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信号与系统（第二版）（英文版）

作译者：Alan V. Oppenheim

出版时间：2020-08 千字数：2047 版次：01-01 页数：984

开本：16开装帧： I S B N ：9787121393327

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内容简介

前言

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CONTENTS  目录
Chapter 1  Signals and Systems  信号与系统
1.0  Introduction  引言
1.1  Continuous-Time and Discrete-Time Signals  连续时间信号和离散时间信号
1.1.1  Examples and Mathematical Representation  举例与数学表示
1.1.2  Signal Energy and Power  信号能量与功率
1.2  Transformations of the Independent Variable  自变量的变换
1.2.1  Examples of Transformations of the Independent Variable  自变量变换举例
1.2.2  Periodic Signals  周期信号
1.2.3  Even and Odd Signals  偶信号与奇信号
1.3  Exponential and Sinusoidal Signals  指数信号与正弦信号
1.3.1  Continuous-Time Complex Exponential and Sinusoidal Signals  连续时间复指数信号与正弦信号
1.3.2  Discrete-Time Complex Exponential and Sinusoidal Signals  离散时间复指数信号与正弦信号
1.3.3  Periodicity Properties of Discrete-Time Complex Exponentials  离散时间复指数序列的周期性质
1.4  The Unit Impulse and Unit Step Functions  单位冲激函数与单位阶跃函数
1.4.1  The Discrete-Time Unit Impulse and Unit Step Sequences  离散时间单位脉冲序列和单位阶跃序列
1.4.2  The Continuous-Time Unit Step and Unit Impulse Functions  连续时间单位阶跃函数和单位冲激函数
1.5  Continuous-Time and Discrete-Time Systems  连续时间系统和离散时间系统
1.5.1  Simple Examples of Systems  简单系统举例
1.5.2  Interconnections of Systems  系统的互联
1.6  Basic System Properties  基本系统性质
1.6.1  Systems with and without Memory  有记忆系统与无记忆系统
1.6.2  Invertibility and Inverse Systems  可逆性与可逆系统
1.6.3  Causality  因果性
1.6.4  Stability  稳定性
1.6.5  Time Invariance  时不变性
1.6.6  Linearity   线性
1.7  Summary  小结
Problems  习题

Chapter 2  Linear Time-Invariant Systems  线性时不变系统
2.0  Introduction  引言
2.1  Discrete-Time LTI Systems: The Convolution Sum  离散时间线性时不变系统：卷积和
2.1.1  The Representation of Discrete-Time Signals in Terms of Impulses  用脉冲表示离散时间信号
2.1.2  The Discrete-Time Unit Impulse Response and the Convolution-Sum Representation of LTI Systems  离散时间线性时不变系统的单位脉冲响应及卷积和表示
2.2  Continuous-Time LTI Systems: The Convolution Integral  连续时间线性时不变系统：卷积积分
2.2.1  The Representation of Continuous-Time Signals in Terms of Impulses  用冲激表示连续时间信号
2.2.2  The Continuous-Time Unit Impulse Response and the Convolution Integral Representation of LTI Systems  连续时间线性时不变系统的单位冲激响应及卷积积分表示
2.3  Properties of Linear Time-Invariant Systems  线性时不变系统的性质
2.3.1  The Commutative Property  交换律性质
2.3.2  The Distributive Property  分配律性质
2.3.3  The Associative Property  结合律性质
2.3.4  LTI Systems with and without Memory  有记忆和无记忆线性时不变系统
2.3.5  Invertibility of LTI Systems  线性时不变系统的可逆性
2.3.6  Causality for LTI Systems  线性时不变系统的因果性
2.3.7  Stability for LTI Systems  线性时不变系统的稳定性
2.3.8  The Unit Step Response of an LTI System  线性时不变系统的单位阶跃响应
2.4  Causal LTI Systems Described by Differential and Difference Equations  用微分方程和差分方程描述的因果线性时不变系统
2.4.1  Linear Constant-Coefficient Differential Equations  线性常系数微分方程
2.4.2  Linear Constant-Coefficient Difference Equations  线性常系数差分方程
2.4.3  Block Diagram Representations of First-Order Systems Described by Differential and Difference Equations  用微分方程和差分方程描述的一阶系统的方框图表示
2.5  Singularity Functions  奇异函数
2.5.1  The Unit Impulse as an Idealized Short Pulse  作为理想化短脉冲的单位冲激
2.5.2  Defining the Unit Impulse through Convolution  通过卷积定义单位冲激
2.5.3  Unit Doublets and Other Singularity Functions  单位冲激偶和其他奇异函数
2.6  Summary  小结
Problems  习题

