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MJ..  I •  ' 
ALAN V. OPPENHEIM 
ALAN S. WILLSKV 
WITH S.  HAMID NAWAB 
• 
PRENTICE HALL SIGNAL PROCESSING SERIES  . 
ALAN V. OPPENHEIM, SERIES EDITOR  ~  ~
SECOND EDITION 
SIGNALS 
&amp; 
SYSTEMS 
v. 
ALAN  OPPENHEIM 
s. w 
ALAN  ILLSKY 
MAssACHUSETTS INsTITUTE OF TEcHNOLOGY 
WITH 
S.IIAMIDNAWAB 
BosToN UNIVERSITY 
PRENTICE-HALL  INTERNATIONAL, INC.
CoNTENTS 
PREFACE  XVII 
AcKNOWLEDGMENTS  xxv 
FoREWORD  xxvii 
1 
1 
SIGNALS  AND  SYSTEMS 
1.0  . Introduction  1 
1.1  Continuous-Time and Discrete-Time Signals  1 
1.1.1  Examples and Mathematical Representation  1 
1.1.2  Signal Energy and Power  5 
1.2  Transformations of the Independent Variable  7 
1.2.1  Examples of Transformations of the Independent Variable  8 
1.2.2  Periodic Signals  11 
1.2.3  Even and Odd Signals  13 
1.3  Exponential and Sinusoidal Signals  14 
1.3.1  Continuous-Time Complex Exponential and Sinusoidal Signals  15 
1.3.2  Discrete-Time Complex Exponential and Sinusoidal Signals  21  · 
1.3.3  Periodicity Properties of Discrete-Time Complex Exponentials  25 
1.4  The Unit Impulse and Unit Step Functions  30 
1.4.1  The Discrete-Time Unit Impulse and Unit Step Sequences  30 
1.4.2  The Continuous-Time Unit Step and Unit Impulse Functions  32 
1.5  Continuous-Time and Discrete-Time Systems  38 
1.5.1  Simple Examples of Systems  39 
1.5.2  Interconnections of Systems  41 
1.6  Basic System Properties  44 
1.6.1  Systems with and without Memory  44 
1.6.2  Invertibility and Inverse Systems  45 
1.6.3  Causality  46 
1.6.4  Stability  48 
1.6.5  Time Invariance  50 
1.6.6  Linearity  53 
1.7  Summary  56 
Problems  57 
2 
74 
LINEAR  TIME-INVARIANT  SYSTEMS 
2.0  Introduction  74 
2.1  Discrete-Time LTI Systems: The Convolution Sum  75 
vii
viii  Contents 
2.1.1  . The Representation of Discrete-Time Signals in Terms 
of Impulses  75 
2.1.2  The Discrete-Time Unit Impulse Response and the Convolution-Sum 
Representation of LTI Systems  77 
2.2  Continuous-Time LTI Systems: The Convolution Integral  90 
2.2.1  The Representation of Continuous-Time Signals in Terms 
of Impulses  90 
2.2.2  The Continuous-Time Unit Impulse Response and the Convolution 
Integral Representation of LTI Systems  94 
2.3  Properties of Linear Time-Invariant Systems  103 
2.3.1  The Commutative Property  104 
2.3.2  The Distributive Property  104 
2.3.3  The Associative Property  107 
2.3.4  LTI Systems with and without Memory  108 
2.3.5  Invertibility of LTI Systems  109 
2.3.6  Causality for LTI Systems  112 
2.3.7  Stability for LTI Systems  113 
2.3.8 • The Unit Step Response of an LTI System  115 
2.4  Causal LTI Systems Described by Differential and Difference 
· Equations  116 
2.4.1  Linear Constant -Coefficient Differential Equations  117 
2.4.2  Linear Constant-Coefficient Difference Equations  121 
2.4.3  Block Diagram Representations of First-Order Systems Described 
by Differential and Difference Equations  124 
2.5  Singularity Functions  127 
2.5.1  The Unit Impulse as an Idealized Short Pulse  128 
2.5.2  Defining the Unit Impulse through Convolution  131 
2.5.3  Unit Doublets and Other Singularity Functions  132 
2.6  Summary  137 
Problems  137 
3 
FoURIER  SERIES  REPRESENTATION  OF 
177 
PERIODIC  SIGNALS 
3.0  Introduction  177 
3.1  A Historical Perspective  178 
3.2  The Response of LTI Systems to Complex Exponentials  182 
3.3  Fourier Series Representation of Continuous-Time 
Periodic Signals  186 
3.3.1  Linear Combinations of Harmonically Related Complex 
Exponentials  186 
3.3.2  Determination of the Fourier Series Representation 
of a Continuous-Time Periodic Signal  190 
3.4  Convergence of the Fourier Series  195
Contents  ix 
3.5  Properties of Continuous-Time Fourier Series  202 
3.5.1  Linearity  202 
3.5.2  Time Shifting  202 
3.5.3  Time Reversal  203 
3.5.4  Time Scaling  204 
3.5.5  Multiplication  204 
3.5.6  Conjugation and Conjugate Symmetry  204 
3.5.7  Parseval's Relation for Continuous-Time Periodic Signals  205 
3.5.8  Summary of Properties of the Continuous-Time Fourier Series  205 
3.5.9  Examples  205 
3.6  Fourier Series Representation of Discrete-Time 
Periodic Signals  211 
3.6.1  Linear Combinations of Harmonically Related Complex 
Exponentials  211 
3.6.2  Determination of the Fourier Series Representation of a · 
Periodic Signal  212 
3.7  Properties of Discrete-Time Fourier Series  221 
3.7.1  Multiplication  222 
3.7.2  First Difference  222 
3.7.3  Parseval's Relation for Discrete-Time Periodic Signals  223 
3.7.4  Examples  223 
3.8  Fourier Series and LTI Systems  226 
3.9  Filtering  231 
3.9.1  Frequency-Shaping Filters  232 
3.9.2  Frequency-Selective Filters  236 
3.10  Examples of Continuous-Time Filters Described by 
Differential Equations  239 
3.10.1  A Simple RC Lowpass Filter  239 
3.10.2  A Simple RC Highpass Filter  241 
3.11  Examples of Discrete-Time Filters Described by 
Difference Equations  244 
3.11.1  First-Order Recursive Discrete-Time Filters  244 
3.11.2  Nonrecursive Discrete-Time Filters  245 
3.12  Summary  249 
Problems  250 
4 
ThE CONTINUOUS-TIME  FOURIER TRANSFORM  284 
4.0  Introduction  284 
4.1  Representation of Aperiodic Signals: The Continuous-Time 
Fourier Transform  285 
4.1.1  Development of the Fourier Transform Representation 
of an Aperiodic Signal  285 
4.1.2  Convergence of Fourier Transforms  289 
4.1.3  Examples of Continuous-Time Fourier Transforms  290
X  Contents 
4.2  The Fourier Transform for Periodic Signals  296 
4.3  Properties of the Continuous-Time Fourier Transform  300 
4.3.1  Linearity  301 
4.3.2  Time Shifting  301 
4.3.3  Conjugation and Conjugate Symmetry  303 
4.3.4  Differentiation and Integration  306 
4.3.5  Time and Frequency Scaling  308 
4.3.6  Duality  309 
4.3.7  Parseval's Relation  312 
4.4  The Convolution Property  314 
4.4.1  Examples  317 
4.5  The Multiplication Property  322 
4.5.1  Frequency-Selective Filtering with Variable Center Frequency  325 
4.6  Tables of Fourier Properties and of Basic Fourier 
Transform Pairs  328  · 
4.7  Systems Characterized by Linear Constant-Coefficient 
Differential Equations  330 
4.8  Summary  333 
Problems  334 
5 
358 
ThE  DISCRETE-TIME  FOURIER  TRANSFORM 
5.0  Introduction  358 
5.1  Representation of Aperiodic Signals: The Discrete-Time 
Fourier Transform  359 
5.1.1  Development of the Discrete-~ime Fourier Transform  359 
5.1.2  Examples of Discrete-Time Fourier Transforms  362 
5 .1.3  Convergence Issues Associated with the Discrete-Time Fourier 
Transform  366 
5.2  The Fourier Transform for Periodic Signals  367 
5.3  Properties of the Discrete-Time Fourier Transform  372 
5.3.1.  Periodicity of the Discrete-Time Fourier Transform  373 
5.3.2  Linearity of the Fourier Transform  373 
5.3.3  Time Shifting and Frequency Shifting  373 
5.3.4  Conjugation and Conjugate Symmetry  375 
5.3.5  Differencing and Accumulation  375 
5.3.6  Time Reversal  376 
5.3.7  Time Expansion  377 
5.3.8  Differentiation in Frequency  380 
5.3.9  Parseval's Relation  380 
5.4  The Convolution Property  382 
5.4.1  Examples  383 
5.5  The Multiplication Property  388 
5.6  Tables of Fourier Transform Properties and Basic Fourier 
Transform Pairs  390
Contents  xi 
5. 7  Duality  390 
5.7.1  Duality in the Discrete-Time Fourier Series  391 
5.7.2  Duality between the Discrete-Time Fourier Transform and the 
Continuous-Time Fourier Series  395 
5.8  Systems Characterized by Linear Constant-Coefficient 
Difference Equations  396 
5.9  Summary  399 
Problems  400 
6 
TIME  AND  FREQUENCY  CHARACTERIZATION 
423 
OF  SIGNALS  AND  SYSTEMS 
6.0  Introduction  423 
6.1  The Magnitude-Phase 'Representation of. the Fourier 
Transform  423 
6.2  The Magnitude-Phase Representation of the Frequency Response 
of LTI Systems  427 
