Discrete-time Signal & Periodicity condition

In summary, the periodicity condition for discrete-time signals, as explained by Oppenheim's Signals & Systems, states that for a DT signal to be periodic, the complex exponential must satisfy the equation ej*w(n+N) = ej*w*n, where w/2*pi = m/N and m/N is a rational number. This is to ensure that ej*w*N = 1 for periodicity. However, even though m=9, N=3, and m/N = 3, which is not a rational number, the equation still yields 1 for ej*2*pi*9, making it a valid instance of a periodic discrete-time signal. This is because 3 is considered a rational number (3/1).
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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.
 
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  • #2
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.
 
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