DTFT Computation Using MATLAB Note The phase spectrum displays a discontinuity of 2T at @=0.72 This discontinuity can be removed using the function unwrap as indicated in the next slide 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 一+ 00.10203040.50.60.70.80.9 0/丌 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 if[n]=xn]@ hnl, then the DTFT Y(el)ofyIn is given by Y(e/0)=X(eo)H(e10) An implication of this result is that the linear convolution yIn] of the sequences xn] and h[n 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 DTET 1)Compute the tfTs X(e)and H(e/o)of the sequences xin] and hn], respectively 2)Form the dTFT Y(elo)=x(e/o)h(e/o) 3)Compute the idFT y[n] of y(e/o) DTET X(ejo y Y(eu) IDTFT y DTFT H 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[nl, defined for 0≤n≤N-1, and its dtft x(e)is obtained by uniformly sampling X(e o)on theO- axis between0≤0≤2at0h=2k/N, 0<k<N-1 From the definition of the dtft we thus have X[k]=X(e/0 2πk/N 0=2πk/N n=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