Problem 5.2 Provide expressions in terms of step functions for the waveforms
displayed in Fig. P5.2.
1(t)
2(t)
6
6
4
4
2
2
2 1
2
1
2
3
4
t (s)
2 1
2
(a) Step
1
2
3
4
3
4
t (s)
(b) Bowl
3(t)
4(t)
6
6
4
4
2
2
2 1
2
1
2
3
4
t (s)
(c) Staircase up
2 1
2
1
2
t
Engineering Signals and Systems
by Fawwaz T. Ulaby and Andrew E. Yagle
Solutions to the Exercises
Fawwaz T. Ulaby and Andrew E. Yagle, Engineering Signals and Systems c 2013 National Technology and Science Press
Chapter 1: Signals
Chapter 2: Linear Time-I
EEL3135-U02: Signals and Systems
Lecture 13
Continuous-time LTI Systems: Convolution Integral
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Review: Discrete-time LTI Systems
1. h[n]: Impulse response
2. For a time-invariant system:
shifted i
EEL3135-U02: Signals and Systems
Lecture 16
LTI System Properties (Cont.)
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Review: LTI System Properties
1. Commutative: =
2. Distributive: (1 + 2 ) = 1 + 2
3. Associative: 1
2 = 2
1 = (1 2
EEL3135-U02: Signals and Systems
Lecture 28
Sampling
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Sampling (Chapter 7)
Represent continuous-time signal with discrete samples
Sampling function: =
= ( )
Sampling period T
Sampling frequen
1/21/2017
EEL3135-U02: Signals and Systems
Lecture 5: Signals
DT Exponentials, Unit Step, & Unit Impulse
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Review: CT Exponential
x(t ) Ce at
1. Non-periodic real exponential signals (C & a are rea
EEL3135-U02: Signals and Systems
Lecture 23
CT Fourier Transform
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Review: Continuous-Time Fourier Transform (CTFT)
CTFT is a representation for general (NON-Periodic) signals in a different domai
EEL3135-U02: Signals and Systems
Lecture 15
LTI System Properties
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Review: LTI Systems
DT LTI:
x[n]
h[n]
y[n]
= =
=
x(t)
h(t)
y(t)
CT LTI:
= =
t t t
t=
2
Review: Properties of LTI Systems
1)
EEL3135-U02: Signals and Systems
Lecture 19
Fourier Series Coefficients
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Review: Linear Combination of Harmonic Components
Harmonically related complex exponentials
=
0
=
(
2
)
, k = 0, 1, 2,
EEL3135-U02: Signals and Systems
Lecture 10:
Discrete-time Linear Time Invariant Systems: Convolution Sum
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Outline
Linear Time-invariant (LTI) discrete-time systems
LTI input-output relationship
EEL3135-U02: Signals and Systems
Lecture 25
Properties of CTFT
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Review: Properties of CT FT
1. Linearity
()
; ()
a +
2. Time Shifting
( 0 )
+
0
3. Conjugation and Conjugate Symmetry
()
1/12/2017
EEL3135-U02: Signals and Systems
Lecture 3: Signals
Time Scaling, Periodic, Even/Odd, Exponential
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Review
Energy and power
T
E limT x(t ) dt
2
T
x(t ) dt
E limN n N x[n] n x[n]
N
Time
EEL3135-U02: Signals and Systems
Lecture 7: Systems
System Properties (Cont.)
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Review: System Properties
1. Memoryless
Memoryless system
System with memory
= []
=
1
2
2. Invertibility
EEL3135-U02: Signals and Systems
Lecture 11
Convolution Sum: Examples
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Review: Discrete-time Linear Time Invariant (LTI) Systems
1. h[n]: Impulse response
2. For a time-invariant system:
shifted i
EEL3135-U02: Signals and Systems
Lecture 20
Properties of Fourier Series Coefficients
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Review: Fourier Series Representation
A periodic signal can be represented in terms of its Fourier series coe
EEL3135-U02: Signals and Systems
Lecture 12
Convolution Sum: More Examples
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
Example 1
= []
= []
x[n]
y[n]
h[n]
= =
=
= []
2
Example 1 (Cont.)
= =
=
= []
=
= [ ]
< 0: =0
(no overlap
and
Complex Numbers
_ _ _ as f0:
It was observed early in history that there are equations which are not L H
satisﬁed by any real number. Examples are ' Syste
2 2 Th
x=—3 or x—10x+40=0.
This led to the invention of complex numbers.1
. and ;
Definition .
A
EEL 3135-U02: Signals and Systems (Spring 2017)
Homework 5: Continuous-time Fourier Transform Representation [12 Points]
Due date: Please, submit your solutions of the following problems to me by Monday Apr. 17th, 2017, 11
am in office (EC 3144).
1. [4 Po
1/8/2017
EEL3135-U02: Signals and Systems
Dr. Ahmed S. Ibrahim
Assistant Professor
aibrahim@fiu.edu
1
What are Signals?
Signals are used to describe a wide variety of physical phenomena
They carry information, represented as a pattern of variations
Cont