hamming window to the impulse response h(t)
instead of the frequency response H(F). Note that this result is specific to the
use of a linear FM waveform.
The output of the resulting filter is overlaid on the matched filter response
i n Fig. 4.36. It has the same general character as the frequency-domain