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 $G(f)$ is proportional to $1/\sqrt{f}$ ; PSD $\propto 1/f$ (``1/f noise'' -- ``equal-loudness noise'')
• brown noise: Filter amplitude response $G(f)$ is proportional to $1/f$ ; PSD $\propto 1/f^2$ (``Brownian motion'' -- ``Wiener process'' -- ``random increments'')

More generally, filtered white noise can be termed colored noise or correlated noise. As long as the filter is linear and time-invariant (LTI), and strictly stable (poles inside and not on the unit circle of the $z$ plane), its output will be a stationary ``colored noise''. We will only consider stochastic processes of this nature.

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 $x(n)$ denote a length $N$ sequence we wish to analyze. Then the Bartlett-windowed acyclic sample autocorrelation of $x$ is $x\star x$ , and the corresponding smoothed sample PSD is $\left\vert X(\omega)\right\vert^2$ (§6.7, §2.3.6).

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

$$x(n) = (h\ast v)(n) \isdef \sum_{m=-\infty}^\infty v(m)h(n-m)$$ (7.29)

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

$$X(\omega) = H(\omega)V(\omega)$$ (7.30)

so that

$$\begin{eqnarray*} x\star x &\longleftrightarrow& \left\vert X(\omega)\right\vert^2 = \left\vert H(\omega)V(\omega)\right\vert^2 = \left\vert H(\omega)\right\vert^2\left\vert V(\omega)\right\vert^2\\ &\longleftrightarrow& (h\star h)\ast (v\star v) \propto h\star h, \end{eqnarray*}$$

since $v\star v \propto \sigma_v^2\delta$ 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 $M$ of observed samples of $x(n) = (h\ast v)(n)$ goes to infinity, the length-$M$ Bartlett-window bias $M-\vert l\vert$ in the autocorrelation $x\star x$ converges to a constant scale factor $M$ at lags such that $M\gg \vert l\vert$ . Therefore, the unbiased autocorrelation can be expressed as

$$\hat{r}_x(l) = \frac{1}{M-\vert l\vert} x\star x \;\to\; \frac{1}{M} x \star x, \quad \hbox{($M\gg l$)}.$$ (7.31)

In the limit, we obtain

$$\lim_{M\to\infty} \frac{1}{M} x \star x = r_x(l)$$ (7.32)

In the frequency domain we therefore have

$$\begin{eqnarray*} S_x(\omega) &=& \lim_{M\to \infty}\frac{1}{M}\vert X(\omega)\vert^2 \;=\; % = \frac{1}{M}\vert H(\omega)\,V(\omega)\vert^2 \vert H(\omega)\vert^2 \cdot \lim_{M\to \infty} \frac{\vert V(\omega)\vert^2}{M} \\ [5pt] &=& \vert H(\omega)\vert^2 S_v(\omega) \;=\; \vert H(\omega)\vert^2\sigma_v^2 \;\longleftrightarrow\; (h\star h) \sigma_v^2 . \end{eqnarray*}$$

In summary, the autocorrelation of filtered white noise $x=h\ast v$ is

$$\zbox {r_x(l) = \sigma_v^2\cdot(h\star h)(l) \;\longleftrightarrow\;\sigma_v^2 \left\vert H(\omega)\right\vert^2}$$ (7.33)

where $\sigma_v^2$ 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 $l$ much less than the number of observed samples $M$ , 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 $\vert H(\omega)\vert^2$ 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.