signals and systems with matlab computing and simulink modeling - steven t. karris

1.2 The Unit Step Function A well known discontinuous function is the unit step function * which is defined as 1.4 It is also represented by the waveform of Figure 1.3.. Waveform for E

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Steven T Karris is the president and founder of Orchard Publications, has undergraduate and

graduate degrees in electrical engineering, and is a registered professional engineer in California and Florida He has more than 35 years of professional engineering experience and more than 30 years of teaching experience as an adjunct professor, most recently at UC Berkeley, California

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

Each chapter contains numerous practical applications supplemented with detailed

instructions for using MATLAB and Simulink to obtain accurate and quick solutions.

with MATLAB ® Computing

and Simulink ® Modeling

Third Edition

M Mooddeelliinngg,, TThhiirrdd EEddiittiioonn, to be a concise and easy-to-learn text It provides complete, clear, and detailed explanations

of the principal analog and digital signal processing concepts and analog and digital filter design illustrated with numerous practical examples.

Signals and Systems

Third Edition

Steven T Karris

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publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher.

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Library of Congress Cataloging-in-Publication Data

Catalog record is available from the Library of Congress

Library of Congress Control Number: 2006932532

ISBN−10: 0−9744239−9−8

ISBN−13: 978−0−9744239−9−9

Copyright TX 5−471−562

This text contains a comprehensive discussion on continuous and discrete time signals andsystems with many MATLAB® and several Simulink® examples It is written for junior andsenior electrical and computer engineering students, and for self−study by working professionals.The prerequisites are a basic course in differential and integral calculus, and basic electric circuittheory.

This book can be used in a two−quarter, or one semester course This author has taught thesubject material for many years and was able to cover all material in 16 weeks, with 2½ lecturehours per week

To get the most out of this text, it is highly recommended that Appendix A is thoroughlyreviewed This appendix serves as an introduction to MATLAB, and is intended for those whoare not familiar with it The Student Edition of MATLAB is an inexpensive, and yet a verypowerful software package; it can be found in many college bookstores, or can be obtained directlyfrom

the Discrete Fourier transform and FFT with the simplest possible explanations Chapter 11contains a thorough discussion to analog and digital filters analysis and design procedures Asmentioned above, Appendix A is an introduction to MATLAB Appendix B is an introduction toSimulink, Appendix C contains a review of complex numbers, and Appendix D is an introduction

to matrix theory

New to the Second Edition

This is an extensive revision of the first edition The most notable change is the inclusion of thesolutions to all exercises at the end of each chapter It is in response to many readers whoexpressed a desire to obtain the solutions in order to check their solutions to those of the author

Another major change is the addition of a rather comprehensive summary at the end of eachchapter Hopefully, this will be a valuable aid to instructors for preparation of view foils forpresenting the material to their class.

New to the Third Edition

The most notable change is the inclusion of Simulink modeling examples The pages where theyappear can be found in the Table of Contents section of this text Another change is theimprovement of the plots generated by the latest revisions of the MATLAB® Student Version,Release 14

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1 Elementary Signals 1−1

1.1 Signals Described in Math Form 1−11.2 The Unit Step Function 1−21.3 The Unit Ramp Function 1−101.4 The Delta Function 1−111.4.1 The Sampling Property of the Delta Function 1−121.4.2 The Sifting Property of the Delta Function 1−131.5 Higher Order Delta Functions 1−141.6 Summary 1−221.7 Exercises 1−231.8 Solutions to End−of−Chapter Exercises 1−24

MATLAB Computing

Pages 1−20, 1−21

Simulink Modeling

Page 1−18

2.1 Definition of the Laplace Transformation 2−12.2 Properties and Theorems of the Laplace Transform 2−22.2.1 Linearity Property 2−32.2.2 Time Shifting Property 2−32.2.3 Frequency Shifting Property 2−42.2.4 Scaling Property 2−42.2.5 Differentiation in Time Domain Property 2−42.2.6 Differentiation in Complex Frequency Domain Property 2−62.2.7 Integration in Time Domain Property 2−62.2.8 Integration in Complex Frequency Domain Property 2−82.2.9 Time Periodicity Property 2−82.2.10 Initial Value Theorem 2−92.2.11 Final Value Theorem 2−102.2.12 Convolution in Time Domain Property 2−112.2.13 Convolution in Complex Frequency Domain Property 2−122.3 The Laplace Transform of Common Functions of Time 2−142.3.1 The Laplace Transform of the Unit Step Function 2−142.3.2 The Laplace Transform of the Ramp Function .2−142.3.3 The Laplace Transform of 2−15

