Discrete-Time Fourier Transform The magnitude and phase of the dtft X(eo)=1/(1-0.5e o)are shown below 0.6 1.5 0.2 三 0 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 11 Discrete-Time Fourier Transform • The magnitude and phase of the DTFT ( ) 1/(1 0.5 ) are shown below w − w = − j j X e e -3 -2 -1 0 1 2 3 0.5 1 1.5 2 w/ Magnitude -3 -2 -1 0 1 2 3 -0.4 -0.2 0 0.2 0.4 0.6 w/ Phase in radians
Discrete-Time Fourier Transform The DtFT X(e/o)of a sequence x[n] is continuous function of o It is also a periodic function of o with a period2π X(e101x)=∑xnl+2x 1=-0 ∑nmne2=∑x n jon Y(e10) 1=-0 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 12 Discrete-Time Fourier Transform • The DTFT of a sequence x[n] is a continuous function of w • It is also a periodic function of w with a period 2: ( ) jw X e ( 2 ) ( 2 ) ( ) [ ] j k j k n n X e x n e w w + − + =− = 2 [ ] j n j kn n x n e e w − − =− = [ ] ( ) j n j n x n e X e w w − =− = =
Discrete-Time Fourier Transform Therefore Y(e0)=∑ xiNle jon 1=-00 represents the Fourier series representation of the periodic function X(eo) As a result, the Fourier coefficients x[n] can be computed from X(e/)using the Fourier integral (n)=1 ∫X( 0)a/0n 13 2几一 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 13 Discrete-Time Fourier Transform • Therefore represents the Fourier series representation of the periodic function • As a result, the Fourier coefficients x[n] can be computed from using the Fourier integral = =− w − w n j j n X (e ) x[n]e ( ) jw X e w = − w w x n X e e d j j n ( ) 2 1 [ ] ( ) jw X e
Discrete-Time Fourier Transform Inverse discrete-time Fourier transform xn]=∫X(eo)eoao 2 Proof: xn=「∑x le oe 2几((= Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 14 Discrete-Time Fourier Transform • Inverse discrete-time Fourier transform: • Proof: w = − w w x n X e e d j j n ( ) 2 1 [ ] w = − w =− − w x n x e e d j j n [] 2 1 [ ]