Name __________________________________________________ CM____________ECE 300Signals and SystemsExam 225 April 2006This exam is closed-book in nature. You are not to use a computer during the exam, but you can use a calculator for simple calculations. Problem 1 ________ / 20 Problem 2 ________ / 30 Problem 3 ________ / 35 Problem 4 ________ / 15 Exam 2 Total Score: _______ / 1001Name __________________________________________________ CM____________1. (20 points)Consider the following system, composed of two subsystems. Assume both systems are initially at rest (no initial enery), and 0c >a) Determine the impulse response for each subsystemb) Determine the impulse response between input ( )x t and output ( )y t( ) ( ) ( )v t v t x t c+ = +&( )x t( )v t( )y t2Name __________________________________________________ CM____________2. (30 points) Consider a causal linear time invariant system with impulse response given by( 1)( ) ( 1)th t e u t- -= -The input to the system is given by( ) ( ) ( 1) ( 3)x t u t u t u t= - - + -Using graphical convolution, determine the output( )y t for 2 5t� �. Note the limited range of t we are interested in !Specifically, you musta) Flip and slide( )h tb) Show graphs displaying both ( )h t l- and ( )x lfor each region of interestc) Determine the range of t for which each part of your solution is validd) Set up any necessary integrals to compute ( )y te) Evaluate the integrals3Name __________________________________________________ CM____________3. (35 points) The spectrum of a periodic signal( )x t, which has period 2T = seconds, is shown below. All angles are multiples of 45 degrees.-3 -2 -1 0 1 2 300.511.52AmplitudeHarmonic-3 -2 -1 0 1 2 3-100-50050100Phase (degrees)Harmonica) Determine an expression for ( )x t in terms of cosines and constants. Your expression must be purely real.b) Determine the average value of ( )x t.c) Determine the average power in ( )x t.Now assume ( )x tis the input to a LTI system with transfer function ( )1jHjwww=+d) Determine the average power in the output, ( )y te) Determine an expression for ( )y t in terms of cosines and constants. Your expression must be purely real.4Name __________________________________________________ CM____________4. (15 points) Assume periodic signal ( )x t has the Fourier series representation( )ojk txkx t c ew=�. ( )x t is the input to an LTI system with the system model( ) 3 ( 2)dy t x tdt= -Determine the Fourier series representation of( )y t, where ( )ojk tyky t c ew=�. Specifically,show how ykc is related to xkc.5Name __________________________________________________ CM____________6Name __________________________________________________ CM____________7Name __________________________________________________ CM____________8Name __________________________________________________ CM____________Some Potentially Useful Relationships( ) ( )T2 2TTE lim x t dt x t dt����- - �= =� �( )T2TT1P lim x t dt2T���-=�( ) ( )jxe cos x jsin x= +j 1= -( )jx jx1cos x e e2-� �= +� �( )jx jx1sin x e e2j-� �= -� �( ) ( )21 1cos x cos 2x2 2= +( ) ( )21 1sin x cos 2x2 2= -00 0t tT Trect u t t u t tT 2 2-� �� � � �= - + - - -� � � �� �� � � ��
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