6.2.1  Linear and Nonlinear Phase  428 
6.2.2  Group Delay  430 
6.2.3  Log-Magnitude and Bode Plots  436 
6.3  Time-Domain Properties of Ideal Frequency-Selective 
Filters  439 
6.4  Time-Domain and Frequency-Domain Aspects of Nonideal 
Filters  444 
6.5  First-Order and Second-Order Continuous-Time Systems  448 
6.5 .1  First -Order Continuous-Time Systems  448 
6.5 .2  Second-Order Continuous-Time Systems  451 
6.5.3  Bode Plots for Rational Frequency Responses  456 
6.6  First-Order and Second-Order Discrete-Time Systems  461 
6.6.1  First-Order Discrete-Time Systems  461 
6.6.2  Second-Order Discrete-Time Systems  465 
6.7  Examples of Time- and Frequency-Domain Analysis 
of Systems  472 
6.7.1  Analysis of an Automobile Suspension System  473 
6.7.2  Examples of Discrete-Time Nonrecursive Filters  476 
6.8  Summary  482 
Problems  483 
7 
514 
SAMPLING 
7.0  Introduction  514 
7.1  Representation of a Continuous-Time Signal by Its Samples: 
The Sampling Theorem  515 
7.1.1  Impulse-Train Sampling  516 
7 .1.2  Sampling with a Zero-Order Hold  520
xii  Contents 
7.2  Reconstruction of a Signal from Its Samples Using 
Interpolation  522 
7.3  The Effect of Undersampling: Aliasing  527 
7.4  Discrete-Time Processing of Continuous-Time Signals  534 
7.4.1  Digital Differentiator  541 
7.4.2  Half-Sample Delay  543 
7.5  Sampling of Discrete-Time Signals  545 
7.5.1  Impulse-Train Sampling  545 
7.5 .2  Discrete-Time Decimation and Interpolation  549 
7.6  Summary  555 
Problems  556 
8 
CoMMUNICATION  SYSTE.M S  582 
8.0  Introduction  582 
8.1  Complex Exponential and Sinusoidal Amplitude Modulation  583 
8.1.1  Amplitude Modulation with a Complex Exponential Carrier  583 
8.1.2  Amplitude Modulation with a Sinusoidal Carrier  585 
8.2  Demodulation for Sinusoidal AM  587 
8.2.1  Synchronous Demodulation  587 
8.2.2  Asynchronous Demodulation  590 
8.3  Frequency-Division Multiplexing  594 
8.4  Single-Sideband Sinusoidal Amplitude Modulation  597 
8.5  Amplitude Modulation with a Pulse-Train Carrier  601 
8.5.1  Modulation of a Pulse-Train Carrier  601 
8.5.2  Time-Division Multiplexing  604 
8.6  Pulse-Amplitude Modulation  604 
8.6.1  Pulse-Amplitude Modulated Signals  604 
8.6.2  Intersymbol Interference in PAM Systems  607 
8.6.3  Digit~l Pulse-Amplitude and Pulse-Code Modulation  610 
8.7  Sinusoidal Frequency Modulation  611 
8.7.1  Narrowband Frequency Modulation  613 
8.7.2  Wideband Frequency Modulation  615 
8.7.3  Periodic Square-Wave Modulating Signal  617 
8.8  Discrete-Time Modulation  619 
8.8.1  Discrete-Time Sinusoidal Amplitude Modulation  619 
8.8.2  Discrete-Time Transmodulation  623 
8.9  Summary  623 
Problems  625 
9 
654 
THE  LAPLACE  TRANSFORM 
9.0  Introduction  654 
9.1  The Laplace Transform  655 
9.2  The Region of Convergence for Laplace Transforms  662
Contents  xiii 
9.3  The Inverse Laplace Transfonn  670 
9.4  Geometric Evaluation of the Fourier Transfonn from the 
Pole-Zero Plot  674 
9.4.1  . First-Order Systems  676 
9.4.2  Second-Order Systems  677 
9.4.3  All-Pass Systems  681 
9.5  Properties of the Laplace Transfonn  682 
9.5.1  Linearity of the Laplace Transform  683 
9.5.2  Time Shifting  684 
9.5.3  Shifting in the s-Domain  685 
9.5 .4  Time Scal1ng  685 
9.5.5  Conjugation  687 
9.5.6  Convolution Property  687 
9.5.7  Differentiation in the Time Domain  688 
9.5.8  Differentiation in the s-Domain  688 
9.5.9  Integration in the Time Domain  690 
9.5.10  The Initial- and Final-Value Theorems  690 
9.5.11  Table of Properties  6?1 
9.6  Some Laplace Transfonn Pairs  692 
9.7  Analysis and Characterization of LTI Systems Using the 
Laplace Transfonn  693 
9.7 .1  Causality  693 
9.7.2  Stability  695 
9.7.3  LTI Systems Characterized by Linear Constant-Coefficient 
Differential Equations  698 
9.7.4  Examples Relating System Behavior to the System Function  701 
9.7.5  Butterworth Filters  703 
9.8  System Function Algebra and Block Diagram 
Representations  706 
9.8.1  System Functions for Interconnections of LTI Systems  707 
9.8.2  Block Diagram Representations for Causal LTI Systems Described 
by Differential Equations and Rational System Functions  708 
9.9  The Unilateral Laplace Transfonn  714 
9.9.1  Examples of Unilateral Laplace Transforms  714 
9.9.2  Properties of the Unilateral Laplace Transform  716 
9.9.3  Solving Differential Equations Using the Unilateral Laplace 
Transform  719 
9.10  Summary  720 
Problems  721 
10 
741 
THE  Z-TRANSFORM 
10.0  Introduction  741 
10.1  The z-Transfonn  741 
10.2  The Region of Convergence for the z-Transfonn.  748
xiv  Contents 
10.3  The Inverse z-Transform  757 
10.4  Geometric Evaluation of the Fourier Transform from the 
Pole-Zero Plot  763 
10.4.1  First-Order Systems  763 
10.4.2  Second-Order Systems  765 
10.5  Properties of the z-Transform  767 
10.5.1  Linearity  767 
10.5.2  Time Shifting  767 
10.5.3  Scaling in the z-Domain  768 
10.5.4  Time Reversal  769 
10.5.5  Time Expansion  769 
10.5.6  Conjugation  770 
10.5.7  The Convolution Property  770 
10.5.8  Differentiation in the z-Domain  772 
10.5.9  The Initial-Value Theorem  773 
10.5.10  Summary of Properties  774 
10.6  Some Common z-Transform Pairs  774 
10.7  Analysis and Characterization of LTI Systems Using 
z-Transforms  774 
10.7.1  Causality  776 
10.7.2  Stability  777 
10.7.3  LTI Systems Characterized by Linear Constant-Coefficient 
Difference Equations  779 
10.7.4  Examples Relating System Behavior to the System Function  781 
10.8  System Function Algebra and Block Diagram 
Representations  783 
10.8.1  System Functions for Interconnections of LTI Systems  784 
10.8.2  Block Diagram Representations for Causal LTI Systems Described 
by Difference Equations and Rational System Functions  784 
10.9  The Unilateral z-Transform  789 
10.9.1  Examples of Unilateral z-Transforms and Inverse Transforms  790 
10.9.2  Properties of the Unilateral z-Transform  792 
10.9.3  Solving Difference Equations Using the Unilateral 
z-Transform  795 
10.10  Summary  796 
Problems  797 
11 
816 
UNEAR  FEEDBACK  SYSTEMS 
11.0  Introduction  816 
11.1  Linear Feedback Systems  819 
11.2  Some Applications and Consequences of Feedback  820 
11.2.1  Inverse System Design  820 
11.2.2  Compensation for Nonideal Elements  821 
11.2.3  Stabilization of Unstable Systems  823</p></div></div></div><div class="jsx-7d4e960818f559df modal-hidden"><div class="jsx-7d4e960818f559df modal-background"></div><div class="jsx-7d4e960818f559df modal-error-header"><div class="jsx-7d4e960818f559df modal-btn"><button class="jsx-7d4e960818f559df btn-close">×</button></div></div><div class="jsx-7d4e960818f559df modal-error-content"><div class="jsx-7d4e960818f559df modal-info"></div></div></div><div class="jsx-7d4e960818f559df ads1"></div><div class="jsx-7d4e960818f559df container"><div class="jsx-7d4e960818f559df detail-container"><div class="jsx-7d4e960818f559df detail-left"><div class="jsx-7d4e960818f559df ebook-frame"><div class="jsx-7d4e960818f559df ebook"><div class="jsx-7d4e960818f559df ebook-left"><img src="https://cdn.pdfdrive.to/media/content/thumbnails/e318dee9-d79b-418f-9800-045783c90f6e.webp" alt="ebook img" width="210" height="300" loading="lazy" title="ebook img" class="jsx-7d4e960818f559df ebook-img"></div><div class="jsx-7d4e960818f559df ebook-right"><h1 class="jsx-7d4e960818f559df ebook-title">Signals and System PDF</h1><div class="jsx-7d4e960818f559df file-info"><span class="jsx-7d4e960818f559df">967<!-- --> Pages</span><span class="jsx-7d4e960818f559df dot">·</span><span class="jsx-7d4e960818f559df">2013</span><span class="jsx-7d4e960818f559df dot">·</span><span class="jsx-7d4e960818f559df">292.34<!