Digital Signal Processing

Determine whether or not the signals below are periodic and, for each signal that is periodic, determine the fundamental period.

1) x(n) = cos(0.125πn)

2) x(n) = cos(0.125πn + 1)

3) x(n) = Re(e12jnπ) + Im(e18jnπ)

4) x(n) = sin(π + 0.2n)

5) x(n) = e16jnπCos(17nπ)

1) x(n) = cos(0.125πn)

Angular Frequency ω=0.125π=8π

For a discrete function to be periodic, there must exist a positive integer N and an integer k that satisfy the following relation:

ωN=2πk → 2πω=Nk(Must be Rational Number)

Hince:

2πω=2π8π=16ππ=161→Nk=161→N=16

So x(n) = cos(0.125πn) is Periodic Function and its period is N=16