![Page 1: EE 350 Continuous-Time Linear Systems Recitation 8](https://reader034.vdocument.in/reader034/viewer/2022051403/627cb80cfa31ed66495e8173/html5/thumbnails/1.jpg)
Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
EE 350
Continuous-Time Linear Systems
Recitation 8
1
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Recitation 8 Topics
• Exam II Review
2
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1• A certain LTI system has the impulse response
1. Is the system causal or noncausal?
2. Is the system BIBO stable?
3
( ) cos(t 1)u(t 1)th t e
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
4
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
5
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2• Consider a LTI system with the input f(t) and zero-
state response y(t) shown below
• Determine the impulse response representation of the system
6
( 1)
( ) 3 ( 1)
( ) 6 ( 1)t
f t u t
y t e u t
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
7
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
8
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3• The system shown below is composed of three LTI
systems whose impulse responses are
• Determine an impulse response h(t) that represents the composite system so that
9
31 2 3( ) ( 2); ( ) sin( ) ( ); h ( ) ( )th t t h t t u t t e u t
( ) ( ) ( )y t f t h t
1( )h t2 ( )h t
3 ( )h t
( )y tf( )t
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
10
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
• Consider a LTI system represented by the ODE
• Determine the impulse response representation h(t) of the system
11
4 4 8 ( )y y y f t
Problem 4
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
12
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
13
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5• A LTI system has the impulse response
• The system is driven by the input
• Determine the zero-state response y(t) by evaluating the convolution integral using the graphical approach
14
( ) 2 ( 2) 4 ( ) 2 ( 2)h t u t u t u t
( 2)f( ) ( 2)tt e u t
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
15
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
16
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
17
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
18
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6• Determine the frequency response function of the
network shown below and pace your answer in the standard form
19
1RL
f t y t
C
2R
11 1 0
11 1 0
( ) ( ) ( )( )( ) ( ) ( )
m mm m
n nn
b b b bHa a a
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6 Solution
20
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6 Solution
21
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6 Solution
22
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 7• Determine the sinusoidal state steady response
23
( )y t0( ) cos( )f t A t LTI( )H j
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 7 Solution
24
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 7 Solution
25
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8• Consider the two following two signals that are
defined over the interval [-1,1] where a and b are nonzero positive real-valued coefficients
1. Are the signals orthogonal?
2. Determine the value of the parameters a and bfor which the signals are orthonormal
26
21
32
( )
( )
t at
t bt
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
27
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
28
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 9• Write a primary MATLAB function that approximates
the exponential function using the first 100 terms in the Taylor series expansion
• The user must be able to call the function using the syntax
• Verify your result using MATLAB29
100
0 0
(x)! !
n nx
n n
x xf en n
>> x = -1;>> f_approx = approx_exp(x);
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
EE 350
Continuous-Time Linear Systems
Recitation 8
1
![Page 31: EE 350 Continuous-Time Linear Systems Recitation 8](https://reader034.vdocument.in/reader034/viewer/2022051403/627cb80cfa31ed66495e8173/html5/thumbnails/31.jpg)
Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Recitation 8 Topics
• Exam II Review
2
![Page 32: EE 350 Continuous-Time Linear Systems Recitation 8](https://reader034.vdocument.in/reader034/viewer/2022051403/627cb80cfa31ed66495e8173/html5/thumbnails/32.jpg)
Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1• A certain LTI system has the impulse response
1. Is the system causal or noncausal?
2. Is the system BIBO stable?
3
![Page 33: EE 350 Continuous-Time Linear Systems Recitation 8](https://reader034.vdocument.in/reader034/viewer/2022051403/627cb80cfa31ed66495e8173/html5/thumbnails/33.jpg)
Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
4
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
5
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2• Consider a LTI system with the input f(t) and zero-
state response y(t) shown below
• Determine the impulse response representation of the system
6
![Page 36: EE 350 Continuous-Time Linear Systems Recitation 8](https://reader034.vdocument.in/reader034/viewer/2022051403/627cb80cfa31ed66495e8173/html5/thumbnails/36.jpg)
Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
7
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
8
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3• The system shown below is composed of three LTI
systems whose impulse responses are
• Determine an impulse response h(t) that represents the composite system so that
9
![Page 39: EE 350 Continuous-Time Linear Systems Recitation 8](https://reader034.vdocument.in/reader034/viewer/2022051403/627cb80cfa31ed66495e8173/html5/thumbnails/39.jpg)
Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
10
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
• Consider a LTI system represented by the ODE
• Determine the impulse response representation h(t) of the system
11
Problem 4
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
12
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
13
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5• A LTI system has the impulse response
• The system is driven by the input
• Determine the zero-state response y(t) by evaluating the convolution integral using the graphical approach
14
![Page 44: EE 350 Continuous-Time Linear Systems Recitation 8](https://reader034.vdocument.in/reader034/viewer/2022051403/627cb80cfa31ed66495e8173/html5/thumbnails/44.jpg)
Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
15
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
16
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
17
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
18
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6• Determine the frequency response function of the
network shown below and pace your answer in the standard form
19
![Page 49: EE 350 Continuous-Time Linear Systems Recitation 8](https://reader034.vdocument.in/reader034/viewer/2022051403/627cb80cfa31ed66495e8173/html5/thumbnails/49.jpg)
Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6 Solution
20
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6 Solution
21
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6 Solution
22
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 7• Determine the sinusoidal state steady response
23
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 7 Solution
24
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 7 Solution
25
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8• Consider the two following two signals that are
defined over the interval [-1,1] where a and b are nonzero positive real-valued coefficients
1. Are the signals orthogonal?
2. Determine the value of the parameters a and bfor which the signals are orthonormal
26
![Page 56: EE 350 Continuous-Time Linear Systems Recitation 8](https://reader034.vdocument.in/reader034/viewer/2022051403/627cb80cfa31ed66495e8173/html5/thumbnails/56.jpg)
Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
27
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Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
28
![Page 58: EE 350 Continuous-Time Linear Systems Recitation 8](https://reader034.vdocument.in/reader034/viewer/2022051403/627cb80cfa31ed66495e8173/html5/thumbnails/58.jpg)
Rec 8.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 9• Write a primary MATLAB function that approximates
the exponential function using the first 100 terms in the Taylor series expansion
• The user must be able to call the function using the syntax
• Verify your result using MATLAB29