u0( ) t

u1( ) t

tnu0( ) t

2.3.4 The Laplace Transform of the Delta Function 2−182.3.5 The Laplace Transform of the Delayed Delta Function 2−182.3.6 The Laplace Transform of 2−192.3.7 The Laplace Transform of 2−192.3.8 The Laplace Transform of 2−202.3.9 The Laplace Transform of 2−202.3.10 The Laplace Transform of 2−212.3.11 The Laplace Transform of 2−222.4 The Laplace Transform of Common Waveforms 2−232.4.1 The Laplace Transform of a Pulse 2−232.4.2 The Laplace Transform of a Linear Segment 2−232.4.3 The Laplace Transform of a Triangular Waveform 2−242.4.4 The Laplace Transform of a Rectangular Periodic Waveform 2−252.4.5 The Laplace Transform of a Half−Rectified Sine Waveform 2−262.5 Using MATLAB for Finding the Laplace Transforms of Time Functions 2−272.6 Summary 2−282.7 Exercises 2−31The Laplace Transform of a Sawtooth Periodic Waveform 2−32The Laplace Transform of a Full−Rectified Sine Waveform 2−322.8 Solutions to End−of−Chapter Exercises 2−33

3.1 The Inverse Laplace Transform Integral 3−13.2 Partial Fraction Expansion 3−13.2.1 Distinct Poles 3−23.2.2 Complex Poles 3−53.2.3 Multiple (Repeated) Poles 3−83.3 Case where F(s) is Improper Rational Function 3−133.4 Alternate Method of Partial Fraction Expansion 3−153.5 Summary 3−193.6 Exercises 3−213.7 Solutions to End−of−Chapter Exercises 3−22

MATLAB Computing

Pages 3−3, 3−4, 3−5, 3−6, 3−8, 3−10, 3−12, 3−13, 3−14, 3−22

4.1 Circuit Transformation from Time to Complex Frequency 4−14.1.1 Resistive Network Transformation 4−14.1.2 Inductive Network Transformation 4−14.1.3 Capacitive Network Transformation 4−1

4.4 Transfer Functions 4−134.5 Using the Simulink Transfer Fcn Block 4−174.6 Summary 4−204.7 Exercises 4−214.8 Solutions to End−of−Chapter Exercises 4−24

MATLAB Computing

Pages 4−6, 4−8, 4−12, 4−16, 4−17, 4−18, 4−26, 4−27, 4−28, 4−29, 4−34

Simulink Modeling

Page 4−17

5.1 Expressing Differential Equations in State Equation Form 5−15.2 Solution of Single State Equations 5−65.3 The State Transition Matrix 5−95.4 Computation of the State Transition Matrix 5−115.4.1 Distinct Eigenvalues 5−115.4.2 Multiple (Repeated) Eigenvalues 5−155.5 Eigenvectors 5−185.6 Circuit Analysis with State Variables 5−225.7 Relationship between State Equations and Laplace Transform 5−305.8 Summary 5−385.9 Exercises 5−415.10 Solutions to End−of−Chapter Exercises 5−43

MATLAB Computing

Pages 5−14, 5−15, 5−18, 5−26, 5−36, 5−48, 5−51

Simulink Modeling

Pages 5−27, 5−37, 5−45

6.1 The Impulse Response in Time Domain 6−16.2 Even and Odd Functions of Time 6−46.3 Convolution 6−76.4 Graphical Evaluation of the Convolution Integral 6−86.5 Circuit Analysis with the Convolution Integral 6−186.6 Summary 6−216.7 Exercises 6−23

