Discrete-Time Fourier Transform We will assume that the phase function A(o) is restricted to the following range of values π≤(0)<兀 called the principal value Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 6 Discrete-Time Fourier Transform • We will assume that the phase function (w) is restricted to the following range of values: called the principal value − (w)
Discrete-Time Fourier Transform The dtFts of some sequences exhibit discontinuities of 2T in their phase responses An alternate type o of phase function that is a continuous function of o is often used It is derived from the original phase function by removing the discontinuities of 2兀 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 7 Discrete-Time Fourier Transform • The DTFTs of some sequences exhibit discontinuities of 2 in their phase responses • An alternate type of phase function that is a continuous function of w is often used • It is derived from the original phase function by removing the discontinuities of 2
Discrete-Time Fourier Transform The process of removing the discontinuities 2 is calle ed unwrapping The continuous phase function generated by unwrapping is denoted as 0c(o) ° In some cases, discontinuities ofπ may be present alter unwrapping Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 8 Discrete-Time Fourier Transform • The process of removing the discontinuities is called “unwrapping” • The continuous phase function generated by unwrapping is denoted as • In some cases, discontinuities of may be present after unwrapping (w) c
Discrete-Time Fourier Transform Example- The dtft of the unit sample sequence 8[n is given by △(e0)=∑8 n]e on=80]=1 n=-0 Example - Consider the causal sequence xm=∞m],a<1 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 9 Discrete-Time Fourier Transform • Example - The DTFT of the unit sample sequence d[n] is given by • Example - Consider the causal sequence ( ) = d[ ] = d[0] =1 − w =− w j n n j e n e x[n] = [n], 1 n
Discrete-Time Fourier Transform Its dtFT is given by XY(e10)=∑oune-~0n=∑ae~on n=- n=0 =∑(e n=0 1-ae J as ae j0=0<1 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 10 Discrete-Time Fourier Transform • Its DTFT is given by as = = = − w =− w − w 0 ( ) [ ] n n j n n j n j n X e n e e − w − = − w = = j e n j n e 1 1 0 ( ) = 1 − jw e