Tài liệu signals and systems with matlab computing and simulink modeling docx

Students and working professionals will find Signals ond Signals and Systems Systems with MATLAB ® Computing and Simlink ® : © Commuting | Modeling Fouth Edfion to be a concise and easy

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—_

Signals and Systems

with MATLAB ® Com ting

and Simulink ® Modeli

Fifth Edition

Steven T, Karris

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Students and working professionals will find Signals ond

Signals and Systems Systems with MATLAB ® Computing and Simlink ®

: © Commuting | Modeling Fouth Edfion to be a concise and easy to-eam with MATLAB ® Computing | tog itprvides complete cleer end deals explenatons and Simulink ® Modeling ~ |) ertne pieipalancleg and dicta sonel processing

Fifth Edition ‘concepts and analog and digital fiter design illustrated

with numerous practical examples This text includes the following chapters and appendices

+ Elementary Signals + The Laplace Transformation + The Inverse Laplace Transformation + Circuit 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

Filters + Introduction to MATLAB © ~ Introduction 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,

ISBN 978-1-98/404.28-2

Visitus on the Intemet www orcharpublieatons.comor email us: Info@orchardpublicatons com

9l7ã1934 leDa2321 |

ISBN-13: 978-1-934404-23-2

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Sigal and Stems with MATLAB® Conon ad Sink Medel, ath ico

‘thou the peor saben peeaisin of the pbs

Ti ll inquses to Orchard Publktins infdtonhandpblitionssonn

MalladsPt lạc Thọ se vu sub ly cm xen and exanaton,wiehou aero ge tralemarks of the Mictosof™ Corporation and The Library of Congress Cataloging-in-Pablication Data

Catalog con ene ra the Library of Conese

brary of Congress Cnt Nonier: 2011939678

ISBN-13; 978-1-934404-23-2

ISBN-10; 4-934404-23-3

Copwricht TXu 1-778-061

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Preface

‘This text contains a comprehensive discussion on continuous and discrete tine sstals atl syste with many MATLAB® and several $i

clectrical and computer engineering students, and for self-study by worhi

prerequisites area basic course in diiercritial and integral vale, and hast electric eiouit theory

anulink® examples, Te is written for jumior and senior

professionals The This book ean be Usel in two-quarter, or oe semester course, This author has taught the subject

lecture hours per

; Appenslix E contains a comprehensive dis describes the cross correlation and autocorrelation funetions, and Appendix G presents at exausple

function

ais theo ion on window fametions, Appendix F

‘ofa nonlinear systeny and derives its describ

New to the Fifth Edition

The mest notable change is the addition of Appendives F and G All chapters and appendises are resvrtten and the MATLAB scripts and Simulink towels are hase on Releate R2011b (MATLAB, Version 7.13, Simulink Version 2.8)

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The author wishes to express his gratitude to the staff of The MathWorks, the developers of MATLAB® and Sinulink®, especially to The MathWorks™ Book Program Tent tor the encouragement and unlimited support they hase provided me with during the production of this and all sther texts by eis publisher

(Our heartielt thanks also « Ms Sally Wright, PE, of Renewable Energy Research Laboratory University of Massachusetts, Amberst, for bringing some errors and suggestions on a previous edition to our attention

Orchant Publications

‘raealpublications.com

‘nfo@orchardpublications com

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Table of Contents

LLL Signals Describes in Math Forms ra L3 The Unie Step Function 2

14.1 The Sampling Property of the Dela Function bit 14.2 The Sitting Peoperty af the Delta Function

2 The Laplace Transformation

Definition of the Laplace Teansfomsation

Properties and Theoreis of the Laplace Transtortn

2.1 Linearity Property Tine Shitting Property Frequency Shifting Property Sealing Property

Dilferentiation in Tite Domain Propesty Ditierentiation in Complex Frequency Domain Property lnteg

28 Inteyration in Complex Frequency Domain Property

9 Tiove Periosicity Property

10 tnitial Valse Theorems 2.11 Final Value Theorem

4 The Laplace Transforns of the Delta Function (1) EU 23.5 The Laplace Tratisforts ofthe Delayed Delta Function ð(t 2 Signals and Sytems ath MATLAB ® Computing and Simulink ® Modeling

