City University of Hong Kong Department of Electronic Engineering
EE3210 Lab 2: Introduction to Complex Exponentials
Prelab: Read the Background section. Verification: The Warm-Up section must be completed during your assigned lab time. The steps marked I
EE 3210 Assignment 4: Solution
Question 1
1
Question 2 The Fourier transform of x(t-t0) is given by
x(t t 0 )e jt dt = x( )e j ( +t0 ) d = e jt0 X ( j ) .
Question 3
Question 4 Using the time-shift property, we have
e
a t 2
F
2a e j 2 . 2 a +
2
Using the
EE3210 Signal and Systems
Quiz 3 (Solution) 1. The answer is: 6 2 5 y[n] = 6 2 0 n=0 n =1 n=2 n=3 n=4 otherwise
Semester A, 2009/10
2. a) The signal is sketched below: 1
3 b) Let y(t) be the systems response to x(t).
5
y (t ) = h( ) x(t )d = e 3 [u (t 3)
EE3210 Signal and Systems
Quiz 2 (Solution) 1. a) Let hp(t) be the impulse response. Then
Semester A, 2009/10
h p (t ) = h1 (t ) + h2 (t ) = 1 + e t u (t ) .
b) The response is
[
]
[1 + e
( t 1)
]u (t 1) .
2. a) Fundamental period = 8. b) x[n] can be rew
EE3210 Tutorial 1 (Solution)
3 1 Evcfw_u[n] = 1 / 2 0 n=0 n = 1 and 1 n = 7, 2, 2 and 7 otherwise n = 1 n =1 n = 2 and 7 n = 7 and 2 otherwise
1.
1 1 Od cfw_u[n] = 1 / 2 1 / 2 0
2.
(a) 7,
(b) Not periodic,
(c) 8
Note that in part (c), the fundamental per
EE3210 Tutorial 1: Basic Concepts of Signals
1. (Problem 1.24b) Determine and sketch the even and odd parts of the following signal: n = 2 1 2 n = 1 u[n] = 3 n=0 1 n=7 0 otherwise (Problem 1.26) Determine whether or not each of the following discrete-time
EE3210 Tutorial 2: Basic Concepts of Systems
1.
Find the input-output relation of the feedback system shown below:
x[n]
+
1 D
y[n]
2.
(Problem 1.16) Consider a discrete-time system with input x[n] and output y[n]. The input-output relationship for this sy
EE3210 Tutorial 3: Fourier Series
1.
(Problem 3.4 in the Textbook) Use the Fourier series analysis equation to calculate the coefficients ak for the continuous-time periodic signal 1.5 0 t < 1 x(t ) = 1.5 1 t < 2 with fundamental frequency 0 = .
2.
The Fo
EE3210 Tutorial 4: Convolution
1. Let
x[ n] = 4 [ n] + 2 [ n 1]
and
h[ n] = 3 [n] + [ n 1].
Use the graphical method to find their convolution sum. Repeat using the superposition method. 2. (Problem 2.1) Let
x[ n] = [ n] + 2 [ n 1] [ n 3]
and
h[ n] = 2 [
EE3210 Tutorial 5: Difference Equations and Fourier Transform
1.
(Problem 2.30) Consider the first-order difference equation
y[ n] + 2 y[n 1] = x[ n] .
Assuming the condition of initial rest, find the impulse response of a system whose input and output ar
EE 3210 Assignment 3: Solution
1.
(Problem 2.6 in the Textbook)
2.
(Problem 2.21(d) in the Textbook) The graphical representation of x[n] and h[n] can be found in the Textbook (Problem 2.21(d), and that of y[n] is shown below:
3.
(Problem 2.22(c) in the T
City University of Hong Kong Department of Electronic Engineering
EE3210 Lab 1: Introduction to MATLAB
Verification: The Warm-Up section must be completed during your assigned lab time. The steps marked Instructor Verification must also be signed off duri
City University of Hong Kong Department of Electronic Engineering EE3210 Lab 3: Network Analysis
Objective
To be familiar with linear network analysis using ordinary differential equation and Laplace Transform.