-- --> MB</span><span class="jsx-7d4e960818f559df dot">·</span><span class="jsx-7d4e960818f559df">english</span></div><div class="jsx-7d4e960818f559df author">by &nbsp;<span class="jsx-7d4e960818f559df"><a href="/author/alan-v-oppenheim" class="jsx-7d4e960818f559df author-name">Alan V. Oppenheim</a></span></div><div class="jsx-7d4e960818f559df tags"><span style="margin-right:5px" class="jsx-7d4e960818f559df"><a href="/top-technique-electronics-signal-processing-books" class="jsx-7d4e960818f559df">#<!-- -->Technique - Electronics: Signal Processing<!-- --> </a></span></div><div class="jsx-7d4e960818f559df btn-container"><div class="jsx-7d4e960818f559df hint-progress invisible"><i class="jsx-7d4e960818f559df">Checking for file health...</i></div><div class="jsx-7d4e960818f559df progress invisible"><div class="jsx-7d4e960818f559df progress-bar "></div></div><div class="jsx-27c21ff6cf33a438 download-content"><div class="jsx-27c21ff6cf33a438 download-group"><a href="/filedownload/signals-and-system" disabled="" class="jsx-27c21ff6cf33a438 btn-download">Download</a></div></div></div></div></div></div><div class="jsx-2ef0af5aad58e0f1 premium-container"><div class="jsx-563764575"><a href="/premium" rel="nofollow" class="jsx-563764575 premium-btn animated-button7"><span class="jsx-563764575"></span><span class="jsx-563764575"></span><span class="jsx-563764575"></span><span class="jsx-563764575"></span><svg id="icon-diamond" viewBox="0 0 32 32" fill="#fff" width="40" height="40" class="jsx-563764575"><path d="M2.103 12.052l13.398 16.629-5.373-16.629h-8.025zM11.584 12.052l4.745 16.663 4.083-16.663h-8.828zM17.051 28.681l12.898-16.629h-7.963l-4.935 16.629zM29.979 10.964l-3.867-6.612-3.869 6.612h7.736zM24.896 3.973h-7.736l3.867 6.839 3.869-6.839zM19.838 10.964l-3.867-6.612-3.868 6.612h7.735zM14.839 3.973h-7.735l3.868 6.839 3.867-6.839zM5.889 4.352l-3.867 6.612h7.735l-3.868-6.612z" class="jsx-563764575"></path></svg>Upgrade Premium</a></div><div class="jsx-2ef0af5aad58e0f1 limit-description-normal">Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.</div></div><div class="jsx-7d4e960818f559df ads3"></div><div class="jsx-7d4e960818f559df mobile-ads"></div><h2 class="jsx-7d4e960818f559df summary-title">Preview <i class="jsx-7d4e960818f559df">Signals and System</i></h2><div class="jsx-7d4e960818f559df summary"><p class="jsx-7d4e960818f559df summary-content">l  •.• 
MJ..  I •  ' 
ALAN V. OPPENHEIM 
ALAN S. WILLSKV 
WITH S.  HAMID NAWAB 
• 
PRENTICE HALL SIGNAL PROCESSING SERIES  . 
ALAN V. OPPENHEIM, SERIES EDITOR  ~  ~
SECOND EDITION 
SIGNALS 
&amp; 
SYSTEMS 
v. 
ALAN  OPPENHEIM 
s. w 
ALAN  ILLSKY 
MAssACHUSETTS INsTITUTE OF TEcHNOLOGY 
WITH 
S.IIAMIDNAWAB 
BosToN UNIVERSITY 
PRENTICE-HALL  INTERNATIONAL, INC.
CoNTENTS 
PREFACE  XVII 
AcKNOWLEDGMENTS  xxv 
FoREWORD  xxvii 
1 
1 
SIGNALS  AND  SYSTEMS 
1.0  . Introduction  1 
1.1  Continuous-Time and Discrete-Time Signals  1 
1.1.1  Examples and Mathematical Representation  1 
1.1.2  Signal Energy and Power  5 
1.2  Transformations of the Independent Variable  7 
1.2.1  Examples of Transformations of the Independent Variable  8 
1.2.2  Periodic Signals  11 
1.2.3  Even and Odd Signals  13 
1.3  Exponential and Sinusoidal Signals  14 
1.3.1  Continuous-Time Complex Exponential and Sinusoidal Signals  15 
1.3.2  Discrete-Time Complex Exponential and Sinusoidal Signals  21  · 
1.3.3  Periodicity Properties of Discrete-Time Complex Exponentials  25 
1.4  The Unit Impulse and Unit Step Functions  30 
1.4.1  The Discrete-Time Unit Impulse and Unit Step Sequences  30 
1.4.2  The Continuous-Time Unit Step and Unit Impulse Functions  32 
1.5  Continuous-Time and Discrete-Time Systems  38 
1.5.1  Simple Examples of Systems  39 
1.5.2  Interconnections of Systems  41 
1.6  Basic System Properties  44 
1.6.1  Systems with and without Memory  44 
1.6.2  Invertibility and Inverse Systems  45 
1.6.3  Causality  46 
1.6.4  Stability  48 
1.6.5  Time Invariance  50 
1.6.6  Linearity  53 
1.7  Summary  56 
Problems  57 
2 
74 
LINEAR  TIME-INVARIANT  SYSTEMS 
2.0  Introduction  74 
2.1  Discrete-Time LTI Systems: The Convolution Sum  75 
vii
viii  Contents 
2.1.1  . The Representation of Discrete-Time Signals in Terms 
of Impulses  75 
2.1.2  The Discrete-Time Unit Impulse Response and the Convolution-Sum 
Representation of LTI Systems  77 
2.2  Continuous-Time LTI Systems: The Convolution Integral  90 
2.2.1  The Representation of Continuous-Time Signals in Terms 
of Impulses  90 
2.2.2  The Continuous-Time Unit Impulse Response and the Convolution 
Integral Representation of LTI Systems  94 
2.3  Properties of Linear Time-Invariant Systems  103 
2.3.1  The Commutative Property  104 
2.3.2  The Distributive Property  104 
2.3.3  The Associative Property  107 
2.3.4  LTI Systems with and without Memory  108 
2.3.5  Invertibility of LTI Systems  109 
2.3.6  Causality for LTI Systems  112 
2.3.7  Stability for LTI Systems  113 
2.3.8 • The Unit Step Response of an LTI System  115 
2.4  Causal LTI Systems Described by Differential and Difference 
· Equations  116 
2.4.1  Linear Constant -Coefficient Differential Equations  117 
2.4.2  Linear Constant-Coefficient Difference Equations  121 
2.4.3  Block Diagram Representations of First-Order Systems Described 
by Differential and Difference Equations  124 
2.5  Singularity Functions  127 
2.5.1  The Unit Impulse as an Idealized Short Pulse  128 
2.5.2  Defining the Unit Impulse through Convolution  131 
2.5.3  Unit Doublets and Other Singularity Functions  132 
2.6  Summary  137 
Problems  137 
3 
FoURIER  SERIES  REPRESENTATION  OF 
177 
PERIODIC  SIGNALS 
3.0  Introduction  177 
3.1  A Historical Perspective  178 
3.2  The Response of LTI Systems to Complex Exponentials  182 
3.3  Fourier Series Representation of Continuous-Time 
Periodic Signals  186 
3.3.1  Linear Combinations of Harmonically Related Complex 
Exponentials  186 
3.3.2  Determination of the Fourier Series Representation 
of a Continuous-Time Periodic Signal  190 
3.4  Convergence of the Fourier Series  195
Contents  ix 
3.5  Properties of Continuous-Time Fourier Series  202 
3.5.1  Linearity  202 
3.5.2  Time Shifting  202 
3.5.3  Time Reversal  203 
3.5.4  Time Scaling  204 
3.5.5  Multiplication  204 
3.5.6  Conjugation and Conjugate Symmetry  204 
3.5.7  Parseval's Relation for Continuous-Time Periodic Signals  205 
3.5.8  Summary of Properties of the Continuous-Time Fourier Series  205 
3.5.9  Examples  205 
3.6  Fourier Series Representation of Discrete-Time 
Periodic Signals  211 
3.6.1  Linear Combinations of Harmonically Related Complex 
Exponentials  211 
3.6.2  Determination of the Fourier Series Representation of a · 
Periodic Signal  212 
3.7  Properties of Discrete-Time Fourier Series  221 
3.7.1  Multiplication  222 
3.7.2  First Difference  222 
3.7.3  Parseval's Relation for Discrete-Time Periodic Signals  223 
3.7.4  Examples  223 
3.8  Fourier Series and LTI Systems  226 
3.9  Filtering  231 
3.9.1  Frequency-Shaping Filters  232 
3.9.2  Frequency-Selective Filters  236 
3.10  Examples of Continuous-Time Filters Described by 
Differential Equations  239 
3.10.1  A Simple RC Lowpass Filter  239 
3.10.2  A Simple RC Highpass Filter  241 
3.11  Examples of Discrete-Time Filters Described by 
Difference Equations  244 
3.11.1  First-Order Recursive Discrete-Time Filters  244 
3.11.2  Nonrecursive Discrete-Time Filters  245 
3.12  Summary  249 
Problems  250 
4 
ThE CONTINUOUS-TIME  FOURIER TRANSFORM  284 
4.0  Introduction  284 
4.1  Representation of Aperiodic Signals: The Continuous-Time 
Fourier Transform  285 
4.1.1  Development of the Fourier Transform Representation 
of an Aperiodic Signal  285 
4.1.2  Convergence of Fourier Transforms  289 
4.1.3  Examples of Continuous-Time Fourier Transforms  290
X  Contents 
4.2  The Fourier Transform for Periodic Signals  296 
4.3  Properties of the Continuous-Time Fourier Transform  300 
4.3.1  Linearity  301 
4.3.2  Time Shifting  301 
