本書是工信部“十四五”規劃教材，闡述了信號與系統的基本概念、核心原理和實用方法，主要包括連續時間與離散時間信號與系統的時域分析、頻域分析、s 域分析、z 域分析、狀態方程與 MIMO 系統等。本書在內容闡述上加強離散時間系統的分析，揭示連續時間與離散時間系統的相互關系以及數字濾波器的核心原理；在體系建構上注重與當今技術發展的適配性，增加了對卡爾曼濾波、擴展卡爾曼濾波，以及人工神經網絡、深度學習等技術領域的基本概念和核心原理的介紹。此外，本書給出了 Matlab 實踐內容和程序示例，可作為與課程配套的實踐教學課程內容。本書可作為通信工程、電子信息工程、電子信息科學與技術等電子信息類專業的本科生教材。提供了多學時配置方案，可滿足不同專業的不同需求，也可供研究生和有關科技工作者參考。
丁志中，合肥工業大學教授，碩士生導師。合肥工業大學計算機與信息學院通信工程系系主任、通信與信息系統研究所所長、IEEE會員及IEEE通信學會會員。安徽省人防、民防信息化建設技術專家。發表論文40余篇，出版教材2部。
目 錄
第 1 章 信號與系統導論 ···················································································.1
1.1 信號的概念 ···························································································.1
1.1.1 信號的定義和描述 ·················································································.1
1.1.2 信號的分類 ·························································································.2
1.1.3 信號的運算和獨立變量變換 ·······································································.6
1.2 基本信號 ·····························································································.10
1.2.1 正弦信號 ···························································································.10
1.2.2 指數信號 ···························································································.12
1.2.3 單位階躍信號 ·····················································································.14
1.2.4 單位沖激信號 ·····················································································.16
1.2.5 抽樣函數 ···························································································.23
1.3 系統 ···································································································.24
1.3.1 系統的基本概念 ···················································································.24
1.3.2 系統特性及 LTI 系統 ··············································································.27
1.3.3 SISO 系統的時域描述 ·············································································.33
1.4 信號與系統的基本問題和基本內容 ····························································.34
*1.5 Matlab 實踐入門 ··················································································.35
1.5.1 Matlab 中信號的表示 ··············································································.35
1.5.2 輸出圖形的修飾 ···················································································.41
1.5.3 應用舉例 ···························································································.43
習題 ··········································································································.45
第 2 章 信號與系統的時域分析 ·········································································.50
2.1 離散時間 LTI 系統的零狀態響應：卷積和 ···················································.50
2.1.1 LTI 系統響應求解的基本思想 ····································································.50
2.1.2 零狀態響應的卷積和求解 ·········································································.50
2.1.3 卷積和的計算 ·····················································································.52
2.1.4 卷積和的性質 ·····················································································.57
2.2 連續時間 LTI 系統的零狀態響應：卷積積分 ················································.60
2.2.1 零狀態響應的卷積積分求解 ······································································.60
2.2.2 卷積積分的計算 ···················································································.61
2.2.3 卷積積分的性質 ···················································································.64
2.3 系統沖激響應的性質 ··············································································.66
2.3.1 系統特性與沖激響應 ··············································································.66
2.3.2 理想系統的沖激響應 ··············································································.67
信號與系統 X
2.3.3 互聯系統的沖激響應 ··············································································.67
*2.4 系統的方框圖表示 ···············································································.69
2.4.1 微分方程描述系統的方框圖表示 ·································································.69
2.4.2 差分方程描述系統的方框圖表示 ·································································.72
*2.5 相關分析 ···························································································.74
2.5.1 連續時間信號相關函數 ···········································································.74
2.5.2 離散時間信號相關函數 ···········································································.81
*2.6 正交分析 ···························································································.82
2.6.1 信號正交的概念和定義 ···········································································.82
2.6.2 信號的正交分解與近似 ···········································································.85
*2.7 Matlab 實踐 ························································································.88
2.7.1 卷積和的計算 ·····················································································.88
2.7.2 系統響應求解 ·····················································································.89
2.7.3 相關函數的計算 ···················································································.91
習題 ··········································································································.92
第 3 章 連續時間信號與系統的傅里葉分析 ··························································.95
3.1 傅里葉分析的意義 ·················································································.95
3.2 周期信號的傅里葉級數分析 ·····································································.96
3.2.1 周期信號的傅里葉級數展開 ······································································.96
3.2.2 復指數展開式系數的基本性質 ·································································.101
3.2.3 周期信號的頻譜 ·················································································.103
*3.2.4 關于傅里葉級數的幾點補充 ···································································.105
3.3 非周期信號的傅里葉變換分析 ································································.106
3.3.1 非周期信號的傅里葉變換表示 ·································································.106
3.3.2 傅里葉變換的性質—X(?)的特點 ·····························································.111
3.3.3 傅里葉變換的性質—信號運算的傅里葉變換 ················································.112
3.3.4 周期信號的傅里葉變換 ·········································································.125
3.3.5 傅里葉變換與信號頻譜 ·········································································.129
*3.3.6 能量譜密度和功率譜密度 ·····································································.133
3.4 連續時間 LTI 系統的頻域分析 ································································.137