Equipment
1. IBM Pentium or compatibles 2. M
EE3210 Assignment 1: Basic Concepts
Due Date: 5p.m., Sep 16, 2009. (Note: Since I will be on leave in Week 3, a box will be put outside my office on Sep 15 and Sep 16. Please put your assignment in the box. The grader of this course will collect it. Late
EE3210 Assignment 2: Fourier Series
Due: (before lecture) Oct 7, 2009 (Marks will be deducted for late submission.) 1. Find the derivative of the following signal: x(t ) = e 2 ( t 1) u (t 1) , where u(t) is the unit step function. (Hint: Use the Chain Rul
EE3210 Assignment 3: Convolution
Due: (before lecture) Oct 14, 2009. 1. Compute the convolution y[n] = x[n] * h[n], where
1 x[n] = u[ n 1] 3
n
and
h[n] = u[n 1].
2.
Consider the following pair of signals: 1 0 n 4 x[n] = 0 otherwise 1 2 n 7, or 11 n 16 h[n
EE3210 Assignment 4: Fourier Transform
Due: (before lecture) Oct 28, 2009 (Marks will be deducted for late submission.)
1.
Determine the Fourier transforms of the following continuous-time signals: a) e 2 ( t 1) u (t 1) b) e Sketch and label the magnitude
EE3210 Assignment 5: Fourier Transform
Due: (before lecture) Nov 18, 2009 (Marks will be deducted for late submission.)
1.
The input and the output of a stable and causal LTI system are related by the differential equation d 2 y (t ) dy (t ) dx(t ) +6 + 8
EE3210 Assignment 6: Laplace Transform
Due: (before lecture) Nov 25, 2009 (Marks will be deducted for late submission.) 1. A causal LTI system with impulse response h(t) has the following properties: When the input to the system is x(t) = e2t for all t, t
EE3210 Assignment 1 (Solution)
1. (a) Periodic, fundamental period = . (b) Periodic, fundamental period = 8.
2.
(a) This is a periodic signal with period T = 2 / . Px = 1 T /2 2 2 T / 2 A cos (t + )dt T A2 T / 2 = [1 + cos(2t + 2 )]dt 2T T / 2 A2 = 2
(b)
EE 3210 Assignment 2: Solution
Question 1 Using the chain rule, we have dx(t ) de 2 (t 1) 2 ( t 1) du (t 1) =e + u (t 1) dt dt dt 2 ( t 1) 2 ( t 1) =e (t 1) 2e u (t 1) = (t 1) 2e 2( t 1) u (t 1) Question 2 It is easy to see that T = 3, 0 = 2 / 3, and a0 =
EE3210 Tutorial 6: Fourier Transform
1.
(Problem 5.21) Compute the Fourier transform of each of the following signals: a) x[n] = u[ n 2] u[n 6]
1 b) x[ n] = u[ n 1] 2
n
2.
Given that x(t) has the Fourier transform X(j), express the Fourier transforms of t
EE3210 Tutorial 7: Fourier Transform and Frequency-Domain Analysis
1. (Problem 4.36 in the Textbook) Consider a continuous-time LTI system whose response to the input
x(t ) = e t + e 3t u (t )
is
[
]
y (t ) = 2e t 2e 4t u (t ) .
[
]
a) b) c)
Find the freq
EE3210 Signal and Systems
Unit 3 Fourier Series
Outline
1. Preliminaries: Complex Exponential Signals 2. Continuous-Time Fourier Series 3. Discrete-Time Fourier Series
Unit 3: Fourier Series
2
Unit 3.1
Preliminaries: Complex Exponential Signals
Exponentia
EE3210 Signals and Systems
Unit 5. Fourier Transform
(Continuous-Time and Discrete-Time)
Objective
Fourier series can be used to represent periodic signals. Q: How to represent aperiodic signals? A: Fourier Transform.
Note: Fourier Transform is more gen
EE3210 Signals and Systems
Unit 6. Linear Time-Invariant Systems (Frequency Domain Analysis)
Unit 6.1
Eigenfunction of LTI System
After studying this unit, you will be able to 1. describe the eigenfunction of LTI systems 2. find the frequency response of
EE 3210 Signals and Systems
Unit 7 Laplace Transform
Unit 7.1
What is Laplace Transform?
After studying this unit, you will be able to 1. describe the definition of Laplace transform and its relationship with Fourier transform 2. find the Laplace transfor
EE 3210 Signals and Systems
Unit 8 z Transform
Unit 8.1
What is z-Transform?
After studying this unit, you will be able to 1. describe the definition of z-transform and its relationship with Fourier transform 2. find the z-transform of some simple signals