Tài liệu signals and systems pdf

Signals and Systems with MATLAB ® Computing and Simulink ® Modeli Fifth Edition Steven T... n ‘Studenis and working professionals wil nd Signals and Signals and Systems Systems with

Signals and Systems

with MATLAB ® Computing

and Simulink ® Modeli

Fifth Edition

Steven T Karris

n ‘Studenis and working professionals wil nd Signals and

Signals and Systems Systems with MATLAB ® Computing ond Simulink ®

and Simulink ® Modeling ~ |e pilslarelog/and dll sil pocessog Fifth Edition opcept andl analog ad egal ie desi Wt

vith numerous practcal examples

This text includes the following chapters and appendices:

+ Elementary Signals + The Laplace Transformation + The Inverse Laplace Transformation + Circut Analysis with Laplace Transforms » State Variables and State Equations + The Impulse Response and Convolution » Fourier Series.» The Fourier Transform + Discrete Time Systems and the Z Transform + The DFT and The FFT Algorithm + Analog and Digital Fiters + Introduction to MATLAB ® + intcoduction to Simulink ® + Review of Complex Numbers + Review of Matrices and Determinants » Window Functions + Correlation

Functions + The Describing Function in Nonlinear systems

Each chapter and appendixcontains numerous practical applications supplemented with detailed instructions for using MATLAB and Simulink to obtain accurate and quick solutions,

Spl a Specs ath MATLAB Computing sa Sain Ming? Fi on

ayy © 202 Olan Piotos A sds mre: aed ne Uninet of

Libeary of Congres Catalogingin Publication Data

itn won aia fo the tay of Conan

'SBN- 13: 978-1-834404-23-Z

ISBN-10: -934404-29-3

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Preface

“This text contains compreienive dacusion on continuous and duct tne signal ad aptems wth mary MATLAB and several Sinulink® examples [tis welten for junior apd senior sfecrcal and computer engineering students, andl lor self-study by working profesional: The prerequisites area basi course in ditlerential ad integral calculus and basi elect ect theory

“This book an he used in a Rh quaer oF one setneter courte, Thi bio ie tau he subjer sateril for many years and was able to cover al mateial in 16 week, with 24 lectute ours pet seek,

To atthe mest cur of tis text tis highly recommended that Appendix As thoroughly reviewed Tis spends serves ar 2 itt to MATLAB, and i Intended for thoee who are Bot fala wath i The Sewent Edicion of MATLAB i an inexpensive, and yet a very powerful software packager ca be fund in aan college bookstores, or exe he obtained deel oan

‘The MathWos™ Inc, 3 Apple Hil Die, Nich, MA 01760-2098,

‘esetibe te crows corrlation and atoortlationfarctions, asd Appendix G presents an cane [s nonlintr ve amllerhesft ckeering[unction

New to the Fifth Edition

The mest notable change ithe addition of Appendixes Fand G All chapters and appends are sesten and the MATLAB seegts and Simulink models ae based on Release R201 (MATLAB, Version 713, Sinulink Version 78.)

“The author wishes to oprer by mantudc to the auẾ of The MathWorks che developers of MATLAR® and Sinulink®, especially to The Mathlors™ Book Program Team, for the srevuragenent and unlimited apport they have povided tae wth durin the prenucion of tie andall ther tems by this publisher

(Our heart thanks also to Ms, Sally Weight, PE, of Renewable Energy Research Laboratory University of Massachisens, Amierst, for bringing some errors and suggestions en previous cion to aur attention

(Orchard Publications

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Table of Contents

JL Elementary Signals

LLL Sion Described in Math Form —

2 The Unit Sep Function sown ans

13 The Unit Ramp Function

[The Dela Fenton

114.1 The Sampling Pope ofthe Bela Fanon

142 The Sitine Popcty of the Bela Faetion

Ls ihr Ones De PUNE 8

2 The Laplace Transformation

1 Definition of the Laplace Transformation

Properties and Theorems ofthe Laphce Translont

ZBL Linearity Property ssn nen

223 Prequeney Shing Proper

326 Dillrentiation in Couples requeney Domain Proper

227 Intention in Tine Donan Proper

228 estan in Complex Frequency Dowain Proper

229 Time Relelein Property sos

2.3.10 Initial Value Theerets cocvoccccccoc co

341 5 Convolution in Conyplex Frequency Donnan Propet

23 The Laplace Tranafonts of Connon Banctons af Tiere

"he Laplace Tranaform of ce Unit Sep Function wy)

