0.5 Random signals and noise

When designing a comm system, it is impossible to know exactly what the signal and noise waveforms will be.But usually we know average characteristics such as energy distribution across frequency.It is exactly these statistics that are most often used for comm system design.

We will consider random waveforms known as “zero-mean wide-sense stationary random processes.”Such an $x\left(t\right)$ is completely completely characterized by its power spectral density $S_x\left(f\right)$ , or average signal power versus frequency f . The technical definition of PSD accounts for the fact that power isdefined as energy per unit time:

Above, $E\{·\}$ denotes expectation , i.e., statistical average.

A common random waveform is “white noise.” Saying that $w\left(t\right)$ is white is equivalent to saying that its PSD is flat:

Broadband noise is often modelled as white random noise. A fundamentally important question is: What does filtering do to the power spectrum of a signal? The answer comes with the aid of the autocorrelation function $R_x\left(\tau \right)$ , defined as

and having the important property

Note that, for white noise $w\left(t\right)$ with PSD N _o , we have

Here we see that $w\left(t_1\right)$ and $w\left(t_2\right)|t_2\ne t_1$ are "uncorrelated" because $E\left\{w\left(t_1\right)w^{*}\left(t_2\right)\right\}=0$ , and that $w\left(t\right)$ has average energy ${\mathcal{E}}_w={E\left\{\right|w\left(t\right)|}^2\}=\infty$ , which may be surprising.We could have found the same via ${\mathcal{E}}_w={\int }_{-\infty }^{\infty }{S}_w\left(f\right)df$ .

Now say that white noise $w\left(t\right)$ is filtered with non-random $h\left(t\right)$ , yielding output $y\left(t\right)={\int }_{-\infty }^{\infty }w\left(q\right)h(t-q)dq$ . We can find $S_y\left(f\right)$ by first finding $R_y\left(\tau \right)$ and then taking the FT. Recall:

Plugging in the expression for $y\left(t\right)$ , we find

Because averaging is a linear operation, and because the $h(·)$ terms are fixed (non-random) quantities, we can write

Since $w\left(t\right)$ is white,

which allows use of the sifting property on the inner integral:

To summarize, the autocorrelation of filtered white noise is

Having $R_y\left(\tau \right)$ , the final step is finding $S_y\left(f\right)=\mathcal{F}\left\{R_y\left(\tau \right)\right\}$ :

To summarize, the power spectrum of filtered white noise is

More generally, the power spectrum of a filtered random process $x\left(t\right)$ can be shown to be

which is quite intuitive.