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
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)
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
3
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
4
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
5
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
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
7
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
8
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
9
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
10
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
11
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
12
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
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
14
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
15
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
16
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
17
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
18
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
19
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
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
21
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
22
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
23
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
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
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
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
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
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
29
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
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)
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
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6 Solution
33
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
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
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
3
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
4
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
5
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
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
7
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
8
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
9
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
10
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
11
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
12
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
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
14
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
15
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
16
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
17
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
18
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
19
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
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
21
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
22
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
23
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
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
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Polar Form and Euler’s Identity
26
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
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
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
29
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
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)
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
Recitation 3.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6 Solution
33