本书是工信部“十四五”规划教材，阐述了信号与系统的基本概念、核心原理和实用方法，主要包括连续时间与离散时间信号与系统的时域分析、频域分析、s 域分析、z 域分析、状态方程与 MIMO 系统等。本书在内容阐述上加强离散时间系统的分析，揭示连续时间与离散时间系统的相互关系以及数字滤波器的核心原理；在体系建构上注重与当今技术发展的适配性，增加了对卡尔曼滤波、扩展卡尔曼滤波，以及人工神经网络、深度学习等技术领域的基本概念和核心原理的介绍。此外，本书给出了 Matlab 实践内容和程序示例，可作为与课程配套的实践教学课程内容。本书可作为通信工程、电子信息工程、电子信息科学与技术等电子信息类专业的本科生教材。提供了多学时配置方案，可满足不同专业的不同需求，也可供研究生和有关科技工作者参考。

丁志中，合肥工业大学教授，硕士生导师。合肥工业大学计算机与信息学院通信工程系系主任、通信与信息系统研究所所长、IEEE会员及IEEE通信学会会员。安徽省人防、民防信息化建设技术专家。发表论文40余篇，出版教材2部。

目 录
第 1 章 信号与系统导论 ···················································································.1
1.1 信号的概念 ···························································································.1
1.1.1 信号的定义和描述 ·················································································.1
1.1.2 信号的分类 ·························································································.2
1.1.3 信号的运算和独立变量变换 ·······································································.6
1.2 基本信号 ·····························································································.10
1.2.1 正弦信号 ···························································································.10
1.2.2 指数信号 ···························································································.12
1.2.3 单位阶跃信号 ·····················································································.14
1.2.4 单位冲激信号 ·····················································································.16
1.2.5 抽样函数 ···························································································.23
1.3 系统 ···································································································.24
1.3.1 系统的基本概念 ···················································································.24
1.3.2 系统特性及 LTI 系统 ··············································································.27
1.3.3 SISO 系统的时域描述 ·············································································.33
1.4 信号与系统的基本问题和基本内容 ····························································.34
*1.5 Matlab 实践入门 ··················································································.35
1.5.1 Matlab 中信号的表示 ··············································································.35
1.5.2 输出图形的修饰 ···················································································.41
1.5.3 应用举例 ···························································································.43
习题 ··········································································································.45
第 2 章 信号与系统的时域分析 ·········································································.50
2.1 离散时间 LTI 系统的零状态响应：卷积和 ···················································.50
2.1.1 LTI 系统响应求解的基本思想 ····································································.50
2.1.2 零状态响应的卷积和求解 ·········································································.50
2.1.3 卷积和的计算 ·····················································································.52
2.1.4 卷积和的性质 ·····················································································.57
2.2 连续时间 LTI 系统的零状态响应：卷积积分 ················································.60
2.2.1 零状态响应的卷积积分求解 ······································································.60
2.2.2 卷积积分的计算 ···················································································.61
2.2.3 卷积积分的性质 ···················································································.64
2.3 系统冲激响应的性质 ··············································································.66
2.3.1 系统特性与冲激响应 ··············································································.66
2.3.2 理想系统的冲激响应 ··············································································.67
信号与系统 X
2.3.3 互联系统的冲激响应 ··············································································.67
*2.4 系统的方框图表示 ···············································································.69
2.4.1 微分方程描述系统的方框图表示 ·································································.69
2.4.2 差分方程描述系统的方框图表示 ·································································.72
*2.5 相关分析 ···························································································.74
2.5.1 连续时间信号相关函数 ···········································································.74
2.5.2 离散时间信号相关函数 ···········································································.81
*2.6 正交分析 ···························································································.82
2.6.1 信号正交的概念和定义 ···········································································.82
2.6.2 信号的正交分解与近似 ···········································································.85
*2.7 Matlab 实践 ························································································.88
2.7.1 卷积和的计算 ·····················································································.88
2.7.2 系统响应求解 ·····················································································.89
2.7.3 相关函数的计算 ···················································································.91
习题 ··········································································································.92
第 3 章 连续时间信号与系统的傅里叶分析 ··························································.95
3.1 傅里叶分析的意义 ·················································································.95
3.2 周期信号的傅里叶级数分析 ·····································································.96
3.2.1 周期信号的傅里叶级数展开 ······································································.96
3.2.2 复指数展开式系数的基本性质 ·································································.101
3.2.3 周期信号的频谱 ·················································································.103
*3.2.4 关于傅里叶级数的几点补充 ···································································.105
3.3 非周期信号的傅里叶变换分析 ································································.106
3.3.1 非周期信号的傅里叶变换表示 ·································································.106
3.3.2 傅里叶变换的性质—X(?)的特点 ·····························································.111
3.3.3 傅里叶变换的性质—信号运算的傅里叶变换 ················································.112
3.3.4 周期信号的傅里叶变换 ·········································································.125
3.3.5 傅里叶变换与信号频谱 ·········································································.129
*3.3.6 能量谱密度和功率谱密度 ·····································································.133
3.4 连续时间 LTI 系统的频域分析 ································································.137
