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Effect of Measurement Noise

In practice, measurements are never perfect. Let $ \hat{\underline{y}}=\underline{y}+\underline{e}$ denote the measured output signal, where $ \underline{e}$ is a vector of ``measurement noise'' samples. Then we have

$\displaystyle \hat{\underline{y}}= \underline{y}+ \underline{e}= \mathbf{x}\underline{h}+ \underline{e}.

By the orthogonality principle [38], the least-squares estimate of $ \underline{h}$ is obtained by orthogonally projecting $ \hat{\underline{y}}$ onto the space spanned by the columns of $ \mathbf{x}$ . Geometrically speaking, choosing $ \underline{h}$ to minimize the Euclidean distance between $ \hat{\underline{y}}$ and $ \mathbf{x}\underline{h}$ is the same thing as choosing it to minimize the sum of squared estimated measurement errors $ \vert\vert\,\underline{e}\,\vert\vert ^2$ . The distance from $ \mathbf{x}\underline{h}$ to $ \hat{\underline{y}}$ is minimized when the projection error $ \underline{e}=\hat{\underline{y}}-\mathbf{x}\underline{h}$ is orthogonal to every column of $ \mathbf{x}$ , which is true if and only if $ \mathbf{x}^T\underline{e}=0$ [84]. Thus, we have, applying the orthogonality principle,

$\displaystyle 0 = \mathbf{x}^T\underline{e}= \mathbf{x}^T(\underline{y}- \mathbf{x}\underline{h}) = \mathbf{x}^T\underline{y}- \mathbf{x}^T\mathbf{x}\underline{h}.

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

It is also straightforward to introduce a weighting function in the least-squares estimate for $ \underline{h}$ by replacing $ \mathbf{x}^T$ in the derivations above by $ \mathbf{x}^TR$ , where $ R$ 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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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-04-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University