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23.6 The Laplace Transforn of “uy

38 The Laplare Transom of snout The Laplace Transfors of te i

39 The Laplace Transloris of sos mt

2.3.10 The Laplare Transtonis of © “snot v0)

23.11 The Laplace Transfony of « Moose! uy()

2.4, The Laplace Tratsfonts of Connon Wavetons

24.1 The Laplace Transform of a Pale

24.2 The Laplace Transfornt of a Linear Sestnent

3/43 The Laplace Transior of a Triangular Wavetors

244 The Laplace Transiorn of a Rectangular Periodic Wavetorn

2.45 ‘The Laplace Transloet of a Hali-Rectified Sine Wavetorn

Using MATLAB for Finling the Laplace Transioens of Time Functions

Sususay

Exerciscs

The Laplace Tranatoee of s Sawtooth Periodic Wavetorin

The Laplace Tranatoroi ofa Pull-Reciied Sine Waveform

Solutions to End-ot-Chapter Exercise

‘The Inverse Laplace Transform,

3.1 The Inverse Laplace Transiorm Inteural

3.2 Partial Fraction Expansion,

3.2.1 Distinct Poles

Complex Poles

3.2.3 Multiple (Repeated) Poles

3 Case wliere Fis) is Improper Rational Function

4 Altesnate Method of Partial Fraction Expansion

Signa cin Systems with MATLAB ® Computing and Simulink ® Madling, Eh Eilitiom

Coprright © Orturd Publications

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45 Using the Simulink Transier Fon Block

F State Variables and State Equations

5.1 Expressing Differential Equations in State Equation Form

5.2 Solution of Single State Equations

5.3 The State Transition Matrix

5.4 Computation of the Stare Transition Matrix

5.4.1 Distinct Bigenvalcs 5.4.2 Multiple (Repeated) Bigerwalucs

55 Bigenvectors

5.9 Circuit Analisis with Stare Varinhles

5.7 Relationship hetween State Equations and Laplace Teanstorm

G The Impulse Response and Convolution

6.1 The Impulse Response in Time Domain

6.2, Byen ang Odd Punetions of Time:

63 Convolution,

(64 Graphical Evaluation of the Convolution Integral

65 Circuit Analysis with the Convolution Integral

Signals and Sytems ath MATLAB ® Computing and Simulink ® Modeling

Consight © Orshard Publications

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7 Fourier Series

TA Wave Analysis

Exaluation of the Coefficients

7.3 Sitntetay itn Trigonmetric Fourier Series

73.1 Ssmmezry in Square Waveform

7.3.2 Ssmmezey in Square Waveforon with Ordinate Axis Shite

7.33 Ssanmetry in Sawtooth Wavetor

734 Ssmmerry in Triangular Waveiorm

7.35 Soumetry in Fondawsental, Second, and Thisd Hasionies

74 Trigonometric Fors of Fourie Series for Common Waveforms

TALL Teigonometric Fourie Series for Square Wavelorm

JA.2 Tngonometnc Fourice Series tor Sawtooth Wavefo

TAB Trigonometric Fourier Seties for Triangular Waseforns

TA44 Trigonometric Fourier Series for Half Wave Rectifier Waveform

TAS Trigonometric Fourier Series for Full-Wave Rectifier Waveform

7.5 Gibhs Phenomenon

7.6 Alternate Forms of the Trigonometric Fourier Series

1.7 Circuit Analysis with Trigonomettic Fourier Series

7.8 The Exponential Forus of the Fourier Series

7.9 Svinnetry in Exponential Fourier Series

7.1L Computation of RMS Values from Fourier Series

7.12, Computation of Average Power trom Fourier Series

7.13 Bxahiation of Fourier Coeflicients Using Excel

2.14 Bsaluation of Fourier Coefliciens Using MATLAB®

8.1 _Deinition and Special Fors 54

82 Special Forins of the Fourier Transform 82

w Signal and Sostems with MATLAB ® Conpating and Sinunk ® Madelng, Eth ition

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3

testis and Theorens of the Fourier Transtar 89

Tinne Seal

Time Shitting Frequency Shitting Time Dilfecenciation Frequency Dillerentiation, Tame Integration

