Latest revision as of 07:20, 14 April 2010

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.

Non 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.