Chapter 3  Fourier Series Representation of Periodic Signals  周期信号的傅里叶级数表示
3.0  Introduction  引言
3.1  A Historical Perspective  历史回顾
3.2  The Response of LTI Systems to Complex Exponentials  线性时不变系统对复指数信号的响应
3.3  Fourier Series Representation of Continuous-Time Periodic Signals  连续时间周期信号的傅里叶级数表示
3.3.1  Linear Combinations of Harmonically Related Complex Exponentials  成谐波关系的复指数信号的线性组合
3.3.2  Determination of the Fourier Series Representation of a Continuous-Time Periodic Signal  连续时间周期信号傅里叶级数表示的确定
3.4  Convergence of the Fourier Series  傅里叶级数的收敛
3.5  Properties of Continuous-Time Fourier Series  连续时间傅里叶级数性质
3.5.1  Linearity  线性性质
3.5.2  Time Shifting  时移性质
3.5.3  Time Reversal  时间反转性质
3.5.4  Time Scaling  时域尺度变换性质
3.5.5  Multiplication  相乘性质
3.5.6  Conjugation and Conjugate Symmetry  共轭与共轭对称性质
3.5.7  Parseval’s Relation for Continuous-Time Periodic Signals  连续时间周期信号的帕塞瓦尔定理
3.5.8  Summary of Properties of the Continuous-Time Fourier Series  连续时间傅里叶级数性质列表
3.5.9  Examples  举例
3.6  Fourier Series Representation of Discrete-Time Periodic Signals  离散时间周期信号的傅里叶级数表示
3.6.1  Linear Combinations of Harmonically Related Complex Exponentials  成谐波关系的复指数信号的线性组合
3.6.2  Determination of the Fourier Series Representation of a Periodic Signal  周期信号傅里叶级数表示的确定
3.7  Properties of Discrete-Time Fourier Series  离散时间傅里叶级数性质
3.7.1  Multiplication  相乘性质
3.7.2  First Difference  一次差分性质
3.7.3  Parseval’s Relation for Discrete-Time Periodic Signals  离散时间周期信号的帕塞瓦尔定理
3.7.4  Examples  举例
3.8  Fourier Series and LTI Systems  傅里叶级数与线性时不变系统
3.9  Filtering  滤波
3.9.1  Frequency-Shaping Filters  频率成形滤波器
3.9.2  Frequency-Selective Filters  频率选择性滤波器
3.10  Examples of Continuous-Time Filters Described by Differential Equations  用微分方程描述的连续时间滤波器举例
3.10.1  A Simple RC Lowpass Filter  简单RC低通滤波器
3.10.2  A Simple RC Highpass Filter  简单RC高通滤波器
3.11  Examples of Discrete-Time Filters Described by Difference Equations  用差分方程描述的离散时间滤波器举例
3.11.1  First-Order Recursive Discrete-Time Filters  一阶递归离散时间滤波器
3.11.2  Nonrecursive Discrete-Time Filters   非递归离散时间滤波器
3.12  Summary  小结
Problems  习题

Chapter 4  The Continuous-Time Fourier Transform  连续时间傅里叶变换
4.0  Introduction  引言
4.1  Representation of Aperiodic Signals: The Continuous-Time Fourier Transform  非周期信号的表示：连续时间傅里叶变换
4.1.1  Development of the Fourier Transform Representation of an Aperiodic Signal  非周期信号傅里叶变换表示的导出
4.1.2  Convergence of Fourier Transforms  傅里叶变换的收敛
4.1.3  Examples of Continuous-Time Fourier Transforms  连续时间傅里叶变换举例
4.2  The Fourier Transform for Periodic Signals  周期信号的傅里叶变换
4.3  Properties of the Continuous-Time Fourier Transform  连续时间傅里叶变换性质
4.3.1  Linearity  线性性质
4.3.2  Time Shifting  时移性质
4.3.3  Conjugation and Conjugate Symmetry  共轭与共轭对称性质
4.3.4  Differentiation and Integration  微分与积分性质
4.3.5  Time and Frequency Scaling  时间与频率的尺度变换性质
4.3.6  Duality  对偶性质
4.3.7  Parseval’s Relation  帕塞瓦尔定理
4.4  The Convolution Property  卷积性质
4.4.1  Examples  举例
4.5  The Multiplication Property  相乘性质
4.5.1  Frequency-Selective Filtering with Variable Center Frequency  具有可变中心频率的频率选择性滤波
4.6  Tables of Fourier Properties and of Basic Fourier Transform Pairs  傅里叶变换性质和基本傅里叶变换对列表
4.7  Systems Characterized by Linear Constant-Coefficient Differential Equations  由线性常系数微分方程表征的系统
4.8  Summary  小结
Problems  习题

Chapter 5  The Discrete-Time Fourier Transform  离散时间傅里叶变换
5.0  Introduction  引言
5.1  Representation of Aperiodic Signals: The Discrete-Time Fourier Transform  非周期信号的表示：离散时间傅里叶变换
5.1.1  Development of the Discrete-Time Fourier Transform  离散时间傅里叶变换的导出
5.1.2  Examples of Discrete-Time Fourier Transforms  离散时间傅里叶变换举例
5.1.3  Convergence Issues Associated with the Discrete-Time Fourier Transform  关于离散时间傅里叶变换的收敛问题
5.2  The Fourier Transform for Periodic Signals  周期信号的傅里叶变换
5.3  Properties of the Discrete-Time Fourier Transform  离散时间傅里叶变换性质
5.3.1  Periodicity of the Discrete-Time Fourier Transform  离散时间傅里叶变换的周期性
5.3.2  Linearity of the Fourier Transform  线性性质
5.3.3  Time Shifting and Frequency Shifting  时移与频移性质
5.3.4  Conjugation and Conjugate Symmetry  共轭与共轭对称性质
5.3.5  Differencing and Accumulation  差分与累加性质
5.3.6  Time Reversal  时间反转性质
5.3.7  Time Expansion  时域扩展性质
5.3.8  Differentiation in Frequency  频域微分性质
5.3.9  Parseval’s Relation  帕塞瓦尔定理
5.4  The Convolution Property  卷积性质
5.4.1  Examples  举例
5.5  The Multiplication Property  相乘性质
5.6  Tables of Fourier Transform Properties and Basic Fourier Transform Pairs  傅里叶变换性质和基本傅里叶变换对列表
5.7  Duality  对偶性质
5.7.1  Duality in the Discrete-Time Fourier Series  离散时间傅里叶级数的对偶性质
5.7.2  Duality between the Discrete-Time Fourier Transform and the Continuous-Time Fourier Series  离散时间傅里叶变换和连续时间傅里叶级数之间的对偶性质
5.8  Systems Characterized by Linear Constant-Coefficient Difference Equations  由线性常系数差分方程表征的系统
5.9  Summary  小结
Problems  习题

