Kronex's Outline of Digital Signals and Syatems Using MATLAB. A Practical Approach

402 Pages·2017·8.594 MB·English

by  Kronex David

Kronex’s Series
Kronex’s Outline of
D igital Signal
&
System
using
MAT LAB
Kronex David
Contents

Chapter 1 Basic Concepts 1
Global Positioning System 1
11 Introduction 2
12 Basic Deinitions 2
13 Classiications of Signals 3
131 Continuous-Time and Discrete-Time Signals 3
132 Periodic and Nonperiodic Signals 5
133 Analog and Digital Signals 6
134 Energy and Power Signals 7
135 Even and Odd Symmetry 8
14 Basic Continuous-Time Signals 13
141 Unit Step Function 13
142 Unit Impulse Function 14
143 Unit Ramp Function 17
144 Rectangular Pulse Function 18
145 Triangular Pulse Function 19
146 Sinusoidal Signal 19
147 Exponential Signal 20
15 Basic Discrete-Time Signals 25
151 Unit Step Sequence 25
152 Unit Impulse Sequence 25
153 Unit Ramp Sequence 26
154 Sinusoidal Sequence 27
155 Exponential Sequence 28
16 Basic Operations on Signals 30
161 Time Reversal 30
162 Time Scaling 31
163 Time Shifting 31
164 Amplitude Transformations 32
17 Classiications of Systems 36
171 Continuous-Time and Discrete-Time Systems 37
172 Causal and Noncausal Systems 37
173 Linear and Nonlinear Systems 39
174 Time-Varying and Time-Invariant Systems 40
175 Systems with and without Memory 41
vii
viii Contents
18 Applications 43
181 Electric Circuit 43
182 Square-Law Device 44
183 DSP System 44
19 Computing with MATLAB® ...................................................45
110 Summary 50
Review Questions 51
Problems 52
Chapter 2 Convolution 63
Enhancing Your Communication Skills63
21 Introduction 64
22 Impulse Response 64
23 Convolution Integral 65
24 Graphical Convolution 70
25 Block Diagram Representation 76
26 Discrete-Time Convolution 78
27 Block Diagram Realization 85
28 Deconvolution 85
29 Computing with MATLAB® ...................................................88
210 Applications 91
2101 BIBO Stability of Continuous-Time Systems 91
2102 BIBO Stability of Discrete-Time Systems 92
2103 Circuit Analysis 93
211 Summary 95
Review Questions 96
Problems 97
Chapter 3 The Laplace Transform 105
Historical Proile 105
31 Introduction 106
32 Deinition of the Laplace Transform 106
33 Properties of the Laplace Transform 110
331 Linearity 110
332 Scaling 111
333 Time Shifting 112
334 Frequency Shifting 113
335 Time Differentiation 113
336 Time Convolution 114
337 Time Integration 115
338 Frequency Differentiation 116
339 Time Periodicity 117
3310 Modulation 118
3311 Initial and Final Values 119
Contents ix
34 The Inverse Laplace Transform 126
341 Simple Poles 127
342 Repeated Poles 128
343 Complex Poles 129
35 Transfer Function 138
36 Applications 143
361 Integro-Differential Equations 143
362 Circuit Analysis 145
363 Control Systems150
37 Computing with MATLAB® 152
38 Summary 157
Review Questions 158
Problems 159