4.3.3  Conjugation and Conjugate Symmetry  303 
4.3.4  Differentiation and Integration  306 
4.3.5  Time and Frequency Scaling  308 
4.3.6  Duality  309 
4.3.7  Parseval's Relation  312 
4.4  The Convolution Property  314 
4.4.1  Examples  317 
4.5  The Multiplication Property  322 
4.5.1  Frequency-Selective Filtering with Variable Center Frequency  325 
4.6  Tables of Fourier Properties and of Basic Fourier 
Transform Pairs  328  · 
4.7  Systems Characterized by Linear Constant-Coefficient 
Differential Equations  330 
4.8  Summary  333 
Problems  334 
5 
358 
ThE  DISCRETE-TIME  FOURIER  TRANSFORM 
5.0  Introduction  358 
5.1  Representation of Aperiodic Signals: The Discrete-Time 
Fourier Transform  359 
5.1.1  Development of the Discrete-~ime Fourier Transform  359 
5.1.2  Examples of Discrete-Time Fourier Transforms  362 
5 .1.3  Convergence Issues Associated with the Discrete-Time Fourier 
Transform  366 
5.2  The Fourier Transform for Periodic Signals  367 
5.3  Properties of the Discrete-Time Fourier Transform  372 
5.3.1.  Periodicity of the Discrete-Time Fourier Transform  373 
5.3.2  Linearity of the Fourier Transform  373 
5.3.3  Time Shifting and Frequency Shifting  373 
5.3.4  Conjugation and Conjugate Symmetry  375 
5.3.5  Differencing and Accumulation  375 
5.3.6  Time Reversal  376 
5.3.7  Time Expansion  377 
5.3.8  Differentiation in Frequency  380 
5.3.9  Parseval's Relation  380 
5.4  The Convolution Property  382 
5.4.1  Examples  383 
5.5  The Multiplication Property  388 
5.6  Tables of Fourier Transform Properties and Basic Fourier 
Transform Pairs  390
Contents  xi 
5. 7  Duality  390 
5.7.1  Duality in the Discrete-Time Fourier Series  391 
5.7.2  Duality between the Discrete-Time Fourier Transform and the 
Continuous-Time Fourier Series  395 
5.8  Systems Characterized by Linear Constant-Coefficient 
Difference Equations  396 
5.9  Summary  399 
Problems  400 
6 
TIME  AND  FREQUENCY  CHARACTERIZATION 
423 
OF  SIGNALS  AND  SYSTEMS 
6.0  Introduction  423 
6.1  The Magnitude-Phase 'Representation of. the Fourier 
Transform  423 
6.2  The Magnitude-Phase Representation of the Frequency Response 
of LTI Systems  427 
6.2.1  Linear and Nonlinear Phase  428 
6.2.2  Group Delay  430 
6.2.3  Log-Magnitude and Bode Plots  436 
6.3  Time-Domain Properties of Ideal Frequency-Selective 
Filters  439 
6.4  Time-Domain and Frequency-Domain Aspects of Nonideal 
Filters  444 
6.5  First-Order and Second-Order Continuous-Time Systems  448 
6.5 .1  First -Order Continuous-Time Systems  448 
6.5 .2  Second-Order Continuous-Time Systems  451 
6.5.3  Bode Plots for Rational Frequency Responses  456 
6.6  First-Order and Second-Order Discrete-Time Systems  461 
6.6.1  First-Order Discrete-Time Systems  461 
6.6.2  Second-Order Discrete-Time Systems  465 
6.7  Examples of Time- and Frequency-Domain Analysis 
of Systems  472 
6.7.1  Analysis of an Automobile Suspension System  473 
6.7.2  Examples of Discrete-Time Nonrecursive Filters  476 
6.8  Summary  482 
Problems  483 
7 
514 
SAMPLING 
7.0  Introduction  514 
7.1  Representation of a Continuous-Time Signal by Its Samples: 
The Sampling Theorem  515 
7.1.1  Impulse-Train Sampling  516 
7 .1.2  Sampling with a Zero-Order Hold  520
xii  Contents 
7.2  Reconstruction of a Signal from Its Samples Using 
Interpolation  522 
7.3  The Effect of Undersampling: Aliasing  527 
7.4  Discrete-Time Processing of Continuous-Time Signals  534 
7.4.1  Digital Differentiator  541 
7.4.2  Half-Sample Delay  543 
7.5  Sampling of Discrete-Time Signals  545 
7.5.1  Impulse-Train Sampling  545 
7.5 .2  Discrete-Time Decimation and Interpolation  549 
7.6  Summary  555 
Problems  556 
8 
CoMMUNICATION  SYSTE.M S  582 
8.0  Introduction  582 
8.1  Complex Exponential and Sinusoidal Amplitude Modulation  583 
8.1.1  Amplitude Modulation with a Complex Exponential Carrier  583 
8.1.2  Amplitude Modulation with a Sinusoidal Carrier  585 
8.2  Demodulation for Sinusoidal AM  587 
8.2.1  Synchronous Demodulation  587 
8.2.2  Asynchronous Demodulation  590 
8.3  Frequency-Division Multiplexing  594 
8.4  Single-Sideband Sinusoidal Amplitude Modulation  597 
8.5  Amplitude Modulation with a Pulse-Train Carrier  601 
8.5.1  Modulation of a Pulse-Train Carrier  601 
8.5.2  Time-Division Multiplexing  604 
8.6  Pulse-Amplitude Modulation  604 
8.6.1  Pulse-Amplitude Modulated Signals  604 
8.6.2  Intersymbol Interference in PAM Systems  607 
8.6.3  Digit~l Pulse-Amplitude and Pulse-Code Modulation  610 
8.7  Sinusoidal Frequency Modulation  611 
8.7.1  Narrowband Frequency Modulation  613 
8.7.2  Wideband Frequency Modulation  615 
8.7.3  Periodic Square-Wave Modulating Signal  617 
8.8  Discrete-Time Modulation  619 
8.8.1  Discrete-Time Sinusoidal Amplitude Modulation  619 
8.8.2  Discrete-Time Transmodulation  623 
8.9  Summary  623 
Problems  625 
9 
654 
THE  LAPLACE  TRANSFORM 
9.0  Introduction  654 
9.1  The Laplace Transform  655 
9.2  The Region of Convergence for Laplace Transforms  662
Contents  xiii 
9.3  The Inverse Laplace Transfonn  670 
9.4  Geometric Evaluation of the Fourier Transfonn from the 
Pole-Zero Plot  674 
9.4.1  . First-Order Systems  676 
9.4.2  Second-Order Systems  677 
9.4.3  All-Pass Systems  681 
9.5  Properties of the Laplace Transfonn  682 
9.5.1  Linearity of the Laplace Transform  683 
9.5.2  Time Shifting  684 
9.5.3  Shifting in the s-Domain  685 
9.5 .4  Time Scal1ng  685 
9.5.5  Conjugation  687 
9.5.6  Convolution Property  687 
9.5.7  Differentiation in the Time Domain  688 
9.5.8  Differentiation in the s-Domain  688 
9.5.9  Integration in the Time Domain  690 
9.5.10  The Initial- and Final-Value Theorems  690 
9.5.11  Table of Properties  6?1 
9.6  Some Laplace Transfonn Pairs  692 
9.7  Analysis and Characterization of LTI Systems Using the 
Laplace Transfonn  693 
9.7 .1  Causality  693 
9.7.2  Stability  695 
9.7.3  LTI Systems Characterized by Linear Constant-Coefficient 
Differential Equations  698 
9.7.4  Examples Relating System Behavior to the System Function  701 
9.7.5  Butterworth Filters  703 
9.8  System Function Algebra and Block Diagram 
Representations  706 
9.8.1  System Functions for Interconnections of LTI Systems  707 
9.8.2  Block Diagram Representations for Causal LTI Systems Described 
by Differential Equations and Rational System Functions  708 
9.9  The Unilateral Laplace Transfonn  714 
9.9.1  Examples of Unilateral Laplace Transforms  714 
9.9.2  Properties of the Unilateral Laplace Transform  716 
9.9.3  Solving Differential Equations Using the Unilateral Laplace 
Transform  719 
9.10  Summary  720 
Problems  721 
10 
741 
THE  Z-TRANSFORM 
10.0  Introduction  741 
10.1  The z-Transfonn  741 
10.2  The Region of Convergence for the z-Transfonn.  748
xiv  Contents 
10.3  The Inverse z-Transform  757 
10.4  Geometric Evaluation of the Fourier Transform from the 
Pole-Zero Plot  763 
10.4.1  First-Order Systems  763 
10.4.2  Second-Order Systems  765 
10.5  Properties of the z-Transform  767 
10.5.1  Linearity  767 
10.5.2  Time Shifting  767 
10.5.3  Scaling in the z-Domain  768 
10.5.4  Time Reversal  769 
10.5.5  Time Expansion  769 
10.5.6  Conjugation  770 
10.5.7  The Convolution Property  770 
10.5.8  Differentiation in the z-Domain  772 
10.5.9  The Initial-Value Theorem  773 
10.5.10  Summary of Properties  774 
10.6  Some Common z-Transform Pairs  774 
10.7  Analysis and Characterization of LTI Systems Using 
z-Transforms  774 
10.7.1  Causality  776 
10.7.2  Stability  777 
10.7.3  LTI Systems Characterized by Linear Constant-Coefficient 
Difference Equations  779 
10.7.4  Examples Relating System Behavior to the System Function  781 
10.8  System Function Algebra and Block Diagram 
Representations  783 
10.8.1  System Functions for Interconnections of LTI Systems  784 