Chapter 6  Time and Frequency Characterization of Signals and Systems  信号与系统的时域和频域特性
6.0  Introduction  引言
6.1  The Magnitude-Phase Representation of the Fourier Transform  傅里叶变换的模和相位表示
6.2  The Magnitude-Phase Representation of the Frequency Response of LTI Systems  线性时不变系统频率响应的模和相位表示
6.2.1  Linear and Nonlinear Phase  线性与非线性相位
6.2.2  Group Delay  群延迟
6.2.3  Log-Magnitude and Bode Plots  对数模和伯德图
6.3  Time-Domain Properties of Ideal Frequency-Selective Filters  理想频率选择性滤波器的时域特性
6.4  Time-Domain and Frequency-Domain Aspects of Nonideal Filters  非理想滤波器的时域和频域特性讨论
6.5  First-Order and Second-Order Continuous-Time Systems  一阶与二阶连续时间系统
6.5.1  First-Order Continuous-Time Systems  一阶连续时间系统
6.5.2  Second-Order Continuous-Time Systems  二阶连续时间系统
6.5.3  Bode Plots for Rational Frequency Responses  有理型频率响应的伯德图
6.6  First-Order and Second-Order Discrete-Time Systems  一阶与二阶离散时间系统
6.6.1  First-Order Discrete-Time Systems  一阶离散时间系统
6.6.2  Second-Order Discrete-Time Systems  二阶离散时间系统
6.7  Examples of Time- and Frequency-Domain Analysis of Systems  系统的时域分析与频域分析举例
6.7.1  Analysis of an Automobile Suspension System  汽车减震系统的分析
6.7.2  Examples of Discrete-Time Nonrecursive Filters  离散时间非递归滤波器举例
6.8  Summary  小结
Problems  习题

Chapter 7  Sampling  采样
7.0  Introduction  引言
7.1  Representation of a Continuous-Time Signal by Its Samples: The Sampling Theorem  用信号样本表示连续时间信号：采样定理
7.1.1  Impulse-Train Sampling  冲激串采样
7.1.2  Sampling with a Zero-Order Hold  零阶保持采样
7.2  Reconstruction of a Signal from Its Samples Using Interpolation  利用内插由样本重建信号
7.3  The Effect of Undersampling: Aliasing  欠采样的效果：混叠现象
7.4  Discrete-Time Processing of Continuous-Time Signals  连续时间信号的离散时间处理
7.4.1  Digital Differentiator  数字微分器
7.4.2  Half-Sample Delay  半采样间隔延迟
7.5  Sampling of Discrete-Time Signals  离散时间信号采样
7.5.1  Impulse-Train Sampling  脉冲串采样
7.5.2  Discrete-Time Decimation and Interpolation  离散时间抽取与内插
7.6  Summary  小结
Problems  习题

Chapter 8  Communication Systems  通信系统
8.0  Introduction  引言
8.1  Complex Exponential and Sinusoidal Amplitude Modulation  复指数与正弦幅度调制
8.1.1  Amplitude Modulation with a Complex Exponential Carrier  复指数载波的幅度调制
8.1.2  Amplitude Modulation with a Sinusoidal Carrier  正弦载波的幅度调制
8.2  Demodulation for Sinusoidal AM  正弦幅度调制的解调
8.2.1  Synchronous Demodulation  同步解调
8.2.2  Asynchronous Demodulation  非同步解调
8.3  Frequency-Division Multiplexing  频分多路复用
8.4  Single-Sideband Sinusoidal Amplitude Modulation  单边带正弦幅度调制
8.5  Amplitude Modulation with a Pulse-Train Carrier  用脉冲串进行载波的幅度调制
8.5.1  Modulation of a Pulse-Train Carrier  脉冲串载波调制
8.5.2  Time-Division Multiplexing  时分多路复用
8.6  Pulse-Amplitude Modulation  脉冲幅度调制
8.6.1  Pulse-Amplitude Modulated Signals  脉冲幅度已调信号
8.6.2  Intersymbol Interference in PAM Systems  脉冲幅度调制系统中的码间干扰
8.6.3  Digital Pulse-Amplitude and Pulse-Code Modulation  数字脉冲幅度调制和脉冲编码调制
8.7  Sinusoidal Frequency Modulation  正弦频率调制
8.7.1  Narrowband Frequency Modulation  窄带频率调制
8.7.2  Wideband Frequency Modulation  宽带频率调制
8.7.3  Periodic Square-Wave Modulating Signals  周期方波调制信号
8.8  Discrete-Time Modulation  离散时间调制
8.8.1  Discrete-Time Sinusoidal Amplitude Modulation  离散时间正弦幅度调制
8.8.2  Discrete-Time Transmodulation  离散时间调制转换
8.9  Summary  小结
Problems  习题