“The Laplace Tranlons ofthe Romp Fanlon e0)

The Laplace Transfor of Mg) The Laplace Transforn of the Delta Bunction Bá) co

The Laplace Transfors ofthe Delayed Deka Function 81)

‘Sgnas and Sens with MATLAB ® Comping and Smulnk © Medehing Fit Edition

(Copyright © Orchasd Paton

“The Laplace Teanaorm of « Malt) +

“The Lagbee Tenudonu c te u/0

“The Laglee Trmudonh o sat tụ

230 The Laplace Trans of emot yl

23.10 The Laplace Trfors of © a0" 00) =

2311 The Lplce Tensor of Meee)

24 The Lalce Trantor of Comnon Waves {2141 The Lplce Transor oa Pub

Pa

243 The Laplace Translorm o a angular Walon

244 The Lplae Talons of a RetanalarPerelie Wasco

245 The Laplace Talon of Hal- Rested Sine Wave

25 Une MATEAB ede ae Fst Tine Rs

26 Sonar

27 Ener The Laplace Transl of 3 Sowtowth Feiole Wave

The Laplace Transform ofa Fall- Resid Sine Water

28 Solutions 0 Bobo Chager Ener co co TC

3A The tnvese Laplace Transform tesa

33 Pan Pracion Empanslon

331 Daeimet de

332 Complex Rde

323 Multiple (Repeated Pal

33 Case where Fas Innproper Rational Fur

1 Atemat Melnloi hon Fasion Egan

44 Circuit Tranatocuaton fot Time to CoMglex EtequenEY

ILL Resistive Network Translorsntion

4.12 Inductive Network Teansforuation

4.13 Capacitive Nework Traeformation

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45 Using the Stlink Tansee Fon Block

5 State Variables and State Equations

S.L_Bapresing Difcrential Equations in State Equation Form:

52 Solution of Single State Equations

5.3 The State Transition Mates

54 Coupusation ofthe State Teanstion Mates

5/41 Ddd Eietnalue

5:42 Multiple Repeated) Bucrvalues

55 Bigewectors

5/6 Circuit Anas with Sate Variables

5.7 Relationship beetcen State Bquations an Laplace Tras

© The Impulse Response and Convolution ot

6.1 The lope Response it Tine Domain

(62 Even and Oxdd Functions of Tine

464 Onplief Boluaton ví tịc Coreoluten me z

5 Circuit Anabsis with the Convolation Intel

7 Foe

mm

1 iia

13 Spe tat Ra Sa BE nh hong

[aginst a etac wos nl ue oa Sie

wee,

CConiputaton of Average Power from Fourer Sei,

3 Braluation of Fourier Coefiients Using Excel nn

14 Gralation of Fourier Cofiiens | ing MATLAB

* Siu and Sytem anh MATLAB ® Compating and Simulink ® Modeling, Fh Eton

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Sl Real Tie Finctons

822 tiaginaey Thue Functions

83 Properties and Theorens ofthe Fourier Transfer

83.13 Atea Under Fea) co

83.14 Parevals Theoret sos

184 Fourier Transforms Pars of Common Funecons

84.1 The Dees Function Pair

842 The Constant Fanetion Pir

843 The Gasine Function Pi

844 TheSine Fonction Par

B45 The Sirus Function Pale

846 The Unit Step Function Pair

847 Thee) Function Pate

B48 The (eese,ptugl Function Pa

849 The Gia) Euneloo Pa

$5 Deiadlonoftbe FauderTramsfam tL Toon

86 Fourier Transonse of Consaion Wavelorns

861 The Tension of 0) = Aisgt+Ti=u0" TÌỊ

(863 TheTrandonn of 1 © AtugÐ=ut~2T)

(861 The Travfemh dÝ f0 = AIgftST)+540<

864 The Trnsionm of fly » Acosesiugt*T)—wlt TH]

[865 The Transform ofa Perioc Time Fanetion with Period T

87 Using MATLAB for Bung the EourleeTranelonh of Tine Functions

(38 The Systen Function and Applications to Ceeut Anahsis

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9.22 Shilo Mule inthe Discrete“Tse Dotan

