EE 350 Continuous-Time Linear Systems Recitation 7
Transcript of EE 350 Continuous-Time Linear Systems Recitation 7
![Page 1: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/1.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
EE 350
Continuous-Time Linear Systems
Recitation 7
1
![Page 2: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/2.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Recitation 7 Topics
• Solved Problems – Convolution– EE 210 Review: Sinusoidal Steady-State Analysis– Relationship between the zero-state response and the
sinusoidal steady-state response
• MATLAB Exercises– Approximation of the convolution integral
2
![Page 3: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/3.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1
• A LTI system with impulse response h(t) is driven by the input f(t)
1. Determine the zero-state response y(t) using graphical convolution
2. Verify that the width property holds for part 1
3
( ) ( ) ( 2)
( ) 2u(t) 2u(t 2)
h t u t u t
f t
![Page 4: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/4.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
4
![Page 5: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/5.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
5
![Page 6: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/6.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
6
![Page 7: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/7.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
7
![Page 8: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/8.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
8
![Page 9: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/9.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Numerical Approximation of the Convolution Integral
• The zero-state response y(t) of a casual LTI system with impulse response h(t) to the causal input f(t) is
• As discussed in lecture, y(t) represents the area under the integrand, and can be approximated by the sum of area under rectangles of width T and height f(kT) h(t-kT) using the Riemann sum
9
0( ) ( ) ( )
ty t f h t d
0
(t) (kT) (t )n
k
y f h kT T
![Page 10: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/10.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
MATLAB conv Function• From the Riemann sum, the zero-state response at t = nT is
• The built-in MATLAB function conv computes the summation
• The syntax of the function conv is
where s1 is a vector of length N1, s2 is a vector of length N2, and the length of the returned vector is N1+N2-1
10
0
(nT) (kT) (nT )n
k
y T f h kT
c = conv(s1, s2)
![Page 11: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/11.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Operation of the conv Function• For example, given the vectors
the function y = conv(f,h) returns the vector
where
11
[ (0), ( ), (2 )]
[ (0),h(T),h(2T)]
f f f T f T
h h
[ (0), ( ) y(2T), y(3 ), (4 )]y y y T T y T
0
0
1
0
(0) ( ) (0 - ) (0) (0),
( ) ( ) ( - ) (0) ( ) ( ) (0),
k
k
y f kT h kT f h
y T f kT h T kT f h T f T h
![Page 12: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/12.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2• As in Problem 1, consider a LTI system with
impulse response h(t) is driven by the input f(t)
• Write an m-file that1. Uses the conv function to approximate y(t) from
t = 0 to t = 4 using T = 0.001 2. Plots the approximate zero-state response y(t)
• Does the numerical result approximate the result obtained in Problem 1?
12
( ) ( ) ( 2)
( ) 2u(t) 2u(t 2)
h t u t u t
f t
![Page 13: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/13.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 m-file
13
![Page 14: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/14.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Numerical Results
14
0 0.5 1 1.5 2 2.5 3 3.5 40
1
2
3
4
5EE 350 Recitation 7 Problem 2
y(t)
t
![Page 15: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/15.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3
• A LTI system with impulse response h(t) is driven by the input f(t)
• Determine the zero-state response y(t) using graphical convolution
15
| | 2( )
h( ) u(t 2) u(t)
tf t e
t
![Page 16: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/16.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
16
![Page 17: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/17.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
17
![Page 18: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/18.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
18
![Page 19: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/19.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
19
![Page 20: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/20.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
• Consider a LTI system with impulse response
• Determine if the system is BIBO stable
20
5( ) 2 2 ( )th t e u t
Problem 4
![Page 21: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/21.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
21
![Page 22: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/22.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5• It is known that a certain physical system with input f(t)
and output y(t) is represented by an ODE of the form
• In order to determine the unknown parameters α and β, the an engineer applies a unit-step input and observes the zero-state response. Using this response, the engineer determines that the impulse response of the system is
• Determine the value of the parameters α and β
22
( ) ( )y y t f t
4( ) 3 ( )th t e u t
![Page 23: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/23.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
23
![Page 24: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/24.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
24
![Page 25: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/25.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
25
![Page 26: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/26.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6• The sinusoidal signal
may be represented by the phasor
• The phasor is a complex-valued constant
• Using Euler’s identity, show that
26
( ) cosf t A t
jF Ae
( ) Re j tf t Fe
![Page 27: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/27.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6 Solution
27
![Page 28: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/28.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 7
• Determine the phasor representation of the following signals
28
(a) ( ) 3sin 10 30
(b) ( ) 2cos 90
f t t
f t t
![Page 29: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/29.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 7 Solution
29
![Page 30: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/30.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8• Consider the RC network with input f(t) and output y(t)
1. Determine the zero-state response for the input
2. Determine the zero-state response for times much larger than the network RC time constant
3. Derive the frequency response function, and use it to determine the sinusoidal steady-state response
4. Compare the results from parts 2 and 330
f(t)R
y(t)C
( ) cos of t A t
![Page 31: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/31.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
31
![Page 32: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/32.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
32
