Alternative Derivation of The Matched Filter
Alternatively, we may solve for the matched filter by solving our maximization problem with a Lagrangian. Again, the matched filter endeavors to maximize the output signal-to-noise ratio of a filtered deterministic signal in stochastic additive noise. The observed sequence, again, is
with the noise covariance matrix,
The signal-to-noise ratio is
Evaluating the expression in the numerator, we have
and in the denominator,
The signal-to-noise ratio becomes
If we now constrain the denominator to be 1, the problem of maximizing is reduced to maximizing the numerator. We can then formulate the problem using a Lagrange multiplier:
which we recognize as a generalized eigenvalue problem
Since is of unit rank, it has only one nonzero eigenvalue. It can be shown that this eigenvalue equals
yielding the following optimal matched filter
This is the same result found in the previous section.
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