113_1_example_problem_chapter13_startedinclass

# 113_1_example_problem_chapter13_startedinclass - EE113:...

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Unformatted text preview: EE113: Digital Signal Processing Huiyu Luo Problems from Chapter 13 13.2 DTFTs (1) Notice x1 (n) = (−1)n sin( π n) 2 Let x(n) = . Then, πn Homework #6 Solutions Spring, 2004 sin( π n) sin( π n) 2 2 = (ejπn ) , πn πn X1 (ejω ) = X (ej (ω−π) ) Figure 1: Magnitude and phase plots of X1 (ejω ). (2) We have x2 (n) = Therefore, sin π n 8 πn The plots are as in Figure 2. x2 (n) = (3) Let y (n) = ej 2 n x2 (n) ∗ x1 (n − 2). ej 2 n x2 (n) ↔ X2 (ej (ω− 2 ) x1 (n − 2) ↔ e−j 2ω X1 (ejω ) 1 π π π sin π n 8 πn 2 = sin π n 8 πn sin π n 8 πn =⇒ X2 (ejω ) = X2 (ejω ) ∗ X2 (ejω ) Figure 2: Magnitude and phase plots of X2 (ejω ). Thus, Y (ejω ) = X2 (ej (ω− 2 ) )e−j 2ω X1 (ejω ) = X2 (ej (ω− 2 ) ) X1 (ejω ) e−j 2ω π π They are plotted in Figure 3. Figure 3: Magnitude and phase plots of Y (ejω ). (4) Deﬁne π y (n) = x2 (n) cos( n) 3 The DTFT of y (n) can be determined as π π 1 1 Y (ejω ) = X2 (ej (ω− 3 ) ) + X2 (ej (ω− 3 ) ) 2 2 The magnitude and phase plots are shown in Figure 4. 2 Figure 4: Magnitude and phase plots of Y (ejω ). 13.3 DTFTs (a) Notice x(n) = = = sin π 1 πn πn 1 3πn sinc + sinc cos 8 8 8 16 16 π π 1 sin 8 n 1 16 sin 16 n 3π n +· cos π π 8 8n 8 16 16 n 16 π sin π n sin 16 n 3π 8 +2 cos n πn πn 16 sin π 8 16 Let x1 (n) = πn and x2 (n) = πn , then X1 (ejω ) is a rectangular pulse between − π 8 π and 8 with an amplitude of 1. Also, n n X2 (ejω ) = DTFT 2x2 (n) cos( 3π n) 16 = X2 (ej (ω− 16 ) ) + X2 (ej (ω+ 16 ) ) 3π 3π which is two rectangular pulses where one of which is located from ω = − π to ω = − π 4 8 and the other located from ω = π to ω = π with the amplitude of 1. Then, 4 8 X (ejω ) = X1 (ejω ) + X2 (ej (ω− 16 ) ) + X2 (ej (ω+ 16 ) ) Which is a rectangular pulse between ω = − π to ω = 4 shown in Figure 5. (b) Deﬁne x1 (n) = sinc πn 8 π 4 3π 3π with the amplitude of one as x2 (n) = x2 (n) 1 1 πn x3 (n) = x2 (n) cos 2 4 πn 1 sinc x4 (n) = 4 4 3 ...
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## This note was uploaded on 02/09/2011 for the course EE 113 taught by Professor Walker during the Spring '08 term at UCLA.

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