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Random Signals and Noise

Module by: Phil Schniter. E-mail the author

Summary: This module describes the basics of random signals and noise as needed for an introductory course on data communications. First it introduces the notion of power spectral density (PSD) and defines "white" noise as a noise with a flat PSD. Next it describes the effect that linear filtering has on a random signal, in both time and frequency domains. For this, the autocorrelation function is introduced, and expectations are used to derive the fundamental properties.

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(t)x(t) is completely completely characterized by its power spectral density Sx(f)Sx(f), or average signal power versus frequency f. The technical definition of PSD accounts for the fact that power is defined as energy per unit time:

S x ( f ) : = lim T 1 2 T E - T T x ( t ) e j 2 π f t d t 2 , S x ( f ) : = lim T 1 2 T E - T T x ( t ) e j 2 π f t d t 2 ,
(1)

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

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

S w ( f ) = N o for all f . S w ( f ) = N o for all f .
(2)

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 Rx(τ)Rx(τ), defined as

R x ( τ ) := E { x ( t ) x * ( t - τ ) } R x ( τ ) := E { x ( t ) x * ( t - τ ) }
(3)

and having the important property

R x ( τ ) F S x ( f ) . R x ( τ ) F S x ( f ) .
(4)

Note that, for white noise w(t)w(t) with PSD No, we have

S w ( f ) = N o R w ( τ ) = N o δ ( τ ) E { w ( t ) w * ( t - τ ) } = N o δ ( τ ) . S w ( f ) = N o R w ( τ ) = N o δ ( τ ) E { w ( t ) w * ( t - τ ) } = N o δ ( τ ) .
(5)

Here we see that w(t1)w(t1) and w(t2)|t2t1w(t2)|t2t1 are "uncorrelated" because E{w(t1)w*(t2)}=0E{w(t1)w*(t2)}=0, and that w(t)w(t) has average energy Ew=E{|w(t)|2}=Ew=E{|w(t)|2}=, which may be surprising. We could have found the same via Ew=-Sw(f)dfEw=-Sw(f)df.

Now say that white noise w(t)w(t) is filtered with non-random h(t)h(t), yielding output y(t)=-w(q)h(t-q)dqy(t)=-w(q)h(t-q)dq. We can find Sy(f)Sy(f) by first finding Ry(τ)Ry(τ) and then taking the FT. Recall:

R y ( τ ) = E { y ( t ) y * ( t - τ ) } . R y ( τ ) = E { y ( t ) y * ( t - τ ) } .
(6)

Plugging in the expression for y(t)y(t), we find

R y ( τ ) = E - w ( q 1 ) h ( t - q 1 ) d q 1 - w * ( q 2 ) h * ( t - τ - q 2 ) d q 2 = E - - w ( q 1 ) w * ( q 2 ) h ( t - q 1 ) h * ( t - τ - q 2 ) d q 1 d q 2 . R y ( τ ) = E - w ( q 1 ) h ( t - q 1 ) d q 1 - w * ( q 2 ) h * ( t - τ - q 2 ) d q 2 = E - - w ( q 1 ) w * ( q 2 ) h ( t - q 1 ) h * ( t - τ - q 2 ) d q 1 d q 2 .
(7)

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

R y ( τ ) = - - E w ( q 1 ) w * ( q 2 ) h ( t - q 1 ) h * ( t - τ - q 2 ) d q 1 d q 2 . R y ( τ ) = - - E w ( q 1 ) w * ( q 2 ) h ( t - q 1 ) h * ( t - τ - q 2 ) d q 1 d q 2 .
(8)

Since w(t)w(t) is white,

E w ( q 1 ) w * ( q 2 ) = E w ( q 1 ) w * q 1 - ( q 1 - q 2 ) = N o δ ( q 1 - q 2 ) , E w ( q 1 ) w * ( q 2 ) = E w ( q 1 ) w * q 1 - ( q 1 - q 2 ) = N o δ ( q 1 - q 2 ) ,
(9)

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

R y ( τ ) = - - N o δ ( q 1 - q 2 ) h ( t - q 1 ) h * ( t - τ - q 2 ) d q 1 d q 2 = N o - h ( t - q 2 ) h * ( t - τ - q 2 ) d q 2 = N o - h ( q ) h * ( q - τ ) ( - d q ) using q : = t - q 2 = N o - h ( q ) h * ( q - τ ) d q . R y ( τ ) = - - N o δ ( q 1 - q 2 ) h ( t - q 1 ) h * ( t - τ - q 2 ) d q 1 d q 2 = N o - h ( t - q 2 ) h * ( t - τ - q 2 ) d q 2 = N o - h ( q ) h * ( q - τ ) ( - d q ) using q : = t - q 2 = N o - h ( q ) h * ( q - τ ) d q .
(10)

To summarize, the autocorrelation of filtered white noise is

R y ( τ ) = N o - h ( q ) h * ( q - τ ) d q . R y ( τ ) = N o - h ( q ) h * ( q - τ ) d q .
(11)

Having Ry(τ)Ry(τ), the final step is finding Sy(f)=F{Ry(τ)}Sy(f)=F{Ry(τ)}:

S y ( f ) = - R y ( τ ) e - j 2 π f τ d τ = - N o - h ( q ) h * ( q - τ ) d q e - j 2 π f τ d τ = N o - h ( q ) e - j 2 π f q - h * ( q - τ ) e j 2 π f ( q - τ ) d τ d q = N o - h ( q ) e - j 2 π f q - h * ( t ) e j 2 π f t d t d q = N o - h ( q ) e - j 2 π f q d q - h ( t ) e - j 2 π f t d t * = N o H ( f ) H * ( f ) = N o | H ( f ) | 2 S y ( f ) = - R y ( τ ) e - j 2 π f τ d τ = - N o - h ( q ) h * ( q - τ ) d q e - j 2 π f τ d τ = N o - h ( q ) e - j 2 π f q - h * ( q - τ ) e j 2 π f ( q - τ ) d τ d q = N o - h ( q ) e - j 2 π f q - h * ( t ) e j 2 π f t d t d q = N o - h ( q ) e - j 2 π f q d q - h ( t ) e - j 2 π f t d t * = N o H ( f ) H * ( f ) = N o | H ( f ) | 2
(12)

To summarize, the power spectrum of filtered white noise is

S y ( f ) = N o | H ( f ) | 2 . S y ( f ) = N o | H ( f ) | 2 .
(13)

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

S y ( f ) = S x ( f ) | H ( f ) | 2 , S y ( f ) = S x ( f ) | H ( f ) | 2 ,
(14)

which is quite intuitive.

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