Page 470 - Fundamentals of Radar Signal Processing

imaginary parts. The linear magnitude requires a square root and is less

computationally convenient.

6.2.4 Other Unknown Parameters
The preceding sections have shown the effect of unknown phase of the received
signal on the optimal detector. However, other parameters of the received signal
are also unknown in practice. The amplitude of the echo depends on all of the
factors in the radar range equation, including especially the unknown target

radar cross section and, at least until it is successfully detected, its range. In
addition, the target may be moving relative to the radar, so that the echo is
modified by a Doppler shift.
The derivation of the magnitude-based detectors of Secs. 6.2.2 and 6.2.3
included an assumption that the received signal amplitude was known.
Specifically, it was assumed that the received signal sample vector was
known, except for its absolute phase. However, as noted the absolute amplitude

is also unknown in general. To determine the effect of an unknown amplitude,
assume that the received signal is , where A is an unknown but deterministic
11
scale factor. The analysis of Sec. 6.2.2 can be repeated under this assumption.
The detector output under hypothesis H is unchanged as would be expected,
0
since the target echo with its unknown amplitude is not present in this case.

Under hypothesis H , the detector output is now . Note that
1
the detector still considers the quantity rather than because the
arises from the matched filter applied to the data and thus does not include the
unknown amplitude factor A of the signal echo. Also, the quantity is
now the energy of the matched filter reference signal, while the actual signal

energy becomes A E.
2
The equivalent of Eq. (6.51) is now

(6.58)

As before, the second argument of Eq. (6.58) can be written in terms of the
probability of false alarm. Furthermore, because the actual signal energy is now

2
A E, the first argument is still . Thus, the detection performance is still given
by Eq. (6.52). The unknown echo amplitude neither requires any change in the
detector structure nor changes its performance.
Despite the unknown amplitude, the sufficient statistic was not changed.
Furthermore, the probability of false alarm could be computed without

knowledge of the amplitude. When both these conditions hold, the detection test
is called a uniformly most powerful (UMP) test (Dudgeon and Johnson, 1993).
A UMP does not exist for the case where the signal delay (range) is