Relations: DFT and DTFT 1.5 2004/06/29 09:04:19 GMT-5 2006/08/02 14:28:40.402 GMT-5 Richard G. Baraniuk richb@rice.edu Richard G. Baraniuk richb@rice.edu Robert Ahlfinger ahlfing@rice.edu Matthew Hutchinson mhutch@rice.edu DFT DTFT ELEC 301 An overview of the relationship between the DFT and the DTFT.

CASE 1: Periodic Signals

Considered as a finite-length signal x n has unnormalized DFT representation x n 1 N k 0 N 1 X k 2 N k n X k DFT u 0 N 1 x n 2 N k n But x n and are also repetions for periodic signal, which is -length. It's DTFT X ω DFT n x n ω n Where x n 1 N k 0 N 1 X k DFT 2 N k n from 1 N n k 0 N 1 X k DFT Assuming we can reverse the order of the sums. 1 N k 0 N 1 X k DFT n 2 N k n ω n Where n 2 N k n ω n is the DTFT of sinusiod with frequency 2 N k 2 δ ω 2 N k so... X ω DFT 2 N k 0 N 1 X k DFT δ ω 2 N k

CASE 2: Finite Length Signals

Assuming finite-length X k DFT n 0 N 1 x n 2 N k n Assuming zero outside 0 N 1 X ω DTFT n 0 N 1 x n ω n and are related by X k DFT X 2 π k / N DTFT ie: DFT N samples of DTFT in 0 2 .

Extension What if we compute zeropadded DFT?

That is, finer sampling