DTFT Computation Using MATLAB Note: The phase spectrum displays a discontinuity of 2nat =0.72 This discontinuity can be removed using the function unwrap as indicated in the next slide 16 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 16 DTFT Computation Using MATLAB • Note: The phase spectrum displays a discontinuity of 2 at = 0.72 • This discontinuity can be removed using the function unwrap as indicated in the next slide
DTFT Computation Using MATLAB Phase Response TttTT 00.102030.40.50.60.70.80.9 0/ 17 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 17 DTFT Computation Using MATLAB
Linear Convolution Using DTET An important property of the dtFT is given by the convolution theorem in Table 3.2 It states that ify(n]=x[n]e h[n then the DTFT Y(eo)of yln] is given by Y(e)=X(e/0)H(e°) An implication of this result is that the linear convolution yn] of the sequences xin and hn can be performed as follows 18 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 18 Linear Convolution Using DTFT • An important property of the DTFT is given by the convolution theorem in Table 3.2 • It states that if y[n] = x[n] h[n], then the DTFT of y[n] is given by • An implication of this result is that the linear convolution y[n] of the sequences x[n] and h[n] can be performed as follows: * ( ) j Y e ( ) ( ) ( ) = j j j Y e X e H e
Linear Convolution Using DTFT 1)Compute the DTFTs X(eo)and H(e/o)of the sequences xn] and hin, respectively 2) Form the dtFt Y(e)=X(e u)H(e o) 3)Compute the IdFTyIn] of y(eo) X(e! DTFT Y(eu) IDTFT y DTFT H(e0) 19 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 19 Linear Convolution Using DTFT 1) Compute the DTFTs and of the sequences x[n] and h[n], respectively 2) Form the DTFT 3) Compute the IDFT y[n] of ( ) ( ) ( ) = j j j Y e X e H e ( ) j X e ( ) j H e ( ) j Y e x[n] h[n] y[n] DTFT DTFT IDTFT ( ) j Y e ( ) j X e ( ) j H e
Discrete Fourier Transform Definition-The simplest relation between a length-N sequence x[n], defined for 0≤n≤N-1, and its dtft x(e0)is obtained by uniformly sampling X(e/o)on the @-axis between0≤o≤2 It at Ok=2兀k/N 0≤k<N-1 From the definition of the dtft we thus have X[k]=Y(e/0 xIne J2Tk/N O=2πk/Nn=0 20 0<k<N-1 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 20 Discrete Fourier Transform • Definition - The simplest relation between a length-N sequence x[n], defined for , and its DTFT is obtained by uniformly sampling on the -axis between at , • From the definition of the DTFT we thus have 0 n N −1 0 k N −1 ( ) j X e ( ) j X e 0 2 k = 2k/ N [ ] ( ) [ ] , 1 0 2 / 2 / = = − = − = N n j k N k N j X k X e x n e 0 k N −1