Page 337 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

326 WORKED OUT EXAMPLES

9.2.4 Matched filtering

This conventional solution is achieved by neglecting the reflections. The
measurements are modelled by a vector z with N elements:

z n ¼ a hðn tÞþ vðn Þ ð9:5Þ

The noise is represented by a random vector v with zero mean and
2
covariance matrix C v ¼ I. Upon introduction of a vector h(t) with
v
elements h(n t) the conditional probability density of z is:
1 1 T
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ðz ahðtÞÞ ðz ahðtÞÞ ð9:6Þ
pðzjtÞ¼ q 2 2
2 N n
ð2 Þ
n
Maximization of this expression yields the maximum likelihood estimate
for t. In order to do so, we only need to minimize the L 2 norm of z ah(t):

T T 2 T T
ðz ahðtÞÞ ðz ahðtÞÞ ¼ z z þ a hðtÞ hðtÞ 2az hðtÞ ð9:7Þ

T
The term z z does not depend on t and can be ignored. The second term
is the signal energy of the direct response. A change of t only causes a
shift of it. But, if the registration period is long enough, the signal energy
is not affected by such a shift. Thus, the second term can be ignored as
well. The maximum likelihood estimate boils down to finding the t that
T
maximizes az h(t). A further simplification occurs if the extent of h(t)is
T
limited to, say, K with K << N. In that case az h(t) is obtained by
cross-correlating z n by a h(n þ t):

K 1
X
yðtÞ¼ a hðk tÞz k ð9:8Þ
k¼0

The value of t which maximizes y(t) is the best estimate. The operator
expressed by (9.8) is called a matched filter or a correlator. Figure 9.8
shows a result of the matched filter. Note that apart from its sign, the
amplitude a does not affect the outcome of the estimate. Hence, the fact
that a is usually unknown doesn’t matter much. Actually, if the nominal
response is given in a non-parametric way, the matched filter doesn’t
have any design parameters.