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Sinusoidal Peak Finding

For each sinusoidal component of a signal, we need to determine its frequency, amplitude, and phase (when needed). As a starting point, consider the windowed complex sinusoid with complex amplitude $ {\cal A}_x$ and frequency $ \omega _x$ :

$\displaystyle x_w(n) = w(n){\cal A}_xe^{j\omega_x nT}$ (11.20)

As discussed in Chapter 5, the transform (DTFT) of this windowed signal is the convolution of a frequency domain delta function at $ \omega _x$ [ $ \delta(\omega - \omega_x) $ ], and the transform of the window function, $ W(\omega)$ , resulting in a shifted version of the window transform $ {\cal A}_xW(\omega-\omega_x)$ . Assuming $ M$ is odd, we can show this as follows:

\begin{eqnarray*}
X_w(\omega) &=& \sum_{n=-\infty}^{\infty}[w(n)x(n)]e^{ -j\omega nT}
\qquad\hbox{(DTFT($x_w$))} \\
&=& \sum_{n=-(M-1)/2}^{(M-1)/2} \left[w(n){\cal A}_xe^{j\omega_xnT}\right]e^{ -j\omega nT}\\
&=& {\cal A}_x\sum_n w(n) e^{-j(\omega-\omega_x)nT} \\
&=& \zbox {{\cal A}_xW(\omega-\omega_x)}
\end{eqnarray*}

Hence,

\begin{eqnarray*}
\vert X_w(\omega) \vert &=& \vert{\cal A}_x\vert \cdot \vert W(\omega-\omega_x)\vert
\qquad \hbox{(see \fref {peak} below)}\\
\angle X_w(\omega) &=& \angle {\cal A}_x+ \angle W(\omega-\omega_x).
\end{eqnarray*}

At $ \omega _x$ , we have

\begin{eqnarray*}
\vert X_w(\omega_x)\vert &=& \vert{\cal A}_x\vert\cdot \vert W(0)\vert \\
\angle X_w(\omega_x)\vert &=& \angle {\cal A}_x+ \angle W(0)
\end{eqnarray*}

If we scale the window to have a dc gain of 1, then the peak magnitude equals the amplitude of the sinusoid, i.e., $ \vert X_w(\omega_x)\vert=\vert{\cal A}_x\vert\isdef a$ , as shown in Fig.10.8.

Figure: Schematic diagram of a window transform amplitude-scaled by $ a$ and frequency-shifted by $ \omega _x$ .
\includegraphics[width=0.8\twidth]{eps/peak}

If we use a zero-phase (even) window, the phase at the peak equals the phase of the sinusoid, i.e., $ \angle X_w(\omega_x) =
\angle {\cal A}_x$ .


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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