10.8.2  Block Diagram Representations for Causal LTI Systems Described 
by Difference Equations and Rational System Functions  784 
10.9  The Unilateral z-Transform  789 
10.9.1  Examples of Unilateral z-Transforms and Inverse Transforms  790 
10.9.2  Properties of the Unilateral z-Transform  792 
10.9.3  Solving Difference Equations Using the Unilateral 
z-Transform  795 
10.10  Summary  796 
Problems  797 
11 
816 
UNEAR  FEEDBACK  SYSTEMS 
11.0  Introduction  816 
11.1  Linear Feedback Systems  819 
11.2  Some Applications and Consequences of Feedback  820 
11.2.1  Inverse System Design  820 
11.2.2  Compensation for Nonideal Elements  821 
11.2.3  Stabilization of Unstable Systems  823</p><div class="jsx-7d4e960818f559df show-more"><span class="jsx-7d4e960818f559df"><i class="jsx-7d4e960818f559df">See more</i></span></div></div><div class="jsx-7d4e960818f559df title"><h2 class="jsx-7d4e960818f559df siminar">The list of books you might like</h2></div><div class="jsx-1ac4e25e396cc2e7 collection"><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.zlibrary.to/media/content/thumbnails/bdb6c691-c928-472f-b015-e0a09200eca3.webp" loading="lazy" alt="book image" title="The Silent Patient" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/the-silent-patient-9" class="jsx-1ac4e25e396cc2e7 title-link">The Silent Patient</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">Alex Michaelides</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.52 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.zlibrary.to/media/content/thumbnails/efb033fd-8b2d-4ed1-8c99-d9414cac1fb5.webp" loading="lazy" alt="book image" title="Corrupt (Devil's Night #1)" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/corrupt-devils-night-1-0" class="jsx-1ac4e25e396cc2e7 title-link">Corrupt (Devil's Night #1)</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">Penelope Douglas</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">518 Pages</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2015</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.74 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.zlibrary.to/media/content/thumbnails/4b204c1d-d091-4ba5-8240-1aee408cc221.webp" loading="lazy" alt="book image" title="Credence" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/credence-3" class="jsx-1ac4e25e396cc2e7 title-link">Credence</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">Penelope Douglas</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">487 Pages</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2020</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.86 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="ads4"></div><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.zlibrary.to/media/content/thumbnails/cf5ae923-f278-40f4-a88f-8d76be0c2a72.webp" loading="lazy" alt="book image" title="The Strength In Our Scars" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/the-strength-in-our-scars-0" class="jsx-1ac4e25e396cc2e7 title-link">The Strength In Our Scars</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">Bianca Sparacino</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2018</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.17 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="mobile-ads"></div><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/16dffc96-269d-4f80-89e9-1f2a821366d0.webp" loading="lazy" alt="book image" title="BELGIUM - Stamp Albums Web" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/belgium-stamp-albums-web" class="jsx-1ac4e25e396cc2e7 title-link">BELGIUM - Stamp Albums Web</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">75 Pages</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2015</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.12 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/7968c211-0ae2-4604-9f5e-671708c5922d.webp" loading="lazy" alt="book image" title="DTIC ADA518756: Reforming Pentagon Decisionmaking" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/dtic-ada518756-reforming-pentagon-decisionmaking" class="jsx-1ac4e25e396cc2e7 title-link">DTIC ADA518756: Reforming Pentagon Decisionmaking</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">Defense Technical Information Center</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.59 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn-2.pdfdrive.to/media/content/thumbnails/1afe085f-7185-4433-9823-3b477746ca51.webp" loading="lazy" alt="book image" title="The Learning Leader: How to Focus School Improvement for Better Results" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/the-learning-leader-how-to-focus-school-improvement-for-better-results" class="jsx-1ac4e25e396cc2e7 title-link">The Learning Leader: How to Focus School Improvement for Better Results</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">Douglas B. Reeves</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">250 Pages</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2006</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">1.562 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/29b8745a-3f5a-4b39-8477-967071d971a7.webp" loading="lazy" alt="book image" title="Boletín Oficial de la República Argentina. 2007 2da sección" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/boletin-oficial-de-la-republica-argentina-2007-2da-seccion-3" class="jsx-1ac4e25e396cc2e7 title-link">Boletín Oficial de la República Argentina. 2007 2da sección</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">República Argentina</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">1.7 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/eaf1e984-1706-46e7-ab24-e06c59d000d7.webp" loading="lazy" alt="book image" title="Exhibition - CMA@MOCA" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/exhibition-cmamoca" class="jsx-1ac4e25e396cc2e7 title-link">Exhibition - CMA@MOCA</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">0.03 MB</span></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="mobile-ads"></div><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/acc9d8c0-975f-40f7-8662-42be20b8f78c.webp" loading="lazy" alt="book image" title="IFRS 15 investment management companies" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/ifrs-15-investment-management-companies-pdf" class="jsx-1ac4e25e396cc2e7 title-link">IFRS 15 investment management companies</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">45 Pages</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2017</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.28 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/8b0b9da9-4de9-450d-9a90-52c75a837df3.webp" loading="lazy" alt="book image" title="Greek Government Gazette: Part 1, 2006 no. 244" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/greek-government-gazette-part-1-2006-no-244" class="jsx-1ac4e25e396cc2e7 title-link">Greek Government Gazette: Part 1, 2006 no. 244</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">The Government of the Hellenic Republic</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2006</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.35 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/d2911ca2-937e-43ea-850e-85e6e7998e08.webp" loading="lazy" alt="book image" title="Capturing the Uncertainty in Adversary Attack Simulations" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/capturing-the-uncertainty-in-adversary-attack-simulations" class="jsx-1ac4e25e396cc2e7 title-link">Capturing the Uncertainty in Adversary Attack Simulations</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">37 Pages</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2008</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.24 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/4c96d6d1-f255-4611-aaef-3621c1e14d06.webp" loading="lazy" alt="book