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Linear Prediction is Peak Sensitive

By Rayleigh's energy theorem (Parseval's theorem):

\begin{eqnarray*}
\sum_{n=-\infty}^{\infty} {\hat e}^2(n)
&=&
\frac{1}{2\pi}\int_{-\pi}^{\pi}\left\vert{\hat E}\left(e^{j\omega}\right)\right\vert^2 d\omega\\
&\mathrel{\stackrel{\mathrm{\Delta}}{=}}&\frac{1}{2\pi}\int_{-\pi}^{\pi}\left\vert{\hat A}\left(e^{j\omega}\right)Y\left(e^{j\omega}\right)\right\vert^2 d\omega\\
&=&\frac{{\hat\sigma}^2_e}{2\pi}\int_{-\pi}^{\pi}\left\vert\frac{Y\left(e^{j\omega}\right)}%
{{\hat Y}\left(e^{j\omega}\right)}\right\vert^2 d\omega
\end{eqnarray*}

From this ``ratio error'' expression in the frequency domain, we can see the following:


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``Cross Synthesis Using Cepstral Smoothing or Linear Prediction for Spectral Envelopes'', by Julius O. Smith III, (From Lecture Overheads, Music 421).
Copyright © 2020-06-27 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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