Page 258 - Fundamentals of Radar Signal Processing

(4.71)

Substitute s′ = s – mT

(4.72)

If the complex ambiguity function of the single simple pulse x (t) is denoted as
p
 (t, F ), the integral in Eq. (4.72) is  (t + (n – m)T, F ). Thus
D
D
p
p

(4.73)

The double sum in Eq. (4.73) is somewhat difficult to deal with.
Obviously, all combinations of m and n having the same difference m – n result
in the same summand in the second sum, but the dependence of the exponential

term on m only prevents straightforward combining of all such terms. Defining
n′ = m – n, it can be shown by simply enumerating all of the combinations that
the double summation of some function f [m, n] can be written (Rihaczek, 1996)

(4.74)

Applying the decomposition of Eq. (4.74) to Eq. (4.73) gives

(4.75)

The geometric series that appears in both halves of the right-hand side of this
equation sums to