Periodic versus non-periodic functions (hw1, ECE301)

Read the instructor's comments here.

A discrete time signal is periodic if there exists T > 0 such that x(t + T) = x(t)

A continuous time signal is periodic if there exists some integer N > 0 such that x[n + N] = x[n]

Periodic Signal

Let $$ x(t) = tan(t) $$

For x(t) to be periodic, the following must hold true:

tan(t) = tan(t + T) for some value of T > 0

Since we know that tan(t) repeats itself after every period of $t = \pi$ , we know that x(t) is a periodic signal.

Let $x[n] = e^{jn}$

For x[n] to be periodic, the following must hold true:

$e^{jn} = e^{j(n+N)}$ for some integer N

$e^{jn} = e^{jn} e^{jN}$

$1 = e^{jN}$

1 = cos(N) + jsin(N)

This equation only holds true if $N = 2\pi$ or some multiple of $2\pi$

Therefore $x[n] = e^{jn}$ is not periodic because $2\pi$ is not an integer.