Discrete-Time Fourier Transform We will assume that the phase function 0(o) is restricted to the following range of values π≤6(0)<T 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 tFTs of some sequences exhibit discontinuities of 2T in their phase responses An alternate type 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丌 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 is called unwrapping The continuous phase function generated by unwrapping is denoted as ec(o) In some cases, discontinuities of Tt may be present after unwrapping 8 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 △(e)=∑8[ n]e=80]=1 1=-00 Example- Consider the causal sequence x[n]=a"u[n] 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 X(e0)=∑a' u[n]e j=∑a'e-~0n n=-0 n=0 =∑ejoy=e/o n=0 as ae Jo=a<1 10 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