This Demonstration creates a white-noise sequence and then uses a low-pass filter to produce a red-noise sequence. The "filter cut-off" is the fractional point in [0, 1] on the spectral frequency axis to apply the filter, as measured from zero frequency. The rate of suppression of frequencies larger than the filter cut-off is given by the "filter roll-off" exponent (1 to 4);larger values of mean greater suppression of the higher frequencies. A new time sequence is generated with the "randomize" button. The amplitude spectrum of the time series is plotted simultaneously, with spectral values on a log scale, out to the Nyquist frequency.
The "white-noise" time sequence is created by sampling from a random normal density. To apply low-pass filtering, the sequence is converted to the frequency domain by a Fourier transform. Then the filtering factor applied to all frequencies in the spectral domain is
where is the cut-off frequency and is the roll-off exponent. Note that the factor is always unity at . The filtered Fourier spectrum is then converted back to the time domain by the inverse Fourier transform. The resultant time sequence is commonly called a "red-noise" time sequence.