Page 468 - Fundamentals of Radar Signal Processing

probability of detection, the envelope detector requires about 0.6 dB higher

SNR than the coherent detector at P = 0.9, and about 0.7 dB more at P = 0.5.
D
D
The extra signal-to-noise ratio required to maintain the detection performance of
the envelope detector compared to the coherent case is called an SNR loss. SNR
losses can result from many factors; this particular one is often called the
detector loss. It represents extra SNR that must be obtained in some way if the
performance of the envelope detector is to match that of the ideal coherent

detector. Increasing the SNR in turn implies one or more of many radar system
changes, such as greater transmitter power, a larger antenna gain, reduced range
coverage, and so forth.
The phenomenon of detector loss illustrates a very important point in
detection theory: the less that is known about the signal to be detected, the
higher must be the SNR to achieve a given combination of P and P . In this
FA
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case, not knowing the absolute phase of the signal has cost about 0.6 dB.
Inconvenient though it may be, this result is intuitively satisfying: the worse the
knowledge of the signal details, the worse the performance of the detector will
be.

6.2.3 Linear and Square-Law Detectors
Equation (6.44) defines the optimal Neyman-Pearson detector for the Gaussian
example with an unknown phase in the data. It was shown that the ln[I (x)]
0
function could be replaced by its argument x without altering the performance. In
Sec. 6.3.2 a simpler detector characteristic than ln[I (·)] will again be desirable
0
for noncoherent integration, but it will not be possible to simply substitute any
monotonic increasing function. It is therefore useful to see what approximations
can be made to the ln[I (·)] function.
0
A standard series expansion for the Bessel function holds that

(6.54)

2
Thus for small x, I (x) ≈ 1 + x /4. Furthermore, one series expansion of the
0
natural logarithm has ln(1 + z) = z – z /2 + z /3 + …. Combining these gives
3
2

(6.55)

Equation (6.55) shows that if x is small, the optimal detector is well
approximated by a matched filter followed by a so-called square law detector,
i.e., a magnitude squaring operation. The factor of four can be incorporated into
the threshold in Eq. (6.44).