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ECE-205 : Dynamical Systems Homework # Due : Friday October 8 at High Noon Exam 2 , Tuesday October 12, 2010 1) Simplify the following expressions as much as possible
a) g ( t ) e (^ t 1) ( t ) b) g t ( ) t 15 e 2(^ t ^ )^ ( 2 ) d c) g t ( ) ^ t ^ 31 e (^ t ^ )^ ( 2) d
d) g t ( ) ^3 t 1 e ( t ^ )^ ( 1) d e) g t ( ) t^2 1 e (^ t ^ ^ ) ( 2 ) d f) g t ( ) 0 te (^ t ^ )^ ( 3 ) d
g) g t ( ) ^ e ^3 ( t ) d h) g t ( ) ^ e^4 ^ u ( 2) ( t 2) d i) g t ( ) ^ e u^ ^ ( 1) ( t 3) d
Scrambled Answers: e 2(^ t^ ^ 2)^ u t ( 7), e ^ (^ t^ 2)^ u ( 1 t ) , e ^3 t^ , et^ ^3 u t ( 4), e 4(^ t^ 2)^ u t ( ), e (^ t^ 3)^ u t ( 3), e (^ t^ 2)^ u t ( 1), e (^ t 1)^ u t ( 2) , e^1 ( t )
2) Determine the impulse response for each of the following systems
a) y t ( ) x t ( 2) ^ te ^ (^ t ^ ^ ) x ( ) d b) y t ( ) ^ te 2(^ t ^ ^ ) x ( 1) d c) y t ( ) ^ e ^ ^ x t ( ) d
d)^12 y t ( ) y t ( ) 6 x t ( 1) e) y t ( ) 3 ( ) y t x t ( 2) f) y t ( ) x t ( 3)
Scrambled Answers: e 2(^ t^ 1)^ u t ( 1), e 3(^ t^ 2)^ u t ( 2), ( u t 3), et , ( t 2) e u t t^^ ( ),12 e 2(^ t 1) u t ( 1)
3) For the following two systems,
a) y t ( ) ^ t e ^ (^ t^ ^ ^ )^ x ( 1) d b) y t ( ) ^ e^2 ^ u ( ) ( xt ) d
i) compute the impulse response directly ii) compute the step response directly ii) show that the derivative of the step response is the impulse response for these two systems. You should gettwo terms for the derivative, but one of the terms is zero (you should be able to show why this is true)! Note that this is a general property of LTI systems, and is important since it is much easier to determine the steprespones of a system than it is to determine the impulse response of an actual system.
4) sure to include any required unit step functions in your answers. For the following inpulse responses and inputs, determine the system output using analytical convolution. Be
a) h t ( ) e tu t ( ), x t () = u t ( -1) b) h t ( ) e t^^ u t ( ) ( ), t x ( ) t = e (^ t 1) u t ( -1)
c) h t ( ) ( ) t ( t 1), x ( ) t = e tu ( t ) d) h t ( ) ( t 1), x t ( ) = ( t 2 )
Scrambled Answers y t ( ) ( t 1), y t ( ) [1 e ^ (^ t^ 1)^ ] ( u t 1), y t ( ) te ^ (^ t^ 1)^ u t ( 1), y t ( ) e u t t^^ ( ) e (^ t 1) u t ( 1) :
5) For LTI systems with the following impulse responses, determine if the system is BIBO stable.
a) h t ( ) ( t ) b) h t ( ) u t ( ) c) h t ( ) e u t t ( ) d) h t ( ) e t^2 u t ( )
Answers: 3 are stable, one is unstable
6) For the following interconnected systems, i) determine the overall impulse response (the impulse response between input x(t) and output y(t) ) and ii) determine if the system is causal.
a) h 1 ( ) t u t ( -1), h 2 ( ) t u ( t 1) b) h 1 ( ) t u t ( -1), h 2 ( ) t ( t 2) c) h 1 ( ) t e (^ t 1)^ u t ( -1 ), h 2 ( ) t ( ) t u ( ) t
Scrambled Answers: h t ( ) u t ( 1), h t ( ) tu t ( ), h t ( ) u t ( 1), h t ( ) u t ( 1) u t ( 1), h t ( ) ( ) t u t ( ) e ( t 1) u t ( 1), h t ( ) u t ( 1) ( t 2)
Three systems not causal
8) Consider a linear time invariant system with impulse response given by h t ( ) t u t [ ( 1) u t ( 2)] and input x t ( ) 2 ( u t 2) 3 ( u t 3) u t ( 4), shown below
Using graphical convolution , determine the output y t ( ) h t ( ) x t ( ) Specifically, you must a) Flip and slide h t ( ), NOT x t ( ) b) Show graphs displaying both h t ( )and x ( )for each region of interest c) Determine the range of t for which each part of your solution is valid
d) Set up any necessary integrals to computethe symbols x ( ) or h t ( )but must contain the actual functions. y t ( ). Your integrals must be complete, in that they cannot contain
e) DO NOT EVALUATE THE INTEGRALS!!
(The answer is at the end of the homework)
3
2
2
x(t)
2
2
h(t)
t
9) Consider a noncausal linear time invariant system with impulse response given by h t ( ) [1 ( t 1)^2 ][ ( ut 1) u t ( 2)] The input to the system is given by x t ( ) u t ( 1) u t ( 3) u t ( 4) u t ( 6)
Using graphical convolution , determine the output y t ( )Specifically, you must Flip and slideShow graphs displaying both h t ( ), NOT x t ( ) h t ( )and x ( )for each region of interest Determine the range ofSet up any necessary integrals to compute t for which each part of your solution is valid y t ( ). Your integrals must be complete, in that they cannot contain the symbolsYour integrals cannot contain any unit step functions^ x (^ )or^ h t (^ ^ )but must contain the actual functions.
-3-2 -1 0 1 2 3 4 5 6 7 8
0
1
Time (sec)
h(t)
-1.5-2 -1 0 1 2 3 4 5 6 7 8
-0.5- 0 0.5^1
Time (sec)
x(t)
