EECS20N: Signals and Systems

Discrete-Time Exponentials and Sinusoids

These are similar to the continuous-time case, except that the DTFT is required to be periodic. Thus, if

x(n) = K eiω0 n ,

then

∀ω ∈ [-π,π],    X(ω) = 2π K δ (ω - ω0 )

This function then periodically repeats with period 2π (as it must to be a DTFT). If

x(n) = cos(ω0 n)

for some real constant ω 0, we can again use Eulers relation to write this as a sum of two complex exponentials, and then use linearity of the DTFT to find

∀ω ∈ [-π,π],    X(ω) = π δ (ω - ω0 ) + π δ (ω + ω 0 )

This function then periodically repeats with period 2π (as it must to be a DTFT).