Lesson 19

Transform of Periodic Sequences

Here we study the DTFT of periodic sequences. We'll start by looking
at the Fourier Series expansion, analogous to what we did in
continuous time. Then we will derive the same result using a
different approach that will lead us into the Discrete Fourier
Transform for finite length sequences.

Recall that for **continuous time periodic signals**, we found the
Fourier transform by first doing a Fourier series expansion

(1) | |||

(2) |

then using the fact that a complex exponential in time transforms to an impulse in the frequency domain

and linearity of the Fourier transform, we get that the CTFT of a periodic signal is made up of harmonically-related impulses with area

**Discrete-time periodic signals** can also be
described by a Fourier Series expansion:

(3) | |||

(4) |

As one would expect, the integral in time goes to a sum. However, there is one more key difference:

First, recall that
. Then since
(since
), the 's are periodic with
period and only terms are needed in the sum.