Smoothed Power Spectral Density

The DTFT of the Bartlett (triangular) window weighting in (5.6) is given by

where is again the number of samples of . We see that equals the sample power spectral density convolved with , or

It turns out that even more smoothing than this is essential for obtaining a stable estimate of the true PSD, as discussed further in §5.11 below.

Since the Bartlett window has no effect on an impulse signal (other than a possible overall scaling), we may use the biased autocorrelation (5.4) in place of the unbiased autocorrelation (5.5) for the purpose of testing for white noise.

The right column of Fig.5.1 shows successively greater averaging of the Bartlett-smoothed sample PSD.

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Order a Hardcopy of Spectral Audio Signal Processing

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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.