8.3.9 Conjugate Tine and Frequency Functions

84 Fourier Trandom Paice of Common Functions

84.1 The Delta Function Pair

84.2 The Constant Function Pair

843 The Cosine Function Pair

3⁄44 The Sine Function Pair 84.5 The Signum Function Pair 84.6 The Unit Step Function Pair

8.5 Derivation of the Fourier Transform from the Laplace Transforus

8.6 Fourier Transforms of Common Wavelorns

8.6.1 The Transom ðf = Alu/t+TJ=ult- T)Ị 8.6.2 The Transform of 11) = Alo/0 - 01-211

63 TheTrmvfotnef Hộ = Aluit*T1tdg0—vi =A 20] c8 28

&64 The enfonecf Nụ = Aeea,fogL+) nút ĐỊ =

865 The Trai oa Peele Tne Function wih PeiodT =

&66 TúeTiaviotacftiePoáele Time Fandio tụ = 4 Ÿ 8u.) „8 30 8:7 Using MATLAB fr Fg the Four Tandon of Tiác Rivtoie

cuit Analisis, 84

88 ‘The System Function and Applications to

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9.2.2 Shit of thonalul in the Discrete-Time Dowsait

9.2.3 Right Shift in the Discrete-Time Domain,

9.24 Lelt Shift in the Diserete-Time Domain

9/35 Mulkiplcation by a in the Disorete—Time Domain

9.2.6 Mulkiplication by &**" in the Discrete-Time Domain,

9.2.7 Muleiplcation by andl n! in the Discrete-Time Domain

9.2.8 Summation in the Discrete-Time Detain

9.2.9 Convolution in the Discrete-Time Domain,

9.2.10 Convolution in the Diensete-Eeluenoy Doruaim

9.2.11 Initial Value Theorem

9.2.12 Final Value Theorem,

9.3 The % Transform of Conon Discrete-Time Functions

9.3.1 The Transfortu of the Geometeie Sequence

9.3.2 The Transori of the Discrete- Tite Unit Step Function

9.3.3 The Transtort of the Discrete-Time Exponential Sequence

9.34 The Transtoruu of the Discrete~Time Cosine and Sine Functions

9.35 The Transort of the Diserete—Tiaie Unit Ramp Fanetion,

9.4 Computation of the % Transform with Contour Integration

9.5 Transformation Between s= and z-Domains

9/6 The Inverse X Transion

8.46.1 Partial Fraction Expansion

9.6.2 The Inversion Integral

9.6.3 Long Division of Polytomials

98.7 The Transler Function of Discrete-Time Systems

9.8 State Equations for DiseveteTime Systems

Signa cin Systems with MATLAB ® Computing and Simulink ® Madling, Eh Eilitiom

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Simulink Modeling

Page 9-40

Excel Plots

Pages 9-32, 9-40

LO The DFT and the FFT Algorithm

IO.L The Discrete Fourier Transform (DET),

10.2 Even and Odd Properties of the DFT

10,3 Common Properties and Theorets of the DET

103/1 Lineacity 103.2 Tite Shit 10.3.3 Frequency Shit

103.4 Tinie Convolution 103.5 Frequency Coneoluion

104 The Sampling Theorem

10.5 Nutaber of Operations Required to Compare the DFT

19.0 The Fast Fourier Transforta (FFD,

AL Analog and Digital Filters

LIAL Filter Types anid Classifications,

11.2 Basic Analoy Filters

11.2.1 RC Low-Pass Fer 1.2.2 RC High Pass Filter 11.23 RLC Band-Pes Bilier 1.24 RLC Band-Elimination Filter 11.3 Love-Pase Analog Biter Pootorvyes

113.1 Boterwosth Analoyg LowPacs Filter Design 113.2 Chebyshev Type [ Analog Low-Pass Filter Desizn 113.3 Chebyshev Type Hl Analog Low-Pase Filter Design|

11.3.4 Bllipgic Analow Low—Pase Filter Design

114 High-Pass, Band-Pass, and Band-Elimination Filter Design

Consight © Orshard Publications th Edition

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164 The Parallel Fort Realization of a Digital Filter H72

