THE DISCRETE FOURIER TRANSFORM Solution 9.1 (a) X(k) N-1 x(n) WNkn = = RN(k) n=0 =RN (k) X (k) = WNkn0 RN (k) (c) X(k) N-1 an WNkn = RN(k) n=0 1-aN W kN 1 N-ak .- a RN (k ) WN 1-a WN) Solution 9.2 x( (n)) 4 0 T? x((n-2)) 4 0 x((n-2)) 4 R4 (n) 0 0 x((-n) ) x((-n)) R (n) 0 Figure S9.2-1 S9.1 Solution 9.3 X1(0) (k) X Since = - = ( (N - RN(k) ((N))N X k) )N RN (k) - = - X (0) (0) = 0 Therefore X Also, for N even, -- ))N RN(k) NN =e - ( X ) = Therefore X() 0. Solution 9.4 x2(M) x2 *1 ()N 7 1 9 1 Figure 59.4-1 Solution 9.5 x 1(M) 0 1 2 3 4 5 x 2 ( (-M)) 0 6 R 6 (M) m Ix 2 ( (2-m)M 0 m x 1(n) Figure S9.5-1 S9.2 x 2 (n) Note that this corresponds to x1 (n) circularly shifted to the right by two points. Solution 9.6 We wish to compute X1 (k) given by 9 X1 (k) =a x(n) zk-n R1 0 (k) n=0 j27k where zk = 0.5 e 10 ej0 so that 9 j-n 1 j2Trk X (k) = x (n) e x(n) e 10 10 e 10 R10 (k) n=0 9 -n jn Tk e 10 R1 0 (k) Thus X (k) is the 10-point DFT of the sequence x (n) = x(n)[j e 10 Solution 9.7 In all of the following equations the DFT computed is valid only in the range O