Page 365 - Fundamentals of Radar Signal Processing

where the {a } are the integration weights. Assume that each data sample y[m]
m
is a (possibly complex) constant A. Then the power of the weighted coherent

sum in the absence of phase error ϕ[m] is

(5.69)

It can be shown that the integrated power when a white Gaussian phase error is
present is (Richards, 2003)

(5.70)

where is the variance of the phase noise in radians. This can be applied to the

two-pulse canceller by letting M = 2 and a = 1, a = –1. Equation (5.70) then
1
0
gives the power at the canceller output as . Since the power of a
clutter sample before the canceller is |A| , the limitation on two-pulse
2
cancellation due to the phase noise becomes .
A Gaussian PDF for phase error is reasonable when the error is small, but
not when it becomes large because the PDF is not confined to the interval [–π,
π]. An extension to this analysis uses instead the Tikhonov distribution for

phase:

(5.71)

As the parameter α varies from zero to infinity, the PDF varies from uniform
random to a fixed value of zero radians, as shown in Fig. 5.14. Using this PDF
produces very similar results to the Gaussian analysis, differing only in
replacing the quantity in Eq. (5.70) and the two-pulse canceller
2
improvement factor limit by the quantity [I (α)/I (α)] , where I (α) and I (α) are
0
1
0
1
the modified Bessel functions of the first kind and orders one and zero,
respectively (Richards, 2011).