Page 121 - Fundamentals of Radar Signal Processing

(2.58)

The complex exponential terms outside the summations cancel. Interchanging
integration and summation and collecting terms then gives

(2.59)

A change of variables K′ = αK makes it clear that the integral has the form of
θ
θ
the inverse discrete-time Fourier transform of a constant spectrum S(K ) = 2π/α.
θ
Therefore, the integral is just the discrete impulse function (2θ/α)δ[n – l]. Using
this fact reduces Eq. (2.59) to a single summation over l that can be evaluated to
give

(2.60)

The decorrelation interval can now be determined by evaluating Eq. (2.60) to
find the value of ΔK which reduces s to a given level. This value of ΔK can
θ
θ
y
then be converted into equivalent changes in frequency or aspect angle.
One criterion is to choose the value of ΔK corresponding to the first zero
θ
of the correlation function, which occurs when the argument of the numerator
equals π. The resulting value is the Rayleigh width of the autocorrelation
function. Using Eq. (2.61) and recalling that L = (2M + 1) Δx gives

(2.61)

Recall that K = (2π/c)F sinθ. The total differential of z is then dK = (2π/c) ·
θ
θ
[sinθ · dF + Fcosθ · dθ]. To determine the decorrelation interval in angle for a
fixed radar frequency, set dF = 0 to give dKθ = (2π/c) · Fcosθ · dθ, so that ΔK θ
≈ (2π/c) · Fcosθ · Δθ. Similarly, the frequency step required to decorrelate the
target is obtained by fixing the aspect angle θ so that dθ = 0, leading to ΔK ≈
θ
(2π/c) · sinθ · ΔF. Combining these relations with Eq. (2.61) then gives the