9.2.3 Right Shit in the DsteteTime Donan

9.34 Lait Shin dhe Disrete-Time Dong,

9.2.5 Multiplication by 2 inthe Disctete-Tinne Domain

{6 Multiplication by € "in the Discrete-Time Denna

9.2.7 Mulipiaton by and nin the Discrete-Time Domain

938 Sunsmation in the Discrete-Tite Donat se

9.29 Cameolunoninthe D&erte-Time Domain

9.2.11 fra Value Theoen

2 Final Vale Theorem

‘R Traneform of Connon Discrete Time Functions 93.1 The Transortn ofthe Groote Sentence

3.2 The Transform ofthe Discrete-Time Unit Step Function 93.3 The Transform ofthe Dsctete-Tine Exponential Sequence

93.4 The Transom ofthe DisceteTinte Cosine and Sine Function

9355 The Transforin ofthe Discete-Tine nit Ranip Function

9.4 Computation of the Teansform with Contoue Integration

9.5 Translornation Between = and 2-Domsaine

96 The Inverse & Transior

9.6.1 Patil Fraction Expansion

9.62 The {nverston Integral

963 Lone Din of Foon

9.2 The Transee Function of Discrete-Time Statens

$4 3a Eat fox Dee Tine Sto

Siu and Sytem anh MATLAB ® Compating and Simulink ® Modeling, Fh Eton

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10 The DFT and the FFT Algorithm

10.1 The Discrete Fourier Transform (DFD)

10.2 ven and Oud Properie ofthe DFT

10.3 Comnon Properties and Theorems ofthe DFP

104 The Sanipling Theor

10.5 Nunber of Operations Requted t9 Compute the DFT

10.6 The Fat Fourier Treen (ET)

LT Analog and Digital Filters

1 Fier Types and Classifiations -

Basie Analog Filters

TZU RC Low-Pae Ber

L122 RC High-Poe Filter

1.23 REC Band-Pas Filters

11.24 REC Band-Blaninaton Filter,

113 LowePae Analog Filter Protonpes

TLS Buterwort Analog Loa Past Per Desig

1132 Chebsshew Tipe | Analog Low-PassFiker Design

133 Chebyshew Type If Analog Low-Pas iter Desist

TLS Ellie Analog Low-Pase Fler Desi,

118 High-Pass Bandl-Pas, and Band- Elimination Flee Design

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109 1-10 leat

“10-11 10-12 Woz

lê 10-13 10-16

10-28 lost

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LS Distal Fer,

116 Distal Fier Design with Simulink

116.1 The Direct Form | Realization ofa Dial ier

166.2 The Direct Fort It Realisation ofa Distal Filer

11.6.3 The Sene Fora Realization of a Digital iter E64 The Parallel Fort Realisation of « Dintal Filter 1.65 The Digital Fier Design Block cớ

[Ad Polmoial Construction Iron Krom Root oo

‘AS Braluaton ofa Polynonial at Specified Vales,

A.G Rational Polonia

AT Using MATLAB to Make Pits,

‘A9_ Muluplication, Dison, and Exponent,

ALLO Scriptand Function Files

MATLAB Computing

Fags AS oA, 8-11, AB ALB, ACTS

APBLACDI,A23, An}, A

MATLAB Computing

Page Bot

Siu ant Sysuyns with MATLAB ® Cơnjdúng and Sali ® Mating, Fifth Baton

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Paes BT, B12, B14, B18

CA Review of Complex Numbers cs

Cl Definition of Coniplex Nunes

C2 Addition and Subtrasicr of Comps Nutaber

£3 Multiplication of Complex Nombers,

Cat Division of Complex Nunhben,

Esgoncntsland Polar Rorteof Conglx Nonbcre

BLL Solution of Stwultanenae Eauations wil Matrices

E.1 Window Rincon Defined ¬

3 Comman Wimlue Finclone E1 E21 ReesweelerWindow Eondtoe =3

£22 Trangulae Window Function ES Sal and Syens with MATLAB ® Compating and Sian © Madea ih Ee “

