Example: Synthesis of 1/F Noise (Pink Noise)

Pink noise^7.10 or ``1/f noise'' is an interesting case because it occurs often in nature [294],^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 $1/\sqrt{f}$ , where $f$ 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

In the next section, we will analyze the noise produced by the above matlab and verify that its power spectrum rolls off at approximately 3 dB per octave.