Page 241 - Fundamentals of Radar Signal Processing

Each integral is just the energy E in x(t), so that

(4.38)

The equality holds only if x(s) = x(s – t)exp(–j2πF s) for all s, which occurs if
D
and only if t = F = 0. Making these substitutions in Eq. (4.38) gives the equality
D
in Eq. (4.33).
The proof of the second property starts by defining the complex conjugate

of the complex ambiguity function, where

(4.39)

The squared magnitude of the ambiguity function can then be written as

(4.40)

The total energy in the ambiguity surface is

(4.41)

Isolating those terms integrated over t and F yields the following two
D
relationships:

(4.42)

(4.43)

Substituting these into Eq. (4.41) yields