Page 240 - Fundamentals of Radar Signal Processing

(4.32)

It is a function of two variables: the time delay relative to the expected matched
filter peak output, and the mismatch between that Doppler shift for which the
filter was designed, and that which is actually received. For example, the AF
evaluated at time t = 0 corresponds to the output of the actual matched filter at

time t = 2R /c + τ for a target at range R . The particular form of the AF is
0
0
determined entirely by the complex waveform x(t).
Three properties of the ambiguity function are of immediate interest. The
first states that if the waveform has energy E, then

(4.33)

Thus, when the filter is matched in Doppler to the echo and is sampled at a
delay corresponding to the target range, the response will be maximum. If the
filter is not matched or is sampled at a different delay, then the response will be

less than or equal to (usually less than) the maximum. The second property
states that total area under any ambiguity function is constant and is given by

(4.34)

This conservation of energy statement implies that, in the design of waveforms,
one cannot remove energy from one portion of the ambiguity surface without
placing it somewhere else; it can only be moved around on the ambiguity
surface. The third property is a symmetry relation:

(4.35)

In order to prove the first property, start with the square of Eq. (4.32)

(4.36)

Applying the Schwartz inequality to Eq. (4.36) yields

(4.37)