Page 458 - Fundamentals of Radar Signal Processing

H
Finally, note the Re{·} operator applied to the matched filter output m y.
Because m and y are complex, one might be concerned that the dot product
H
could be purely imaginary or nearly so, such that Re{m y} ≈ 0. The measured
data y would then have little or no effect on the threshold test. For this example,

m = 0 under hypothesis H and the Re{·} operator is inconsequential. Under
N
0
hypothesis H each element of m is a complex number . If the target is
1
actually present the elements of the measured data vector y = m + w will be of

the form where w is a zero mean complex Gaussian noise sample.
n
It follows that

(6.31)

The first term is again the energy E in the signal m; this is real-valued and
therefore unaffected by the Re{·} operator. The second term is simply weighted
and integrated noise samples. The phase of this noise component and therefore
the effect of the Re{·} operator is random. Its effect on the phase of the sum will
be large when the SNR is low, but minimal when the SNR is high.

It is evident by inspection of Eq. (6.30) that the sufficient statistic is now
H
Re{m y}. Expressing the LRT in its sufficient statistic form for the complex
case gives

(6.32)

H
Note that if m = m1 , the term Re{m y} = m Σy and Eq. (6.32) is very similar
n
N
to Eq. (6.13).
To complete consideration of the complex Gaussian case, its performance,
i . e . , P and P , must be determined. The sufficient statistic
D
FA
is just a sum of Gaussian random variables and so will
also be Gaussian. To determine the performance of the coherent detector, the
PDF of ϒ must be determined under each hypothesis. To do so it is useful to
H
first consider the quantity z = m y, which will be a complex Gaussian. First
suppose hypothesis H is true. In this case the {y } are zero mean and therefore
0
n
so is z. Because the {y } are independent the variance of z is just the sum of the
n
variances of the individual weighted samples: