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 e^{iw₀ n} ,

then

"w Î [-p, p],    X(w) = 2p K d (w - w ₀ )

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

x(n) = cos(w ₀ n)

for some real constant w 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

"w Î [-p, p],    X(w) = p d (w - w ₀ ) + p d (w + w ₀ )

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