- pink noise: Filter amplitude response is proportional to ; PSD (``1/f noise'' -- ``equal-loudness noise'')
- brown noise: Filter amplitude response is proportional to ; PSD (``Brownian motion'' -- ``Wiener process'' -- ``random increments'')
In the preceding sections, we have looked at two ways of analyzing noise: the sample autocorrelation function in the time or ``lag'' domain, and the sample power spectral density (PSD) in the frequency domain. We now look at these two representations for the case of filtered noise.
The DTFT of is then, by the convolution theorem (§2.3.5),
since for white noise. Thus, we have derived that the autocorrelation of filtered white noise is proportional to the autocorrelation of the impulse response times the variance of the driving white noise.
Let's try to pin this down more precisely and find the proportionality constant. As the number of observed samples of goes to infinity, the length- Bartlett-window bias in the autocorrelation converges to a constant scale factor at lags such that . Therefore, the unbiased autocorrelation can be expressed as
In the limit, we obtain
In the frequency domain we therefore have
In summary, the autocorrelation of filtered white noise is
where is the variance of the driving white noise.
In words, the true autocorrelation of filtered white noise equals the autocorrelation of the filter's impulse response times the white-noise variance. (The filter is of course assumed LTI and stable.) In the frequency domain, we have that the true power spectral density of filtered white noise is the squared-magnitude frequency response of the filter scaled by the white-noise variance.
For finite number of observed samples of a filtered white noise process, we may say that the sample autocorrelation of filtered white noise is given by the autocorrelation of the filter's impulse response convolved with the sample autocorrelation of the driving white-noise sequence. For lags much less than the number of observed samples , the driver sample autocorrelation approaches an impulse scaled by the white-noise variance. In the frequency domain, we have that the sample PSD of filtered white noise is the squared-magnitude frequency response of the filter scaled by a sample PSD of the driving noise.
where is unit-variance white noise.
for nonnegative lags ( ). More completely, we can write
Thus, the autocorrelation of is a triangular pulse centered on lag 0. The true (unbiased) autocorrelation is given by
The true power spectral density (PSD) is then
Figure 6.3 shows a collection of measured autocorrelations together with their associated smoothed-PSD estimates.
Example: Synthesis of 1/F Noise (Pink Noise)
Pink noise7.10 or ``1/f noise'' is an interesting case because it occurs often in nature ,7.11is often preferred by composers of computer music, and there is no exact (rational, finite-order) filter which can produce it from white noise. This is because the ideal amplitude response of the filter must be proportional to the irrational function , where denotes frequency in Hz. However, it is easy enough to generate pink noise to any desired degree of approximation, including perceptually exact.
The following Matlab/Octave code generates pretty good pink noise:
Nx = 2^16; % number of samples to synthesize B = [0.049922035 -0.095993537 0.050612699 -0.004408786]; A = [1 -2.494956002 2.017265875 -0.522189400]; nT60 = round(log(1000)/(1-max(abs(roots(A))))); % T60 est. v = randn(1,Nx+nT60); % Gaussian white noise: N(0,1) x = filter(B,A,v); % Apply 1/F roll-off to PSD x = x(nT60+1:end); % Skip transient response
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 ? 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.)
Welch's Method with Windows