3.4.1 连续时间 LTI 系统的频域表示··································································.137
3.4.2 理想传输系统和滤波器 ·········································································.139
*3.4.3 因果稳定系统的频率响应特性 ································································.141
3.5 信号的抽样 ························································································.142
3.5.1 时域抽样分析和时域抽样定理 ·································································.142
*3.5.2 模拟信号的数字处理系统 ·····································································.149
*3.5.3 频域抽样分析和频域抽样定理 ································································.152
*3.6 希尔伯特变换 ···················································································.153
3.6.1 希尔伯特变换及其有关概念 ····································································.153
*3.6.2 解析信号 ·······················································································.157
目 录 XI
3.7 Matlab 实践 ························································································.159
3.7.1 周期信号的分解与合成—傅里叶级数 ·······················································.159
3.7.2 非周期信号的频谱—傅里叶变换 ····························································.161
3.7.3 信号时域抽样 ···················································································.162
3.7.4 希尔伯特变换在 AM 信号解调中的应用—包络检波 ·······································.165
附录 ········································································································.166
习题 ········································································································.168
第 4 章 离散时间信号与系统的傅里叶分析 ························································.174
4.1 周期序列的傅里叶级数分析 ···································································.174
4.1.1 离散傅里叶级数（DFS） ·······································································.174
4.1.2 DFS 的性质—周期序列频谱 ck的特点 ······················································.176
4.1.3 复指数谐波序列及其性质 ·······································································.178
4.1.4 周期序列的频谱及其特征 ·······································································.180
4.2 非周期序列的傅里叶变换分析 ································································.181
4.2.1 离散时间傅里叶变换（DTFT） ································································.181
4.2.2 DTFT 的性质—频谱函数 j X (e ) ? 的特点 ····················································.184
4.2.3 DTFT 的性质—变换的性质 ··································································.187
4.2.4 非周期序列的频谱及其特征 ····································································.192
*4.2.5 周期序列的傅里叶变换 ········································································.194
*4.2.6 序列内插零和序列抽取的频谱分析 ···························································.196
4.3 离散傅里叶变换（DFT） ······································································.201
4.3.1 序列频谱的离散化和 DFT 的定义 ······························································.201
4.3.2 DFT 和 DFS 的关系 ·············································································.203
*4.3.3 周期卷积与圆周卷积 ··········································································.208
4.3.4 DFT 的性质—X[k]的特性 ····································································.213
*4.3.5 DFT 的性质—变换的性质 ··································································.215
*4.3.6 DFT 的谱线间隔分析— j X (e ) ? 的频域抽样 ···············································.224
*4.3.7 快速傅里叶变换（FFT） ······································································.225
4.4 离散时间系统与连续时间系统 ································································.228
4.4.1 离散时间系统频率响应 ·········································································.229
4.4.2 连续时间频率和离散时间频率 ·································································.232
4.4.3 连续时间和离散时间信号频谱之间的关系 ····················································.234
4.4.4 连续时间和离散时间频率响应函数之间的关系 ···············································.235
*4.5 数字滤波—FIR 滤波器 ·····································································.236
4.5.1 数字滤波的核心原理—脉冲响应不变法 ····················································.236
4.5.2 FIR 滤波器 ·······················································································.238
*4.6 Matlab 实践 ······················································································.243
4.6.1 fft 函数调用和谱线分析 ·········································································.243
4.6.2 数字滤波函数调用和信号的数字滤波 ··························································.246
习题 ········································································································.250
信号与系统 XII
第 5 章 连续时间信号与系统的 s 域分析 ···························································.254
5.1 拉普拉斯变换 ·····················································································.254
5.1.1 拉普拉斯变换的定义及收敛域 ·································································.254
5.1.2 拉普拉斯变换的性质 ············································································.263
5.1.3 拉普拉斯逆变换求解 ············································································.274
*5.1.4 单边周期信号的拉普拉斯变换 ································································.281
5.2 系统响应的 s 域求解 ············································································.282
5.2.1 微分方程描述系统的响应求解 ·································································.283
*5.2.2 电路的 s 域分析 ················································································.285
*5.2.3 自由响应与强迫响应、稳态响应与暂态响应 ·················································.288
5.3 系统的 s 域分析 ··················································································.290
5.3.1 系统函数与系统特性 ············································································.290
5.3.2 系统函数与系统频率响应特性 ·································································.293
*5.3.3 全通系统和最小相位系统 ·····································································.298
*5.3.4 物理可实现系统 ················································································.301
*5.4 模拟滤波器设计 ················································································.303
5.4.1 巴特沃斯低通滤波器 ············································································.303
*5.4.2 切比雪夫低通滤波器 ··········································································.308
5.4.3 低通到高通、带通和带阻滤波器的变换 ·······················································.310
*5.5 Matlab 实践 ······················································································.313
5.5.1 拉普拉斯变换求解 ··············································································.313
