Periodic versus non-periodic functions (hw1, ECE301)

Read the instructor's comments here.

Periodic Signal

In discrete time, a signal x[n] is considered a periodic signal if there exists a natural number N such that for all integers n, x[n+N] = x[n]. (i.e., $ \exists N \in \mathbb{N}, s.t. \forall n \in \mathbb{Z}, x[n+N] = x[n]. $ An example function would be:

$ f[n] \begin{cases} 1, &\text{if n is odd;}\\ 0, &\text{if n is even.} \end{cases} $

In this case, N = 2 would satisfy the definition for the function being periodic, since f[n+2] = f[n] for any given n. (N = 4, N = 22, or N = [any even number] would also work. N = 2 is called the fundamental period because it is the smallest of all possible N.)

Non-Periodic Signal

In discrete time, x[n] is said to be a non-periodic signal if it is not a periodic signal -- that is, if there does not exist any N that would satisfy the criteria in the definition. An example would be:

$ f[n] \begin{cases} 1, &\text{if n is prime;}\\ 0, &\text{if n is not prime.} \end{cases} $

Since there is no N that can satisfy the definition*, then the signal must be non-periodic.

(*Since 2 and 3 are both prime, either 2+N or 3+N must be composite for any natural number N, since one of them will be even and thus divisible by 2. (Note: We will consider natural numbers to be integers greater than 0.) Therefore, no N can satisfy f[n + N] = f[n] for all n.)