1165 The Dighal Filter Design Block tt

Ad Polynomial Construction from Known Roots A4

AS Evaluation of a Polynomial at Specitied Values AS

A9 Mulhipleaten, Dtdon, and Esponentiation ACT

MATLAB Computing

Pages A-3 through A-D, ALL, A-13, A-15, A-I6,

A-D0, A-1, A-23, A~26, A-27

tals aad Systoms with MATLAB ® Camping a Simin ® Meveling, Fifth Eilition

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Simulink Modeling

Pages 8-7, B-I2, B-14, BIS

CA Review of Complex Numbers

C.1 Definition ofa Complex Number,

C.2 Addition and Subtraction of Complex Nurabers ° C3 Multiplication of Complex Numbers é

°

e

C4 Division of Complex Numbers

C.5 Exponential and Polar Forms of Complex Numbers

D_ Matrices and Determinants

Dil Matrix Definition,

D2 Matrix Operations

D.3_ Special Foenis of Matrices

D4 Determinants

DS Minors and

Dio Cramer's Rule

D7 Gaussian Elimination Method

D8 The Adioint of a Matrix,

D9 Singular and Non-Singular Matrices

D.10 The Inverse ofa Matrix

iLL Solution of Situltancous Equations with Mattices

E Window Functions

E.L Window Function Detined

E2 Commision Window Bunctions

EZ Rectangular Window Punction E22 Triangular Window Function

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E23 Hanning Window Function

£24 Hamming Window Function E25 Blackman Window Function E.26 Kaiser Family of Winslow Functions E3 Other Window Functions

4 Fourier Series Method lor Approximaning an FIR Annplituce Response

tals aad Systoms with MATLAB ® Camping a Simin ® Meveling, Fifth Eilition

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Chapter 1 Elementary Signals

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

“AC -AMCLST) -Au 1)

Figure 18, Other rs ofthe unt ste fntton

Unit step fimotions can he used to represent other time-varying functions such as the rectangular pulse shown in Figure 19,

The unit step lunetion offers a convenient mietliod of describing the siden application of a voltage

‘or eurvent sauce, For example, a constant voltage source of 26 V applied at t = 0 can be denoted

as Muy(t) VI, Likewise, a sinusoidal voltage sourre v(t) = Veost V that ie applied toa cout at

1 > ty, cam be described as v(t) = (Vj c0s@t}uy(t—ty) V- Also if te excitation in a cieuit isa

teetangtlar, or triangular, or sawtooth, or anyother recurritw pulse, it ean he represented as a sui (iffeence) of unit sep lunetions

14 nals and Sater with MATLAB ® Computing and Simslink ® Modeling, Fifth Elton

Capsright © Onsared Pbleatons

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Higher Order Delta Functions

8 Todeaga point along the x-axis, we select that point, and we hold down the Shift key while dag

syms k a':% Define symbolic variables

U=K*sym(Heaviside(ta))% Create unit step function at

The MATLAB heaviside function can be used! to plot the unit step, unit impulse, and unit ramp

in Figure 1.30, and the unit

Signals and Systems with MATLAB ® Compacting and Simudink ® Model

Coparight © Orchard Publeations Fifi Editon 119

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Chapter 2: The Laplace Transformation

2.2.3 Frequency Shifting Property

‘The feynoncy shifting jsoferty tates that iTwe multiply a tinue dowain function f(t) by at exponen

rial imecion ef where a isan arbitrary positive constant, this multiplication will predluce a shilt

of the s variable in the complex recuency domain by a units Thus,

eye F(s+ a) (14)

⁄t£“RÓ) = fretane ta = fine? a = rere)

Note 2:

A clianige of scale ts represented by wsultiplication ofthe tite variable ( by a positive scaling lactor

4, Thuis the funetion f(t) after scaling the tinte axis, becomes fat)

and Ìetling E= 9/4, e ehtain

2.2.5 Differentiation in Time Domain Property

‘The diffeentiation in tine domain property states that differentiation in the time domain corte spond to multiplication by i the complex frequency domain, minus the intial value of RQ) at

1 = 0, Thus,

24 ‘Siguals and Sutems with MATLAB Ÿ Computing and Sindink ® Modeling, Fit Edition

