Page 375 - Fundamentals of Radar Signal Processing

multiplying the data by a truncated, asymmetric window (the portion of the

actual window that overlaps the M nonzero data points), resulting in greatly
increased sidelobes. Applying a shortened K-point window to a turned data
sequence results in DFT samples that do not equal samples of the DTFT of the
windowed M-point original data sequence.
Because the DFT is a sampled version of the DTFT, the peak value of the
DFT obtained for a pure sinusoidal signal is greatest when the Doppler

frequency coincides exactly with one of the DFT sample frequencies, and
decreases when the target signal is between DFT frequencies. This reduction in
amplitude is called a Doppler straddle loss. The amount of loss depends on the
particular window used and the ratio K/M. For a given signal length M, the
straddle loss is always greatest for signal frequencies exactly halfway between
DFT sample frequencies. The calculation can be simplified by assuming that the
signal frequency is F = 0 so that y[m] = w[m] and then evaluating Eq. (5.81)
D
with k = 0 (DFT sample on the “sinusoid” peak) and k = 1/2 (1/2 bin away from
the sinusoid peak). To be explicit, consider the rectangular window case; then

(5.82)

Assuming K ≥ M and evaluating at k = 1/2 gives

(5.83)

The last step was obtained by assuming that K is large enough to allow a small
angle approximation to the sine function in the denominator. Y[0] is obtained
either by applying L’HÔpital’s rule to the second form of Eq. (5.82) or
computing it explicitly from the first form; the result is Y [0] = M. The maximum
straddle loss for the DFT filterbank with no windowing is

(5.84)

This equation verifies that the loss depends on the ratio K/M. The worst case

occurs when the sinc term is minimized; this happens when K = M. In decibels,
this worst-case loss for a rectangular window is sinc (1/2) = sin(π/2)/(π/2) =
2/π, equivalent to –3.92 dB.
The straddle loss for shaped windows varies somewhat with M as can be
seen in Table 5.2. A calculation for the Hamming window similar to that