Processing of Signals - Digital Signal Processing: A Practitioner's Approach - page 5

probability density function (pdf). For the same signal we have estimated the pdf
by a simple histogram (Figure 1.4(b)). For this signal we have actually generated
the pdf from the given time series using a numerical technique; notice how
closely it matches with the theoretical bell-shaped curve representing a normal
distribution.
Statistical or random signals can also be characterised only by their moments,
such as first moment (mean

2 ) and higher-order
moments [8]. In general, a statistical signal is completely defined by its pdf and in
turn this pdf is uniquely related to the moments. In some cases the entire pdf can be
generated using only

), second moment (variance

2 , as in the case of a Gaussian distribution, where the
pdf of a Gaussian random variable x is given by

and

1
2

e ð x Þ 2

2

= 2

f x ð x Þ¼ p :

The pdf does not have to be static; it could vary with time, making the signal non-
stationary.

1.2.3 Periodic Signals
A signal x ð t Þ is said to be periodic if

x ð t Þ¼ x ð t þ nT Þ where T is a real number and n is an integer ð 1 : 3 Þ

The periodicity of x ð t Þ is min ð nT Þ , which is T. If we rewrite this using the notation
t ¼ k t is a real number, 3 t, where k is an integer and then

T

x ð k t Þ¼ xk t þ n t ð 1 : 4 Þ

t

:

¼ xk t þ n ½ N þ t ð 1 : 5 Þ

t ¼ is a real number 4 less than 1 and

We write the quantity T = N þ , where N is

an integer. The practising engineer's periodicity in a loose sense is N; in a strict

N þ Þ ¼ N must be an integer. N could mathematical sense the periodicity min n ð

be very different from

N or may not exist. This illustrates that periodicity is not
preserved while moving from the continuous domain to the discrete domain in all
cases in a strict sense.

3

This quantity is known as sampling time.

4

A proper choice of t can make this quantity almost zero or zero.