6.8 Solutions to End−of−Chapter Exercises 6−25

MATLAB Applications

Pages 6−12, 6−15, 6−30

7.1 Wave Analysis 7−17.2 Evaluation of the Coefficients 7−27.3 Symmetry in Trigonometric Fourier Series 7−67.3.1 Symmetry in Square Waveform 7−87.3.2 Symmetry in Square Waveform with Ordinate Axis Shifted 7−87.3.3 Symmetry in Sawtooth Waveform 7−97.3.4 Symmetry in Triangular Waveform 7−97.3.5 Symmetry in Fundamental, Second, and Third Harmonics 7−107.4 Trigonometric Form of Fourier Series for Common Waveforms 7−107.4.1 Trigonometric Fourier Series for Square Waveform 7−117.4.2 Trigonometric Fourier Series for Sawtooth Waveform 7−147.4.3 Trigonometric Fourier Series for Triangular Waveform 7−167.4.4 Trigonometric Fourier Series for Half−Wave Rectifier Waveform 7−177.4.5 Trigonometric Fourier Series for Full−Wave Rectifier Waveform 7−207.5 Gibbs Phenomenon 7−247.6 Alternate Forms of the Trigonometric Fourier Series 7−247.7 Circuit Analysis with Trigonometric Fourier Series 7−287.8 The Exponential Form of the Fourier Series 7−317.9 Symmetry in Exponential Fourier Series 7−337.9.1 Even Functions 7−337.9.2 Odd Functions 7−347.9.3 Half-Wave Symmetry 7−347.9.4 No Symmetry 7−347.9.5 Relation of to 7−347.10 Line Spectra 7−367.11 Computation of RMS Values from Fourier Series 7−417.12 Computation of Average Power from Fourier Series 7−447.13 Evaluation of Fourier Coefficients Using Excel® 7−467.14 Evaluation of Fourier Coefficients Using MATLAB® 7−477.15 Summary 7−507.16 Exercises 7−537.17 Solutions to End−of−Chapter Exercises 7−55

MATLAB Computing

Pages 7−38, 7−47

C–n Cn

8 The Fourier Transform 8−1

8.1 Definition and Special Forms 8−18.2 Special Forms of the Fourier Transform 8−28.2.1 Real Time Functions 8−38.2.2 Imaginary Time Functions 8−68.3 Properties and Theorems of the Fourier Transform 8−98.3.1 Linearity 8−98.3.2 Symmetry 8−98.3.3 Time Scaling 8−108.3.4 Time Shifting 8−118.3.5 Frequency Shifting 8−118.3.6 Time Differentiation 8−128.3.7 Frequency Differentiation 8−138.3.8 Time Integration 8−138.3.9 Conjugate Time and Frequency Functions 8−138.3.10 Time Convolution 8−148.3.11 Frequency Convolution 8−158.3.12 Area Under 8−158.3.13 Area Under 8−158.3.14 Parseval’s Theorem 8−168.4 Fourier Transform Pairs of Common Functions 8−188.4.1 The Delta Function Pair 8−188.4.2 The Constant Function Pair 8−188.4.3 The Cosine Function Pair 8−198.4.4 The Sine Function Pair 8−208.4.5 The Signum Function Pair 8−208.4.6 The Unit Step Function Pair 8−228.4.7 The Function Pair 8−248.4.8 The Function Pair 8−248.4.9 The Function Pair 8−258.5 Derivation of the Fourier Transform from the Laplace Transform 8−258.6 Fourier Transforms of Common Waveforms 8−278.6.1 The Transform of 8−278.6.2 The Transform of 8−28

sin t ( ) u ( 0t )

f t ( ) = A [ u0( t + T ) u – 0( t – T ) ]

f t ( ) = A [ u0( ) u t – 0( t – 2T ) ]

f t ( ) = A [ u0( t + T ) u + 0( ) u t – 0( t – T ) – u0( t – 2T ) ]

8.6.4 The Transform of 8−308.6.5 The Transform of a Periodic Time Function with Period T 8−318.6.6 The Transform of the Periodic Time Function 8−328.7 Using MATLAB for Finding the Fourier Transform of Time Functions 8−338.8 The System Function and Applications to Circuit Analysis 8−348.9 Summary 8−428.10 Exercises 8−478.11 Solutions to End−of−Chapter Exercises 8−49