Chapter 9  The Laplace Transform  拉普拉斯变换
9.0  Introduction  引言
9.1  The Laplace Transform  拉普拉斯变换
9.2  The Region of Convergence for Laplace Transforms  拉普拉斯变换的收敛域
9.3  The Inverse Laplace Transform  拉普拉斯逆变换
9.4  Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot  利用零极点图对傅里叶变换进行几何求值
9.4.1  First-Order Systems  一阶系统
9.4.2  Second-Order Systems  二阶系统
9.4.3  All-Pass Systems  全通系统
9.5  Properties of the Laplace Transform  拉普拉斯变换的性质
9.5.1  Linearity of the Laplace Transform  线性性质
9.5.2  Time Shifting  时移性质
9.5.3  Shifting in the s-Domain  s域平移性质
9.5.4  Time Scaling  时域尺度变换性质
9.5.5  Conjugation  共轭性质
9.5.6  Convolution Property  卷积性质
9.5.7  Differentiation in the Time Domain  时域微分性质
9.5.8  Differentiation in the s-Domain  s域微分性质
9.5.9  Integration in the Time Domain  时域积分性质
9.5.10 The Initial-and Final-Value Theorems  初值定理与终值定理
9.5.11 Table of Properties  性质列表
9.6  Some Laplace Transform Pairs   常用拉普拉斯变换对
9.7  Analysis and Characterization of LTI Systems Using the Laplace Transform  用拉普拉斯变换分析与表征线性时不变系统
9.7.1  Causality  因果性
9.7.2  Stability  稳定性
9.7.3  LTI Systems Characterized by Linear Constant-Coefficient Differential Equations  由线性常系数微分方程表征的线性时不变系统
9.7.4  Examples Relating System Behavior to the System Function  系统特性与系统函数的关系举例
9.7.5  Butterworth Filters  巴特沃思滤波器
9.8  System Function Algebra and Block Diagram Representations  系统函数的代数属性与方框图表示
9.8.1  System Functions for Interconnections of LTI Systems  线性时不变系统互联的系统函数
9.8.2  Block Diagram Representations for Causal LTI Systems Described by Differential Equations and Rational System Functions  由微分方程和有理系统函数描述的因果线性时不变系统的方框图表示
9.9  The Unilateral Laplace Transform  单边拉普拉斯变换
9.9.1  Examples of Unilateral Laplace Transforms  单边拉普拉斯变换举例
9.9.2  Properties of the Unilateral Laplace Transform  单边拉普拉斯变换性质
9.9.3  Solving Differential Equations Using the Unilateral Laplace Transform  利用单边拉普拉斯变换求解微分方程
9.10  Summary  小结
Problems  习题

Chapter 10  The z-Transform  z变换
10.0  Introduction  引言
10.1  The z-Transform  z变换
10.2  The Region of Convergence for the z-Transform  z变换的收敛域
10.3  The Inverse z-Transform  z逆变换
10.4  Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot  利用零极点图对傅里叶变换进行几何求值
10.4.1  First-Order Systems  一阶系统
10.4.2  Second-Order Systems  二阶系统
10.5  Properties of the z-Transform  z变换的性质
10.5.1  Linearity  线性性质
10.5.2  Time Shifting  时移性质
10.5.3  Scaling in the z-Domain  z域尺度变换性质
10.5.4  Time Reversal  时间反转性质
10.5.5  Time Expansion  时间扩展性质
10.5.6  Conjugation  共轭性质
10.5.7  The Convolution Property  卷积性质
10.5.8  Differentiation in the z-Domain  z域微分性质
10.5.9  The Initial-Value Theorem  初值定理
10.5.10  Summary of Properties  性质小结
10.6  Some Common z-Transform Pairs  常用z变换对
10.7  Analysis and Characterization of LTI Systems Using z-Transforms  利用z变换分析与表征线性时不变系统
10.7.1  Causality  因果性
10.7.2  Stability  稳定性
10.7.3  LTI Systems Characterized by Linear Constant-Coefficient Difference Equations  由线性常系数差分方程表征的线性时不变系统
10.7.4  Examples Relating System Behavior to the System Function  系统特性与系统函数的关系举例
10.8  System Function Algebra and Block Diagram Representations  系统函数的代数属性与方框图表示
10.8.1  System Functions for Interconnections of LTI Systems  线性时不变系统互联的系统函数
10.8.2  Block Diagram Representations for Causal LTI Systems Described by Difference Equations and Rational System Functions  由差分方程和有理系统函数描述的因果线性时不变系统的方框图表示
10.9  The Unilateral z-Transform  单边z变换
10.9.1  Examples of Unilateral z-Transforms and Inverse Transforms  单边z变换和单边z逆变换举例
10.9.2  Properties of the Unilateral z-Transform  单边z变换性质
10.9.3  Solving Difference Equations Using the Unilateral z-Transform  利用单边z变换求解差分方程
10.10  Summary  小结
Problems  习题