3.4.1 連續時間 LTI 系統的頻域表示··································································.137
3.4.2 理想傳輸系統和濾波器 ·········································································.139
*3.4.3 因果穩定系統的頻率響應特性 ································································.141
3.5 信號的抽樣 ························································································.142
3.5.1 時域抽樣分析和時域抽樣定理 ·································································.142
*3.5.2 模擬信號的數字處理系統 ·····································································.149
*3.5.3 頻域抽樣分析和頻域抽樣定理 ································································.152
*3.6 希爾伯特變換 ···················································································.153
3.6.1 希爾伯特變換及其有關概念 ····································································.153
*3.6.2 解析信號 ·······················································································.157
目 錄 XI
3.7 Matlab 實踐 ························································································.159
3.7.1 周期信號的分解與合成—傅里葉級數 ·······················································.159
3.7.2 非周期信號的頻譜—傅里葉變換 ····························································.161
3.7.3 信號時域抽樣 ···················································································.162
3.7.4 希爾伯特變換在 AM 信號解調中的應用—包絡檢波 ·······································.165
附錄 ········································································································.166
習題 ········································································································.168
第 4 章 離散時間信號與系統的傅里葉分析 ························································.174
4.1 周期序列的傅里葉級數分析 ···································································.174
4.1.1 離散傅里葉級數（DFS） ·······································································.174
4.1.2 DFS 的性質—周期序列頻譜 ck的特點 ······················································.176
4.1.3 復指數諧波序列及其性質 ·······································································.178
4.1.4 周期序列的頻譜及其特征 ·······································································.180
4.2 非周期序列的傅里葉變換分析 ································································.181
4.2.1 離散時間傅里葉變換（DTFT） ································································.181
4.2.2 DTFT 的性質—頻譜函數 j X (e ) ? 的特點 ····················································.184
4.2.3 DTFT 的性質—變換的性質 ··································································.187
4.2.4 非周期序列的頻譜及其特征 ····································································.192
*4.2.5 周期序列的傅里葉變換 ········································································.194
*4.2.6 序列內插零和序列抽取的頻譜分析 ···························································.196
4.3 離散傅里葉變換（DFT） ······································································.201
4.3.1 序列頻譜的離散化和 DFT 的定義 ······························································.201
4.3.2 DFT 和 DFS 的關系 ·············································································.203
*4.3.3 周期卷積與圓周卷積 ··········································································.208
4.3.4 DFT 的性質—X[k]的特性 ····································································.213
*4.3.5 DFT 的性質—變換的性質 ··································································.215
*4.3.6 DFT 的譜線間隔分析— j X (e ) ? 的頻域抽樣 ···············································.224
*4.3.7 快速傅里葉變換（FFT） ······································································.225
4.4 離散時間系統與連續時間系統 ································································.228
4.4.1 離散時間系統頻率響應 ·········································································.229
4.4.2 連續時間頻率和離散時間頻率 ·································································.232
4.4.3 連續時間和離散時間信號頻譜之間的關系 ····················································.234
4.4.4 連續時間和離散時間頻率響應函數之間的關系 ···············································.235
*4.5 數字濾波—FIR 濾波器 ·····································································.236
4.5.1 數字濾波的核心原理—脈沖響應不變法 ····················································.236
4.5.2 FIR 濾波器 ·······················································································.238
*4.6 Matlab 實踐 ······················································································.243
4.6.1 fft 函數調用和譜線分析 ·········································································.243
4.6.2 數字濾波函數調用和信號的數字濾波 ··························································.246
習題 ········································································································.250
信號與系統 XII
第 5 章 連續時間信號與系統的 s 域分析 ···························································.254
5.1 拉普拉斯變換 ·····················································································.254
5.1.1 拉普拉斯變換的定義及收斂域 ·································································.254
5.1.2 拉普拉斯變換的性質 ············································································.263
5.1.3 拉普拉斯逆變換求解 ············································································.274
*5.1.4 單邊周期信號的拉普拉斯變換 ································································.281
5.2 系統響應的 s 域求解 ············································································.282
5.2.1 微分方程描述系統的響應求解 ·································································.283
*5.2.2 電路的 s 域分析 ················································································.285
*5.2.3 自由響應與強迫響應、穩態響應與暫態響應 ·················································.288
5.3 系統的 s 域分析 ··················································································.290
5.3.1 系統函數與系統特性 ············································································.290
5.3.2 系統函數與系統頻率響應特性 ·································································.293
*5.3.3 全通系統和最小相位系統 ·····································································.298
*5.3.4 物理可實現系統 ················································································.301
*5.4 模擬濾波器設計 ················································································.303
5.4.1 巴特沃斯低通濾波器 ············································································.303
*5.4.2 切比雪夫低通濾波器 ··········································································.308
5.4.3 低通到高通、帶通和帶阻濾波器的變換 ·······················································.310
*5.5 Matlab 實踐 ······················································································.313
5.5.1 拉普拉斯變換求解 ··············································································.313