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Laplace Transforms of Common Fanctions of Time

|ð [Time Priodiey f(t+nT) on J ioe “a vn

10 [onal Ve Tico tin FO Be lim SPC) = 0)

we

AL | Final Value Theorem Tim fc) bo vi lim sF(s) = (==)

| Tine Comin en Fore

B[ Fane Gonolaren — [q00 a5 FuwFas) pers

Signals and Systems with MATLAB © Computing and Simulink ® Modeling, Fifth Edition 213

`

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Chapter 3

‘The Inverse Laplace Transformation

his chapter tsa continuation to the Laplace transioemation topic ofthe previous chapter and presents several methods of finding the Inverse Laplace Transformation, The partial fraction, expansion mith is explained thoroughly and iis illustrated with several examples

3.1 The Inverse Laplace ‘Transform Integral

The Inverse Laplace Transform Integral was stated in the previous elapters tis repeated here for

3.2 Partial Fraction Expansion

Quite often the Laplace transform expressions are not in recognizable forts, but in tiost eases appear itv a rational form of Seat is,

Ina proper tational function, the roots of N(s) i (3.3) are found by setting N(s) = 0; these are called the zac of F(s) The roots of Bs), found by setting Dis) = 0, are called the pues of F(s)

‘We assume that F(s) iy 3.3) iea proper rational function, Tlen, itis customary and very cone riient to make the eoeicient of unity; thus, we eowrite PS) ác

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Partial Fraction Expansion

‘The value of the residue ess can also be found without differentiation as follows:

Substitution ofthe already known values of rand ry into (3.44), and letting s = 0°, we obsain

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Chapter 3 The Inverse Laplace Transformation

(Check with MATLAB:

3 Muliplies polynomials dt and d2 to express the

% denominator D(s} of F(s) as a polynomial Irp.KIEresidue(Ns Da)

We observe that there isa pole of wuliplicty 3 at š = =1, and a pole of multiplicity 2 at s = 2

Then, in partial fraction expansion forms, F,(s) is written as

jp

The resichtes are

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Chapter 4 Circuit Analysis with Laplace Transforms

Figure 4.10, Plot of Yay (0) forthe eit of Example 4.3

4.2 Complex Impedance Zis)

Consider the domain RLC series circuit of Figure 4.11, where the initial conditions are assured to be 2et,

and delinins the rao V,(S)/1(S) as 2¢3), 9 obtain

48 Signalsand Systems with MATLAB * Compasing and Simink ® Madeling, Ff Edin

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‘Using the Simulink Transfer Fen Block

W=1:10:10000; Gs=~1.(2 6"10.A-6) "w22-8 "J*10.^(-8) "w+8);

‘semilogx(w.abs(Gs)); xlabel(Radian Frequency w), vlabel(|VoutVin[),

tile Magnitude Vout/in vs Radian Frequency’; grid

“The plot is shown in Figure 4.22, We observe that the given op amp circuit isa second order low pass filter whose cutoff frequency (-3 4B) occurs at about 700 £/s

Magnitude Vout/Vin ve Racin Frequency

GO" VAG) RICK FURR TC GRC] |

Signals an Stems with MATLAB ® Computing and Simulink ® Modeling, Fifth Edition +

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‘Chapter 5 State Variables and State Equations

This citi contains only one ened

state variable, We choose the state variable to denote the wt storing device, the capacitor Therefore, we need only one arrose the capacitor as show in

54 ‘Signals and Systems with MATLAB ® Computing an! Sonadink ® Moaeling, Fi Edition

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Computation of the State Transition Matrix

"xổ ta, yet = eM

“(|

‘We use as many equations as the number of the eigenvalues, and we solve for the coefficients

4, Wesubstitute the a, coeilicients into the state tsansition matrix of (5.54), and we snap Example 5.7

Compute the state transition mattix ¢™ given that

ancl expansion ofthis determinant yields the polynomial

‘We will use MATLAB roots(p) function to obtain the roots of (5.57)

p={1 -6 11-6}; rroots(o: forint ìn); fprinti(lambdat = 96.21, r(1)

frin(lambda2 = %5:24' (2); fprint(lambdas = %5.2F,r3))

