Periodicity

A discrete-time signal may always be classified as either periodic or aperiodic. A signal $x\left[n\right]$ is said to be periodic if, for some positive real integer $N$ the following relation holds: $$x\left[n\right]=x[n+N].$$ 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 $x\left[n\right]=x[n+N]$. If this equation is not satisfied for any integer $N$, the signal $x\left[n\right]$ is said to be an aperiodic signal.

Examples of periodic signals are: $x\left[n\right]=\sin \left(n\pi /4\right)$ and $x\left[n\right]=e^{jn\frac{\pi }{8}}$. 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: $x\left[n\right]=\sin \left(n\right)$ and $x\left[n\right]=a^n\cdot u\left[n\right]$ since there is no integer $N$ for which the signal repeats.