MATLAB Computing

Pages 8−33, 8−34, 8−50, 8−54, 8−55, 8−56, 8−59, 8−60

9.1 Definition and Special Forms of the Z Transform 9−19.2 Properties and Theorems of the Z Transform 9−39.2.1 Linearity 9−39.2.2 Shift of in the Discrete−Time Domain 9−39.2.3 Right Shift in the Discrete−Time Domain 9−49.2.4 Left Shift in the Discrete−Time Domain 9−59.2.5 Multiplication by in the Discrete−Time Domain 9−69.2.6 Multiplication by in the Discrete−Time Domain 9−69.2.7 Multiplication by and in the Discrete−Time Domain 9−69.2.8 Summation in the Discrete−Time Domain 9−79.2.9 Convolution in the Discrete−Time Domain 9−89.2.10 Convolution in the Discrete−Frequency Domain 9−99.2.11 Initial Value Theorem 9−99.2.12 Final Value Theorem 9−109.3 The Z Transform of Common Discrete−Time Functions 9−119.3.1 The Transform of the Geometric Sequence 9−119.3.2 The Transform of the Discrete−Time Unit Step Function 9−149.3.3 The Transform of the Discrete−Time Exponential Sequence 9−169.3.4 The Transform of the Discrete−Time Cosine and Sine Functions 9−169.3.5 The Transform of the Discrete−Time Unit Ramp Function 9−189.4 Computation of the Z Transform with Contour Integration 9−209.5 Transformation Between s− and z−Domains 9−229.6 The Inverse Z Transform 9−25

9.6.3 Long Division of Polynomials 9−369.7 The Transfer Function of Discrete−Time Systems 9−389.8 State Equations for Discrete−Time Systems 9−459.9 Summary 9−489.10 Exercises 9−539.11 Solutions to End−of−Chapter Exercises 9−55

10.1 The Discrete Fourier Transform (DFT) 10−110.2 Even and Odd Properties of the DFT 10−910.3 Common Properties and Theorems of the DFT 10−1010.3.1 Linearity 10−1010.3.2 Time Shift 10−1110.3.3 Frequency Shift 10−1210.3.4 Time Convolution 10−1210.3.5 Frequency Convolution 10−1310.4 The Sampling Theorem 10−1310.5 Number of Operations Required to Compute the DFT 10−1610.6 The Fast Fourier Transform (FFT) 10−1710.7 Summary 10−2810.8 Exercises 10−3110.9 Solutions to End−of−Chapter Exercises 10−33

MATLAB Computing

Pages 10−5, 10−7, 10−34

Excel Analysis ToolPak

Pages 10−6, 10−8

11 Analog and Digital Filters

11.1 Filter Types and Classifications 11−111.2 Basic Analog Filters 11−2

11.2.1 RC Low−Pass Filter 11−211.2.2 RC High−Pass Filter 11−411.2.3 RLC Band−Pass Filter 11−711.2.4 RLC Band−Elimination Filter 11−811.3 Low−Pass Analog Filter Prototypes 11−1011.3.1 Butterworth Analog Low−Pass Filter Design 11−1411.3.2 Chebyshev Type I Analog Low−Pass Filter Design 11−2511.3.3 Chebyshev Type II Analog Low−Pass Filter Design 11−3811.3.4 Elliptic Analog Low−Pass Filter Design 11−3911.4 High−Pass, Band−Pass, and Band−Elimination Filter Design 11−4111.5 Digital Filters 11−5111.6 Digital Filter Design with Simulink 11−7011.6.1 The Direct Form I Realization of a Digital Filter 11−7011.6.2 The Direct Form II Realization of a Digital Filter 11−7111.6.3 The Series Form Realization of a Digital Filter 11−7311.6.4 The Parallel Form Realization of a Digital Filter 11−7511.6.5 The Digital Filter Design Block 11−7811.7 Summary 11−8711.8 Exercises 11−9111.9 Solutions to End−of−Chapter Exercises 11−97

C.1 Definition of a Complex Number C−1C.2 Addition and Subtraction of Complex Numbers C−2C.3 Multiplication of Complex Numbers C−3C.4 Division of Complex Numbers C−4C.5 Exponential and Polar Forms of Complex Numbers C−4

Elementary Signals

his chapter begins with a discussion of elementary signals that may be applied to electricnetworks The unit step, unit ramp, and delta functions are then introduced The samplingand sifting properties of the delta function are defined and derived Several examples forexpressing a variety of waveforms in terms of these elementary signals are provided Throughoutthis text, a left justified horizontal bar will denote the beginning of an example, and a right justi-fied horizontal bar will denote the end of the example These bars will not be shown whenever anexample begins at the top of a page or at the bottom of a page Also, when one example followsimmediately after a previous example, the right justified bar will be omitted

