pcshw1

# pcshw1 - . ±ind the average power in the signal x(t), 1 T...

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THE STATE UNIVERSITY OF NEW JERSEY RUTGERS College of Engineering Department of Electrical and Computer Engineering 332:322 Principles of Communications Systems Spring 2003 Problem Set 1 1. Derive the convolution intergal from Frst principles (as outlined in class) given by y ( t ) = R -∞ x ( τ ) h ( t - τ ) , where h ( t ) is the impulse response of a LTI system, x ( t ) the input and y ( t ) the output. 2. ±or each of the systems described by the input output relationships below, determine which of the following properties apply to the system: Memoryless(M), causal(C), linear(L), time- invariant(TI), stable(S). Justify your answers. (a) y ( t ) = sin( t + 1) x ( t ) (b) y [ n ] = x [2 - n ] + 1 3. Consider a continuous time system with the following input x ( t ) and impulse response h ( t ) , x ( t ) = 1 0 < t < 2 - 1 2 < t < 4 0 otherwise and h ( t ) = exp ( - 2 t ) u ( t ) (a) Compute the output of the system y ( t ) = x ( t ) * h ( t ) (b) Is the system stable? Is the system causal? 4. A signal x ( t ) is periodic with period T = 10 - 3 . The ±ourier series coeffecients for x(t) are given by a k = ( 1 2 j ) k k > 0 0 k = 0 ( 1 - 2 j ) - k otherwise

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Unformatted text preview: . ±ind the average power in the signal x(t), 1 T R T | x ( t ) | 2 . Hint:Use Parseval’s relationship! 5. Let x ( t ) = exp (-at ) u ( t ) ,where u(t) is the unit step function. ±ind the ±ourier transform of the following signals. (a) x ( t ) (b) y ( t ) = x ( t + 5) (c) z ( t ) = x ( t ) sin(2 π 40 t ) 1 6. Evaluate the Fourier transform of the damped sinusoidal wave g ( t ) = exp(-t ) sin(2 πf c t ) u ( t ) , where u ( t ) is the unit step function. 7. Show that the overall system function H ( s ) for the feedback system in FIGURE 7 is given by H ( s ) = F ( s ) 1-F ( s ) G ( s ) . + F(s) G(s) 8. A signal x ( t ) of ±nite energy is applied to a sqaure-law device whose output is de±ned by y ( t ) = x 2 ( t ) . The spectrum of x ( t ) is limited to the frequency interval-W ≤ f ≤ W . Hence show that the spectrum of y ( t ) is limited to-2 W ≤ f ≤ 2 W . 2...
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## This note was uploaded on 04/03/2008 for the course ECE ECE332 taught by Professor Rose during the Spring '08 term at Rutgers.

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pcshw1 - . ±ind the average power in the signal x(t), 1 T...

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