image" title="ERIC EJ1069138: Taking a Closer Look at Reading Online" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/eric-ej1069138-taking-a-closer-look-at-reading-online" class="jsx-1ac4e25e396cc2e7 title-link">ERIC EJ1069138: Taking a Closer Look at Reading Online</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">ERIC</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2006</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.9 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/5b5b5ebc-dbe6-4165-be4b-5c432d64edb2.webp" loading="lazy" alt="book image" title="Appalachian Regional Commission" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/appalachian-regional-commission" class="jsx-1ac4e25e396cc2e7 title-link">Appalachian Regional Commission</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">60 Pages</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2012</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">1.17 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="mobile-ads"></div><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/ea5ba8ff-da26-45a2-b729-45e883bf7a99.webp" loading="lazy" alt="book image" title="DTIC ADA458978: Valuing Programmed Depot Maintenance Speed: An Analysis of F-15 PDM" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/dtic-ada458978-valuing-programmed-depot-maintenance-speed-an-analysis-of-f-15-pdm" class="jsx-1ac4e25e396cc2e7 title-link">DTIC ADA458978: Valuing Programmed Depot Maintenance Speed: An Analysis of F-15 PDM</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">Defense Technical Information Center</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.54 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="/default_image.png" loading="lazy" alt="book image" title="böbrek nakli alıcı adayı değerlendirmesi" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/bobrek-nakli-alc-aday-degerlendirmesi" class="jsx-1ac4e25e396cc2e7 title-link">böbrek nakli alıcı adayı değerlendirmesi</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">112 Pages</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2012</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">7.95 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/29d56f7e-40f1-4bd9-8756-5463ecb03ec7.webp" loading="lazy" alt="book image" title="Socio-economic indicators in integrated coastal zone and community-based fisheries management [electronic resource] : case studies from the Caribbean" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/socio-economic-indicators-in-integrated-coastal-zone-and-community-based-fisheries-management-electronic-resource-case-studies-from-the-caribbean" class="jsx-1ac4e25e396cc2e7 title-link">Socio-economic indicators in integrated coastal zone and community-based fisheries management [electronic resource] : case studies from the Caribbean</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">Tietze, Uwe, 1950-</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2006</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">7.6 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/904bf504-d27c-4a98-8fd8-22bb9f0c7a78.webp" loading="lazy" alt="book image" title="Greek Government Gazette: Part 7, 2006 no. 674" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/greek-government-gazette-part-7-2006-no-674" class="jsx-1ac4e25e396cc2e7 title-link">Greek Government Gazette: Part 7, 2006 no. 674</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">The Government of the Hellenic Republic</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2006</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.19 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/90660cfd-6c74-4564-9b35-233a0fc41884.webp" loading="lazy" alt="book image" title="ERIC EJ1095001: FLAP: Personal Writing as a Tool for Physician Well-Being" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/eric-ej1095001-flap-personal-writing-as-a-tool-for-physician-well-being" class="jsx-1ac4e25e396cc2e7 title-link">ERIC EJ1095001: FLAP: Personal Writing as a Tool for Physician Well-Being</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">ERIC</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2006</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.18 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/c874e669-7709-4c08-8737-55a818d3ef59.webp" loading="lazy" alt="book image" title="Canada Ancillary Restraints Article" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/canada-ancillary-restraints-article" class="jsx-1ac4e25e396cc2e7 title-link">Canada Ancillary Restraints Article</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">Patrick English</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">21 Pages</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2010</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.1 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/a52b8214-9b71-41c6-88e7-3396c27bdb3b.webp" loading="lazy" alt="book image" title="Greek Government Gazette: Part 8, 2006 no. 118" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/greek-government-gazette-part-8-2006-no-118" class="jsx-1ac4e25e396cc2e7 title-link">Greek Government Gazette: Part 8, 2006 no. 118</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">The Government of the Hellenic Republic</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2006</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.28 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/321fdb32-e610-4636-8863-a44be39285c0.webp" loading="lazy" alt="book image" title="Aydın Çalışma ve İş Kurumu İl Müdürlüğü Resmi Web Sitesi" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/aydn-calsma-ve-is-kurumu-il-mudurlugu-resmi-web-sitesi" class="jsx-1ac4e25e396cc2e7 title-link">Aydın Çalışma ve İş Kurumu İl Müdürlüğü Resmi Web Sitesi</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">118 Pages</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2013</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">3.73 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/cdcec227-98e7-4082-b9aa-75ddf4292941.webp" loading="lazy" alt="book image" title="ERIC ED617122: Financial Literacy in Education Process: Literature Study" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/eric-ed617122-financial-literacy-in-education-process-literature-study" class="jsx-1ac4e25e396cc2e7 title-link">ERIC ED617122: Financial Literacy in Education Process: Literature Study</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">ERIC</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2022</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">0.51 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div><div class="jsx-1ac4e25e396cc2e7"><div class="jsx-1ac4e25e396cc2e7 collection-item"><img width="70" height="100" src="https://cdn.pdfdrive.to/media/content/thumbnails/c6764255-df7a-4a69-a2b6-ee0acb6849f1.webp" loading="lazy" alt="book image" title="by Carmen Agra Deedy illustrated by Renato Alarcão The Frog Princess" class="jsx-1ac4e25e396cc2e7 collection-img"><div class="jsx-1ac4e25e396cc2e7 information"><h3 class="jsx-1ac4e25e396cc2e7 title-info"><a href="/dl/by-carmen-agra-deedy-illustrated-by-renato-alarcao-the-frog-princess" class="jsx-1ac4e25e396cc2e7 title-link">by Carmen Agra Deedy illustrated by Renato Alarcão The Frog Princess</a></h3><div class="jsx-1ac4e25e396cc2e7 file-info"><span class="jsx-d61d22f45bf3720e">30 Pages</span><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">2014</span></div><div class="jsx-d61d22f45bf3720e"><span class="jsx-d61d22f45bf3720e dot">·</span><span class="jsx-d61d22f45bf3720e">4.13 MB</span></div></div><p class="jsx-1ac4e25e396cc2e7 desc"></p></div></div></div></div><div class="jsx-7d4e960818f559df ads5"></div></div><div class="jsx-7d4e960818f559df detail-right"><div class="jsx-17865f972f87972f category-container"><div class="jsx-17865f972f87972f premium"><div class="jsx-563764575"><a href="/premium" rel="nofollow" class="jsx-563764575 premium-btn animated-button7"><span class="jsx-563764575"></span><span class="jsx-563764575"></span><span class="jsx-563764575"></span><span class="jsx-563764575"></span><svg id="icon-diamond" viewBox="0 0 32 32" fill="#fff" width="40" height="40" 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Oppenheim","description":null,"page":967,"release_year":2013,"file_size":292.34,"downloaded_count":1096,"language":"english","summary":"l  •.