Page 232 - Fundamentals of Radar Signal Processing

The corresponding matched filter impulse response is

(4.16)

where T > τ for causality. Because x(t) is a much simpler function than its
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Fourier transform (a sinc function), it is easier to work with the correlation
interpretation of Eq. (4.10) to compute the output. Figure 4.2 illustrates the two
terms in the integrand, helping to establish the regions of integration. Part a of
the figure shows that

FIGURE 4.2 Convolution of simple pulse and its matched filter: (a) T – τ ≤ t ≤
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T , (b) T ≤ t ≤ T + τ.
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M
M

(4.17)

while part b is useful in identifying the next two regions

(4.18)