Jambéat = 3.00 lambda2 = 2.00 lanbda3 = 1.00

andl thos the eigenvalues are

2, Since A isa 33 matris, we use the first 3 terms of (5.54), thạt

Signals and Sens with MATLAB ® Coputing aed Sinalnk ® Madetng Fifth Eaton 53

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‘Chapter 5 State Variables and State Equations

We can obtain the plot of Figure 5.8 with the Simulink State-Space block with the unit step func tion as the input using the Step hlock, and the capacitor voltage as the output displayed on the Scope block as shown in the model of Figure 5.9 where for the State-Space block Function Block

naranicters dialog box we have entered

‘We compute the eigenvalues from

5-26 Signals snd Systeme with MATLAB ® Computing and Simulink ® Modeling, Eifh Ein

Coppi

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‘Chapter 6 The Impulse Response and Convolution

so = J “eX *bucride = of “ebuceyee (62)

6.4 Graphical Evaluation of the Convolution Integral

The convolution integral is more conventently evaluated by the graphical evaluation, The procedure

is beat illustrated with the followin exaunples

where + ia duasny variable, thats, u(x) and! f(s), ate conserel tobe the same as ult) an

1) We form u(t—t) by fiest constructing the itiage of u(t) this i shown as a(-s) in Figure 6.7

Figuce 6.8 Formation of u(t—t) for Example 64

68 Signals and Sytems with MATLAB ® Contin and Sinking ® Mein, Bifth Eton

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‘Chapter 6 The Impulse Response and Convolution

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Evaluation of the Coefficients

Also ifm and m are different intewers, then,

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‘Trigonometric Form of Fourier Series for Common Waveforms ics will be present since this wavelorm has also half-wave sywnmetty However, we will compute all coetficients to verify this Als, tor brevity, we will assume that «2

Figore 712, Square waveform as ol fein

‘The a, coefficients are fund from

and since n isan integer (positive or negative) or seto, the terms inside the parentheses on the see-

o, as expected since the square

cond line of (7.19) are sen and therefore, all a, coelficients at

waveform has odd symmetry Also, by inspection, the average (DC } value is 2et0, but if we attenspe

to verte this using (7.19), we will obtain the indeterminate forns 0/0 To work ateund this prob-

“The b, soefkientx are fount from (LM), Page 7-6, thats

by 2 [ff asinnta= fe aysionat = A Ceosmiis ~ cosm?*)

ĐỀ Ị 20

+ ung — chưng) = Â (I-Begnx cam) For n = even, (7.20 vekls

+= Aaron

as expected since the square waveform has half-wave symmetry

Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fifth Eiition TAL

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Chapter 7 Fourier Series

7.8 The Exponential Form of the Fourier Series

The Fourier series ae often expressed in exponential form, The advantage ofthe exponential foron

ro performs one integration rather than ‘wo, one for the a, andl another fo

into A(t) Thus,

Signals und Sytems with MATLAB Computing and Simalink ® Modeling, Fil Edition

c 7-30

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Chapter 8 The Fourier Transform

main Canerponence (Completed Ta

and Fig(-O) = =FjmlA) — T0) = Real (8.32)

Now, if F(@) of some function of tine f(e) és known, and Fh) issuch that F(-@) = F*(@), can

we conclude that ((0) is real! The answer is yes; we ean very this with (814), Page 8-3, which is repeated liere foe eonvenience

fat) ~ Ef UF snot + Fi ycorenide (835)

We olerve that the integrand of (8.35) i eto since its an edd function with respect to @ erase

both prolactin the bracket ate oak fonetions

Therefore, fg) = 0, thats (4) is real

Accordingly, we can state that « necessary and sufficient condition for f(t) to be teal, és that FC) = Fre)

má

ve id a Fg iene and Fig) Wo

88 Siqualsand Systems with MATLAB ® Computing and Simuink ® Madeling, if Edin

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Fourier Transform Pairs of Common Functions

‘TABLE 8.8 Fourier Transform Propernes and Theorems

NỈ Time Shatin s Tw to Flole Fa

Tim Tin tay 7 gar"Frey )

a Frequency Dileoutiation auto N

inne Conon DI) Fo Fe)

Frequenay Common | TF) Le cerrst

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