5.5.2 部分分式展开及逆变换求解 ····································································.314
5.5.3 系统零极点和系统频率响应 ····································································.315
5.5.4 模拟滤波器设计 ·················································································.317
附录 ········································································································.320
习题 ········································································································.322
第 6 章 离散时间信号与系统的 z 域分析 ···························································.327
6.1 z 变换 ·······························································································.327
6.1.1 z 变换的定义及收敛域 ··········································································.327
6.1.2 z 变换的性质 ····················································································.333
6.1.3 逆 z 变换的求解 ·················································································.340
*6.1.4 单边周期序列的 z 变换 ········································································.346
6.2 系统响应的 z 域求解 ············································································.349
6.2.1 差分方程描述系统的响应求解 ·································································.349
6.2.2 方框图描述系统的响应求解 ····································································.351
*6.2.3 完全响应的分解及响应分量的 z 域求解 ······················································.351
6.3 系统的 z 域分析 ··················································································.354
6.3.1 系统函数与系统特性 ············································································.354
6.3.2 系统函数与系统频率响应特性 ·································································.357
*6.3.3 全通系统和最小相位系统 ·····································································.359
目 录 XIII
*6.3.4 因果稳定性和可实现性 ········································································.361
*6.4 z 变换与拉普拉斯变换 ········································································.362
6.4.1 从拉普拉斯变换到 z 变换 ·······································································.362
6.4.2 s 平面和 z 平面之间的映射关系 ································································.363
*6.5 数字滤波—IIR 滤波器 ······································································.364
6.5.1 双线性变换法 ···················································································.364
6.5.2 IIR 滤波器的设计与实现 ········································································.367
*6.6 Matlab 实践 ······················································································.369
6.6.1 离散时间系统频率响应 ·········································································.369
6.6.2 IIR 数字滤波器设计与实现 ·····································································.370
附录 ········································································································.372
习题 ········································································································.373
第 7 章 状态方程与 MIMO 系统 ······································································.375
7.1 系统的状态方程描述 ············································································.375
7.1.1 离散时间状态方程 ··············································································.375
7.1.2 连续时间状态方程 ··············································································.377
*7.1.3 状态方程描述的特点及相关问题的讨论 ······················································.378
7.2 卡尔曼滤波 ························································································.379
7.2.1 卡尔曼滤波的基本原理和迭代公式 ····························································.379
7.2.2 卡尔曼增益和卡尔曼滤波的核心概念 ··························································.387
7.2.3 协方差矩阵和误差传播 ·········································································.389
7.2.4 一维卡尔曼滤波的收敛性分析 ·································································.390
7.2.5 卡尔曼滤波的初始赋值 ·········································································.393
*7.2.6 卡尔曼滤波和频域滤波 ········································································.395
*7.3 贝叶斯滤波 ······················································································.395
7.3.1 状态值的贝叶斯估计 ············································································.396
7.3.2 贝叶斯滤波算法 ·················································································.397
7.3.3 概率滤波和高斯滤波 ············································································.401
7.3.4 贝叶斯滤波框架下的线性卡尔曼滤波 ··························································.402
*7.4 非线性卡尔曼滤波 ·············································································.403
7.4.1 扩展卡尔曼滤波（EKF） ·······································································.403
7.4.2 无迹卡尔曼滤波（UKF） ······································································.406
7.5 MIMO 系统 ························································································.412
7.5.1 MIMO 系统的描述 ··············································································.412
7.5.2 MIMO LTI 系统的变换域求解 ··································································.417
*7.6 MIMO 与神经网络 ·············································································.420
7.6.1 人脑神经系统的 MIMO 建模 ···································································.420
7.6.2 神经网络的基本原理 ············································································.422
7.6.3 多层神经网络的反向传播算法 ·································································.426
7.6.4 神经网络与分类 ·················································································.429
信号与系统 XIV
7.6.5 深度学习 ·························································································.432
7.6.6 卷积神经网络 CNN ·············································································.434
*7.7 Matlab 实践 ······················································································.435
7.7.1 卡尔曼滤波应用一：匀速平面行驶车辆的位置跟踪 ··········································.436
7.7.2 卡尔曼滤波应用二：系统控制设计 ····························································.438
附录 ········································································································.439
附录 A 矢量和矩阵的基本概念及公式 ·······························································.439
附录 B 随机变量和高斯分布的基本概念及公式 ·····················································.444
习题 ········································································································.451
参考文献 ·····································································································.453