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 orthogonality principle [38], the least-squares estimate of
Solving for
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].