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New York University - EL 6113hw07_solution EXCELLENT SOLUTIONS

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1.13 Consider the following 8-point signals, 0  n  7. (a) [1, 1, 1, 0, 0, 0, 1, 1] (b) [1, 1, 0, 0, 0, 0, "1, "1] (c) [0, 1, 1, 0, 0, 0, "1, "1] (d) [0, 1, 1, 0, 0, 0, 1, 1] Which of these signals have a real-valu ...

1.13 Consider the following 8-point signals, 0  n  7. (a) [1, 1, 1, 0, 0, 0, 1, 1] (b) [1, 1, 0, 0, 0, 0, "1, "1] (c) [0, 1, 1, 0, 0, 0, "1, "1] (d) [0, 1, 1, 0, 0, 0, 1, 1] Which of these signals have a real-valued 8-point DFT? Which of these signals have a imaginary-valued 8-point DFT? Do not use MATLAB or any computer to solve this problem and do not explicitly compute the DFT; instead use the properties of the DFT. Solution Signals (a) and (d) both have purely real-valued DFT. Signal (c) has a purly imaginary-valued DFT. 131.30 The following MATLAB commands define two ten-point signals and the DFT of each. x1 = cos([0:9]/9*2*pi); x2 = cos([0:9]/10*2*pi); X1 = fft(x1); X2 = fft(x2); (a) Roughly sketch each of the two signals, highlighting the distinction between them. (b) Which of the following four graphs illustrates the DFT |X1(k)|? Explain your answer. Which graph illustrates the DFT |X2(k)|? 0 1 2 3 4 5 6 7 8 9 6 4 2 0 GRAPH A 0 1 2 3 4 5 6 7 8 9 6 4 2 0 GRAPH C 0 1 2 3 4 5 6 7 8 9 6 4 2 0 GRAPH D 0 1 2 3 4 5 6 7 8 9 6 4 2 0 GRAPH B 0 1 2 3 4 5 6 7 8 9 6 4 2 0 GRAPH E 0 1 2 3 4 5 6 7 8 9 6 4 2 0 GRAPH F 0 1 2 3 4 5 6 7 8 9 6 4 2 0 GRAPH G 0 1 2 3 4 5 6 7 8 9 6 4 2 0 GRAPH H Give your answer by completing the table and provide an explanation for your answer. Note that there are more graphs shown than needed. Your answer will use only two of the eight graphs. Provide a brief explanation for your answer. You should be able to do this exercise without using MATLAB (this was an exam question). DFT Graph X1 X2 Solution Note that x1(9) = cos(9/9 · 2 · ⇡) = 1; this signal is slightly more than a full cycle of a cosine signal (it it were extended into a periodic signal by repeating the signal, then the ‘1’ would appear twice in a row.). On the other hand, the signal x2(n) is exactly one cycle of a cosine signal. 0 1 2 3 4 5 6 7 8 9 −1 −0.5 0 0.5 1 x1 0 1 2 3 4 5 6 7 8 9 −1 −0.5 0 0.5 1 x2 Therefore, signal x2(n) has the DFT illustrated in graph A. The DFT of signal x1(n) is similar but leakage is visible – which is visible in graph C. DFT Graph X1

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