Colorado State University, Ft. Collins
Spring 2016
ECE 312: Linear Systems Analysis II (Signal and Systems)
Homework 2
Assigned on: 02/16/2016, Due by: 03/03/2016
2.1
A causal LTI system with impulse response ht  has the following properties:
1. When the input to the system is xt   e 2t for all t , the output is y t  
1 2t
e for
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all t .
2. The impulse response ht  satisfies the differential equation
dht 
 2ht   e 4t u t   bu t 
dt
where b is an unknown constant.
Determine the system function H s  of the system, consistent with the information
above. There should be no unknown constant in your answer; that is, the constant b
should not appear in the answer.
2.2
Consider the system S characterized by the differential equation
d 3 yt 
d 2 yt 
dy t 

6
 11
 6 yt   xt  .
3
2
dt
dt
dt
(a)
(b)
Determine the zero-state response of this system for the input
xt   e 4t u t  .
Determine the zero-input response of the system for t  0 , given that
y0  1 ,
d 2 yt 
dyt 
 1.
 1 ,
dt t 0
dt 2 t 0
1
(c)
Determine the output of S when the input is xt   e 4t u t  and the initial
conditions are the same as those specified in part (b).
2.3
Find the inverse Laplace transform of the function
F s  
s9
ss  1
for the following regions of convergence:
(a)
(b)
(c)
(d)
Res   1
Res   0
 1  Res   0
Give the final values of the functions of Parts (a), (b), and (c).
2.4
A pressure gauge that can be modeled as an LTI system has a time response to a unit
step input given by 1  e t  te t ut  . For a certain input xt  , the output is
observed to be 2  3e t  e 3t ut  . For this observed measurement, determine the
true pressure input to the gauge as a function of time.




2.5
Let H s  represent the system function of a causal, stable system. The input to the
system consists of the sum of three terms, one of which is an impulse  t  and
another a complex exponential of the form e s0t , where s0 is a complex constant.
The output is
y t   6e t u t  
4 4t
18
e cos 3t  e 4t sin 3t   t 
34
34
Determine H s  , consistently with this information.
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