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E23 Hanning Window Function

E24 Hannning Window Function

E4 Feoner Soiex Mehod or Apptueinating an EIR Arpt

Chapter 1 Elementary Signals

(Other fora ofthe unit step function are shown in Figure 18,

Ths, the puke of Figure L9G) isthe sum of the unit sep fumed

seis represented a (0= mát sof Figures 1906) and 1910 and

‘The unitstep function offers a convenient metho of describing the den applition of voltage

tr cute ute For example, conatane voltage souree of 24 V applied a = 0, eat be dence

as 24ytt) V Likewise, sinsoidal oltage source ¥(2) > Vgeont ¥ that isapplied toa cect at

iv ean be describes as VQ) = (Vyeosentut-ty) V- Also ithe exitation in a circuit isa rectangular angular, or saute, oe anyother seursng pulse, teas be represented a3 sunt dilecence) of unt tep fanetions

14 Sicals and Sotens with MATLAB ® Computing and Simink © Modeling, AR iin

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Higher Order Delta Functions

8 Todraga pot along the ams we sels that point, ad we ll deen the Shale key wie drag

9 When we select line segienton the tne axis (x-ais) we observe that a che lower el of the waveform clplay window the Left Poi and Right Point 1

reshape the given saelor by specfring the Tine CP) and Amplitude (¥) points become vile, We ean then

The te postive spikes that exer at = 2, and ¢ = 7 ae cealy shown in Figut 1.26

MATL

the names of the mathematicians who used thet in their work The unit te Function u(t) is AB hs huis functions fo dhe unit step, and the delta feta, These are denoted by

‘elered to as Heaviside), al he delta function 84) ls eee o a Diath Tlie use i ie sys k at% Define symbolic variables

UPk*=ym(Hoaviside(tal}% Create unit step function att = a

The MATLAB heaviside lun

Gunetions as trated in Figures

sap in Figure 13

be use to plo the uni step, unit imple an unit ramp

the sit inp in gute £39, ad the unit

‘Chapter 2 The Laplace Transformation

2.2.3 Frequency Shifting Property

The ñepeno hing propery sates that fe apy a ne domain function ft) byan exponen Wal fonction €™ where a 6am ahirary pole constant, this saulipation wl produess MU

the arable nthe comple frequency dena by writs Thus,

A change of sai represent by mhiplention ofthe time variable hy a postive sealing lator

1, Ths the function ft) ater caine the ie ais, besos f(a)

andleting t= #/a, we obtain

seaman = [noe aE) = 1ƒ re 224G)

Note 3:

[Generally the intl value of f(t) stalin at 1 = 07 to include any dlscontinnity at maybe pre cntat t= 0, five knoan that nosich dacontnnity exists at ¢~ 0°, we snp interpret FU") 38 10)

2.2.5 Differentiation in Time Domain Property

ty stater that difereniatin inthe time cain core

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‘Syanvion melod is explained thereuhly an ti ikastrated with several examples

3.1 The Inverse Laplace Transform Integral

“The Inwene Laplace Tiaifoen Iceval vn arta in the prevous chapter: ite rpeatel hee or

“hủ tncegeal leu wo evaluate because It rules contour integration using cnnplex variables theory, Fortunately, fr nos engineers probleie we can refer to Tables of Proertes, and Com tom Laplace tanaforn pr to fookyp the Inverse Laplace trafic

3.2 Partial Fraction Expansion

se form, but in moe cases

(Quite often the Laplare transform expressions are notin son

Ina proper rational function, the roots of Nés) in (3.3) are found by setting NOS) ~ 05 the

lt the su of F(2) The roots of Ds), fund by setting Dts} = 0, ate calle the ple of CS)

‘We asm that F(2) in G3) isa proper rational uneion Then, t ẽcteoMary and very conve

lent to make te coeisint of 3? unin thus, we teste Fes) a8

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‘We observe that there is pole of miulrplicity 2 at ¢ = 1, and thus in partial inaction expansion form, Fs) ie written as

(Chapter 3 The Inverse Laplace Transformation

(Check with MATLAB,

Se Coetcents of (+ 2) term i DIS)

% Muliptes polynomials d and 22 to exprass the

°% denominator Dis) of Fs) a8 & polynomial

Figs 4.10, Poot vag (W) fore dc of Engle AS

4.2 Complex Impedance Zis)

where the initial conditions ace

and defining the ratio V,t8)/16) as 28) ie obtain

48 Sigal and Stans with MATLAB ® Contin and Smlink ® Medlin, Fth Etim

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(Using the Simulink Transfer Fen Block

\W=:10:10000; Gs=-1.2 5.°10.%-6) -w?2-5.)10.%-3) ~we5)

