Discrete-time Signal & Periodicity condition

[wirefree]

TL;DR 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.

e^j*w(n+N) = e^j*w*n,

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

Above is simply to satisfy the condition that e^j*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 e^j*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.

 

[marcusl]

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.