# Discrete-time Signal & Periodicity condition

Summary:
I seek a clarification on the periodicity of discrete-time (DT) signals.
Namaste

I seek a clarification on the periodicity condition of discrete-time (DT) signals.

As stated in Oppenheim’s Signals & Systems, for a DT signal, for example the complex exponential, to be periodic, i.e.

ej*w(n+N) = ej*w*n,

w/2*pi = m/N, where m/N must be a rational number.

Above is simply to satisfy the condition that ej*w*N = 1.

Please help me see why m=9, N=3, and, thereby, m/N = 3, which is not a rational number, yet still yielding 1 for ej*2*pi*9, not be considered a valid instance of a periodic discrete-time signal.

Would greatly appreciate a refutation and a chance to stand corrected.

Thank you.

## Answers and Replies

marcusl
Science Advisor
Gold Member
3 *is* a rational number (3/1).

Your first equation above is satisfied for m = integer, which is a different way of expressing your rational condition that might be clearer.

• wirefree