• \nMJ..  I •  ' \nALAN V. OPPENHEIM \nALAN S. WILLSKV \nWITH S.  HAMID NAWAB \n• \nPRENTICE HALL SIGNAL PROCESSING SERIES  . \nALAN V. OPPENHEIM, SERIES EDITOR  ~  ~\nSECOND EDITION \nSIGNALS \n\u0026 \nSYSTEMS \nv. \nALAN  OPPENHEIM \ns. w \nALAN  ILLSKY \nMAssACHUSETTS INsTITUTE OF TEcHNOLOGY \nWITH \nS.IIAMIDNAWAB \nBosToN UNIVERSITY \nPRENTICE-HALL  INTERNATIONAL, INC.\nCoNTENTS \nPREFACE  XVII \nAcKNOWLEDGMENTS  xxv \nFoREWORD  xxvii \n1 \n1 \nSIGNALS  AND  SYSTEMS \n1.0  . Introduction  1 \n1.1  Continuous-Time and Discrete-Time Signals  1 \n1.1.1  Examples and Mathematical Representation  1 \n1.1.2  Signal Energy and Power  5 \n1.2  Transformations of the Independent Variable  7 \n1.2.1  Examples of Transformations of the Independent Variable  8 \n1.2.2  Periodic Signals  11 \n1.2.3  Even and Odd Signals  13 \n1.3  Exponential and Sinusoidal Signals  14 \n1.3.1  Continuous-Time Complex Exponential and Sinusoidal Signals  15 \n1.3.2  Discrete-Time Complex Exponential and Sinusoidal Signals  21  · \n1.3.3  Periodicity Properties of Discrete-Time Complex Exponentials  25 \n1.4  The Unit Impulse and Unit Step Functions  30 \n1.4.1  The Discrete-Time Unit Impulse and Unit Step Sequences  30 \n1.4.2  The Continuous-Time Unit Step and Unit Impulse Functions  32 \n1.5  Continuous-Time and Discrete-Time Systems  38 \n1.5.1  Simple Examples of Systems  39 \n1.5.2  Interconnections of Systems  41 \n1.6  Basic System Properties  44 \n1.6.1  Systems with and without Memory  44 \n1.6.2  Invertibility and Inverse Systems  45 \n1.6.3  Causality  46 \n1.6.4  Stability  48 \n1.6.5  Time Invariance  50 \n1.6.6  Linearity  53 \n1.7  Summary  56 \nProblems  57 \n2 \n74 \nLINEAR  TIME-INVARIANT  SYSTEMS \n2.0  Introduction  74 \n2.1  Discrete-Time LTI Systems: The Convolution Sum  75 \nvii\nviii  Contents \n2.1.1  . The Representation of Discrete-Time Signals in Terms \nof Impulses  75 \n2.1.2  The Discrete-Time Unit Impulse Response and the Convolution-Sum \nRepresentation of LTI Systems  77 \n2.2  Continuous-Time LTI Systems: The Convolution Integral  90 \n2.2.1  The Representation of Continuous-Time Signals in Terms \nof Impulses  90 \n2.2.2  The Continuous-Time Unit Impulse Response and the Convolution \nIntegral Representation of LTI Systems  94 \n2.3  Properties of Linear Time-Invariant Systems  103 \n2.3.1  The Commutative Property  104 \n2.3.2  The Distributive Property  104 \n2.3.3  The Associative Property  107 \n2.3.4  LTI Systems with and without Memory  108 \n2.3.5  Invertibility of LTI Systems  109 \n2.3.6  Causality for LTI Systems  112 \n2.3.7  Stability for LTI Systems  113 \n2.3.8 • The Unit Step Response of an LTI System  115 \n2.4  Causal LTI Systems Described by Differential and Difference \n· Equations  116 \n2.4.1  Linear Constant -Coefficient Differential Equations  117 \n2.4.2  Linear Constant-Coefficient Difference Equations  121 \n2.4.3  Block Diagram Representations of First-Order Systems Described \nby Differential and Difference Equations  124 \n2.5  Singularity Functions  127 \n2.5.1  The Unit Impulse as an Idealized Short Pulse  128 \n2.5.2  Defining the Unit Impulse through Convolution  131 \n2.5.3  Unit Doublets and Other Singularity Functions  132 \n2.6  Summary  137 \nProblems  137 \n3 \nFoURIER  SERIES  REPRESENTATION  OF \n177 \nPERIODIC  SIGNALS \n3.0  Introduction  177 \n3.1  A Historical Perspective  178 \n3.2  The Response of LTI Systems to Complex Exponentials  182 \n3.3  Fourier Series Representation of Continuous-Time \nPeriodic Signals  186 \n3.3.1  Linear Combinations of Harmonically Related Complex \nExponentials  186 \n3.3.2  Determination of the Fourier Series Representation \nof a Continuous-Time Periodic Signal  190 \n3.4  Convergence of the Fourier Series  195\nContents  ix \n3.5  Properties of Continuous-Time Fourier Series  202 \n3.5.1  Linearity  202 \n3.5.2  Time Shifting  202 \n3.5.3  Time Reversal  203 \n3.5.4  Time Scaling  204 \n3.5.5  Multiplication  204 \n3.5.6  Conjugation and Conjugate Symmetry  204 \n3.5.7  Parseval's Relation for Continuous-Time Periodic Signals  205 \n3.5.8  Summary of Properties of the Continuous-Time Fourier Series  205 \n3.5.9  Examples  205 \n3.6  Fourier Series Representation of Discrete-Time \nPeriodic Signals  211 \n3.6.1  Linear Combinations of Harmonically Related Complex \nExponentials  211 \n3.6.2  Determination of the Fourier Series Representation of a · \nPeriodic Signal  212 \n3.7  Properties of Discrete-Time Fourier Series  221 \n3.7.1  Multiplication  222 \n3.7.2  First Difference  222 \n3.7.3  Parseval's Relation for Discrete-Time Periodic Signals  223 \n3.7.4  Examples  223 \n3.8  Fourier Series and LTI Systems  226 \n3.9  Filtering  231 \n3.9.1  Frequency-Shaping Filters  232 \n3.9.2  Frequency-Selective Filters  236 \n3.10  Examples of Continuous-Time Filters Described by \nDifferential Equations  239 \n3.10.1  A Simple RC Lowpass Filter  239 \n3.10.2  A Simple RC Highpass Filter  241 \n3.11  Examples of Discrete-Time Filters Described by \nDifference Equations  244 \n3.11.1  First-Order Recursive Discrete-Time Filters  244 \n3.11.2  Nonrecursive Discrete-Time Filters  245 \n3.12  Summary  249 \nProblems  250 \n4 \nThE CONTINUOUS-TIME  FOURIER TRANSFORM  284 \n4.0  Introduction  284 \n4.1  Representation of Aperiodic Signals: The Continuous-Time \nFourier Transform  285 \n4.1.1  Development of the Fourier Transform Representation \nof an Aperiodic Signal  285 \n4.1.2  Convergence of Fourier Transforms  289 \n4.1.3  Examples of Continuous-Time Fourier Transforms  290\nX  Contents \n4.2  The Fourier Transform for Periodic Signals  296 \n4.3  Properties of the Continuous-Time Fourier Transform  300 \n4.3.1  Linearity  301 \n4.3.2  Time Shifting  301 \n4.3.3  Conjugation and Conjugate Symmetry  303 \n4.3.4  Differentiation and Integration  306 \n4.3.5  Time and Frequency Scaling  308 \n4.3.6  Duality  309 \n4.3.7  Parseval's Relation  312 \n4.4  The Convolution Property  314 \n4.4.1  Examples  317 \n4.5  The Multiplication Property  322 \n4.5.1  Frequency-Selective Filtering with Variable Center Frequency  325 \n4.6  Tables of Fourier Properties and of Basic Fourier \nTransform Pairs  328  · \n4.7  Systems Characterized by Linear Constant-Coefficient \nDifferential Equations  330 \n4.8  Summary  333 \nProblems  334 \n5 \n358 \nThE  DISCRETE-TIME  FOURIER  TRANSFORM \n5.0  Introduction  358 \n5.1  Representation of Aperiodic Signals: The Discrete-Time \nFourier Transform  359 \n5.1.1  Development of the Discrete-~ime Fourier Transform  359 \n5.1.2  Examples of Discrete-Time Fourier Transforms  362 \n5 .1.3  Convergence Issues Associated with the Discrete-Time Fourier \nTransform  366 \n5.2  The Fourier Transform for Periodic Signals  367 \n5.3  Properties of the Discrete-Time Fourier Transform  372 \n5.3.1.  Periodicity of the Discrete-Time Fourier Transform  373 \n5.3.2  Linearity of the Fourier Transform  373 \n5.3.3  Time Shifting and Frequency Shifting  373 \n5.3.4  Conjugation and Conjugate Symmetry  375 \n5.3.5  Differencing and Accumulation  375 \n5.3.6  Time Reversal  376 \n5.3.7  Time Expansion  377 \n5.3.8  Differentiation in Frequency  380 \n5.3.9  Parseval's Relation  380 \n5.4  The Convolution Property  382 \n5.4.1  Examples  383 \n5.5  The Multiplication Property  388 \n5.6  Tables of Fourier Transform Properties and Basic Fourier \nTransform Pairs  390\nContents  xi \n5. 