Filtered White Noise

When a white-noise sequence is filtered, successive samples generally
become correlated.^{7.8} Some of these filtered-white-noise signals have
names:

*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.

Let denote a length sequence we wish to analyze. Then the Bartlett-windowed acyclic sample autocorrelation of is , and the corresponding smoothed sample PSD is (§6.7, §2.3.6).

For filtered white noise, we can write as a convolution of white noise and some impulse response :

(7.29) |

The DTFT of is then, by the convolution theorem (§2.3.5),

(7.30) |

so that

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

(7.31) |

In the limit, we obtain

(7.32) |

In the frequency domain we therefore have

In summary, the autocorrelation of filtered white noise is

(7.33) |

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.

We reiterate that every stationary random process may be defined, for
our purposes, as *filtered white noise*.^{7.9} As we
see from the above, all correlation information is embodied in the
filter used.

- Example: FIR-Filtered White Noise
- Example: Synthesis of 1/F Noise (Pink Noise)
- Example: Pink Noise Analysis

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