Chapter 11  Linear Feedback Systems  线性反馈系统
11.0  Introduction  引言
11.1  Linear Feedback Systems  线性反馈系统
11.2  Some Applications and Consequences of Feedback  反馈的某些应用及结果
11.2.1  Inverse System Design  逆系统设计
11.2.2  Compensation for Nonideal Elements  非理想元件的补偿
11.2.3  Stabilization of Unstable Systems  不稳定系统的稳定
11.2.4  Sampled-Data Feedback Systems  采样数据反馈系统
11.2.5  Tracking Systems  跟踪系统
11.2.6  Destabilization Caused by Feedback  反馈引起的不稳定
11.3  Root-Locus Analysis of Linear Feedback Systems  线性反馈系统的根轨迹分析法
11.3.1  An Introductory Example  一个例子
11.3.2  Equation for the Closed-Loop Poles  闭环极点方程
11.3.3  The End Points of the Root Locus: The Closed-Loop Poles for K = 0 and |K| =∞  根轨迹的端点：K = 0和|  K  | =∞时的闭环极点
11.3.4  The Angle Criterion  角判据
11.3.5  Properties of the Root Locus  根轨迹的性质
11.4  The Nyquist Stability Criterion  奈奎斯特稳定判据
11.4.1  The Encirclement Property  围线性质
11.4.2  The Nyquist Criterion for Continuous-Time LTI Feedback Systems  连续时间线性时不变反馈系统的奈奎斯特判据
11.4.3  The Nyquist Criterion for Discrete-Time LTI Feedback Systems  离散时间线性时不变反馈系统的奈奎斯特判据
11.5  Gain and Phase Margins  增益和相位裕度
11.6  Summary  小结
Problems  习题
Appendix  Partial-Fraction Expansion  部分分式展开
Bibliography  文献清单
Answers  基本题答案
Index  索引