‘emiogx(wabs(Gs) xabel(Racian Frequency w/; ylabei Vout)

tile( Magnitude Voutvin vs Raglan Froquency) gd

The pet is shown in Figure 4.22 We observe that the gven opamp cru is second onde on pas filter

ve cut fegensy (FAB) osc a abot 78 #7,

Magrtude Voutvin , Red Frequency

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8 Axebe G19

Rị ¬ ` tì safe R A=|9 6 1 OL 0001 eal, ks pall andy = ay

ate viable

‘We ean sla atthe sate elon nel iom gieh lute We cies th

represent inductor currents al eapacitor voltages [nother words, we assign state variables to nena toring devices The exanpleshelew leas he broeslue

Taample 53

Write tate equation() for the cru of Phyure 5.2, ven that ve(0") =O, and y(t) thế

Fell) = Saul)

Figure $2, Che fr ample 5.3 Solutions

This crew contains only one enenm-sorln SG Thasele ire ee al ote

ahle, We choose the sate sriahleto denote the wltage across the capacitor a shown Sn

54 Signal rnd Stans with MATLAB ® Computing nd Silink * Mideling Fifth

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‘Computation of the State Transition Matrix

We use as many equations asthe nuniber ofthe egerwales, and we soe fr the set

4, Wesubsierethe s, gocfilent im the sae anton mate of 8.54), and we simpli

‘Weill ue MATLAB rootap) funtion to cba the

pelt -6 11 -6} eroois(p): pant in): pint lamba = %68 2", 1)

5201 2) pret(lambdad = 55 2F, 3) lanbdai = 2.00 lanbda2 = 2.00 3anbdai = 3.06

and thus the eigenvalues ate

aye G59)

2, Since A tea 353 mann, cu th is

Signals and Ssrems with MATLAB F Computing and Siudink ® Madeling Pith Eon s8,

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Chapter 5 State Variables and State Equations

‘We can obtain the plot of Figure 5.8 with the Saul State-Space

thon a the inp sing the ourput displayed om the kwnh the iste Fan

Chapter 6 The apute Response and Convolution

yt = Teh “ha giết = Y ha tác on

6.4 Graphical Evaluation of the Convolution Integral

‘The convolution inte 1s more contenienty evaluate by the graphical evaluation, The procedure Inet tated wth te fllowing examples

68 Sign nl Stems with MATLAB * Conguting and Sink ® Malling, ih ion

Copii © Onan Palins

gute 0.33, Borman of w(t) or Bea 67

618 “Seneb sml Sen eúi MATLAB ® Computing ad Simulink ™ Malling ith Ein

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Evaluation of the Coefficients

Aloo ifm and w ate dlferen integers then,

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“Trigonometric Form of Fourier Series for Common Waveforms ies willbe present since this waveform has alo lall-wave mmr However, we will compute al

-clcent to ef this, Ako foe bret, well ance tt =

A Giang -0~ sinn2e + sine) = 2 2sina~ sino2x)

are snoe iin intent Celie neste) rane the eta rade the ponents on these ond Fine of (7.19) ace se and therefore alla cotliclents ae ser, ae expected tice the ue

The, coefficients ate fa rom (7.19, Page 7-0, hat,

cua]>Re-e- Rao đo

a expected, since the square waveform as half-wave spinner

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© Oral Plains

Chapter7 Fourier Series

Fate 2:30, Walon retin (80)

Xi 4 rkneotary presentation ofthe capac

ve fweakd ascent number of

The waver of Fie

ite 1.27 However, fil in

gore 731 Similnk model for the sot of ge

78 The Exponential Form of the Fourier Series

Nel [ imasinary | Comores [fren [Os

Fatt) ~ Ef UF torsinar + Fislsosouite 639)

‘We crv thatthe integrand of (6.38) is eo since it ia dl unto wie repose tb esas Desh products nse the backers ate ol functions

‘Therefore 0,1 = O.that i, A) real

‘Accondingls, we ean state that a necessary und sufficient condition fur Q) to be wal, that

Fourier Transform Pairs of Common Functions TABLES FourterTansform Properties and Thcorms

TTT ff soar 7 Oe fe HHỜNM

rae acre Po mm

KG Tơ) Fier Rwy

Faia Cama — a Ta Thesim

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