Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 36

0.07

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−4 −2 0 2 4

(a) (b)

Figure 2.12 Amplitude statistics of u k

2.4.2 Fourier Transform and pdf of Noise
We have constructed the pdf of a noise signal u k that is white and Gaussian. It is
given in Figure 2.12 for a WGN.
If the autocorrelation function r k ¼ k , then the noise is white. This is because the
power spectrum of the noise is flat or the power level at all frequencies is constant.
This whiteness (Figure 2.12(a)) is independent from the pdf of the noise
(Figure 2.12(b)). The implication of this statement is that the pdf can be of any
form. When the amplitude statistics of the given time series have a Gaussian
distribution, and the power spectrum of the same signal has power levels constant
and uniform across all the frequencies, then it is called white Gaussian noise, which
essentially means that r k ¼ k .

2.5 Propagation of Noise in Linear Systems
When a WGN (Figure 2.12) is passed through a filter, the output is a coloured noise
as in Figure 2.13(a). Here the amplitude statistics continue to remain Gaussian, as in
Figure 2.13(b).

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(a) (b)

Figure 2.13 Amplitude statistics and power spectrum of y k