PREFACE
This book is the second edition of a text designed for undergraduate courses in signals and systems. While such courses are frequently found in electrical engineering curricula, the concepts and techniques that form the core of the subject are of fundamental importance in all engineering disciplines. In fact, the scope of potential and actual applications of the methods of signal and system analysis continues to expand as engineers are confronted with new challenges involving the synthesis or analysis of complex processes. For these reasons we feel that a course in signals and systems not only is an essential element in an engineering program but also can be one of the most rewarding, exciting, and useful courses that engineering students take during their undergraduate education.
Our treatment of the subject of signals and systems in this second edition maintains the same general philosophy as in the first edition but with significant rewriting, restructuring, and additions. These changes are designed to help both the instructor in presenting the subject material and the student in mastering it. In the preface to the first edition we stated that our overall approach to signals and systems had been guided by the continuing developments in technologies for signal and system design and implementation, which made it increasingly important for a student to have equal familiarity with techniques suitable for analyzing and synthesizing both continuous-time and discrete-time systems. As we write the preface to this second edition, that observation and guiding principle are even more true than before. Thus, while students studying signals and systems should certainly have a solid foundation in disciplines based on the laws of physics, they must also have a firm grounding in the use of computers for the analysis of phenomena and the implementation of systems and algorithms. As a consequence, engineering curricula now reflect a blend of subjects, some involving continuous-time models and others focusing on the use of computers and discrete representations. For these reasons, signals and systems courses that bring discrete-time and continuous-time concepts together in a unified way play an increasingly important role in the education of engineering students and in their preparation for current and future developments in their chosen fields.
It is with these goals in mind that we have structured this book to develop in parallel the methods of analysis for continuous-time and discrete-time signals and systems. This approach also offers a distinct and extremely important pedagogical advantage. Specifically, we are able to draw on the similarities between continuous- and discrete-time methods in order to share insights and intuition developed in each domain. Similarly, we can exploit the differences between them to sharpen an understanding of the distinct properties of each. In organizing the material both originally and now in the second edition, we have also considered it essential to introduce the student to some of the important uses of the basic methods that are developed in the book. Not only does this provide the student with an appreciation for the range of applications of the techniques being learned and for directions for further study, but it also helps to deepen understanding of the subject. To achieve this goal we include introductory treatments on the subjects of filtering, communications, sampling, discrete-time processing of continuous-time signals, and feedback. In fact, in one of the major changes in this second edition, we have introduced the concept of frequency-domain filtering very early in our treatment of Fourier analysis in order to provide both motivation for and insight into this very important topic. In addition, we have again included an up-to-date bibliography at the end of the book in order to assist the student who is interested in pursuing additional and more advanced studies of the methods and applications of signal and system analysis.
The organization of the book reflects our conviction that full mastery of a subject of this nature cannot be accomplished without a significant amount of practice in using and applying the tools that are developed. Consequently, in the second edition we have significantly increased the number of worked examples within each chapter. We have also enhanced one of the key assets of the first edition, namely the end-of-chapter homework problems. As in the first edition, we have included a substantial number of problems, totaling more than 600 in number. A majority of the problems included here are new and thus provide additional flexibility for the instructor in preparing homework assignments. In addition, in order to enhance the utility of the problems for both the student and the instructor we have made a number of other changes to the organization and presentation of the problems. In particular, we have organized the problems in each chapter under several specific headings, each of which spans the material in the entire chapter but with a different objective. The first two sections of problems in each chapter emphasize the mechanics of using the basic concepts and methods presented in the chapter. For the first of these two sections, which has the heading Basic Problems with Answers, we have also provided answers (but not solutions) at the end of the book. These answers provide a simple and immediate way for the student to check his or her understanding of the material. The problems in this first section are generally appropriate for inclusion in homework sets. Also, in order to give the instructor additional flexibility in assigning homework problems, we have provided a second section of Basic Problems for which answers have not been included.
A third section of problems in each chapter, organized under the heading of Advanced Problems, is oriented toward exploring and elaborating upon the foundations and practical implications of the material in the text. These problems often involve mathematical derivations and more sophisticated use of the concepts and methods presented in the chapter. Some chapters also include a section of Extension Problems which involve extensions of material presented in the chapter and/or involve the use of knowledge from applications that are outside the scope of the main text (such as advanced circuits or mechanical systems). The overall variety and quantity of problems in each chapter will hopefully provide students with the means to develop their understanding of the material and instructors with considerable flexibility in putting together homework sets that are tailored to the specific needs of their students. A solutions manual is also available to instructors through the publisher.
Another significant additional enhancement to this second edition is the availability of the companion book Explorations in Signals and Systems Using MATLAB by Buck, Daniel, and Singer. This book contains MATLAB-based computer exercises for each topic in the text, and should be of great assistance to both instructor and student. Students using this book are assumed to have a basic background in calculus as well as some experience in manipulating complex numbers and some exposure to differential equations. With this background, the book is self-contained. In particular, no prior experience with system analysis, convolution, Fourier analysis, or Laplace and z-transforms is assumed. Prior to learning the subject of signals and systems most students will have had a course such as basic circuit theory for electrical engineers or fundamentals of dynamics for mechanical engineers. Such subjects touch on some of the basic ideas that are developed more fully in this text. This background can clearly be of great value to students in providing additional perspective as they proceed through the book.
The Foreword, which follows this preface, is written to offer the reader motivation and perspective for the subject of signals and systems in general and our treatment of it in particular. We begin Chapter 1 by introducing some of the elementary ideas related to the mathematical representation of signals and systems. In particular we discuss transformations (such as time shifts and scaling) of the independent variable of a signal. We also introduce some of the most important and basic continuous-time and discrete-time signals, namely real and complex exponentials and the continuous-time and discrete-time unit step and unit impulse. Chapter 1 also introduces block diagram representations of interconnections of systems and discusses several basic system properties such as causality, linearity and time-invariance. In Chapter 2 we build on these last two properties, together with the sifting property of unit impulses to develop the convolution-sum representation for discrete-time linear, time-invariant (LTI) systems and the convolution integral representation for continuous-time LTI systems. In this treatment we use the intuition gained from our development of the discrete-time case as an aid in deriving and understanding its continuous-time counterpart. We then tum to a discussion of causal, LTI systems characterized by linear constant-coefficient differential and difference equations. In this introductory discussion we review the basic ideas involved in solving linear differential equations (to which most students will have had some previous exposure) and we also provide a discussion of analogous methods for linear difference equations. However, the primary focus of our development in Chapter 2 is not on methods of solution, since more convenient approaches are developed later using transform methods. Instead, in this first look, our intent is to provide the student with some appreciation for these extremely important classes of systems, which will be encountered often in subsequent chapters. Finally, Chapter 2 concludes with a brief discussion of singularity functions-steps, impulses, doublets, and so forth-in the context of their role in the description and analysis of continuous-time LTI systems. In particular, we stress the interpretation of these signals in terms of how they are defined under convolution-that is, in terms of the responses of LTI systems to these idealized signals.
