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 , 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 . Thus, we have, applying the orthogonality principle,
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 .