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Smoothed Power Spectral Density

The DTFT of the Bartlett (triangular) window weighting in (6.16) is given by

$\displaystyle N^2\cdot\hbox{asinc}^2(\omega)\isdef \left[\frac{\sin(N\omega/2)}{\sin(\omega/2)}\right]^2,$ (7.17)

where $ N$ is again the number of samples of $ v(n)$ . We see that $ \left\vert V(\omega)\right\vert^2$ equals the sample power spectral density convolved with $ N^2\cdot\hbox{asinc}_N^2(\omega)$ , or

$\displaystyle \left\vert V(\omega)\right\vert^2 = N^2\cdot \hbox{asinc}_N^2 \ast {\hat S}_{v,N}(\omega).$ (7.18)

It turns out that even more smoothing than this is essential for obtaining a stable estimate of the true PSD, as discussed further in §6.11 below.

Since the Bartlett window has no effect on an impulse signal (other than a possible overall scaling), we may use the biased autocorrelation (6.14) in place of the unbiased autocorrelation (6.15) for the purpose of testing for white noise.

The right column of Fig.6.1 shows successively greater averaging of the Bartlett-smoothed sample PSD.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University