Chapters 3 through 6 present a thorough and self-contained development of the methods of Fourier analysis in both continuous and discrete time and together represent the most significant reorganization and revision in the second edition. In particular, as we indicated previously, we have introduced the concept of frequency-domain filtering at a much earlier point in the development in order to provide motivation for and a concrete application of the Fourier methods being developed. As in the first edition, we begin the discussions in Chapter 3 by emphasizing and illustrating the two fundamental reasons for the important role Fourier analysis plays in the study of signals and systems in both continuous and discrete time: (1) extremely broad classes of signals can be represented as weighted sums or integrals of complex exponentials; and (2) the response of an LTI system to a complex exponential input is the same exponential multiplied by a complex-number characteristic of the system. However, in contrast to the first edition, the focus of attention in Chapter 3 is on Fourier series representations for periodic signals in both continuous time and discrete time. In this way we not only introduce and examine many of the properties of Fourier representations without the additional mathematical generalization required to obtain the Fourier transform for aperiodic signals, but we also can introduce the application to filtering at a very early stage in the development. In particular, taking advantage of the fact that complex exponentials are eigenfunctions of LTI systems, we introduce the frequency response of an LTI system and use it to discuss the concept of frequency-selective filtering, to introduce ideal filters, and to give several examples of nonideal filters described by differential and difference equations. In this way, with a minimum of mathematical preliminaries, we provide the student with a deeper appreciation for what a Fourier representation means and why it is such a useful construct.
Chapters 4 and 5 then build on the foundation provided by Chapter 3 as we develop first the continuous-time Fourier transform in Chapter 4 and, in a parallel fashion, the discrete-time Fourier transform in Chapter 5. In both chapters we derive the Fourier transform representation of an aperiodic signal as the limit of the Fourier series for a signal whose period becomes arbitrarily large. This perspective emphasizes the close relationship between Fourier series and transforms, which we develop further in subsequent sections and which allows us to transfer the intuition developed for Fourier series in Chapter 3 to the more general context of Fourier transforms. In both chapters we have included a discussion of the many important properties of Fourier transforms, with special emphasis placed on the convolution and multiplication properties. In particular, the convolution property allows us to take a second look at the topic of frequency-selective filtering, while the multiplication property serves as the starting point for our treatment of sampling and modulation in later chapters. Finally, in the last sections in Chapters 4 and 5 we use transform methods to determine the frequency responses of LTI systems described by differential and difference equations and to provide several examples illustrating how Fourier transforms can be used to compute the responses for such systems. To supplement these discussions (and later treatments of Laplace and z-transforms) we have again included an Appendix at the end of the book that contains a description of the method of partial fraction expansion.
Our treatment of Fourier analysis in these two chapters is characteristic of the parallel treatment we have developed. Specifically, in our discussion in Chapter 5, we are able to build on much of the insight developed in Chapter 4 for the continuous-time case, and toward the end of Chapter 5 we emphasize the complete duality in continuous-time and discrete-time Fourier representations. In addition, we bring the special nature of each domain into sharper focus by contrasting the differences between continuous- and discrete-time Fourier analysis.
As those familiar with the first edition will note, the lengths and scopes of Chapters 4 and 5 in the second edition are considerably smaller than their first edition counterparts. This is due not only to the fact that Fourier series are now dealt with in a separate chapter but also to our moving several topics into Chapter 6. The result, we believe, has several significant benefits. First, the presentation in three shorter chapters of the basic concepts and results of Fourier analysis, together with the introduction of the concept of frequency-selective filtering, should help the student in organizing his or her understanding of this material and in developing some intuition about the frequency domain and appreciation for its potential applications. Then, with Chapters 3-5 as a foundation, we can engage in a more detailed look at a number of important topics and applications. In Chapter 6 we take a deeper look at both the time- and frequency-domain characteristics of LTI systems. For example, we introduce magnitude-phase and Bode plot representations for frequency responses and discuss the effect of frequency response phase on the time domain characteristics of the output of an LTI system. In addition, we examine the time- and frequency-domain behavior of ideal and nonideal filters and the tradeoffs between these that must be addressed in practice. We also take a careful look at first- and second-order systems and their roles as basic building blocks for more complex system synthesis and analysis in both continuous and discrete time. Finally, we discuss several other more complex examples of filters in both continuous and discrete time. These examples together with the numerous other aspects of filtering explored in the problems at the end of the chapter provide the student with some appreciation for the richness and flavor of this important subject. While each of the topics in Chapter 6 was present in the first edition, we believe that by reorganizing and collecting them in a separate chapter following the basic development of Fourier analysis, we have both simplified the introduction of this important topic in Chapters 3-5 and presented in Chapter 6 a considerably more cohesive picture of time and frequency-domain issues.
In response to suggestions and preferences expressed by many users of the first edition we have modified notation in the discussion of Fourier transforms to be more consistent with notation most typically used for continuous-time and discrete-time Fourier transforms. Specifically, beginning with Chapter 3 we now denote the continuous-time Fourier transform as X(jw) and the discrete-time Fourier transform as X(ejw)_ As with all options with notation, there is not a unique best choice for the notation for Fourier transforms. However, it is our feeling, and that of many of our colleagues, that the notation used in this edition represents the preferable choice.
Our treatment of sampling in Chapter 7 is concerned primarily with the sampling theorem and its implications. However, to place this subject in perspective we begin by discussing the general concepts of representing a continuous-time signal in terms of its samples and the reconstruction of signals using interpolation. After using frequency-domain methods to derive the sampling theorem, we consider both the frequency and time domains to provide intuition concerning the phenomenon of aliasing resulting from undersampling. One of the very important uses of sampling is in the discrete-time processing of continuous-time signals, a topic that we explore at some length in this chapter. Following this, we tum to the sampling of discrete-time signals. The basic result underlying discrete-time sampling is developed in a manner that parallels that used in continuous time, and the applications of this result to problems of decimation and interpolation are described. Again a variety of other applications, in both continuous and discrete time, are addressed in the problems.
