Periodic signals and fundamental period

A discrete signal is periodic if x[n + N] = x[n] for all n, for some smallest positive integer N (the fundamental period).

i) x[n] = cos(πn²/8)

Need cos(π(n + N)²/8) = cos(πn²/8) for every n. Expanding, the phase changes by

π(2nN + N²)/8 = nNπ/4 + N²π/8.

This must be a multiple of 2π for all n:

• nNπ/4 = 2πk for every n requires N to be a multiple of 8.
• Try N = 8: N²π/8 = 64π/8 = 8π, a multiple of 2π. Satisfied.

So the signal is periodic with fundamental period N = 8. Verified numerically: cos(πn²/8) for n = 0 … 7 repeats for n = 8 … 15.

ii) x[n] = cos(n/2) cos(πn/4)

Use cos A cos B = ½[cos(A + B) + cos(A − B)]:

x[n] = ½ cos((½ + π/4)n) + ½ cos((½ − π/4)n)

Frequencies: ω₁ = ½ + π/4 and ω₂ = ½ − π/4. For periodicity, ω/2π must be rational:

ω₁/2π = 1/(4π) + 1/8, which is irrational (because of π).

Similarly cos(n/2) alone has f = 1/(4π), which is irrational. So no integer N exists.

The signal is not periodic.