Periodic Extension to DTFS 2.8 2002/08/01 2007/07/19 16:08:33.136 GMT-5 Roy Ha rha@alumni.rice.edu Prashant Singh prash@ece.rice.edu Richard G. Baraniuk richb@rice.edu Mariyah Poonawala mariyah@rice.edu discrete time fourier series dtfs fourier fourier series period periodic This module looks at the periodic extensions of the DTFS coefficients and how the coefficients play a critical role in manipulating signals.

Introduction Now that we have an understanding of the discrete-time Fourier series (DTFS), we can consider the periodic extension of c k (the Discrete-time Fourier coefficients). shows a simple illustration of how we can represent a sequence as a periodic signal mapped over an infinite number of intervals.

vectors periodic sequences

Why does a periodic extension to the DTFS coefficients c k make sense? Aliasing: b k 2 N k n b k + N 2 N k N n 2 N k n 2 n 2 N n b k → DTFS coefficients are also periodic with period N.

Examples Discrete time square wave

Calculate the DTFS c k using: c k 1 N n 0 N 1 f n 2 N k n Just like continuous time Fourier series, we can take the summation over any interval, so we have c k 1 N n N 1 N 1 2 N k n Let m n N 1 (so we can get a geometric series starting at 0) c k 1 N m 0 2 N 1 2 N m N 1 k 1 N 2 N k m 0 2 N 1 2 N m k Now, using the "partial summation formula" n 0 M a n 1 a M 1 1 a c k 1 N 2 N N 1 k m 0 2 N 1 2 N k m 1 N 2 N N 1 k 1 2 N 2 N 1 1 1 k 2 N Manipulate to make this look like a sinc function (distribute): c k 1 N k 2 2 N k 2 N N 1 1 2 k 2 N N 1 1 2 k 2 2 N k 2 N 1 2 k 2 N 1 2 1 N 2 k N 1 1 2 N k N digital sinc It's periodic! , , and show our above function and coefficients for various values of N 1 .

Plot of f n . Plot of c k . N 1 1

Plot of f n . Plot of c k . N 1 3

Plot of f n . Plot of c k . N 1 7