Once again the reader acquainted with our first edition will note a change, in this case involving the reversal in the order of the presentation of sampling and communications. We have chosen to place sampling before communications in the second edition both because we can call on simple intuition to motivate and describe the processes of sampling and reconstruction from samples and also because this order of presentation then allows us in Chapter 8 to talk more easily about forms of communication systems that are closely related to sampling or rely fundamentally on using a sampled version of the signal to be transmitted.
Our treatment of communications in Chapter 8 includes an in -depth discussion of continuous-time sinusoidal amplitude modulation (AM), which begins with the straightforward application of the multiplication property to describe the effect of sinusoidal AM in the frequency domain and to suggest how the original modulating signal can be recovered. Following this, we develop a number of additional issues and applications related to sinusoidal modulation, including frequency-division multiplexing and single-sideband modulation. Many other examples and applications are described in the problems. Several additional topics are covered in Chapter 8. The first of these is amplitude modulation of a pulse train and time-division multiplexing, which has a close connection to the topic of sampling in Chapter 7. Indeed we make this tie even more explicit and provide a look into the important field of digital communications by introducing and briefly describing the topics of pulse-amplitude modulation (PAM) and intersymbol interference. Finally, our discussion of frequency modulation (FM) provides the reader with a look at a nonlinear modulation problem. Although the analysis of FM systems is not as straightforward as for the AM case, our introductory treatment indicates how frequency-domain methods can be used to gain a significant amount of insight into the characteristics of FM signals and systems. Through these discussions and the many other aspects of modulation and communications explored in the problems in this chapter we believe that the student can gain an appreciation both for the richness of the field of communications and for the central role that the tools of signals and systems analysis play in it.
Chapters 9 and 10 treat the Laplace and z-transforms, respectively. For the most part, we focus on the bilateral versions of these transforms, although in the last section of each chapter we discuss unilateral transforms and their use in solving differential and difference equations with nonzero initial conditions. Both chapters include discussions on: the close relationship between these transforms and Fourier transforms; the class of rational transforms and their representation in terms of poles and zeros; the region of convergence of a Laplace or z-transform and its relationship to properties of the signal with which it is associated; inverse transforms using partial fraction expansion; the geometric evaluation of system functions and frequency responses from pole-zero plots; and basic transform properties. In addition, in each chapter we examine the properties and uses of system functions for LTI systems. Included in these discussions are the determination of system functions for systems characterized by differential and difference equations; the use of system function algebra for interconnections of LTI systems; and the construction of cascade, parallel- and direct-form block-diagram representations for systems with rational system functions.
The tools of Laplace and z-transforms form the basis for our examination of linear feedback systems in Chapter 11. We begin this chapter by describing a number of the important uses and properties of feedback systems, including stabilizing unstable systems, designing tracking systems, and reducing system sensitivity. In subsequent sections we use the tools that we have developed in previous chapters to examine three topics that are of importance for both continuous-time and discrete-time feedback systems. These are root locus analysis, Nyquist plots and the Nyquist criterion, and log-magnitude/phase plots and the concepts of phase and gain margins for stable feedback systems.
The subject of signals and systems is an extraordinarily rich one, and a variety of approaches can be taken in designing an introductory course. It was our intention with the first edition and again with this second edition to provide instructors with a great deal of flexibility in structuring their presentations of the subject. To obtain this flexibility and to maximize the usefulness of this book for instructors, we have chosen to present thorough, in -depth treatments of a cohesive set of topics that forms the core of most introductory courses on signals and systems. In achieving this depth we have of necessity omitted introductions to topics such as descriptions of random signals and state space models that are sometimes included in first courses on signals and systems. Traditionally, at many schools, such topics are not included in introductory courses but rather are developed in more depth in follow-on undergraduate courses or in courses explicitly devoted to their investigation. Although we have not included an introduction to state space in the book, instructors of introductory courses can easily incorporate it into the treatments of differential and difference equations that can be found throughout the book. In particular, the discussions in Chapters 9 and 10 on block diagram representations for systems with rational system functions and on unilateral transforms and their use in solving differential and difference equations with initial conditions form natural points of departure for the discussions of state-space representations.
A typical one-semester course at the sophomore-junior level using this book would cover Chapters 1-5 in reasonable depth (although various topics in each chapter are easily omitted at the discretion of the instructor) with selected topics chosen from the remaining chapters. For example, one possibility is to present several of the basic topics in Chapters 6-8 together with a treatment of Laplace and z-transforms and perhaps a brief introduction to the use of system function concepts to analyze feedback systems. A variety of alternate formats are possible, including one that incorporates an introduction to state space or one in which more focus is placed on continuous-time systems by de-emphasizing Chapters 5 and 10 and the discrete-time topics in Chapters 3, 7, 8, and 11.
In addition to these course formats this book can be used as the basic text for a thorough, two-semester sequence on linear systems. Alternatively, the portions of the book not used in a first course on signals and systems can, together with other sources, form the basis for a subsequent course. For example, much of the material in this book forms a direct bridge to subjects such as state space analysis, control systems, digital signal processing, communications and statistical signal processing. Consequently, a follow-on course can be constructed that uses some of the topics in this book together with supplementary material in order to provide an introduction to one or more of these advanced subjects. In fact, a new course following this model has been developed at MIT and has proven not only to be a popular course among our students but also a crucial component of our signals and systems curriculum.
As it was with the first edition, in the process of writing this book we have been fortunate to have received assistance, suggestions, and support from numerous colleagues, students and friends. The ideas and perspectives that form the heart of this book have continued to evolve as a result of our own experiences in teaching signals and systems and the influences of the many colleagues and students with whom we have worked. We would like to thank Professor Ian T. Young for his contributions to the first edition of this book and to thank and welcome Professor Hamid Nawab for the significant role he played in the development and complete restructuring of the examples and problems for this second edition. We also express our appreciation to John Buck, Michael Daniel and Andrew Singer for writing the MATLAB companion to the text. In addition, we would like to thank Jason Oppenheim for the use of one of his original photographs and Vivian Berman for her ideas and help in arriving at a cover design. Also, as indicated on the acknowledgment page that follows, we are deeply grateful to the many students and colleagues who devoted a significant number of hours to a variety of aspects of the preparation of this second edition.
We would also like to express our sincere thanks to Mr. Ray Stata and Analog Devices, Inc. for their generous and continued support of signal processing and this text through funding of the Distinguished Professor Chair in Electrical Engineering. We also thank MIT for providing support and an invigorating environment in which to develop our ideas.
The encouragement, patience, technical support, and enthusiasm provided by Prentice-Hall, and in particular by Marcia Horton, Tom Robbins, Don Fowley, and their predecessors and by Ralph Pescatore of TKM Productions and the production staff at Prentice-Hall, have been crucial in making this second edition a reality.

Alan V. Oppenheim
Alan S. Willsky
Cambridge, Massachusetts

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