Page 230 - Fundamentals of Radar Signal Processing

(4.9)

This particular choice of the receiver filter frequency or impulse response is
called the matched filter, because the response is “matched” to the signal
waveform. Thus, the waveform and the receiver filter needed to maximize the
output SNR are a matched pair. If the radar changes waveforms, it must also
change the receiver filter response in order to stay in a matched condition. The
impulse response of the matched filter is obtained by time-reversing and

conjugating the complex waveform. The gain constant α is often set equal to
unity; it has no impact on the achievable SNR, as seen later in this chapter. The
time T at which the SNR is maximized is arbitrary. However, T ≥ τ is
M
M
required for h(t) to be causal.
Given an input signal x′(t) consisting of both target and noise components,
the output of the matched filter is given by the convolution

(4.10)

The second line of Eq. (4.10) is recognized as the cross-correlation of the

target-plus-noise signal x′(t) with the transmitted waveform x(t), evaluated at
lag T – t. Thus, the matched filter implements a correlator with the transmitted
M
waveform as the reference signal.
It is useful to determine the maximum value of SNR achieved by the
matched filter. Using H(Ω) = αX*(Ω) exp(–jΩT ) in Eq. (4.6)
M

(4.11)

The energy in the signal x(t) is