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Chapters

Spectral Audio Signal Processing
    Spectrum Analysis of Sinusoids
       The Rectangular Window
          Frequency Resolution
             Two Cosines (``In-Phase'' Case)

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Two Cosines (``In-Phase'' Case)

Figure 1.13 shows a spectrum analysis of two cosines

$\displaystyle x(n) = \cos(\omega_1 n) + \cos(\omega_2 n), \quad n=0,1,\ldots,M-1, $

where $ \omega_1 = \pi/2$ and $ \omega_2 = \omega_1 + \Delta\omega$, and the frequency separation $ \Delta \omega = \omega_2-\omega_1$ is $ 2\pi/40$ radians per sample. The zero-padded Fourier analysis uses rectangular windows of lengths $ M=20$, $ 30$, $ 40$, and $ 80$ ( $ \Delta\omega = \frac{1}{2}\Omega_M, \frac{3}{4}\Omega_M, \Omega_M, 2\Omega_M$, where $ \Omega_M\isdef 2\pi/M$). The length $ N=1024$ FFT output is divided by $ M$ so that the ideal height of each spectral peak is $ \max_{\omega_k}\{\vert X(\omega_k)\vert\}=1/2$.

Figure 1.13: DTFT of two closely spaced in-phase sinusoids, various rectangular-window lengths $ M$.
\includegraphics[width=\textwidth]{eps/resolvedSines}

The longest window ($ M=80$) resolves the sinusoids very well, while the shortest case ($ M=20$) does not resolve them at all (only one ``lump'' appears in the spectrum analysis). In difference-frequency cycles, the analysis windows are two cycles and half a cycle in these cases, respectively. It can be debated whether or not the other two cases are resolved, and we will return to them shortly.

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Previous: Frequency Resolution
Next: One Sine and One Cosine ``Phase Quadrature'' Case
written by 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.


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