1.1 Signals Described in Math Form

Consider the network of Figure 1.1 where the switch is closed at time

Figure 1.1 A switched network with open terminals

We wish to describe in a math form for the time interval To do this, it is nient to divide the time interval into two parts, , and

conve-For the time interval , the switch is open and therefore, the output voltage is zero

In other words,

(1.1)For the time interval , the switch is closed Then, the input voltage appears at theoutput, i.e.,

(1.2)Combining (1.1) and (1.2) into a single relationship, we obtain

We can express (1.3) by the waveform shown in Figure 1.2.

The waveform of Figure 1.2 is an example of a discontinuous function A function is said to be

dis-continuous if it exhibits points of discontinuity, that is, the function jumps from one value to

another without taking on any intermediate values

1.2 The Unit Step Function

A well known discontinuous function is the unit step function * which is defined as

(1.4)

It is also represented by the waveform of Figure 1.3

Figure 1.3 Waveform for

In the waveform of Figure 1.3, the unit step function changes abruptly from to at But if it changes at instead, it is denoted as In this case, its waveform anddefinition are as shown in Figure 1.4 and relation (1.5) respectively

Figure 1.4 Waveform for

Consider the network of Figure 1.6, where the switch is closed at time

Figure 1.6 Network for Example 1.1

Express the output voltage as a function of the unit step function, and sketch the appropriatewaveform

Solution:

For this example, the output voltage for , and for Therefore,

(1.7)and the waveform is shown in Figure 1.7

Figure 1.7 Waveform for Example 1.1

Other forms of the unit step function are shown in Figure 1.8

Figure 1.8 Other forms of the unit step function

Unit step functions can be used to represent other time−varying functions such as the rectangularpulse shown in Figure 1.9

Figure 1.9 A rectangular pulse expressed as the sum of two unit step functions

−Τ

−Τ Τ

Thus, the pulse of Figure 1.9(a) is the sum of the unit step functions of Figures 1.9(b) and 1.9(c)and it is represented as

The unit step function offers a convenient method of describing the sudden application of a age or current source For example, a constant voltage source of applied at , can be

volt-denoted as Likewise, a sinusoidal voltage source that is applied to

a circuit at , can be described as Also, if the excitation in acircuit is a rectangular, or triangular, or sawtooth, or any other recurring pulse, it can be repre-sented as a sum (difference) of unit step functions

0

A –

Thus, the square waveform of Figure 1.10 can be expressed as the summation of (1.8) through(1.11), that is,

(1.12)

Combining like terms, we obtain

(1.13)

Example 1.3

Express the symmetric rectangular pulse of Figure 1.11 as a sum of unit step functions

Figure 1.11 Symmetric rectangular pulse for Example 1.3

Express the symmetric triangular waveform of Figure 1.12 as a sum of unit step functions

Figure 1.12 Symmetric triangular waveform for Example 1.4

T – ⁄ 2 0 T 2 ⁄

i t ( )

2 - +

2 - –

2 - +

2 - –

v t ( )

T 2 ⁄

We first derive the equations for the linear segments and shown in Figure 1.13.

Figure 1.13 Equations for the linear segments of Figure 1.12

For line segment ,

(1.15)and for line segment ,

(1.16)Combining (1.15) and (1.16), we obtain

(1.17)

Example 1.5

Express the waveform of Figure 1.14 as a sum of unit step functions

Figure 1.14 Waveform for Example 1.5

2 T - t 1 +

2 - –

2 - +

T -

2 - –

– +

=

1 2

3

As in the previous example, we first find the equations of the linear segments linear segments and shown in Figure 1.15.

Figure 1.15 Equations for the linear segments of Figure 1.14

Following the same procedure as in the previous examples, we obtain

Multiplying the values in parentheses by the values in the brackets, we obtain

and combining terms inside the brackets, we obtain

(1.18)

Two other functions of interest are the unit ramp function, and the unit impulse or delta function.

