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Example: Pink Noise Analysis

Let's test the pink noise generation algorithm presented in §6.14.2. We might want to know, for example, does the power spectral density really roll off as $ 1/f$ ? Obviously such a shape cannot extend all the way to dc, so how far does it go? Does it go far enough to be declared ``perceptually equivalent'' to ideal 1/f noise? Can we get by with fewer bits in the filter coefficients? Questions like these can be answered by estimating the power spectral density of the noise generator output.

Figure 6.4 shows a single periodogram of the generated pink noise, and Figure 6.5 shows an averaged periodogram (Welch's method of smoothed power spectral density estimation). Also shown in each log-log plot is the true 1/f roll-off line. We see that indeed a single periodogram is quite random, although the overall trend is what we expect. The more stable smoothed PSD estimate from Welch's method (averaged periodograms) gives us much more confidence that the noise generator makes high quality 1/f noise.

Note that we do not have to test for stationarity in this example, because we know the signal was generated by LTI filtering of white noise. (We trust the randn function in Matlab and Octave to generate stationary white noise.)

Figure 6.4: Periodogram of pink noise.

Figure 6.5: Estimated power spectral density of pink noise.

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Mathematical Definition of the STFT
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Example: Synthesis of 1/F Noise (Pink Noise)