### Effect of Measurement Noise

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

*orthogonality principle*[38], the least-squares estimate of is obtained by orthogonally projecting onto the space spanned by the columns of . Geometrically speaking, choosing to minimize the Euclidean distance between and is the same thing as choosing it to minimize the sum of squared estimated measurement errors . The distance from to is minimized when the

*projection error*is orthogonal to every column of , which is true if and only if [84]. Thus, we have, applying the orthogonality principle,

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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