7  Duality  390 \n5.7.1  Duality in the Discrete-Time Fourier Series  391 \n5.7.2  Duality between the Discrete-Time Fourier Transform and the \nContinuous-Time Fourier Series  395 \n5.8  Systems Characterized by Linear Constant-Coefficient \nDifference Equations  396 \n5.9  Summary  399 \nProblems  400 \n6 \nTIME  AND  FREQUENCY  CHARACTERIZATION \n423 \nOF  SIGNALS  AND  SYSTEMS \n6.0  Introduction  423 \n6.1  The Magnitude-Phase 'Representation of. the Fourier \nTransform  423 \n6.2  The Magnitude-Phase Representation of the Frequency Response \nof LTI Systems  427 \n6.2.1  Linear and Nonlinear Phase  428 \n6.2.2  Group Delay  430 \n6.2.3  Log-Magnitude and Bode Plots  436 \n6.3  Time-Domain Properties of Ideal Frequency-Selective \nFilters  439 \n6.4  Time-Domain and Frequency-Domain Aspects of Nonideal \nFilters  444 \n6.5  First-Order and Second-Order Continuous-Time Systems  448 \n6.5 .1  First -Order Continuous-Time Systems  448 \n6.5 .2  Second-Order Continuous-Time Systems  451 \n6.5.3  Bode Plots for Rational Frequency Responses  456 \n6.6  First-Order and Second-Order Discrete-Time Systems  461 \n6.6.1  First-Order Discrete-Time Systems  461 \n6.6.2  Second-Order Discrete-Time Systems  465 \n6.7  Examples of Time- and Frequency-Domain Analysis \nof Systems  472 \n6.7.1  Analysis of an Automobile Suspension System  473 \n6.7.2  Examples of Discrete-Time Nonrecursive Filters  476 \n6.8  Summary  482 \nProblems  483 \n7 \n514 \nSAMPLING \n7.0  Introduction  514 \n7.1  Representation of a Continuous-Time Signal by Its Samples: \nThe Sampling Theorem  515 \n7.1.1  Impulse-Train Sampling  516 \n7 .1.2  Sampling with a Zero-Order Hold  520\nxii  Contents \n7.2  Reconstruction of a Signal from Its Samples Using \nInterpolation  522 \n7.3  The Effect of Undersampling: Aliasing  527 \n7.4  Discrete-Time Processing of Continuous-Time Signals  534 \n7.4.1  Digital Differentiator  541 \n7.4.2  Half-Sample Delay  543 \n7.5  Sampling of Discrete-Time Signals  545 \n7.5.1  Impulse-Train Sampling  545 \n7.5 .2  Discrete-Time Decimation and Interpolation  549 \n7.6  Summary  555 \nProblems  556 \n8 \nCoMMUNICATION  SYSTE.M S  582 \n8.0  Introduction  582 \n8.1  Complex Exponential and Sinusoidal Amplitude Modulation  583 \n8.1.1  Amplitude Modulation with a Complex Exponential Carrier  583 \n8.1.2  Amplitude Modulation with a Sinusoidal Carrier  585 \n8.2  Demodulation for Sinusoidal AM  587 \n8.2.1  Synchronous Demodulation  587 \n8.2.2  Asynchronous Demodulation  590 \n8.3  Frequency-Division Multiplexing  594 \n8.4  Single-Sideband Sinusoidal Amplitude Modulation  597 \n8.5  Amplitude Modulation with a Pulse-Train Carrier  601 \n8.5.1  Modulation of a Pulse-Train Carrier  601 \n8.5.2  Time-Division Multiplexing  604 \n8.6  Pulse-Amplitude Modulation  604 \n8.6.1  Pulse-Amplitude Modulated Signals  604 \n8.6.2  Intersymbol Interference in PAM Systems  607 \n8.6.3  Digit~l Pulse-Amplitude and Pulse-Code Modulation  610 \n8.7  Sinusoidal Frequency Modulation  611 \n8.7.1  Narrowband Frequency Modulation  613 \n8.7.2  Wideband Frequency Modulation  615 \n8.7.3  Periodic Square-Wave Modulating Signal  617 \n8.8  Discrete-Time Modulation  619 \n8.8.1  Discrete-Time Sinusoidal Amplitude Modulation  619 \n8.8.2  Discrete-Time Transmodulation  623 \n8.9  Summary  623 \nProblems  625 \n9 \n654 \nTHE  LAPLACE  TRANSFORM \n9.0  Introduction  654 \n9.1  The Laplace Transform  655 \n9.2  The Region of Convergence for Laplace Transforms  662\nContents  xiii \n9.3  The Inverse Laplace Transfonn  670 \n9.4  Geometric Evaluation of the Fourier Transfonn from the \nPole-Zero Plot  674 \n9.4.1  . First-Order Systems  676 \n9.4.2  Second-Order Systems  677 \n9.4.3  All-Pass Systems  681 \n9.5  Properties of the Laplace Transfonn  682 \n9.5.1  Linearity of the Laplace Transform  683 \n9.5.2  Time Shifting  684 \n9.5.3  Shifting in the s-Domain  685 \n9.5 .4  Time Scal1ng  685 \n9.5.5  Conjugation  687 \n9.5.6  Convolution Property  687 \n9.5.7  Differentiation in the Time Domain  688 \n9.5.8  Differentiation in the s-Domain  688 \n9.5.9  Integration in the Time Domain  690 \n9.5.10  The Initial- and Final-Value Theorems  690 \n9.5.11  Table of Properties  6?1 \n9.6  Some Laplace Transfonn Pairs  692 \n9.7  Analysis and Characterization of LTI Systems Using the \nLaplace Transfonn  693 \n9.7 .1  Causality  693 \n9.7.2  Stability  695 \n9.7.3  LTI Systems Characterized by Linear Constant-Coefficient \nDifferential Equations  698 \n9.7.4  Examples Relating System Behavior to the System Function  701 \n9.7.5  Butterworth Filters  703 \n9.8  System Function Algebra and Block Diagram \nRepresentations  706 \n9.8.1  System Functions for Interconnections of LTI Systems  707 \n9.8.2  Block Diagram Representations for Causal LTI Systems Described \nby Differential Equations and Rational System Functions  708 \n9.9  The Unilateral Laplace Transfonn  714 \n9.9.1  Examples of Unilateral Laplace Transforms  714 \n9.9.2  Properties of the Unilateral Laplace Transform  716 \n9.9.3  Solving Differential Equations Using the Unilateral Laplace \nTransform  719 \n9.10  Summary  720 \nProblems  721 \n10 \n741 \nTHE  Z-TRANSFORM \n10.0  Introduction  741 \n10.1  The z-Transfonn  741 \n10.2  The Region of Convergence for the z-Transfonn.  748\nxiv  Contents \n10.3  The Inverse z-Transform  757 \n10.4  Geometric Evaluation of the Fourier Transform from the \nPole-Zero Plot  763 \n10.4.1  First-Order Systems  763 \n10.4.2  Second-Order Systems  765 \n10.5  Properties of the z-Transform  767 \n10.5.1  Linearity  767 \n10.5.2  Time Shifting  767 \n10.5.3  Scaling in the z-Domain  768 \n10.5.4  Time Reversal  769 \n10.5.5  Time Expansion  769 \n10.5.6  Conjugation  770 \n10.5.7  The Convolution Property  770 \n10.5.8  Differentiation in the z-Domain  772 \n10.5.9  The Initial-Value Theorem  773 \n10.5.10  Summary of Properties  774 \n10.6  Some Common z-Transform Pairs  774 \n10.7  Analysis and Characterization of LTI Systems Using \nz-Transforms  774 \n10.7.1  Causality  776 \n10.7.2  Stability  777 \n10.7.3  LTI Systems Characterized by Linear Constant-Coefficient \nDifference Equations  779 \n10.7.4  Examples Relating System Behavior to the System Function  781 \n10.8  System Function Algebra and Block Diagram \nRepresentations  783 \n10.8.1  System Functions for Interconnections of LTI Systems  784 \n10.8.2  Block Diagram Representations for Causal LTI Systems Described \nby Difference Equations and Rational System Functions  784 \n10.9  The Unilateral z-Transform  789 \n10.9.1  Examples of Unilateral z-Transforms and Inverse Transforms  790 \n10.9.2  Properties of the Unilateral z-Transform  792 \n10.9.3  Solving Difference Equations Using the Unilateral \nz-Transform  795 \n10.10  Summary  796 \nProblems  797 \n11 \n816 \nUNEAR  FEEDBACK  SYSTEMS \n11.0  Introduction  816 \n11.1  Linear Feedback Systems  819 \n11.2  Some Applications and Consequences of Feedback  820 \n11.2.1  Inverse System Design  820 \n11.2.2  Compensation for Nonideal Elements  821 \n11.2.3  Stabilization of Unstable Systems  823","status":1,"price":null,"rent_price":null,"isbn":null,"redirect_url":"book"},"myLikesProps":null,"myDriveProps":null,"requestedBooksProps":null,"user":null},"__N_SSP":true},"page":"/book/[detail]","query":{"detail":"signals-and-system"},"buildId":"0yngpp-mRDH7Bzgj1Hnuf","isFallback":false,"gssp":true,"scriptLoader":[]}</script><next-route-announcer><p aria-live="assertive" id="__next-route-announcer__" role="alert" style="border: 0px; 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