Page 237 - Fundamentals of Radar Signal Processing

(4.24)

Thus, the matched filter for x′(t) can be obtained by simply shifting the center
frequency of the matched filter for x(t) to the expected Doppler shift.

A more interesting situation occurs when the velocity is not known in
advance so that the receiver filter is not matched to the target Doppler shift.
More generally, suppose the filter is matched to some Doppler shift Ω radians
i
per second but the actual Doppler shift of the echo is Ω . Choosing T = 0 for
M
D
simplicity, the matched filter output will be zero for | t | > τ. For 0 ≤ t ≤ τ the
response is

(4.25)

If the filter is in fact matched to the actual Doppler shift, Ω = Ω , the output
D
i
becomes

(4.26)

The analysis is similar for negative t, –τ ≤ t ≤ 0. The complete result is

(4.27)

Thus, | y(t) | is the usual triangular function, peaking as expected at t = 0.
If there is a Doppler mismatch, Ω ≠ Ω , the response at the expected peak
D
i
time t = 0 is

(4.28)