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hw5 - ECE220 Signals and Information Spring 2008 Homework 5...

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ECE220 Signals and Information Spring 2008 Homework 5: Due Monday, March 10, at 10:08am Drop your homework in the collection box marked ”ECE220 Spring 2008, homework” , located on the second floor of Phillips at the south entrance to 219 Phillips. Print your name, NetId, and lab section in the top right corner on all pages. NOTE: This homework is based on lectures 12-16 and Chapters 9 and 10 in the textbook. Problem 1 Consider narrow pulses δ Δ ( t ) and δ Δ (2 t ) depicted in Figure 1. δ (2 t 29 2∆ 1 t 2 2 δ ( t 29 2∆ 1 t Figure 1. We argued informaly in class that lim Δ 0 δ Δ ( t ) = δ ( t ) (a) What is lim Δ 0 δ Δ (2 t )? (b) What is lim Δ 0 δ Δ ( αt ) for α negationslash = 0? Problem 2 Consider the function x ( t ) = e - 2 t u ( t 3). On the same graph plot functions (a) x ( t )
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(b) x (2 t ) (c) x ( T t ) where T is some nonzero number. For what values of T in (c) is the product x ( t ) · x ( T t ) nonzero? Problem 3 Consider functions x ( t ) and y ( t ) defined by the graphs shown in Figure 2. 5t 1 3t 2 t 2 x(t) y(t) t 1 Figure 2. Let denote the convolution operation. (a) Let z ( t ) = x ( t ) y ( t ). For what values of t will we have z ( t ) = 0? (b) In general, if x ( t ) negationslash = 0 for T 11 t T 12 and y ( t ) negationslash = 0 for
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