EE 350 Continuous-Time Linear Systems Recitation...
Transcript of EE 350 Continuous-Time Linear Systems Recitation...
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
EE 350
Continuous-Time Linear Systems
Recitation 3
1
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1• Consider the following circuit that contains an ideal diode
1. Sketch the output y(t) as a function of the input f(t)2. Is the system zero-state linear or nonlinear?3. Is the system instantaneous (memoryless) or dynamic?
2
R R
f(t)
2V
y(t)
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
3
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
4
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
5
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2• Consider the circuit below with input f(t) and output y(t)
1. Is the system linear or nonlinear? 2. Is the system time-invariant or time-varying?3. Is the system instantaneous (memoryless) or dynamic?4. Is the system causal or noncausal?
6
Rf(t)
y(t)C
v ( )C
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
7
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Recitation 3.School of Electrical Engineering and Computer Science
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Problem 2 Solution
8
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Recitation 3.School of Electrical Engineering and Computer Science
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Problem 2 Solution
9
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
10
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
11
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
12
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3• The relationship between the input f(t) and zero-state
response y(t) of two systems are given below. Determine if each system is
1. Zero-state linear or nonlinear2. Time-invariant or time-varying3. Causal or noncausal4. Instantaneous (memoryless) or dynamic
13
| | 2(a) ( ) ( 1)
(b) ( ) ( )e
ty t e f t
y t f t d
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
14
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
15
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Problem 3 Solution
16
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
17
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Recitation 3.School of Electrical Engineering and Computer Science
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Problem 3 Solution
18
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Problem 3 Solution
19
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4• Consider a linear time-invariant (LTI) system with input f(t)
and zero-state response y(t)
1. Can we use f1(t) = 2f(t/2) to find y1(t) in terms of y(t)?2. Sketch the zero-state response y1(t) to the input f1(t)3. Sketch the input f2(t) yielding the zero-state response y2(t)
20
f(t)
t
1
10
y(t)
t
1
1
LTISystem
f(t)y(t)
0
1f (t)
t2
20
2y (t)
t
3
11
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
21
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
22
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
23
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Complex Number Review• A complex number is a number that can be expressed in
the form a + j b, where a and b are real numbers and jsatisfies the equation j2 = 1
• Notation
• Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part
24
Complex Number:
Real part of Re , Imaginary part of Im
z a jb
z z a z z b
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Rectangular and Polar Form• A complex number z can be represented either in
rectangular or polar form
25
z
Re
Im
a
b
Rectangular Form z a jb
z
Re
Im
Polar Form r z
r
2 2
1
cos( ) | |
sin( ) = z = Tan
a r r z a bbb ra
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Polar Form and Euler’s Identity
26
Rectangular Form:
cos( ) sin( )
r cos( ) sin( )
Polar Form: j
z a jb
z r jr
z j
z re
Euler's Identity: cos( ) sin( )je j
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Operations on Complex Numbers• Use rectangular form for addition and subtraction
• Use polar form for multiplication and division
• The complex conjugate of the complex number z = a + jb is defined as z* = a – jb
27
1 2
1 2
1 2
If and , then z ( ) ( ) z ( ) ( )
z a jb z c jdz a c j b dz a c j b d
1 2
1 2 1 2
11 2
2
1 1 2 2
( )1 2 1 2 1 2
( )1 1 1
2 2 2
If e and e , then z e e
e e
j j
j j j
jj
j
z r z r
z r r r r e
z r r ez r r
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5
• Given z = a + jb = r ej show that1. z* = r e-j
2. z + z* = 2a = 2Re{z}3. z z* = 2jb = j2Im{z}4. z z* = a2 + b2 = r2
28
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
29
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Entering Complex Numbersin MATLAB
30
• The complex number z = 2 + j3 may be entered into MATLAB as
>> z = 2 + 3*i>> z = 2 + 3i
>> z = 2 + 3*j>> z = 2 + 3j
>> z = 2 + 3*1i
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Recitation 3.School of Electrical Engineering and Computer Science
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Complex Functions in MATLAB
31
Operation Functionmagnitude (r) of z abs(z)angle (q) of z in radians angle(z)construct z from a and b complex(a,b)conjugate of z conj(z)imaginary part of z imag(z)real part of z real(z)
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6• Use MATLAB to determine
• Explain the difference between the results in parts 3 and 4
32
1 21. The rectangular form of 1 2
2 2 32. The polar form of
1 3
3. Execute the MATLAB command >>exp(pi/2*i)
4. Execute the MATLAB command >>exp(pi/2i)
jj
j j
j j
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6 Solution
33
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
EE 350
Continuous-Time Linear Systems
Recitation 3
1
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1• Consider the following circuit that contains an ideal diode
1. Sketch the output y(t) as a function of the input f(t)2. Is the system zero-state linear or nonlinear?3. Is the system instantaneous (memoryless) or dynamic?
2
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
3
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
4
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
5
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2• Consider the circuit below with input f(t) and output y(t)
1. Is the system linear or nonlinear? 2. Is the system time-invariant or time-varying?3. Is the system instantaneous (memoryless) or dynamic?4. Is the system causal or noncausal?
6
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
7
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Recitation 3.School of Electrical Engineering and Computer Science
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Problem 2 Solution
8
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
9
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Recitation 3.School of Electrical Engineering and Computer Science
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Problem 2 Solution
10
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
11
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
12
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3• The relationship between the input f(t) and zero-state
response y(t) of two systems are given below. Determine if each system is
1. Zero-state linear or nonlinear2. Time-invariant or time-varying3. Causal or noncausal4. Instantaneous (memoryless) or dynamic
13
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
14
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
15
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
16
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
17
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
18
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
19
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4• Consider a linear time-invariant (LTI) system with input f(t)
and zero-state response y(t)
1. Can we use f1(t) = 2f(t/2) to find y1(t) in terms of y(t)?2. Sketch the zero-state response y1(t) to the input f1(t)3. Sketch the input f2(t) yielding the zero-state response y2(t)
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
21
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
22
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
23
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Complex Number Review• A complex number is a number that can be expressed in
the form a + j b, where a and b are real numbers and jsatisfies the equation j2 = 1
• Notation
• Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Rectangular and Polar Form• A complex number z can be represented either in
rectangular or polar form
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Polar Form and Euler’s Identity
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Operations on Complex Numbers• Use rectangular form for addition and subtraction
• Use polar form for multiplication and division
• The complex conjugate of the complex number z = a + jb is defined as z* = a – jb
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5
• Given z = a + jb = r ej show that1. z* = r e-j
2. z + z* = 2a = 2Re{z}3. z z* = 2jb = j2Im{z}4. z z* = a2 + b2 = r2
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
29
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Entering Complex Numbersin MATLAB
30
• The complex number z = 2 + j3 may be entered into MATLAB as
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Complex Functions in MATLAB
31
Operation Functionmagnitude (r) of z abs(z)angle (q) of z in radians angle(z)construct z from a and b complex(a,b)conjugate of z conj(z)imaginary part of z imag(z)real part of z real(z)
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6• Use MATLAB to determine
• Explain the difference between the results in parts 3 and 4
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Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6 Solution
33