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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 10 37 Copyright © Orchard Publications Solutions to End of Chapter Exercises We are asked to compute only. However, we will derive all equations as we did in Exam- ple 10.5. Column 1 (C=1): (1) Column 2 (C=2): (2) Column 3 (C=3): (3) (4) F3 () Y01 , Y00 , Y40 , + = Y11 , Y10 , Y50 , + = Y21 , Y20 , Y60 , + = Y31 , Y30 , Y70 , + = Y41 , [] W N 0 , Y 40 , = Y51 , W N 1 , Y 50 , = Y61 , W N 2 , Y 60 , = Y71 , W N 3 , Y 70 , = Y02 , , , + = Y12 , , , + = Y22 , W N 0 , Y 21 , = Y32 , W N 2 , Y 31 , = Y42 , , , + = Y52 , , , + = Y62 , W N 0 , Y 61 , = Y72 , W N 2 , Y 71 , = Y03 , , , + = Y13 , W N 0 , Y 12 , [ ] = Y23 , , , + = Y33 , W N 0 , Y 32 , [ ] = Y43 , , , + = Y53 , W N 0 , Y 52 , [ ] = Y63 , , , + = Y73 , W N 0 , Y 72 , [ ] = X3 , , , + == =

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Chapter 10 The DFT and the FFT Algorithm 10 38 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications where and From (1) and by substitution into (4) (5) From Exercise 4, (6) Since , and (see proof below), we see that , , , , , and . Therefore, (5) and (6) are the same. Proof that : 6 . The rectangular pulse is produced with the MATLAB script below. x=[linspace( 2, 1,100) linspace( 1,1,100) linspace(1,2,100)];. .. y=[linspace(0,0,100) linspace(1,1,100) linspace(0,0,100)]; plot(x,y) and the FFT is produced with plot(x, fft(y)) The Inverse FFT is produced with plot(x,ifft(fft(y))) Y62 , () W N 0 Y41 , Y 61 , [] = Y72 , W N 2 Y51 , Y 71 , = , W N 0 Y00 , Y 40 , = , W N 1 Y10 , Y 50 , = Y61 , W N 2 Y20 , Y 60 , = Y71 , W N 3 Y30 , Y 70 , = F3 X3 W N 0 , W N 3 , W N 2 , W N 5 , + == W N 0 Y40 , W N 3 Y50 , W N 2 Y60 , W N 5 Y70 , + + x0 W N 3 x1 W N 6 x2 W N 9 x3 +++ +W N 12 x4 W N 15 x5 W N 18 x6 W N 21 x7 Yk0 , xk = W 8 8i n + W 8 n = W 8 n4 ± W 8 n = W 8 6 W 8 2 = W 8 9 W 8 5 = W 8 12 W 8 0 = W 8 15 W 8 3 = W 8 18 W 8 2 = W 8 21 W N 5 = W 8 ± W 8 n = W 8 ± W 8 n W 8 4 ± W 8 n e j2 π 8 4 ± W 8 n ππ sin + cos W 8 n = =
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 10 39 Copyright © Orchard Publications Solutions to End of Chapter Exercises The original rectangular pulse, its FFT, and the Inverse FFT are shown below.

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