In practice, measurements are never perfect. Let denote the measured output signal, where is a vector of ``measurement noise'' samples. Then we have

By the

Solving for yields Eq. (F.8) as before, but this time we have derived it as the least squares estimate of in the presence of output measurement error.

It is also straightforward to introduce a *weighting function* in
the least-squares estimate for
by replacing
in the
derivations above by
, where
is any positive definite
matrix (often taken to be diagonal and positive). In the present
time-domain formulation, it is difficult to choose a
weighting function that corresponds well to *audio perception*.
Therefore, in audio applications, frequency-domain formulations are
generally more powerful for linear-time-invariant system
identification. A practical example is the frequency-domain
equation-error method described in §I.4.4 [78].

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