# 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 ^{iω0
n}* ,

then

∀*ω
*∈*
*[-*π*,*π*], *X*(*ω*)
= 2*π* *K δ
*(*ω -* *ω*_{0}
)

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

*x*(*n*) = cos(*ω*_{0}*
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 -* *ω*_{0}
) + *π* * δ
*(*ω *+
*ω *_{0}
)

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