Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 40

and

n o ¼E n ð t Þ no ¼ r 2 :

var ð z Þ¼E ð z ð t Þ m z Þ 2

Hence, the pdf of the signal z(t) at any time t is given by:

p ð z Þ¼ 1
r

ð 2

p e 2

z a

2p

While the above result was derived above for the case where s(t) is a constant, its
general form is actually valid for any time signal s(t). For example, if

s(t) = sin(x o t), then z = m z = s(t) = sin(x o t) while the variance is still r 2 .

1.3.3.3 Power Spectral Density of Random Signals

A random signal n(t) can be classified as a power signal, and like other power
signals, has a PSD. This PSD is often denoted by G n (f), and is defined as:

1
T EF n ð t Þ P T ð t Þ

1

j 2 ¼ lim
T !1

T E N T ð f j j 2 ;

G n ð f Þ¼ lim
T !1

j f g

where E denotes the expected value.

1.3.3.4 Stationary Random Signals

Signals whose statistics (i.e., mean, variance and other higher order moments) do
not change with time are referred to as stationary.

1.3.3.5 The Autocorrelation Function of Random Signals

The autocorrelation function of a random signal x(t) is defined by:

R x ð t 1 ; t 2 Þ¼Ef x ð t 1 Þ x ð t 2 Þg:
In the above definition, the E denotes expected value, and this expected value
needs to be obtained by doing many experiments and averaging the results of all
those experiments. This kind of average is referred to as an ensemble average.
R x (t 1 ,t 2 ) provides an indication of how strongly the signal values at two dif-
ferent time instants are related to one another.

1.3.3.6 Wide-Sense Stationary Signals

A random signal is WSS if its mean and autocorrelation are time-invariant, i.e.,