Page 461 - Fundamentals of Radar Signal Processing

3. ξ is deterministic but unknown.

Different techniques are used to handle each of these cases. The first is the most

important because it has the greatest effect on the structure of the optimal
Neyman-Pearson detector.
To illustrate the approach for handling a random parameter with a known
PDF, consider yet again the complex Gaussian case. The optimal detector
H
implemented a matched filter operation m y followed by the Re{·} operator.
The success of the matched filter structure depended on knowing exactly the
constant component of y = m + w under hypothesis H , so that the filter
1
coefficients could be set equal to m and the filter output would be real-valued.
Recall that, when applied to radar, y under H is considered to consist only of
0
samples w of receiver noise, and under H to consist of noisy samples m + w of
1
the echoes from a radar target over multiple pulses, or alternately successive
fast-time samples of the waveform of one pulse echo from a target.
Claiming perfect knowledge of m implies knowing the range to the target
very precisely, since a variation in one-way range of only λ/4 causes the
received echo phase to change by 180°. A quarter-wavelength is only 30 cm at

L band and 3.16 mm at 95 GHz. Because this precision is usually unrealistic, it
is more reasonable to assume m is known only to within a phase factor exp(jθ),
where the phase angle θ is considered to be a random variable distributed
uniformly over (0, 2π] and independent of the random variables {m }. In other
n
words, , where is known exactly but θ is a random phase. Note
that the energy in is the same as that in m, that is, . This “unknown

phase” assumption cannot usually be avoided in radar. What is its effect on the
optimal detector and its performance?
The goal remains to carry out the LRT, so it is necessary to return to its
basic definition of Eq. (6.6) and determine p (y|H ) and p (y|H ), both of which
1
y
0
y
now presumably depend on θ, and use the technique known as the Bayesian
approach for random parameters with known PDFs (Kay, 1998). Specifically,
10
compute the PDF under H by averaging the conditional PDFs p (y|H, θ) over θ
y
i
i
(6.37)

The unconditional PDFs p (y|H) are then used to define the likelihood ratio in
i
y
the usual way.
As an example of the Bayesian approach for random parameters, consider
again the complex Gaussian case, but now with an unknown phase in the data,
. The conditional PDF of the observations y becomes, under each of

the two hypotheses,