0.10 Models of noise  (Page 2/3)

Sometimes it is (erroneously) written as $S\left(f\right)={\left|E\left(f\right)\right|}^2$ . erroneously because it should be averaged: $S\left(f\right)={\mathrm{I}\mathrm{E}\left|E\left(f\right)\right|}^2$ . The famous Wiener-Khinchine theorem says, that

where $r\left(k\right)=\mathrm{I}\mathrm{E}[{\epsilon }_n·{\epsilon }_{n-k}]=\mathrm{I}\mathrm{E}[{\epsilon }_k·{\epsilon }_0]$ is the auto-correlation of the process. Note $S\left(f\right)=S(f+1)$ .

Connection to Discrete Fourier transform (DFT) In practice, a finite length signal ${\epsilon }_0,...,{\epsilon }_{K-1}$ is interpreted as periodically repeated (with period $K$ ). Its Fourier transform is then a periodic series again, called DFT, which can be computed viaan algorithm called FFT in only $Klog\left(K\right)$ computations according to the formula

Recall that Matlab starts indices always with 1, thus interpreting the first entry of the vector $\hat{\epsilon }$ as corresponding to frequency 0 (see matlab help fft ).

If we set $K=2L+1$ we find that ${\hat{\epsilon }}_n$ appears on the right side of [link] . If we assume that $L$ is large so that we can neglect the limit, we find that $S\left(f\right)$ indicates what to expect of the squared FFT coefficients (i.e., the FFT-power-spectrum) on average:

for $n=1,..,K$ . Vice versa, the samples of the power spectrum $S\left(f\right)$ for $f=k/K$ can be estimated from averaging the FFT coefficients over several noisesignals each of the form ${\epsilon }_1,...,{\epsilon }_{2L+1}$ :

for $k=-L,...,0,...,L$ (it is more natural to represent $S$ over a symmetric interval; recall that ${\hat{\epsilon }}_{-L}={\hat{\epsilon }}_{L+1},...,{\hat{\epsilon }}_0={\hat{\epsilon }}_{2L+1}$ ).

Flat spectrum of zero mean white noise

The power spectrum $S\left(f\right)$ of white noise with zero mean ( $\mathrm{I}\mathrm{E}\left[e_k\right]=0$ ) can be computed as follows. First we find

by independence. By [link] we find

for all $f$ . Note that the power spectrum of white noise is flat . Its constant value is equal to the variance of the zero mean noise.The constant value must clearly be equal to the power. (Also: $P_{\epsilon }=1/K\sum {\epsilon }_n^2\simeq \mathrm{I}\mathrm{E}\left[{\epsilon }_0^2\right]={\sigma }^2$ from statistics.)

Consequently, since the mean of the DFT, i.e. $\mathrm{I}\mathrm{E}|{\hat{\epsilon }}_n{|}^2$ is constant, the average of the DFT when computed over many realizations of the noise is nearly flat matlab demo: plot(mean(abs(fft(0.1*(rand(1000,1000)-0.5),[],2). $\stackrel{ˆ}{}$ 2)));axis([0 1000 0 0.01]) ; however,the FFT of any single realization will show large oscillations (see [link] 3rd).

Nevertheless, the power can be estimated from one realization: using [link] , using ergodicity to estimate the expected value $\mathrm{I}\mathrm{E}[...]$ simply as the average over all samples, and using [link] we get

This computation also confirms that the spectrum of white noise is flat and that its constant value is equal to the power.

In fact, a direct computation from [link] shows that the FFT output ${\widehat{\epsilon }}_n$ is a white noise as well, however with variance $K{\sigma }^2$ and so [link] (the first approximation in [link] ) holds for white noise exactly,not only approximatively.

Intuitively, the formula $S\left(f\right)={\sigma }^2$ means that all frequencies are present in white noise with equal overall-amplitude. The spectrum being flat isa direct consequence of the independence between the noise terms. We should recall, though, that strictly speaking the spectrumis flat only as an average over many noise realizations.