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Question 5: The Discrete and Fast Fourier Transform This question is concerned with the discrete Fourier transform, and the Fast Fourier transform. Answer all parts of this question. The use of mathematical software, such as MATLAB, is permitted, but the answers expected will be numbers or mathematical expressions. The following definition is used in this question: W = exp j2 N where N is assumed to be a positive power of 2 (N = 2'). The conjugate of WW! Please note that for this question, complex numbers in should be entered using the imaginary number i as in i, 1+2i, and a bi All numbers should be rounded to 3 decimal places unless otherwise advised.

where Ww= exp("*") 12 N The complex function W forn 1 and N8 is illustrated in Figure Q5(a) where we note that 0 = 2#/8 = */4 WS w W. W W: 0 = We W 12(-8) W Figure Q5(a) - The function wherek-1,2.... 7 Use the results of parts 11-d) to complete Table 05 below. Enter the imaginary number as in 1, 1+21. Enter numerical values to 3 decimal places. W Table Q5: Tabulation of Work 0,1...8 W! radians Value 0 1 1 W 8.787-0.7073 2 w 1/2 3 W3 -0.707-9.7071 4 W -pi 15 WA -0.707-0.7073 6 WS 3pi/2 7 W 0.707-0.7071 8 291 BEBE **

Q5(ii) Properties of Wk N A property of the geometry of Wk is that if Nis a power of 2, any value Wy" for mn > 7 maps to W where k = nim 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 nxm=7 x 6 when N = 8 Round your answer to the nearest integer Submit part 1 mark Unanswered c) Give the angle Olin radians) of the complex number W, 5x2). Submit part 1 mark Unanswered d) Give the equivalent value of W., k < 8 illustrated in Figure Q5(a) for W. (67) Note, We should be entered as w_8^k, Submit part

Q5(iii) - The Discrete Fourier Transform (DFT) - e) An 8-point sequence is defined as [n] = (0, 0, 1, 1, 1, 1, 0, 0) Use the results of parts Question 500 parts a)-d), or some other method, to compute the Discrete Fourier Transform X[4) Enter complex numbers using the imaginary numberi as in i. 1-21, and a bi. A Submit part 1 mark Unanswered f) Use the MATLAB ift function to compute the DFT of the 8-point sequence 1 = (1, 2, 1, 2, 1, 2, 1, 2). Enter complex values in the form 1+2i. Enter numbers to 3 places of decimals. X[0] = X[1] X[2] X[3] X[4] X[5] X[6] X17) 11 Il Submit part 2 marks Unanswered

Q5(iv) The Fast-Fourier Transform (FFT) g) The FFT is applied to a sequence that is 25 samples long. Approximately how many complex operations are required for this operation? An integer expected. Round your answer to the nearest integer Submit part 1 mark Unanswered h) h) How much more efficient is the 215-point FFT compared to the DFT defined in Q5(1)? Give your answer as the ratio of the approximate number of complex operations needed to compute the FFT versus the approximate number of complex operations needed to compute the DFT. Give your answer as a percentage. Round your answer to 3 significant figures. Submit part 1 mark Unanswered