113_1_example_problem_chapter13_startedinclass

113_1_example_problem_chapter13_startedinclass - EE113:...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

This note was uploaded on 02/09/2011 for the course EE 113 taught by Professor Walker during the Spring '08 term at UCLA.

Ask a homework question - tutors are online