- The Following Definition Is Used In This Question W Exp J21 N Where N Is Assumed To Be A Positive Power Of 2 N 1 (60.94 KiB) Viewed 51 times
Q5(i) DFT The discrete Fourier Transform (DFT) of a discrete-time signal z[n] is given by N-1 X[m] = a[n]W* nim N 10 where j2 WN = exp N The complex function Win for n = 1 and N = 8 is illustrated in Figure Q5(a) where we note that 8 = 27/8 = 7/4. WS WS W3 W. W @= = wa W 12(-0) W3 Figure Q5(a) - The function where k=1,2,... 7.
a) Use the results of parts 1)-d) to complete Table Q5 below. Enter the imaginary number i as in i, 1+2i. Enter numerical values to 3 decimal places. 8 0 @ 1 Table Q5: Tabulation of W for k=0,1,...8 k W radians Value W: 1 w 0.707-0.707i 2 W? -pi/2 3 w -0.707-0.707i 4 WA -pi 5 -0.707+0.707i w W6 6 3*pi/2 7 w 0.707+0.7071 00 W - 2*pi
TIT A property of the geometry of Wis that if N is a power of 2, any value in for mn > 7 maps to WĄ where k = nm mod N and a mod bis the integer remainder of a/b. For example, for nm = 27, k = 27 mod 8 = 3 (27/8 = 3r3). Hint the MATLAB function for modulo division a mod bis mod(a,b). b) Use the property to give the equivalent k for n x m = 5 x 1 when N = 8. Round your answer to the nearest integer. Submit part 1 mark Unanswered c) Give the angle 0 (in radians) of the complex number W.7*3) 8 Submit part 1 mark Unanswered d) Give the equivalent value of W., k<8 illustrated in Figure Q5(a) for W. 6x4) Note, should be entered as w_84k. Submit part 1 mark