2.Matrix Relations 2.Matrix Relations 2.Matrix Relations ·where X=[X(O)X() .X(W- .Likewise.the IDFT relations can be Obviously,the relation between the two x=[xOx)…xW-I expressed in x=D'X coefficient matrices can be expressed as follows 「11 1 1 1 1 1 W W … E 1 W w Wtw-n D时=D N 1 Dx= 1 W w时 Waw-b Dy=- W N 在 x-b Therefore,the algorithms designed for DFT are applicable to IDFT 1 Ww-WaN-D -w- 1 Ww-）W2- N-IXN-3) 3.DFT Computation 3.DFT Computation 4.Relations between DTFT and Using MATLAB Using MATLAB DFT and their inverses The functions to compute the DFT and the IDFT are fft and ifft DTFT from DFT by interpolation These functions make use of FFT algorithms 。Sampling the DTFT which are computationally highly efficient Numerical computation of the DTFT using compared with the direct computation DFT Figure in the next slide shows the DFT and DTFT of the sequence X(k) interpolation) cos(6xm/160≤n≤15 sampling2. Matrix Relations where  (0) (1) ( 1)T X    X X XN   (0) (1) ( 1)T x   x x xN   12 1 11 1 1 1 N WW W        2 4 2( 1) 1 1 NN N N N NN N WW W WW W         D 1 2( 1) ( 1)( 1) 1 N N NN WW W             19 ( ) ( )( ) 1 WW W NN N N N     2. Matrix Relations Likewise, the IDFT relations can be d i 1 expressed in   11 1 1 1 N  x DX  1 2 ( 1) 11 1 1 1 N WW W NN N          1 2 4 2( 1) 1 1 NN N N N NN N WW W N             D ( 1) 2( 1) ( 1)( 1) 1 N N NN NN N N WW W               20 1  WW W NN N N N 2. Matrix Relations Obviously, the relation between the two coefficient matrices can be expressed as follows 1 * 1 N N N  D D  Therefore, the algorithms designed for DFT N are applicable to IDFT 21 3. DFT Computation Using MATLAB The functions to compute the DFT and the IDFT are fft and ifft These functions make use of FFT algorithms These functions make use of FFT algorithms which are computationally highly efficient compared with the direct computation compared with the direct computation Figure in the next slide shows the DFT and DTFT of the sequence DTFT of the sequence cos(6 /16) 0 15 n n   22 3. DFT Computation Using MATLAB 9 DFT 6 7 8 DTFT 4 5 6 Magnitude 1 2 3 0 0.2 0.4 0.6 0.8 1 0 1 / 23 4. Relations between DTFT and DFT and their inverses DTFT from DFT by interpolation Samp g lin the DTFT Numerical computation of the DTFT using DFT ( ) j X k( ) X interpolation ( ) j X k( ) X e sampling 24