We will introduce them with the examples that follow

Figure 1.16 Network for Example 1.6

Solution:

The current through the capacitor is , and the capacitor voltage is

where is a dummy variable

Since the switch closes at , we can express the current as

(1.20)and assuming that for , we can write (1.19) as

(1.21)

or

(1.22)

Therefore, we see that when a capacitor is charged with a constant current, the voltage across it is

a linear function and forms a ramp with slope as shown in Figure 1.17.

Figure 1.17 Voltage across a capacitor when charged with a constant current source

* Since the initial condition for the capacitor voltage was not specified, we express this integral with at the lower limit of integration so that any non-zero value prior to would be included in the integration.

t

∫

=

∞ –

1.3 The Unit Ramp Function

The unit ramp function, denoted as , is defined as

(1.23)where is a dummy variable

We can evaluate the integral of (1.23) by considering the area under the unit step function from as shown in Figure 1.18

Figure 1.18 Area under the unit step function from

Therefore, we define as

(1.24)Since is the integral of , then must be the derivative of , i.e.,

(1.25)

Higher order functions of can be generated by repeated integration of the unit step function Forexample, integrating twice and multiplying by , we define as

(1.26)Similarly,

(1.27)and in general,

(1.28)Also,

u1( ) t

∞ –

t

∫

= τ

t

∫

=

(1.31)Therefore, we can write (1.30) as

(1.32)But, as we know, is constant ( or ) for all time except at where it is discontinuous.

Since the derivative of any constant is zero, the derivative of the unit step has a non−zerovalue only at The derivative of the unit step function is defined in the next section

1.4 The Delta Function

The unit impulse or delta function, denoted as , is the derivative of the unit step It is alsodefined as

(1.33)and

t

∫ = u0( ) t

To better understand the delta function , let us represent the unit step as shown in ure 1.20 (a)

Fig-Figure 1.20 Representation of the unit step as a limit

The function of Figure 1.20 (a) becomes the unit step as Figure 1.20 (b) is the derivative ofFigure 1.20 (a), where we see that as , becomes unbounded, but the area of the rect-angle remains Therefore, in the limit, we can think of as approaching a very large spike orimpulse at the origin, with unbounded amplitude, zero width, and area equal to

Two useful properties of the delta function are the sampling property and the sifting property

1.4.1 The Sampling Property of the Delta Function

The sampling property of the delta function states that

(1.35)

or, when ,

(1.36)that is, multiplication of any function by the delta function results in sampling the func-tion at the time instants where the delta function is not zero The study of discrete−time systems isbased on this property

−ε ε

1 2ε

Figure (a)

Figure (b) Area =1

The first integral on the right side of (1.39) contains the constant term ; this can be writtenoutside the integral, that is,

(1.40)The second integral of the right side of (1.39) is always zero because

and

Therefore, (1.39) reduces to

(1.41)Differentiating both sides of (1.41), and replacing with , we obtain

(1.42)

1.4.2 The Sifting Property of the Delta Function

The sifting property of the delta function states that

∞ –

t

∞ –

t

∞ –

t

∞ –

t

∞ –

sub-stitution into (1.44), we obtain

(1.47)

We have assumed that ; therefore, for , and thus the first term of theright side of (1.47) reduces to Also, the integral on the right side is zero for , and there-fore, we can replace the lower limit of integration by We can now rewrite (1.47) as

(1.48)

1.5 Higher Order Delta Functions

An nth-order delta function is defined as the derivative of , that is,

(1.49)

The function is called doublet, is called triplet, and so on By a procedure similar to the

derivation of the sampling property of the delta function, we can show that

(1.50)Also, the derivation of the sifting property of the delta function can be extended to show that

c The given expression contains the doublet; therefore, we use the relation

Then, for this example,

Figure 1.21 Waveform for Example 1.9

Solution:

a We begin with the derivation of the equations for the linear segments of the given waveform asshown in Figure 1.22

Figure 1.22 Equations for the linear segments of Figure 1.21

Next, we express in terms of the unit step function , and we obtain

(1.52)

−1

−2

−1 1 2 3

0

V ( )

b The derivative of is

(1.53)

From the given waveform, we observe that discontinuities occur only at , , and

these delta functions vanish Also, by application of the sampling property,

and by substitution into (1.53), we obtain

(1.54)The plot of is shown in Figure 1.23

Figure 1.23 Plot of the derivative of the waveform of Figure 1.21

We observe that a negative spike of magnitude occurs at , and two positive spikes ofmagnitude occur at , and These spikes occur because of the discontinuities atthese points

