Periodicity

A discrete-time signal may always be classified as either periodic or aperiodic. A signal $x[n]$ is said to be periodic if, for some positive real integer $N$ the following relation holds: $$ x[n]=x[n+N]. $$ This is equivalent to saying that the sequence repeats itself every $N$ samples. If a signal is periodic with period $N$, it is also periodic with $2N$, period $3N$, and all other integer multiples of $N$. The fundamental period, which will be denoted by $N$, is the smallest positive integer for which $x[n]=x[n+N]$. If this equation is not satisfied for any integer $N$, the signal $x[n]$ is said to be an aperiodic signal.

Examples of periodic signals are: $x[n]=\sin(n\pi/4)$ and $x[n] = e^{jn \frac{\pi}{8}}$. The first signal has a fundamental period of $N=8$ samples and the second signal has a period of $N=16$ samples. Examples of aperiodic signals are: $x[n]=\sin(n)$ and $x[n]=a^n \cdot u[n]$ since there is no integer $N$ for which the signal repeats.

Example

Determine for which value(s) of the relative frequency $\theta$ the signal $x[n]= A \sin(n \theta + \phi)$ is periodic and calculate the period(s).
The function $x[n]$ is periodic for $\theta = 2 \pi K/N$, in which both $K$ and $N$ are some positive real integers. Thus an equation for the period of $x[n]$ is given by $N = 2 \pi K_{min}/\theta$, in which $K_{min}$ represents the smallest possible value of $K$. For example for $\theta = 0.23 \pi$ we have $K_{min}=23$ and $N=200$ while for $\theta = 0.24 \pi$ we have $K_{min}=3$ and $N=25$.

Example

Determine whether or not the complex signal $x[n] = e^{jn}$ is periodic or not.
Both real and imaginary parts of the signal $x[n] = e^{jn} = \cos (n) + j \sin (n)$ are aperiodic, since there is no integer $N$ for which $\cos(n)=\cos(n+N)$ and/or $\sin(n)=\sin(n+N)$.
Last updated on Mar 7, 2020