![Page 33: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/33.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
33
![Page 34: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/34.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
34
![Page 35: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/35.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
35
![Page 36: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/36.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
EE 350
Continuous-Time Linear Systems
Recitation 7
1
![Page 37: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/37.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Recitation 7 Topics
• Solved Problems – Convolution– EE 210 Review: Sinusoidal Steady-State Analysis– Relationship between the zero-state response and the
sinusoidal steady-state response
• MATLAB Exercises– Approximation of the convolution integral
2
![Page 38: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/38.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1
• A LTI system with impulse response h(t) is driven by the input f(t)
1. Determine the zero-state response y(t) using graphical convolution
2. Verify that the width property holds for part 1
3
![Page 39: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/39.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
4
![Page 40: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/40.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
5
![Page 41: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/41.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
6
![Page 42: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/42.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
7
![Page 43: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/43.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
8
![Page 44: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/44.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Numerical Approximation of the Convolution Integral
• The zero-state response y(t) of a casual LTI system with impulse response h(t) to the causal input f(t) is
• As discussed in lecture, y(t) represents the area under the integrand, and can be approximated by the sum of area under rectangles of width T and height f(kT) h(t-kT) using the Riemann sum
9
![Page 45: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/45.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
MATLAB conv Function• From the Riemann sum, the zero-state response at t = nT is
• The built-in MATLAB function conv computes the summation
• The syntax of the function conv is
where s1 is a vector of length N1, s2 is a vector of length N2, and the length of the returned vector is N1+N2-1
10
![Page 46: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/46.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Operation of the conv Function• For example, given the vectors
the function y = conv(f,h) returns the vector
where
11
![Page 47: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/47.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2• As in Problem 1, consider a LTI system with
impulse response h(t) is driven by the input f(t)
• Write an m-file that1. Uses the conv function to approximate y(t) from
t = 0 to t = 4 using T = 0.001 2. Plots the approximate zero-state response y(t)
• Does the numerical result approximate the result obtained in Problem 1?
12
![Page 48: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/48.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 m-file
13
![Page 49: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/49.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Numerical Results
14
![Page 50: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/50.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3
• A LTI system with impulse response h(t) is driven by the input f(t)
• Determine the zero-state response y(t) using graphical convolution
15
![Page 51: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/51.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
16
![Page 52: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/52.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
17
![Page 53: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/53.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
18
![Page 54: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/54.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
19
![Page 55: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/55.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
• Consider a LTI system with impulse response
• Determine if the system is BIBO stable
20
Problem 4
![Page 56: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/56.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
21
![Page 57: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/57.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5• It is known that a certain physical system with input f(t)
and output y(t) is represented by an ODE of the form
• In order to determine the unknown parameters α and β, the an engineer applies a unit-step input and observes the zero-state response. Using this response, the engineer determines that the impulse response of the system is
• Determine the value of the parameters α and β
22
![Page 58: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/58.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
23
![Page 59: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/59.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
24
![Page 60: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/60.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
25
![Page 61: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/61.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6• The sinusoidal signal
may be represented by the phasor
• The phasor is a complex-valued constant
• Using Euler’s identity, show that
26
![Page 62: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/62.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6 Solution
27
![Page 63: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/63.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 7
• Determine the phasor representation of the following signals
28
![Page 64: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/64.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 7 Solution
29
![Page 65: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/65.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8• Consider the RC network with input f(t) and output y(t)
1. Determine the zero-state response for the input
2. Determine the zero-state response for times much larger than the network RC time constant
3. Derive the frequency response function, and use it to determine the sinusoidal steady-state response
4. Compare the results from parts 2 and 330
![Page 66: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/66.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
31
![Page 67: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/67.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
32
![Page 68: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/68.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
33
![Page 69: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/69.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
34
![Page 70: EE 350 Continuous-Time Linear Systems Recitation 7](https://reader034.fdocuments.us/reader034/viewer/2022051102/627700355a88e9103c075466/html5/thumbnails/70.jpg)
Recitation 7.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 8 Solution
35