Chapter 4 Fourier Series 171
Historical Proile 171
41 Introduction 172
42 Trigonometric Fourier Series 172
43 Exponential Fourier Series 181
44 Properties of Fourier Series 188
441 Linearity 188
442 Time Shifting 189
443 Time Reversal 189
444 Time Scaling 190
445 Even and Odd Symmetries 190
446 Parseval’s Theorem 192
45 Truncated Complex Fourier Series 196
46 Applications 197
461 Circuit Analysis 197
462 Spectrum Analyzers 200
463 Filters 200
47 Computing with MATLAB® .................................................204
48 Summary 208
Review Questions 210
Problems 211
Chapter 5 Fourier Transform 221
Career in Control Systems 221
51 Introduction 222
52 Deinition of the Fourier Transform 222
53 Properties of Fourier Transform 229
531 Linearity 229
532 Time Scaling 230
533 Time Shifting 230
x Contents
534 Frequency Shifting 231
535 Time Differentiation 232
536 Frequency Differentiation 232
537 Time Integration 233
538 Duality 233
539 Convolution 234
54 Inverse Fourier Transform 239
55 Applications 240
551 Circuit Analysis 241
552 Amplitude Modulation 244
553 Sampling 247
56 Parseval’s Theorem 250
57 Comparing the Fourier and Laplace Transforms 253
58 Computing with MATLAB® .................................................254
59 Summary 257
Review Questions 258
Problems 259
Chapter 6 Discrete Fourier Transform 271
Career in Communications Systems 271
61 Introduction 272
62 Discrete-Time Fourier Transform272
63 Properties of DTFT 277
631 Linearity 277
632 Time Shifting and Frequency Shifting 278
633 Time Reversal and Conjugation 279
634 Time Scaling 280
635 Frequency Differentiation 281
636 Time and Frequency Convolution 282
637 Accumulation 283
638 Parseval’s Relation 284
64 Discrete Fourier Transform 289
65 Fast Fourier Transform 294
66 Computing with MATLAB® .................................................295
67 Applications 298
671 Touch-Tone Telephone 298
672 Windowing 299
68 Summary 301
Review Questions 302
Problems 303
Contents xi
Chapter 7 z-Transform 309
Codes of Ethics 309
71 Introduction 310
72 Deinition of the z-Transform 311
73 Region of Convergence 313
74 Properties of the z-Transform 315
741 Linearity 316
742 Time-Shifting 316
743 Frequency Scaling 317
744 Time Reversal 318
745 Modulation 318
746 Accumulation 319
747 Convolution 320
748 Initial and Final Values 320
75 Inverse z-Transform 327
751 Long Division Expansion 327
752 Partial Fraction Expansion 329
76 Applications 332
761 Linear Difference Equation 333
762 Transfer Function 335
77 Computing with MATLAB® .................................................339
78 Summary 343
Review Questions 344
Problems 345
Selected Bibliography 353
Appendix A: Mathematical Formulas 355
Appendix B: Complex Numbers 367
Appendix C: Introduction to MATLAB® ..........................................................375
Appendix D: Answers to Odd-Numbered Problems 389
Index 415

