Page 377 - Fundamentals of Radar Signal Processing

spectrum (DTFT or DFT) sample will have variance . If the slow-time data
2
is windowed, the variance becomes , where E = ∑ |w[m]| is the energy of
w
w
the window sequence. Because the DTFT and DFT samples are complex
Gaussian, it also follows that the magnitude of a spectrum sample has a
Rayleigh PDF, the magnitude-squared has an exponential PDF, and the phase

has a uniform PDF over [–π, π]. The PDF and its parameters are important in
setting detection thresholds as will be seen in Chap. 6. These and additional
properties of the DTFT and DFT of noise are derived in Richards (2007).
DTFT and DFT values at different frequencies or indices for random
inputs are, in general, correlated. In the unwindowed case the normalized
autocorrelation function in frequency is an asinc function of peak amplitude 1
and a zero spacing of 2π/M radians per sample. For the windowed case, the
normalized autocorrelation function follows the shape of the DTFT of the

window function. In the case of an unwindowed DFT of size K = M the DFT
sample spacing matches the zero spacing of the asinc function so that the DFT
samples will be uncorrelated. Because the complex samples are Gaussian, they
are also independent in this case.

5.3.4 Pulse Doppler Processing Gain
The DTFT and DFT represent forms of coherent integration in the slow-time

domain and result in a processing gain. Equation (5.74) shows that each sample
o f y[m] is phase-adjusted by multiplication with the DTFT kernel and then
integrated. If no window is used, then at the value of F that matches the signal
frequency, F = F , the kernel values exactly compensate the phase history of the
D
data such that all of the signal samples add in phase. At this frequency Y(F ) =
D
A·M. The value of the noise component at the same frequency value will be .

The SNR is therefore a processing gain of a factor of
G = M compared to the SNR in the slow-time signal before the DTFT. The
sp
same result applies for the DFT. Note that in the DFT case, the processing gain

is determined by M, the number of slow-time samples and not by K, the DFT
size. Computing a DFT larger than the number of time domain samples, K > M,
does not increase the processing gain.
Various issues can reduce G . When a window is used with either a DTFT
sp
or DFT, G is decreased by the processing loss PL described earlier [Eq.
sp
(5.80)]. In this case, the range equation signal processing gain due to Doppler
processing becomes G p = M/PL. When a DFT is used, G p is also reduced by
s
s
any straddle loss that may occur.

5.3.5 Matched Filter and Filterbank Interpretations of Pulse Doppler
Processing with the DFT
Equation (5.13) defined the coefficients of the matched Doppler filter. In MTI
filtering, it is assumed that the target Doppler shift is unknown. The resulting