Consider a complex exponential sequence \({e^{j{\omega _0}n}}\) with frequency ω₀. Suppose ω₀ = 1, then 

Concept:

Any discrete signal is said to be periodic if:

\(\frac{{{\omega _0}}}{{2\pi }}\) is a rational number \(\left( {\frac{M}{N}} \right)\)

Calculation:

Given signal is \({e^{j{\omega _0}n}}\), ω₀ = 1 rad / sec

\(\frac{{{\omega _0}}}{{2\pi }} = \frac{1}{{2\pi }}\) 

Since this is not a rational number, so it is not periodic at all.

Note:

Continuous sinusoidal and complex sinusoids are periodic for every value of ‘ω₀’, but discrete-time signals are periodic only if \(\frac{{{\omega _0}}}{{2\pi }}\) is a rational number \(\frac{M}{N}\).