Simple Operations and properties of Sequences

Signals in Natural Domain

Chapter 2 : Simple Operations and properties of Sequences

Periodic Signals:

An important class of signals that we encounter frequently is the class of periodic signals. We say that a signal {x[n] is periodic with period N, where N is a positive integer, if the signal is unchanged by the time shift of N ie.,

{x[n]} = {x[n + N]}

or x[n] = x[ n + N ] for all n.

Since {x[n]} is same as {x[n+N]} , it is also periodic so we get

{x[n]} = {x[n+N]} = {x[n+N+N]} = {x[n+2N]}

Generalizing this we get {x[n]} = {x[n+kN]}, where k is a positive integer. From this we see that {x[n]} is periodic with 2N, 3N,... The fundamental period N₀ is the smallest positive value N for which the signal is periodic.

The signal illustrated below is periodic with fundamental period N₀ = 4

FIGURE

By change of variable we can write {x[n]} = {x[n+N]} as {x[m-N]} = {x[m]} and then arguing as before, we see that

{x[n]} = {x[n+kN]},

for all integer values of k positive, negative or zero. By definition, period of a signal is always a positive integer N.

Except for a all zero signal all periodic signals have infinite energy. They may have finite power. Let {x[n]} be periodic with period N, then the power P_x is given by

$\displaystyle P-x=\lim\limits_{M\rightarrow \infty}\frac{1}{(2M+1)}\sum\limits_{n=-M}^{M}\vert x[n]\vert^2$

where k is largest integer such that kN -1 ≤ M. Since the signal is periodic, sum over one period will be same for all terms. We see that k is approximately equal to M/N (it is integer part of this) and for large M we get 2M/N terms and limit 2M/(2M +1) as M goes to infinite is one we get

$\displaystyle P_x=\frac{1}{N}\sum\limits_{n=0}^{N-1}\vert x[n]\vert^2$