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

Figure 5.7 shows a spectrum analysis of two cosines

$\displaystyle x(n) = \cos(\omega_1 n) + \cos(\omega_2 n), \quad n=0,1,\ldots,M-1,$ (6.17)

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: DTFT of two closely spaced in-phase sinusoids, various rectangular-window lengths $ M$ .
\includegraphics[width=\twidth]{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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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA