Discrete-Time Fourier Transform The magnitude and phase of the dtFt X(eo)=1/(1-0.5e o)are shown below 02 0. 0/r 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(eo)of a sequence x[n]is a continuous function of o It is also a periodic function of o with a period2π X( j(a+2k )=∑x10+2x n=-00 ∑ Xinle JonnY 2=∑xnem2=X(em) n=-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 X(e/0)=∑ xn]e y0n represents the Fourier series representation of the periodic function X(e/o) As a result, the Fourier coefficients x[n can be computed from X(e/o)using the Fourier integral xm]=∫X(eo)eodo 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 / X(e/0) an n)2一 Proof: xn=了|2x-0|0o 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 [ ]