5.5.2 部分分式展開及逆變換求解 ····································································.314
5.5.3 系統零極點和系統頻率響應 ····································································.315
5.5.4 模擬濾波器設計 ·················································································.317
附錄 ········································································································.320
習題 ········································································································.322
第 6 章 離散時間信號與系統的 z 域分析 ···························································.327
6.1 z 變換 ·······························································································.327
6.1.1 z 變換的定義及收斂域 ··········································································.327
6.1.2 z 變換的性質 ····················································································.333
6.1.3 逆 z 變換的求解 ·················································································.340
*6.1.4 單邊周期序列的 z 變換 ········································································.346
6.2 系統響應的 z 域求解 ············································································.349
6.2.1 差分方程描述系統的響應求解 ·································································.349
6.2.2 方框圖描述系統的響應求解 ····································································.351
*6.2.3 完全響應的分解及響應分量的 z 域求解 ······················································.351
6.3 系統的 z 域分析 ··················································································.354
6.3.1 系統函數與系統特性 ············································································.354
6.3.2 系統函數與系統頻率響應特性 ·································································.357
*6.3.3 全通系統和最小相位系統 ·····································································.359
目 錄 XIII
*6.3.4 因果穩定性和可實現性 ········································································.361
*6.4 z 變換與拉普拉斯變換 ········································································.362
6.4.1 從拉普拉斯變換到 z 變換 ·······································································.362
6.4.2 s 平面和 z 平面之間的映射關系 ································································.363
*6.5 數字濾波—IIR 濾波器 ······································································.364
6.5.1 雙線性變換法 ···················································································.364
6.5.2 IIR 濾波器的設計與實現 ········································································.367
*6.6 Matlab 實踐 ······················································································.369
6.6.1 離散時間系統頻率響應 ·········································································.369
6.6.2 IIR 數字濾波器設計與實現 ·····································································.370
附錄 ········································································································.372
習題 ········································································································.373
第 7 章 狀態方程與 MIMO 系統 ······································································.375
7.1 系統的狀態方程描述 ············································································.375
7.1.1 離散時間狀態方程 ··············································································.375
7.1.2 連續時間狀態方程 ··············································································.377
*7.1.3 狀態方程描述的特點及相關問題的討論 ······················································.378
7.2 卡爾曼濾波 ························································································.379
7.2.1 卡爾曼濾波的基本原理和迭代公式 ····························································.379
7.2.2 卡爾曼增益和卡爾曼濾波的核心概念 ··························································.387
7.2.3 協方差矩陣和誤差傳播 ·········································································.389
7.2.4 一維卡爾曼濾波的收斂性分析 ·································································.390
7.2.5 卡爾曼濾波的初始賦值 ·········································································.393
*7.2.6 卡爾曼濾波和頻域濾波 ········································································.395
*7.3 貝葉斯濾波 ······················································································.395
7.3.1 狀態值的貝葉斯估計 ············································································.396
7.3.2 貝葉斯濾波算法 ·················································································.397
7.3.3 概率濾波和高斯濾波 ············································································.401
7.3.4 貝葉斯濾波框架下的線性卡爾曼濾波 ··························································.402
*7.4 非線性卡爾曼濾波 ·············································································.403
7.4.1 擴展卡爾曼濾波（EKF） ·······································································.403
7.4.2 無跡卡爾曼濾波（UKF） ······································································.406
7.5 MIMO 系統 ························································································.412
7.5.1 MIMO 系統的描述 ··············································································.412
7.5.2 MIMO LTI 系統的變換域求解 ··································································.417
*7.6 MIMO 與神經網絡 ·············································································.420
7.6.1 人腦神經系統的 MIMO 建模 ···································································.420
7.6.2 神經網絡的基本原理 ············································································.422
7.6.3 多層神經網絡的反向傳播算法 ·································································.426
7.6.4 神經網絡與分類 ·················································································.429
信號與系統 XIV
7.6.5 深度學習 ·························································································.432
7.6.6 卷積神經網絡 CNN ·············································································.434
*7.7 Matlab 實踐 ······················································································.435
7.7.1 卡爾曼濾波應用一：勻速平面行駛車輛的位置跟蹤 ··········································.436
7.7.2 卡爾曼濾波應用二：系統控制設計 ····························································.438
附錄 ········································································································.439
附錄 A 矢量和矩陣的基本概念及公式 ·······························································.439
附錄 B 隨機變量和高斯分布的基本概念及公式 ·····················································.444
習題 ········································································································.451
參考文獻 ·····································································································.453
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