# Filtering a White-Noise Sequence

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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.

Contributed by: David von Seggern (February 2008)

Open content licensed under CC BY-NC-SA

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## Details

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.

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