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 $\omega _x$ :

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

As discussed in Chapter 4, 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 is odd, we can show this as follows:

$\begin{eqnarray*} X_w(\omega) &=& \sum_{n=-\infty}^{\infty}[w(n)x(n)]e^{ -j\omeg... ...mega-\omega_x)nT} \\ &=& \zbox {{\cal A}_xW(\omega-\omega_x)} \end{eqnarray*}$

Hence,

$\begin{eqnarray*} \vert X_w(\omega) \vert &=& \vert{\cal A}_x\vert \cdot \vert W... ...le X_w(\omega) &=& \angle {\cal A}_x+ \angle W(\omega-\omega_x). \end{eqnarray*}$

At $\omega _x$ , we have

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

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.9.8.

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

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

Previous: Following Spectral Peaks
Next: Tracking Sinusoidal Peaks in a Sequence of FFTs

Order a Hardcopy of Spectral Audio Signal Processing

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

Sign in

Search Online Books

Free Online Books

Chapters

See Also

Spectral Audio Signal Processing
    Applications of the STFT
       Additive Synthesis Analysis
          Sinusoidal Peak Finding