1
 Basic Concepts
GLOBAL POSITIONING SYSTEM
Artist interpretation of GPS satellite. (Image courtesy of NASA, Washington, DC; Global Positioning System.
http://en.wikipedia.org/wiki/Global_Positioning_System.)
The global positioning system (GPS) is a typical illustration of what signals and
systems are all about GPS is a satellite-based navigation system made up of a
network of 24 satellites placed into orbit by the US Department of Defense GPS
was originally designed for military use, but in the 1980s, the government made
the system available for civilian use
The 24 satellites that make up the GPS space segment are orbiting the earth about
12,000 miles above us These satellites travel at speeds of roughly 7,000 miles/h
GPS satellites transmit two low power radio signals The signals travel by line of
sight, meaning they will pass through clouds, glass, and plastic but will not go
through most solid objects, such as buildings and mountains
A GPS signal contains three different bits of information—a pseudorandom code,
ephemeris data, and almanac data The pseudorandom code identiies which satel-
lite is transmitting information You can view this number on your GPS unit’s sat-
ellite page Ephemeris data contains important information about the status of the
1
2
satellite, current date, and time This part of the signal is essential for determining
a position The almanac data tells the GPS receiver where each GPS satellite should
be at any time throughout the day Each satellite transmits almanac data showing the
orbital information for that satellite and for every other satellite in the system
A GPS receiver calculates its position by precisely timing the signals sent by GPS
satellites The receiver uses the messages it receives to determine the transit time of
each message and computes the distance to each satellite These distances along with
the satellites’ locations are used to compute the position of the receiver
1.1 INTRODUCTION
The idea of signals and systems arises in different disciplines such as science, engi-
neering, economics, politics, and medicine Scientists, mathematicians, inancial
analysts, cardiologists, and engineers all use the concepts of systems and signals
because they are the foundation on which we build many things in our daily lives
Typical examples of systems include radio and television, telephone networks,
radar systems, computer networks, wireless communication, military surveillance
systems, and satellite communication systems The theory of signals and systems
provides a solid foundation for control systems, communication systems, power sys-
tems, and networking, to name a few
Our objective in this book is to present an introductory, yet comprehensive, treat-
ment of signals and systems with an emphasis on computing using MATLAB® The
knowledge of a broad range of signals and systems is of practical value in describing
human experience It is also important because engineers must be familiar with sig-
nal and system concepts and apply the knowledge to analyze some speciic signals
and systems they will deal with in their professional life
In this chapter, we begin by discussing some of the basic concepts in signals and
systems We introduce the mathematical representations of signals, their properties,
and applications We also discuss different systems and how the material covered in
this chapter is used in some applications We inally learn how to use MATLAB to
process signals
1.2 BASIC DEFINITIONS
To avoid any misconception, it is expedient that we deine at the outset what we mean
by signals and systems
A signal x(t) is a set of data or function of time that represents a variable
of interest
A signal typically contains information about the nature of a phenomenon
Examples of signals include the atmospheric temperature, humidity, human voice,
television images, a dog’s bark, and birdsongs More generally, a signal may be a
function of more than one independent variable (time) For example, pictures are
signals that depend on two independent variables (horizontal and vertical positions)
Basic Concepts 3
Excitation Response
x(t) System y(t)
(Input signal) (Output signal)
FIGURE 1.1 A simple system
and may be regarded as two-dimensional signals However, in this text, we will con-
sider only one-dimensional signals with time as the independent variable
A system is a collection of devices that operate on input signal x(t) (or excita-
tion) to produce an output signal y(t) (or response)
A system may also be regarded as a mathematical model of a physical process
that relates the input signal to the output signal Examples of systems include elec-
tric circuits, computer programs, the stock market, weather, and the human body
A system may have several mathematical models or representations The variables
in the mathematical model are described as signals, which may be current, voltage,
or displacement In electrical systems, signals are often represented as currents and
voltages In mechanical systems, they are often represented as temperatures, forces,
and velocities In hydraulic systems, signals may be displacements and pressures
Figure 11 illustrates the block diagram of a single-input single-output system We
classify the signals that enter the system as input signals, while the signals produced
by the system as outputs For example, we may regard voltages and currents as func-
tions of time in an electric circuit as signals, while the circuit itself is regarded as a
system In engineering systems, signals may carry energy or information
1.3 CLASSIFICATIONS OF SIGNALS
There are many ways of classifying signals: continuous-time or discrete-time, peri-
odic or nonperiodic, energy or power, analog or digital, random or nonrandom, real
or complex, etc
1.3.1 CONTINUOUS-TIME AND DISCRETE-TIME SIGNALS
A signal x(t) that is deined at all instants of time is known as a continuous-time signal
A continuous-time signal takes a value at every instant of time t
An example of a continuous-time signal x(t) is shown in Figure 12a
A discrete-time signal is deined only at particular instants of time
A discrete-time signal is usually identiied as a sequence of numbers, denoted
by x[n], where n is an integer It may represent a phenomenon for which the
4
x(t)
0 t
(a)
x[n]
2
1
(b) –3 –2 –1 0 1 2 3 n
FIGURE 1.2 Typical examples of (a) continuous-time and (b) discrete-time signals
independent variable n is inherently discrete An example of discrete-time signal
x[n] is shown in Figure 12b
Since time is naturally continuous, most physical systems are continuous-time
systems Discrete-time signals are often obtained from continuous-time signals
through sampling
As typically shown in Figure 13, the continuous-time signal x(t) in Figure 13a is
sampled uniformly with sampling period T to produce the discrete-time signal x[n]
in Figure 13b We simplify notation by letting x(kT)≜x[k] A discrete-time signal
x(t)
t
(a)
x[k]
kT
T
(b)
FIGURE 1.3 Obtaining x[k] from x(t) through sampling