4.3 Numerical Computation of the 4.3 Numerical Computation of the DTFT using DFT DTFT using DFT -艺m-艺nc学 Thus X(ei)is essentially an M-point DFT X(k)of the length-M sequence x(n) ●Define a new sequence The DFT X(k)can be computed very efficiently using the FFT algorithm if Mis an [x(n)0≤n≤N-1 x(n)= integer power of 2 0 N≤n≤M-1 The function freqz employs this approach 。Then -空a学 to evaluate the frequency response at a prescribed set of frequencies of a DTFT expressed as a rational function in e4.3 Numerical Computation of the DTFT using DFT       1 1 2         k k N N j kn j jn M X e xne xne Define a new sequence         0 0   n n X e xne xne Define a new sequence      0 1    xn n N x n Then   0 1     e x n NnM Then     1 2 k M j kn j M X e x ne e     37     0 e n  4.3 Numerical Computation of the DTFT using DFT Thus is essentially an M-point DFT X (k) of the length M sequence x (n) ( ) k j X e Xe(k) of the length-M sequence xe(n) The DFT Xe(k) can be computed very efficiently using the FFT algorithm if M is an integer power of 2 The function freqz employs this approach to evaluate the frequency response at a to evaluate the frequency response at a prescribed set of frequencies of a DTFT expressed as a rational function in j e 38 expressed as a rational function in e