Page 259 - Fundamentals of Radar Signal Processing

(4.76)

Using this result in Eq. (4.75) and combining the two remaining sums over n′
into one while renaming the index of summation as m gives

(4.77)

Equation (4.77) expresses the complex ambiguity function of the coherent pulse
train in terms of the complex ambiguity function of its constituent simple pulses
and the PRI.
Recall that the support in the delay axis of  (t, F ) is |t| ≤ τ. If T > 2τ,
p
D
which is almost always the case, the replications of  in Eq. (4.77) will not
p
overlap and the magnitude of the sum of the terms as m varies will be equal to
the sum of the magnitude of the individual terms. The ambiguity function of the
pulse burst waveform can then be written as

(4.78)

To understand this ambiguity function, it is convenient to first look at the
zero Doppler and zero delay responses. The zero Doppler response is obtained
by setting F = 0 in Eq. (4.78) and recalling that A (t, 0) = 1 – |t|/τ :
D
p

(4.79)

Equation (4.79) describes the triangular output of the single-pulse matched
filter, repeated every T seconds and weighted by an overall triangular function

M – |m|. Figure 4.18 illustrates this function for the case M = 5 and T = 4τ. The
ambiguity function has been normalized by the signal energy E so that it has a
maximum value of 1.0. Note that, as with any waveform, the maximum of the AF
occurs at t = 0 and the duration is twice the total waveform duration (2MT in
this case). The local peaks every T seconds represent the range ambiguities
discussed previously in Sec. 4.5.3 and illustrated in Fig. 4.15a. If the

transmitted waveform were extended by P pulses while the reference waveform