It would be interesting to observe the given signal and its derivative on the Scope block of theSimulink®* model of Figure 1.24 They are shown in Figure 1.25

Figure 1.24 Simulink model for Example 1.9

The waveform created by the Signal Builder block is shown in Figure 1.25

* A brief introduction to Simulink is presented in Appendix B For a detailed procedure for generating piece-wise linear functions with Simulink’s Signal Builder block, please refer to Introduction to Simulink with Engineering

−1

−1 1 2

0

2δ t 1 ( + ) –

dv dt - (V s⁄ )

δ t 2 ( – ) δ t 7 ( – )

t s ( )

Figure 1.25 Piece−wise linear waveform for the Signal Builder block in Figure 1.24

The waveform in Figure 1.25 is created with the following procedure:

1 We open a new model by clicking on the new model icon shown as a blank page on the left

cor-ner of the top menu bar Initially, the name Untitled appears on the top of this new model We

save it with the name Figure_1.25 and Simulink appends the .mdl extension to it

2 From the Sources library, we drag the Signal Builder block into this new model We also drag the Derivative block from the Continuous library, the Bus Creator block from the Com- monly Used Blocks library, and the Scope block into this model, and we interconnect these

blocks as shown in Figure 1.24

3 We double−click on the Signal Builder block in Figure 1.24, and on the plot which appears as asquare pulse, we click on the y−axis and we enter Minimum: −2.5, and Maximum: 3.5 Like-wise we right−click anywhere on the plot and we specify the Change Time Range at Min time:

−2, and Max time: 8

4 To select a particular point, we position the mouse cursor over that point and we left−click Acircle is drawn around that point to indicate that it is selected

5 To select a line segment, we left−click on that segment That line segment is now shown as athick line indicating that it is selected To deselect it, we press the Esc key

6 To drag a line segment to a new position, we place the mouse cursor over that line segment andthe cursor shape shows the position in which we can drag the segment.

7 To drag a point along the y−axis, we move the mouse cursor over that point, and the cursorchanges to a circle indicating that we can drag that point Then, we can move that point in adirection parallel to the x−axis

8 To drag a point along the x−axis, we select that point, and we hold down the Shift key whiledragging that point

9 When we select a line segment on the time axis (x−axis) we observe that at the lower end of

the waveform display window the Left Point and Right Point fields become visible We can then reshape the given waveform by specifying the Time (T) and Amplitude (Y) points.

Figure 1.26 Waveforms for the Simulink model of Figure 1.24

The two positive spikes that occur at , and , are clearly shown in Figure 1.26

MATLAB* has built-in functions for the unit step, and the delta functions These are denoted bythe names of the mathematicians who used them in their work The unit step function isreferred to as Heaviside(t), and the delta function is referred to as Dirac(t) Their use is illus-trated with the examples below

int(d) % Integrate the delta function

ans =

Heaviside(t-a)*k

1.6 Summary

• The unit step function is defined as

• The unit step function offers a convenient method of describing the sudden application of avoltage or current source

• The unit ramp function, denoted as , is defined as

• The unit impulse or delta function, denoted as , is the derivative of the unit step It isalso defined as

and

• The sampling property of the delta function states that

or, when ,

• The sifting property of the delta function states that

• The sampling property of the doublet function states that

∞

b Using the result of part (a), compute the derivative of , and sketch its waveform This

waveform cannot be used with Sinulink’s Function Builder block because it contains thedecaying exponential segment which is a non−linear function

tδ

6 - –

4 - –

2 - –

∞ –

∞

2 - –

v t ( )

1.8 Solutions to End−of−Chapter Exercises

Dear Reader:

The remaining pages on this chapter contain the solutions to the exercises

You must, for your benefit, make an honest effort to solve the problems without first looking atthe solutions that follow It is recommended that first you go through and solve those you feel thatyou know For the exercises that you are uncertain, review this chapter and try again If yourresults do not agree with those provided, look over your procedures for inconsistencies and com-putational errors Refer to the solutions as a last resort and rework those problems at a later date.You should follow this practice with the exercises on all chapters of this book