Page 463 - Fundamentals of Radar Signal Processing

(6.43)

or in sufficient statistic form

(6.44)

Equation (6.44) defines the signal processing required for optimum
detection in the presence of an unknown phase. It calls for taking the magnitude
of the matched filter output , passing it through the memoryless nonlinearity
ln[I (·)], and comparing the result to a threshold. This result is appealing in that
0
the matched filter is still applied to utilize the internal phase structure of the
known signal and maximize the integration gain, but then a magnitude operation

is applied because the absolute phase of the result cannot be known. Also, note
that the argument of the Bessel function is the energy in the matched filter output
divided by half the noise power; again, a signal-to-noise ratio. Only half of the
noise power appears because the total noise power in the complex case is split
between the real and imaginary channels.
As a practical matter, it is desirable to avoid having to compute the natural

logarithm and Bessel function for every threshold test, since these might occur
millions of times per second in some systems. Because the function ln[I (·)] is
0
monotonically increasing, the same detection results can be obtained by simply
comparing its argument to a modified threshold. Equation (6.44) then
becomes simply

(6.45)

Figure 6.6 illustrates the optimal detector for the coherent detector with an
unknown phase.