From 68d312469a3611e7f6fa11b1f926dec9d653e72a Mon Sep 17 00:00:00 2001 From: dgsyrc <1779252249@qq.com> Date: Sun, 16 Jun 2024 01:50:24 +0800 Subject: [PATCH] Create note --- .gitignore | 9 + .vscode/settings.json | 116 ++++++ Analysis of Signals and Linear Systems.pdf | Bin 0 -> 1695171 bytes Analysis of Signals and Linear Systems.tex | 43 ++ Chapter01.tex | 18 + Chapter01A.tex | 23 ++ Chapter01B.tex | 273 +++++++++++++ Chapter01C.tex | 89 +++++ Chapter01D.tex | 198 ++++++++++ Chapter01E.tex | 76 ++++ Chapter01F.tex | 101 +++++ Chapter02.tex | 6 + Chapter02A.tex | 181 +++++++++ Chapter02B.tex | 80 ++++ Chapter02C.tex | 68 ++++ Chapter02D.tex | 123 ++++++ Chapter03.tex | 8 + Chapter03A.tex | 0 Chapter03B.tex | 0 Chapter03C.tex | 0 Chapter03D.tex | 0 Chapter03E.tex | 0 Chapter03F.tex | 0 Chapter04.tex | 11 + Chapter04A.tex | 81 ++++ Chapter04B.tex | 183 +++++++++ Chapter04C.tex | 82 ++++ Chapter04D.tex | 201 ++++++++++ Chapter04E.tex | 188 +++++++++ Chapter04F.tex | 51 +++ 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+ +\usepackage{Local} + +\begin{document} + +% \setkeys{Gin}{draft} + +\maketitle + +\frontmatter +%\makesymb + +\tableofcontents +\listoffigures +%\listoftables + +\mainmatter + +\input{Chapter01.tex} +\input{Chapter02.tex} +\input{Chapter03.tex} +\input{Chapter04.tex} +\input{Chapter05.tex} +\input{Chapter06.tex} +\input{Chapter07.tex} +\input{Chapter08.tex} + +\appendix + +\backmatter + +\input{Reference.tex} + +\end{document} \ No newline at end of file diff --git a/Chapter01.tex b/Chapter01.tex new file mode 100644 index 0000000..fbe47b1 --- /dev/null +++ b/Chapter01.tex @@ -0,0 +1,18 @@ +\chapter{信号与系统} + +学习要点: + +\begin{itemize} + \item 认识本课程领域的一些名词、术语。 + \item 学习信号运算规律、熟悉表达式与波形的对应关系。 + \item 理解冲激信号的特性 + \item 了解本课程研究范围、学习目标 + \item 初步了解本课程用到的主要方法和手段 +\end{itemize} + +\input{Chapter01A.tex} +\input{Chapter01B.tex} +\input{Chapter01C.tex} +\input{Chapter01D.tex} +\input{Chapter01E.tex} +\input{Chapter01F.tex} \ No newline at end of file diff --git a/Chapter01A.tex b/Chapter01A.tex new file mode 100644 index 0000000..8d2c5d7 --- /dev/null +++ b/Chapter01A.tex @@ -0,0 +1,23 @@ +\section{绪论} + +\subsection{信号的概念} +信号\footnote{指电信号}是\uwave{信息的载体} + +辨析: +\begin{itemize} + \item 消息:人们常常把来自外界的各种报道统称为消息。 + \item 信息\footnote{本课程中对“信息”和“消息”两词不加严格区分}:通常把消息中有意义的内容称为信息。 + \item 信号:信号是信息的载体,通过信号传递信息。 +\end{itemize} + +\subsection{系统的概念} + +\begin{Figure}[系统组成] + \includegraphics[width=100mm]{visio/1.1.pdf} +\end{Figure} + +信号处理:对信号进行加工/变换 + +信号传输:通信(长距离) + + diff --git a/Chapter01B.tex b/Chapter01B.tex new file mode 100644 index 0000000..603984d --- /dev/null +++ b/Chapter01B.tex @@ -0,0 +1,273 @@ +\section{信号的描述和分类} + +\subsection{信号的描述} + +信号:信息的物理体现,随时间变化 + +分类:电信号/非电信号 + +基本形式:随时间变化的电压$v(t)$/电流$i(t)$\footnote{本课程内不区分电压信号和电流信号} + +描述:时间的函数\footnote{本课程内信号和函数说法上等价},图形表示为波形 + +\subsection{信号的分类} + +\begin{itemize} + \item 用途:电视信号/雷达信号\dots + \item 时间特性:确定/随机信号、连续\footnote{时间和幅值均连续}/离散\footnote{时间和幅值均离散,又称序列}信号、模拟/数字信号、周期/非周期信号、能量/功率信号\dots +\end{itemize} + +\subsubsection{连续信号和离散信号} + +\begin{Figure}[连续信号] + \includegraphics[width=80mm]{visio/1.2.pdf} +\end{Figure} + +对于连续时间信号,其要求定义域连续,可包含间断点,值域可以不连续 + +\begin{Figure}[离散信号] + \includegraphics[width=40mm]{visio/1.3.pdf} +\end{Figure} + +对于离散信号,其仅在离散时刻有定义,且离散点间隔可以不等,通常取间隔$T$,表示为$f(kT)$,简写为$f(k)$。 + +等间隔的离散信号称为序列,其中$k$为序号。 + +\xref{fig:离散信号}可以表示为以下形式: + +\begin{equation} + f(k)=\left\{ + \begin{aligned} + 1 & , & k=-1 \\ + 2 & , & k=0 \\ + -1.5 & , & k=1 \\ + 2 & , & k=2 \\ + 0 & , & k=3 \\ + 1 & , & k=4 \\ + 0 & , & \text{其他}k + \end{aligned} + \right. +\end{equation} + +\subsubsection{模拟信号、抽样信号和数字信号} + +\begin{Figure}[模拟、抽样、数字信号] + \begin{FigureSub}[模拟信号] + \includegraphics[width=40mm]{visio/1.4-a.pdf} + \end{FigureSub} + \begin{FigureSub}[抽样后信号] + \includegraphics[width=40mm]{visio/1.4-b.pdf} + \end{FigureSub} + \begin{FigureSub}[数字信号] + \includegraphics[width=40mm]{visio/1.4-c.pdf} + \end{FigureSub} +\end{Figure} + +对于模拟信号,其时间和幅值均连续,是连续时间信号,经过抽样后变换为抽样信号。 + +抽样信号时间离散但幅值连续,经过量化后变换为数字信号。 + +数字信号时间和幅值均离散,是离散时间信号。 + +\subsubsection{周期信号} + +\begin{BoxDefinition}[周期信号] + 定义在$(-\infty,\infty)$区间,每隔一定时间$T$(或整数$N$),按相同规律重复变化的信号。 + + 连续周期信号$f(t)$满足 + \begin{Equation} + f(t) = f(t+mt), m=0,\pm 1,\pm 2,\dots + \end{Equation} + 离散周期信号$f(t)$满足 + \begin{Equation} + f(k) = f(k+mN), m=0,\pm 1,\pm 2,\dots + \end{Equation} + + 满足上述关系的最小$T$(或整数$N$)称为该信号的周期。 +\end{BoxDefinition} + +不具有周期性的信号称为非周期信号。 + +\begin{BoxProperty}[连续周期信号的周期] + 两个周期信号$x(t),y(t)$的周期分别为$T_1$和$T_2$,若其周期之比$T_1/T_2$为有理数,则其和信号$x(t)+y(t)$仍然是周期信号,其周期为$T_1$和$T_2$的最小公倍数。 +\end{BoxProperty} + +\begin{BoxProperty}[正弦序列的周期] + 对于离散周期信号$f(k) = \sin(\beta k)$。 + + 仅当$\frac{2\pi}{\beta}$为整数时,正弦序列才具有周期 + \begin{Equation} + N=\frac{2\pi}{\beta} + \end{Equation} + 当$\frac{2\pi}{\beta}$为有理数时,正弦序列仍具有周期性,其周期为 + \begin{Equation} + N=M\cdot\frac{2\pi}{\beta} + \end{Equation} + 其中$M$取使$N$为整数的最小整数。 + + 当$\frac{2\pi}{\beta}$为无理数时,为非周期序列。 +\end{BoxProperty} + +容易得知,两个周期信号的和不一定是周期信号,但两周期序列之和一定是周期序列。 + +\subsubsection{能量信号和功率信号} + +\begin{BoxDefinition}[能量信号] + 满足以下条件的连续信号称为能量信号 + \begin{Equation} + E = \int_{-\infty}^{\infty}\left|f(t)\right|^2 dt < \infty + \end{Equation} + 满足以下条件的离散信号称为能量信号 + \begin{Equation} + E = \sum\limits_{k=-\infty}^{\infty}\left|f(k)\right|^2 < \infty + \end{Equation} + 即能量有界,此时有$P = 0$ +\end{BoxDefinition} + +\begin{BoxDefinition}[功率信号] + 满足以下条件的连续信号称为功率信号 + \begin{Equation} + P = \lim_{T \rightarrow \infty} \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}}\left|f(t)\right|^2 dt < \infty + \end{Equation} + 满足以下条件的离散信号称为功率信号 + \begin{Equation} + P = \lim_{N \rightarrow \infty} \frac{1}{N} \sum\limits_{k=-\frac{N}{2}}^{\frac{N}{2}}\left|f(k)\right|^2 < \infty + \end{Equation} + 即能量有界,此时有$E = 0$ +\end{BoxDefinition} + +一般周期信号为功率信号,时限信号\footnote{有限时间区间不为零的非周期信号}为能量信号。 + +一些非周期信号也是非能量信号,例如:$\varepsilon (t)$ 是功率信号,$t\varepsilon (t)$、$e^t$是非功率非能量信号,$\delta (t)$是无定义的非功率非能量信号。 + +\subsubsection{一维信号和多维信号} + +\begin{BoxDefinition}[一维信号] + 只由一个自变量描述的信号,如语音信号。 +\end{BoxDefinition} + +\begin{BoxDefinition}[多维信号] + 由多个自变量描述的信号,如图像信号。 +\end{BoxDefinition} + +\subsection{几种典型确定性信号} + +\subsubsection{指数信号} + +\begin{Figure}[指数信号] + \includegraphics[width=40mm]{visio/1.5.pdf} +\end{Figure} + +\begin{BoxDefinition}[指数信号] + 形如以下形式的信号为指数信号 + \begin{Equation} + f(t)=Ke^{\alpha t} + \end{Equation} + $\alpha = 0$时为直流(常数) + + $\alpha < 0$时指数衰减 + + $\alpha > 0$时指数增长 +\end{BoxDefinition} + +\begin{BoxDefinition}[单边衰减指数信号] + \begin{Equation} + f(t)=\left\{ + \begin{aligned} + 0 & , &t<0 \\ + e^{-\frac{t}{\tau}} & , &t \geq 0 + \end{aligned} + \right. + \end{Equation} + 时间常数,代表信号的衰减速度 + \begin{Equation} + \tau = \frac{1}{|\alpha|} + \end{Equation} +\end{BoxDefinition} + + +\subsubsection{正弦信号} + +\begin{BoxDefinition}[正弦信号] + 形如以下形式的信号为正弦信号 + \begin{Equation} + f(t)=K\sin(\omega t +\theta) + \end{Equation} + 周期 + \begin{Equation} + T = \frac{2\pi}{\omega} + \end{Equation} + 角频率 + \begin{Equation} + \omega = 2\pi f + \end{Equation} +\end{BoxDefinition} + +\begin{BoxDefinition}[衰减正弦信号] + \begin{Equation} + f(t)=\left\{ + \begin{aligned} + Ke^{-\alpha t}\sin(\omega t) & , &t\geq 0 \\ + 0 & , &t < 0 + \end{aligned} + \right. + \quad (\alpha > 0) + \end{Equation} +\end{BoxDefinition} + +\subsubsection{复指数信号} + +\begin{BoxDefinition}[复指数信号] + 复指数信号 + \begin{Equation} + f(t)=Ke^{(\sigma + \mathrm{j} \omega)t}=Ke^{\sigma t}\cos(\omega t) + \mathrm{j}Ke^{\sigma t}\sin(\omega t) + \end{Equation} + 复频率 + \begin{Equation} + s=\sigma + \mathrm{j} \omega + \end{Equation} + 其中$\sigma$的量纲为$1/s$,$\omega$的量纲为$rad/s$ +\end{BoxDefinition} + +特点:不能产生,用来描述各种信号,用于信号分析及运算简化。 + +当$\sigma = 0, \omega = 0$时,为直流信号 + +当$\sigma > 0, \omega = 0$时,为升指数信号 + +当$\sigma < 0, \omega = 0$时,为衰减指数信号 + +当$\sigma = 0, \omega \neq 0$时,为等幅振荡 + +当$\sigma > 0, \omega \neq 0$时,为增幅振荡 + +当$\sigma < 0, \omega \neq 0$时,为衰减振荡 + +\subsubsection{抽样信号} + +\begin{Figure}[抽样信号] + \includegraphics[width=100mm]{visio/1.6.pdf} +\end{Figure} + +\begin{BoxDefinition}[抽样信号] + 抽样信号 + \begin{Equation} + Sa(t)=\frac{\sin t }{t} + \end{Equation} +\end{BoxDefinition} + +\begin{BoxProperty}[抽样信号的性质] + 抽样信号有如下性质 + \begin{Equation} + \begin{array}{l} + Sa(-t) = Sa(t) \\ + t=0,Sa(t)=1 \quad (\lim\limits_{t\rightarrow 0}Sa(t)=1)\\ + Sa(t)=0, t=\pm n\pi, n=1,2,3\cdots \\ + \int_0^{\infty}\frac{\sin t}{t}dt = \frac{\pi}{2}, \int_{-\infty}^{\infty}\frac{\sin t}{t}dt=\pi \\ + \lim\limits_{t\rightarrow\pm\infty}Sa(t)=0\\ + \mathrm{sinc}(t)=\frac{\sin(\pi t)}{\pi t} + \end{array} + \end{Equation} + +\end{BoxProperty} + diff --git a/Chapter01C.tex b/Chapter01C.tex new file mode 100644 index 0000000..dc7fbd8 --- /dev/null +++ b/Chapter01C.tex @@ -0,0 +1,89 @@ +\section{信号的基本运算} + +\subsection{信号的加法和乘法} + +同一瞬时两信号对应值相加(相乘) + +离散序列相加、乘对应值相加(相乘) + +\subsection{信号的时间变换} + +\subsubsection{信号的反转} + +\begin{Figure}[信号的反转] + \begin{FigureSub}[反转前原始信号$f(t)$] + \includegraphics[width=40mm]{visio/1.7-a.pdf} + \end{FigureSub} + \begin{FigureSub}[反转信号$f(-t)$] + \includegraphics[width=40mm]{visio/1.7-b.pdf} + \end{FigureSub} +\end{Figure} + +\begin{BoxDefinition}[信号的反转] + 将$f(t)\rightarrow f(-t),f(k)\rightarrow f(-k)$称为对信号$f(\cdot)$的反转或反折。 +\end{BoxDefinition} + +\subsubsection{信号的平移} + +\begin{Figure}[信号的平移] + \begin{FigureSub}[平移前原始信号$f(t)$] + \includegraphics[width=40mm]{visio/1.8-a.pdf} + \end{FigureSub} + \begin{FigureSub}[平移信号$f(t-1)$] + \includegraphics[width=40mm]{visio/1.8-b.pdf} + \end{FigureSub} +\end{Figure} + +\begin{BoxDefinition}[信号的平移] + 将$f(t)\rightarrow f(t-t_0),f(k)\rightarrow f(k-k_0)$称为对信号$f(\cdot)$的平移。 + + 若$t_0$(或$k_0$)$>0$,则将$f(\cdot)$右移,否则左移。 +\end{BoxDefinition} + +\subsubsection{信号的展缩} + +\begin{Figure}[信号的展缩] + \begin{FigureSub}[压缩前原始信号$f(t)$] + \includegraphics[width=40mm]{visio/1.9-a.pdf} + \end{FigureSub} + \begin{FigureSub}[压缩信号$f(2t)$] + \includegraphics[width=40mm]{visio/1.9-b.pdf} + \end{FigureSub} +\end{Figure} + + +\begin{BoxDefinition}[信号的展缩] + 将$f(t)\rightarrow f(at)$称为对信号$f(t)$的尺度变换。 + + 若$a>1$,则波形沿横坐标压缩。 + + 若$0 0 \\ + 0 & , & t < 0 + \end{aligned} + \right. + \end{Equation} + $t=0$处发生跳变,或认为$u(t)=\frac{1}{2}$ +\end{BoxDefinition} + +对单位阶跃函数作平移可得延迟单位阶跃信号。 + +\begin{BoxProperty}[单位阶跃函数的性质] + 可以通过平移和加减运算表示某些函数 + + 可以表示某些函数的区间(乘上阶跃函数的组合) + + 阶跃函数的积分 + + \begin{Equation} + \int_{-\infty}^{t} \varepsilon(\tau) d\tau = t\varepsilon(t) + \end{Equation} +\end{BoxProperty} + +\subsection{单位冲激函数} + +\begin{Figure}[单位冲激函数] + \includegraphics[width=40mm]{visio/1.13.pdf} +\end{Figure} + +\begin{BoxDefinition}[单位冲激函数] + 设一矩形脉冲$p_{\tau}(t)$的宽为$\tau$,高为$\frac{1}{\tau}$,面积为$1$ + \begin{Equation} + \delta(t) = \lim\limits_{\tau\rightarrow 0} p_{\tau}(t) + \end{Equation} + 或由狄拉克定义 + \begin{Equation} + \left\{ + \begin{array}{ll} + \delta (t) = 0 & (t \neq 0) \\ + \int_{-\infty}^{\infty} \delta (t)dt=1 + \end{array} + \right. + \end{Equation} + 且 + \begin{Equation} + \int_{-\infty}^{\infty} \delta (t)dt= \int_{0_{-}}^{0_{+}} \delta (t)dt = 1 + \end{Equation} + $t=0$时,$\delta(t)\rightarrow \infty$,为无界函数。 +\end{BoxDefinition} + +\subsection{冲激函数的性质} + +\begin{BoxProperty}[冲激函数的取样性] + 冲激函数的取样性 + \begin{Equation} + f(t)\delta(t)=f(0)\delta(t) \\ + \end{Equation} + \begin{Equation} + \int_{-\infty}^{\infty} f(t)\delta(t)dt = f(0) + \end{Equation} +\end{BoxProperty} + +\begin{BoxProperty}[冲激函数的奇偶性] + 冲激函数为偶函数 + \begin{Equation} + \delta(-t)=\delta(t) + \end{Equation} +\end{BoxProperty} + +\begin{BoxProperty}[冲激函数的比例性] + 冲激函数的展缩变换有以下性质 + \begin{Equation} + \delta(at)=\frac{1}{|a|}\delta(t) + \end{Equation} + 结合平移 + \begin{Equation} + \delta(at-t_0)=\frac{1}{|a|}\delta(t-\frac{t_0}{a}) + \end{Equation} +\end{BoxProperty} + +\begin{BoxProperty}[冲激函数的微积分性质] + 冲激函数与阶跃函数有以下关系 + \begin{Equation} + \varepsilon (t) = \int_{-\infty}^{t} \delta (\tau) d\tau + \end{Equation} + 或 + \begin{Equation} + \delta (t) = \frac{d\varepsilon(t)}{dt} + \end{Equation} +\end{BoxProperty} + +引入冲激函数后,间断点导数也连续。 + +\begin{BoxProperty}[复合函数形式的冲激函数] + 设$f(t)$有$n$个不相等的实根$t_i(i=1,2,\dots,n)$,则 + \begin{Equation} + \delta[f(t)]=\sum\limits_{i=1}^n\frac{1}{|f'(t_i)|}\delta(t-t_i) + \end{Equation} + 若$f(t)$有重根,则$\delta[f(t)]$无意义。 +\end{BoxProperty} + +\begin{Figure}[冲激偶] + \includegraphics[width=60mm]{visio/add1.pdf} +\end{Figure} + +\begin{BoxProperty}[冲激偶的性质] + 冲激偶即冲激函数的导数,有以下性质 + \begin{Equation} + f(t)\delta'(t)=f(0)\delta'(t)-f'(0)\delta(t) + \end{Equation} + \begin{Equation} + \int_{-\infty}^{\infty} f(t)\delta'(t)dt = -f'(0) + \end{Equation} + \begin{Equation} + \int_{-\infty}^{t} \delta'(t)dt = \delta(t) + \end{Equation} + \begin{Equation} + \int_{-\infty}^{\infty} \delta'(t)dt = 0 + \end{Equation} + \begin{Equation} + \delta^{(n)}(at)=\frac{1}{|a|}\cdot\frac{1}{a^n}\delta^{(n)}(t) + \end{Equation} +\end{BoxProperty} + +\subsection{单位样值序列和单位阶跃序列} + +\subsubsection{单位样值序列} + +\begin{Figure}[单位样值序列] + \includegraphics[width=40mm]{visio/1.14.pdf} +\end{Figure} + +\begin{BoxDefinition}[单位样值序列] + 单位样值序列 + \begin{Equation} + \varepsilon (k) \overset{\mathrm{def}}{=} + \left\{ + \begin{aligned} + 1 & , & k = 0 \\ + 0 & , & k \neq 0 + \end{aligned} + \right. + \end{Equation} +\end{BoxDefinition} + + +\begin{BoxProperty}[单位样值序列的取样性] + 单位样值序列的取样性 + \begin{Equation} + f(k)\delta(k-k_0)=f(k_0)\delta(k-k_0) + \end{Equation} + \begin{Equation} + \sum_{k=-\infty}^{\infty}f(k)\delta(k-k_0) = f(k_0) + \end{Equation} +\end{BoxProperty} + +\subsubsection{单位阶跃序列} + +\begin{Figure}[单位阶跃序列] + \includegraphics[width=60mm]{visio/1.15.pdf} +\end{Figure} + +\begin{BoxDefinition}[单位阶跃序列] + 单位阶跃序列 + \begin{Equation} + \varepsilon (k) \overset{\mathrm{def}}{=} + \left\{ + \begin{aligned} + 1 & , & k \geq 0 \\ + 0 & , & k < 0 + \end{aligned} + \right. + \end{Equation} +\end{BoxDefinition} + +\begin{BoxProperty}[单位样值序列和单位阶跃序列的关系] + 单位样值序列和单位阶跃序列的关系 + \begin{Equation} + \delta (k) = \varepsilon(k) - \varepsilon(k-1) + \end{Equation} + \begin{Equation} + \varepsilon(k) = \delta(k) + \delta(k-1) + \dots + \end{Equation} +\end{BoxProperty} + + diff --git a/Chapter01E.tex b/Chapter01E.tex new file mode 100644 index 0000000..631cbc7 --- /dev/null +++ b/Chapter01E.tex @@ -0,0 +1,76 @@ +\section{系统的描述} + +系统是由若干个有相互关联的单元组合而成的具有特定功能的整体。 + +\subsection{系统的分类} + +\begin{itemize} + \item 连续(时间)系统/离散(时间)系统/混合系统(连续和离散系统的组合) + \item 动态系统/即时系统(记忆/无记忆系统) +\end{itemize} + +\subsection{系统的数学模型} + +连续系统解析描述:微分方程 + +离散系统解析描述:差分方程 + +\subsection{系统的框图描述} + +\begin{Figure}[连续系统的基本单元] + \begin{FigureSub}[加法器] + \includegraphics[width=40mm]{visio/1.16-a.pdf} + \end{FigureSub} + \begin{FigureSub}[乘法器] + \includegraphics[width=40mm]{visio/1.16-b.pdf} + \end{FigureSub} + \begin{FigureSub}[积分器] + \includegraphics[width=40mm]{visio/1.16-c.pdf} + \end{FigureSub} + \begin{FigureSub}[延时器] + \includegraphics[width=40mm]{visio/1.16-d.pdf} + \end{FigureSub} + \begin{FigureSub}[数乘器] + \includegraphics[width=40mm]{visio/1.16-e.pdf} + \end{FigureSub} +\end{Figure} + +微分器用积分器表示 + +\begin{Figure}[离散系统的基本单元] + \begin{FigureSub}[离散系统的加法器] + \includegraphics[width=40mm]{visio/1.17-a.pdf} + \end{FigureSub} + \begin{FigureSub}[离散系统的延时器] + \includegraphics[width=40mm]{visio/1.17-b.pdf} + \end{FigureSub} + \begin{FigureSub}[离散系统的数乘器] + \includegraphics[width=40mm]{visio/1.17-c.pdf} + \end{FigureSub} +\end{Figure} + +根据微分方程画系统框图: + +例:$ay''(t)+by'(t)+cy(t) = df'(t)+ef(t)$ + +设辅助函数$f(t)=ax''(t)+bx'(t)+cx(t)$ + +\begin{figure}[H] + \centering + \includegraphics[width=80mm]{visio/sol1.pdf} +\end{figure} + +由LTI特性:$y(t)=ex(t)+dx'(t)$ + +辅助函数移项可得:$x''(t)=f(t)-\frac{b}{a}x'(t)-\frac{c}{a}x(t)$ + + +\begin{figure}[H] + \centering + \includegraphics[width=100mm]{visio/sol2.pdf} +\end{figure} + +对于框图求微分方程,逆向过程即可 + + + diff --git a/Chapter01F.tex b/Chapter01F.tex new file mode 100644 index 0000000..19f3714 --- /dev/null +++ b/Chapter01F.tex @@ -0,0 +1,101 @@ +\section{系统的特性与分析方法} + +\subsection{系统的特性} + +\subsubsection{线性} + +\begin{BoxProperty}[线性系统的性质] + 设$y(\cdot)$为系统的响应,$f(\cdot)$为系统的激励,即$y(\cdot) = T[f(\cdot)]$ + + 齐次性 + \begin{Equation} + f(\cdot) \rightarrow y(\cdot) \Rightarrow af(\cdot)\rightarrow ay(\cdot) + \end{Equation} + 可加性 + \begin{Equation} + \left\{ + \begin{aligned} + f_1(\cdot) \rightarrow y_1(\cdot) \\ + f_2(\cdot) \rightarrow y_2(\cdot) + \end{aligned} + \right. + \Rightarrow + f_1(\cdot) + f_2(\cdot) \rightarrow y_1(\cdot) + y_2(\cdot) + \end{Equation} + 即线性性质 + \begin{Equation} + af_1(\cdot) + bf_2(\cdot) \rightarrow ay_1(\cdot) + by_2(\cdot) + \end{Equation} +\end{BoxProperty} + +\begin{BoxProperty}[线性系统的条件] + 判断一个系统是否属于线性系统,需要满足以下三个条件 + + 可分解性 + \begin{Equation} + y(\cdot) = y_{zi}(\cdot)+y_{zs}(\cdot) + \end{Equation} + 零状态线性 + \begin{Equation} + T[\{af_1(t)+bf_2(t)\},\{0\}] = aT[\{f_1(\cdot)\},\{0\}]+bT[\{f_2(\cdot)\},\{0\}] + \end{Equation} + 零输入线性 + \begin{Equation} + T[\{0\},\{ax_1(0)+bx_2(0)\}] = aT[\{0\},\{x_1(0)\}]+bT[\{0\},\{x_2(0)\}] + \end{Equation} +\end{BoxProperty} + +\subsubsection{时不变性} + +\begin{BoxProperty}[线性时不变系统的性质] + 线性时不变系统满足以下性质 + \begin{Equation} + y_{zs}(t-t_d) = T[\{f(t-t_d)\},\{0\}] + \end{Equation} +\end{BoxProperty} + +直观判断方法:$f(\cdot)$前出现变系数,或有反转、展缩变换,则系统为时变系统 + +\subsubsection{微分与积分特性} + +\begin{BoxProperty}[LTI系统的微分和积分特性] + 微分特性 + \begin{Equation} + f(t) \rightarrow y_{zs}(t) \Rightarrow f'(t) \rightarrow y_{zs}'(t) + \end{Equation} + 积分特性 + \begin{Equation} + f(t) \rightarrow y_{zs}(t) \Rightarrow \int_{-\infty}^{t} f(x)dx \rightarrow \int_{-\infty}^{t} y_{zs}(t) dx + \end{Equation} +\end{BoxProperty} + +\subsubsection{因果性} +\begin{BoxDefinition}[因果系统] + 零状态响应不会出现在激励之前的系统即因果系统,即$t=t_0$时$f(t)$加入,当$t0$时激励$f(t)$的形式确定 + + 例如 + + \begin{Equation} + f(t) = \varepsilon(t) \quad (t>0) \Rightarrow f(t) = 1 \Rightarrow y_p(t) = C + \end{Equation} + + 零状态响应已有初始条件 + + \begin{Equation} + y_{zs}(0_-) = y_{zs}'(0_-) = 0 + \end{Equation} + + 将初始条件及零状态响应代入原方程可解出特解系数 + + 剩余系数可通过\xref{ppt:冲激函数匹配法}冲激函数匹配法求出$y_{zs}(0_+),y_{zs}'(0_+)$,再代入$t>0$时的方程求解 + + 冲激函数匹配法过程 + + 根据激励的冲激项及其系数列出$y_{zs}(t)$各阶导数项,假设激励的冲激项为$a\delta(t)$,响应的最高阶导数为$y''(t)$ + + \begin{Equation} + \left\{ + \begin{array}{ll} + y_{zs}''(t) = a\delta(t) + r_1(t) \\ + y_{zs}'(t) = r_2(t) \\ + y_{zs}(t) = r_3(t) + \end{array} + \right. + \end{Equation} + + 其中$r_i(t)$为不含$\delta(t)$的某函数 + + 方程组系数代入原微分方程根据系数平衡解出 + + 两侧同时从$0_-$到$0_+$积分可得$y_{zs}(0_+),y_{zs}'(0_+)$ + + 积分满足 + \begin{Equation} + \begin{array}{ll} + \int_{0_-}^{0_+}y_{zs}''(t)dt = y_{zs}'(0_+) - y_{zs}'(0_-) \\ + \int_{0_-}^{0_+}r(t)dt = 0 \\ + \int_{0_-}^{0_+}a\delta (t)dt = a + \end{array} + \end{Equation} + $y_{zs}(t)$定义域为$t\geq0$ +\end{BoxFormula} \ No newline at end of file diff --git a/Chapter02B.tex b/Chapter02B.tex new file mode 100644 index 0000000..c0eadd8 --- /dev/null +++ b/Chapter02B.tex @@ -0,0 +1,80 @@ +\section{冲激响应和阶跃响应} + +\subsection{冲激响应} + +\begin{BoxDefinition}[冲激响应] + 由单位冲激函数$\delta(t)$所引起的零状态响应称为单位冲激响应 + \begin{Equation} + h(t) = T[\{0\},\delta(t)] + \end{Equation} +\end{BoxDefinition} + +\begin{BoxDefinition}[冲激响应的数学模型] + LTI系统的冲激响应的数学模型由$n$阶微分方程表示 + \begin{Equation} + \begin{array}{ll} + \frac{d^{n}h(t)}{dt^n}+a_{n-1}\frac{d^{n-1}h(t)}{dt^{n-1}}+\dots+a_1\frac{dh(t)}{dt}+a_0h(t)=b_m\frac{d^m\delta(t)}{dt^m}+b_{m-1}\frac{d^{m-1}\delta(t)}{dt^{m-1}}\\ + +\dots+b_1\frac{d\delta(t)}{dt}+b_0\delta(t) + \end{array} + \end{Equation} +\end{BoxDefinition} + +\begin{BoxFormula}[冲激响应的求解-形式] + 由于$t\geq0$时等式右边为$0$,因此冲激函数响应的齐次解形式相同。 + + 解与特征根有关,对于特征根均为单根的情况 + \begin{Equation} + h(t) = \left[\sum\limits_{i=1}^{n}c_i e^{\lambda_i t} \right]\varepsilon(t) + \end{Equation} + 与$n,m$相对大小有关 + \begin{itemize} + \item 当$n>m$时,$h(t)$不含$\delta(t)$及其各阶导数 + \item 当$n=m$时,$h(t)$包含$\delta(t)$ + \item 当$n0$时图像右移,$t<0$时图像左移}) + \begin{Equation} + f_2(\tau) \rightarrow f_2(-\tau) \rightarrow f_2(t-\tau) + \end{Equation} + 第三步将信号重叠部分相乘(分类讨论区间),列出对应式子。 + + 第四步按分类讨论的区间将相乘后的图形进行积分。 + + 若求某一时刻的卷积积分值,反转平移步骤直接平移对应单位后相乘积分即可。 + +\end{BoxProperty} diff --git a/Chapter02D.tex b/Chapter02D.tex new file mode 100644 index 0000000..3ccac70 --- /dev/null +++ b/Chapter02D.tex @@ -0,0 +1,123 @@ +\section{卷积积分的性质} + +\subsection{卷积代数运算} + +\begin{BoxProperty}[卷积积分的交换律] + 卷积积分满足交换律 + \begin{Equation} + f_1(t)*f_2(t) = f_2(t) * f_1(t) + \end{Equation} +\end{BoxProperty} + +\begin{BoxProperty}[卷积积分的分配律] + 卷积积分满足分配律 + \begin{Equation} + f_1(t)*\left[f_2(t) + f_3(t)\right]= f_1(t)*f_2(t) + f_1(t)*f_3(t) + \end{Equation} + 即系统并联时总的响应为子系统响应之和。 +\end{BoxProperty} + +\begin{BoxProperty}[卷积积分的结合律] + 卷积积分满足结合律 + \begin{Equation} + \left[f(t)*f_1(t)\right]*f_2(t) = f(t)*\left[f_1(t)*f_2(t)\right] + \end{Equation} + 即系统级联时总的响应等于子系统响应的卷积。 +\end{BoxProperty} + +\begin{BoxProperty}[卷积积分的时移特性] + 若$f_1(t)*f_2(t) = f(t)$,则 + \begin{Equation} + f_1(t-t_1)*f_2(t-t_2) = f_1(t-t_2)*f_2(t-t_1) = f(t-t_1-t_2) + \end{Equation} +\end{BoxProperty} + +\subsection{与冲击函数或阶跃函数的卷积} + +\begin{BoxProperty}[与冲激函数卷积的筛选性] + $f(t)$与$\delta(t)$卷积积分满足 + \begin{Equation} + f(t)*\delta(t) = \delta(t)*f(t) = f(t) + \end{Equation} + 即\xref{def:信号的时域分解} + \begin{Equation} + f(t)*\delta(t) = \int_{-\infty}^{\infty} f(\tau)\delta(t-\tau)d\tau = f(t) + \end{Equation} + 推广 + \begin{Equation} + f(t)*\delta(t-t_0) = f(t-t_0) + \end{Equation} + 根据\xref{ppt:卷积积分的时移特性}再次推广 + \begin{Equation} + f(t-t_1)*\delta(t-t_2) = f(t-t_1-t_2) + \end{Equation} +\end{BoxProperty} + +\begin{BoxProperty}[与阶跃函数卷积] + 与阶跃函数卷积满足 + \begin{Equation} + f(t)*\varepsilon(t) = \int_{-\infty}^{\infty} f(\tau)\varepsilon(t-\tau)d\tau = \int_{-\infty}^{t} f(\tau) d\tau + \end{Equation} + 推广 + \begin{Equation} + \varepsilon(t) * \varepsilon(t) = t\varepsilon(t) + \end{Equation} + 但$\varepsilon(t)*\varepsilon(-t)$不存在。 + 该性质可简记为求原函数的积分上限函数。 +\end{BoxProperty} + +\subsection{卷积的微积分性质} + +\begin{BoxProperty}[卷积的微分性质] + 若$f(t) = f_1(t)*f_2(t) = f_2(t)*f_1(t)$,则: + \begin{Equation} + f^{(1)}(t) = f_1^{(1)}(t) * f_2(t) = f_1(t) * f_2^{(1)}(t) + \end{Equation} + 证明: + \begin{Equation} + f^{(1)}(t) = \frac{d}{dt}\int_{-\infty}^{\infty} f_1(\tau)f_2(t-\tau)d\tau = \int_{-\infty}^{\infty} f_1(\tau) \frac{d}{dt}f_2(t-\tau) = f_1(t)*f_2^{(1)}(t) + \end{Equation} + $f_1^{(1)}(t) * f_2(t)$同理。 +\end{BoxProperty} + +\begin{BoxProperty}[卷积的积分性质] + 若$f(t) = f_1(t)*f_2(t) = f_2(t)*f_1(t)$,则: + \begin{Equation} + \int_{-\infty}^{t}\left[f_1(\tau)*f_2(\tau)\right]d\tau = \left[\int_{-\infty}^{t}f_1(\tau)d\tau\right]*f_2(t) = f_1(t)*\left[\int_{-\infty}^{t}f_2(\tau)d\tau\right] + \end{Equation} +\end{BoxProperty} + +\begin{BoxProperty}[卷积的微积分性质] + 在$f_1(-\infty) = 0$\footnote{先微分再积分}或$f_2^{(-1)}(\infty) = 0$\footnote{先积分再微分}前提下,有 + \begin{Equation} + f_1(t)*f_2(t) = f_1^{(1)}(t)*f_2^{(-1)}(t) + \end{Equation} + 条件推导: + + 该式成立要以原式进行一次微分再积分能够还原为条件,即 + \begin{Equation} + \int_{-\infty}^{t}\frac{d\left[f_1(\tau)*f_2(\tau)\right]}{d\tau} d\tau = f_1(t)*f_2(t) - \lim\limits_{t\rightarrow -\infty} \left[f_1(t)*f_2(t)\right] + \end{Equation} + 即满足 + \begin{Equation} + \lim\limits_{t\rightarrow -\infty} \left[f_1(t)*f_2(t)\right]=0 + \end{Equation} + 即 + \begin{Equation} + f_1(-\infty) = f_2(-\infty) = 0 + \end{Equation} + + 性质推广: + \begin{Equation} + f^{(i)}(t) = f_1^{(j)}(t)*f_2^{(i-j)}(t) + \end{Equation} +\end{BoxProperty} + +因此,对于求解连续时间系统的零状态响应还可以使用杜阿密积分。 + +\begin{BoxFormula}[杜阿密积分] + 杜阿密积分 + \begin{Equation} + y_{zs}(t) = f(t)*h(t) = f^{(1)}*h^{(-1)}(t) = f^{(1)}(t)*g(t) = \int_{-\infty}^{\infty} f^{(1)}(\tau)g(t-\tau)d\tau + \end{Equation} +\end{BoxFormula} \ No newline at end of file diff --git a/Chapter03.tex b/Chapter03.tex new file mode 100644 index 0000000..7ce4967 --- /dev/null +++ b/Chapter03.tex @@ -0,0 +1,8 @@ +\chapter{离散系统的时域分析} + +%\input{Chapter03A.tex} +%\input{Chapter03B.tex} +%\input{Chapter03C.tex} +%\input{Chapter03D.tex} +%\input{Chapter03E.tex} +%\input{Chapter03F.tex} \ No newline at end of file diff --git a/Chapter03A.tex b/Chapter03A.tex new file mode 100644 index 0000000..e69de29 diff --git a/Chapter03B.tex b/Chapter03B.tex new file mode 100644 index 0000000..e69de29 diff --git a/Chapter03C.tex b/Chapter03C.tex new file mode 100644 index 0000000..e69de29 diff --git a/Chapter03D.tex b/Chapter03D.tex new file mode 100644 index 0000000..e69de29 diff --git a/Chapter03E.tex b/Chapter03E.tex new file mode 100644 index 0000000..e69de29 diff --git a/Chapter03F.tex b/Chapter03F.tex new file mode 100644 index 0000000..e69de29 diff --git a/Chapter04.tex b/Chapter04.tex new file mode 100644 index 0000000..0021dd3 --- /dev/null +++ b/Chapter04.tex @@ -0,0 +1,11 @@ +\chapter{傅里叶变换和系统的频域分析} + +\input{Chapter04A.tex} +\input{Chapter04B.tex} +\input{Chapter04C.tex} +\input{Chapter04D.tex} +\input{Chapter04E.tex} +\input{Chapter04F.tex} +\input{Chapter04G.tex} +\input{Chapter04H.tex} +\input{Chapter04I.tex} \ No newline at end of file diff --git a/Chapter04A.tex b/Chapter04A.tex new file mode 100644 index 0000000..182ffb0 --- /dev/null +++ b/Chapter04A.tex @@ -0,0 +1,81 @@ +\section{信号分解为正交函数} + +\subsection{矢量正交与正交分解} + +\begin{BoxDefinition}[矢量正交] + 矢量正交指矢量$V_x=(v_{x1},v_{x2},v_{x3})$与$V_y=(v_{y1},v_{y2},v_{y3})$的内积为零,即 + \begin{Equation} + \boldsymbol{V}_x\boldsymbol{V}_y^{T} = \sum\limits_{i=1}^{3}v_{xi}v_{yi} = 0 + \end{Equation} +\end{BoxDefinition} + +\begin{BoxDefinition}[矢量正交集与正交分解] + 矢量正交集指由两两正交的矢量组成的矢量集合。 + + 正交分解即任意向量用矢量正交集中的向量表示。 +\end{BoxDefinition} + +\subsection{信号正交与正交函数集} + +\begin{BoxDefinition}[信号正交] + 定义在$(t_1,t_2)$区间的$\varphi_1(t)$和$\varphi_2(t)$满足 + \begin{Equation} + \int_{t_1}^{t_2}\varphi_1(t)\varphi_2(t)dt=0 + \end{Equation} + 则称$\varphi_1(t)$和$\varphi_2(t)$在区间$(t_1,t_2)$正交。 + + 可简记为函数内积为$0$。 +\end{BoxDefinition} + +\begin{BoxDefinition}[正交函数集] + 若$n$个函数$\varphi_1(t),\varphi_2(t),\dots,\varphi_n(t)$构成一个函数集,这些函数在区间$(t_1,t_2)$满足 + \begin{Equation} + \int_{t_1}^{t_2}\varphi_i(t)\varphi_j(t)dt = \left\{\begin{aligned} + 0 & , & i\neq j \\ + K_i \neq 0 & , & i = j + \end{aligned} + \right. + \end{Equation} + 则称此函数集为在区间$(t_1,t_2)$的正交函数集。 + + 若该正交函数集之外不存在函数与集合内函数正交,则称此函数集为完备正交函数集。 + + 常见的两个在区间$(t_0,t_0+T)(T=\frac{2\pi}{\Omega})$上的完备正交函数集: + + 三角函数集 + \begin{Equation} + \left\{1,\cos(n\Omega t),\sin(n\Omega t), n=1,2,\dots\right\} + \end{Equation} + 虚指数函数集 + \begin{Equation} + \left\{e^{\mathrm{j}n\Omega t},n=0,\pm 1,\pm 2,\dots \right\} + \end{Equation} +\end{BoxDefinition} + +\subsection{信号的正交分解} + +\begin{BoxDefinition}[信号的正交分解] + 设有$n$个函数$\varphi_1(t),\varphi_2(t),\dots,\varphi_n(t)$在区间$(t_1,t_2)$构成一个正交函数空间。将任一函数$f(t)$用这$n$个正交函数的线性组合来近似,可表示为 + \begin{Equation} + f(t)\approx C_1\varphi_1(t)+C_2\varphi_2(t)+\dots+\varphi_n(t) + \end{Equation} + 当近似函数与原函数的均方误差为$0$时 + \begin{Equation} + f(t) = \sum\limits_{i=1}^{\infty} C_i\varphi_i(t) + \end{Equation} + \begin{Equation} + C_i = \frac{1}{K_i}\int_{t_1}^{t_2}f(t)\varphi_i(t)dt + \end{Equation} + \begin{Equation} + K_i = \int_{t_1}^{t_2} \varphi_i^2(t)dt + \end{Equation} +\end{BoxDefinition} + +\begin{BoxFormula}[巴塞瓦尔能量公式] + 巴塞瓦尔能量公式 + \begin{Equation} + \int_{t_1}^{t_2} f^2(t) dt= \sum\limits_{i=1}^{\infty}C_i^2K_i + \end{Equation} + 表示在区间$(t_1,t_2)$上$f(t)$所含能量恒等于$f(t)$在完备正交函数集中分解的各正交分量能量之和。 +\end{BoxFormula} + diff --git a/Chapter04B.tex b/Chapter04B.tex new file mode 100644 index 0000000..e106e7d --- /dev/null +++ b/Chapter04B.tex @@ -0,0 +1,183 @@ +\section{傅里叶级数} + +\subsection{傅里叶级数的三角式} + +\begin{BoxDefinition}[三角函数集] + 三角函数集 + \begin{Equation} + \left\{1,\cos(n\Omega t),\sin(n\Omega t), n=1,2,\dots\right\} + \end{Equation} + 在一个周期内是一个完备的正交函数集。 + + 由积分可知 + \begin{Equation} + \int_{-\frac{T}{2}}^{\frac{T}{2}}\cos(n\Omega t )\cdot\sin(m\Omega t)dt = 0 + \end{Equation} + \begin{Equation} + \int_{-\frac{T}{2}}^{\frac{T}{2}}\cos(n\Omega t )\cdot\cos(m\Omega t)dt = \left\{ + \begin{aligned} + \frac{T}{2} & , & m=n \\ + 0 & , & m\neq n + \end{aligned} + \right. + \end{Equation} + \begin{Equation} + \int_{-\frac{T}{2}}^{\frac{T}{2}}\sin(n\Omega t )\cdot\sin(m\Omega t)dt = \left\{ + \begin{aligned} + \frac{T}{2} & , & m=n \\ + 0 & , & m\neq n + \end{aligned} + \right. + \end{Equation} +\end{BoxDefinition} + +\begin{BoxDefinition}[傅里叶级数的三角函数形式] + 设$f(t) = f(t+mT)$,即$f(t)$为周期信号,且$\Omega = \frac{2\pi}{T}$,满足狄里赫利(Dirichlet)条件,可分解为以下三角级数,称为傅里叶级数。 + \begin{Equation} + f(t) = \frac{a_0}{2} + \sum\limits_{n=1}^{\infty}a_n\cos(n\Omega t)+ \sum\limits_{n=1}^{\infty}b_n \sin(n\Omega t) + \end{Equation} + 其中系数$a_n$和$b_n$称为傅里叶系数,满足 + \begin{Equation} + a_n = \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}}f(t)\cos(n\Omega t)dt + \end{Equation} + \begin{Equation} + b_n = \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}}f(t)\sin(n\Omega t)dt + \end{Equation} + 且$a_n$是$n$的偶函数,$b_n$是$n$的奇函数。 + + 合并同频率项 + \begin{Equation} + f(t) = \frac{A_0}{2} + \sum\limits_{n=1}^{\infty}A_n\cos(n\Omega t + \varphi_n) + \end{Equation} + 其中 + \begin{Equation} + \begin{array}{ll} + A_0 = a_0 \\ + A_n = \sqrt{a_n^2 + b_n^2} \\ + \varphi_n = - \arctan\frac{b_n}{a_n} + \end{array} + \end{Equation} + 可见$A_n$是$n$的偶函数,$\varphi_n$是$n$的奇函数。 + + 且有 + \begin{Equation} + a_n = A_n\cos\varphi_n,\ b_n = -A_n\sin\varphi_n,\ n = 1,2,\dots + \end{Equation} + 合并同频率项式中的$\frac{A_0}{2}$称为直流分量,$A_1\cos(\Omega t + \varphi_1)$称为基波或一次谐波, + + $A_n\cos(n\Omega t + \varphi_n)$称为$n$次谐波。 +\end{BoxDefinition} + +\subsection{波形的对称性与谐波特性} + +\begin{BoxProperty}[波形的对称性] + 波形为偶函数时满足 + \begin{Equation} + f(t)=f(-t) + \end{Equation} + 此时$b_n=0$,展开为余弦级数。 + + 波形为奇函数时满足 + \begin{Equation} + f(t)=-f(-t) + \end{Equation} + 此时$a_n=0$,展开为正弦级数。 +\end{BoxProperty} + +\begin{BoxProperty}[波形的谐波特性] + \begin{Figure}[奇谐函数] + \includegraphics[width=40mm]{visio/4.1.pdf} + \end{Figure} + $f(t)$为奇谐函数$f(t)=-f(t\pm \frac{T}{2})$时,傅里叶级数满足 + \begin{Equation} + a_0=a_2=\dots=b_2=b_4=0 + \end{Equation} + 即只含奇次谐波分量,不含偶次谐波分量。 + \begin{Figure}[偶谐函数] + \includegraphics[width=40mm]{visio/4.2.pdf} + \end{Figure} + $f(t)$为偶谐函数$f(t)=f(t\pm \frac{T}{2})$时,傅里叶级数满足 + \begin{Equation} + a_1=a_3=\dots=b_1=b_3=0 + \end{Equation} + 即只含偶次谐波分量,不含奇次谐波分量。 +\end{BoxProperty} + +\subsection{傅里叶级数的指数形式} + +\begin{BoxDefinition}[傅里叶级数的指数形式] + 虚指数函数集 + \begin{Equation} + \left\{e^{jn\Omega t},n=0,\pm 1,\pm 2, \dots\right\} + \end{Equation} + 傅里叶级数的指数形式 + \begin{Equation} + f(t) = \sum\limits_{n=-\infty}^{\infty}F_n e^{jn\Omega t} + \end{Equation} + 傅里叶系数 + \begin{Equation} + F_n = \frac{1}{T}\int_{-\infty}^{\infty}f(t)e^{-jn\Omega t}dt + \end{Equation} + 由欧拉公式 + \begin{Equation} + e^{i\theta} = \cos \theta + i sin \theta + \end{Equation} + 可推导出 + \begin{Equation} + \cos x = \frac{e^{\mathrm{j}x}+e^{-\mathrm{j}x}}{2} + \end{Equation} + \begin{Equation} + \sin x = \frac{e^{\mathrm{j}x}-e^{-\mathrm{j}x}}{2\mathrm{j}} + \end{Equation} + 则三角函数形式的傅里叶级数可推导为 + \begin{Equation} + \begin{aligned} + f(t) & = \frac{A_0}{2}+\sum\limits_{n=1}^{\infty} A_n \cos(n\Omega t +\varphi_n) \\ + & = \frac{A_0}{2}+\sum\limits_{n=1}^{\infty} \frac{A_n}{2}\left[e^{\mathrm{j}(n\Omega t+\varphi_n)}+e^{-\mathrm{j}(n\Omega t+\varphi_n)}\right] \\ + & = \frac{A_0}{2}+\frac{1}{2}\sum\limits_{n=1}^{\infty}A_ne^{\mathrm{j}\varphi_n}e^{\mathrm{j}n\Omega t}+\frac{1}{2}\sum\limits_{n=1}^{\infty}A_ne^{-\mathrm{j}\varphi_n}e^{-\mathrm{j}n\Omega t} + \end{aligned} + \end{Equation} + 令$A_{-n}=A_n,\varphi_{-n}=\varphi_n,A_0=A_0e^{\mathrm{j}\varphi_0}e^{\mathrm{j}0\Omega t},\varphi_0=0$,则 + \begin{Equation} + \begin{aligned} + f(t) & = \frac{A_0}{2}+\frac{1}{2}\sum\limits_{n=1}^{\infty}A_ne^{\mathrm{j}\varphi_n}e^{\mathrm{j}n\Omega t}+\frac{1}{2}\sum\limits_{n=-1}^{-\infty}A_ne^{\mathrm{j}\varphi_n}e^{\mathrm{j}n\Omega t} \\ + & = \frac{1}{2}\sum\limits_{n=-\infty}^{\infty}A_ne^{\mathrm{j}\varphi_n}e^{\mathrm{j}n\Omega t} + \end{aligned} + \end{Equation} +\end{BoxDefinition} + +\begin{BoxProperty}[傅里叶系数之间的关系] + 傅里叶系数之间满足以下关系 + \begin{Equation} + F_n = \frac{1}{2}A_ne^{\mathrm{j}\varphi_n} = |F_n|e^{\mathrm{j}\varphi_n} = \frac{1}{2}(a_n-\mathrm{j}b_n) + \end{Equation} + \begin{Equation} + F_0 = \frac{A_0}{2} + \end{Equation} + \begin{Equation} + |F_0| = \frac{1}{2}\sqrt{a_n^2+b_n^2} = \frac{1}{2}A_n + \end{Equation} + \begin{Equation} + \varphi_n = \arctan(-\frac{b_n}{a_n}) + \end{Equation} + \begin{Equation} + a_n = A_n\cos\varphi_n + \end{Equation} + \begin{Equation} + b_n = -A_n\sin\varphi_n + \end{Equation} + 其中$n$的奇函数有:$a_n,A_n,|F_n|$ + + $n$的偶函数有:$b_n,\varphi_n$ +\end{BoxProperty} + + +\subsection{周期信号的功率} + +\begin{BoxFormula}[Parseval等式] + 周期信号一般为功率信号,其平均功率为 + \begin{Equation} + P = \frac{1}{T}\int_{0}^{T}f^2(t)dt=\left(\frac{A_0}{2}\right)^2+\sum\limits_{n=1}^{\infty}\frac{1}{2}A_n^2=\sum\limits_{n=-\infty}^{\infty}|F_n|^2 + \end{Equation} + 其中$\left(\frac{A_0}{2}\right)^2$为直流功率,$\sum\limits_{n=1}^{\infty}\frac{1}{2}A_n^2$为各次谐波功率和。 +\end{BoxFormula} diff --git a/Chapter04C.tex b/Chapter04C.tex new file mode 100644 index 0000000..579aaf6 --- /dev/null +++ b/Chapter04C.tex @@ -0,0 +1,82 @@ +\section{周期信号的频谱} + +\subsection{周期信号频谱的概念} + +对于傅里叶级数$f(t) = \frac{A_0}{2}+\sum\limits_{n=1}^{\infty}A_n\cos(n\Omega t+ \varphi_n)$ + +\begin{Figure}[振幅频谱] + \includegraphics[width=100mm]{visio/4.3.pdf} +\end{Figure} + +振幅频谱为$A_n\sim\omega$曲线时为单边谱线,$|F_n|\sim\omega$曲线时为双边谱线。 + +\begin{Figure}[相位频谱] + \includegraphics[width=100mm]{visio/4.4.pdf} +\end{Figure} + +求某傅里叶级数的频谱图,先求出其基波角频率,再根据各次谐波分量作图。 + +求某傅里叶级数的双边频谱图,先化为指数形式,写出$F_n$各值再做图。 + +\subsection{周期信号频谱的特点} + +以幅度为$1$,脉冲宽度为$\tau$的周期矩形脉冲为例,其周期为$T$ + +\begin{Figure}[周期矩形脉冲] + \includegraphics[width=40mm]{visio/4.5.pdf} +\end{Figure} + +其频谱$F_n$满足 + +\begin{Equation} + F_n=\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}f(t)e^{-\mathrm{j}n\Omega t}dt=\frac{1}{T}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}e^{-\mathrm{j}n\Omega t}dt=\frac{1}{T}\left.\frac{e^{-\mathrm{j}n\Omega t}}{-\mathrm{j}n\Omega}\right|^{\frac{\tau}{2}}_{-\frac{\tau}{2}}=\frac{\tau}{T}\frac{\sin\frac{n\Omega\tau}{2}}{\frac{n\Omega\tau}{2}} +\end{Equation} + +令$Sa(x)=\frac{\sin(x)}{x}$(取样函数),则 + +\begin{Equation} + F_n=\frac{\tau}{T}Sa(\frac{n\Omega\tau}{2})=\frac{\tau}{T}Sa(\frac{n\pi\tau}{T}),n=0,\pm 1,\pm 2,\dots +\end{Equation} + +设$T=5\tau$,则 + +\begin{Figure}[周期矩形脉冲的幅频曲线] + \includegraphics[width=80mm]{visio/4.6.pdf} +\end{Figure} + +特点: +\begin{itemize} + \item 周期信号的频谱具有谐波(离散)性,谱线位置是基频$\Omega$的整数倍 + \item 频谱一般具有收敛性,总趋势减小 + \item $T$一定时,$\tau$变小,此时$\Omega=\frac{2\pi}{T}$(谱线间隔)不变,但两零点间谱线数目$\frac{\omega_1}{\Omega}=\frac{\frac{2\pi}{\tau}}{\frac{2\pi}{T}}=\frac{T}{\tau}$增多 + \item $\tau$一定时,$T$增大,间隔$\Omega$减小,频谱变密,幅度减小($T\rightarrow\infty$时,谱线间隔趋于零,过渡到非周期信号的连续频谱,各频率分量幅度趋于无穷小) +\end{itemize} + + +\subsection{频带宽度} + +由频谱的收敛性可知,信号的功率集中在低频段。 + +\begin{BoxFormula}[部分功率公式] + \begin{Equation} + P_{in}=F_0^2+|F_1|^2+\dots+|F_{i-1}|^2+|F_{-1}|^2+\dots+|F_{-(i-1)}|^2 + \end{Equation} +\end{BoxFormula} + +\begin{BoxDefinition}[频带宽度] + 在满足一定失真条件下,信号可以用某段频率范围内的信号表示,此频率范围称为频率宽度。 + + 对于矩形脉冲信号一般把第一个零点作为信号的频带宽度,记为 + \begin{Equation} + B_\omega = \frac{2\pi}{\tau} + \end{Equation} + 或 + \begin{Equation} + B_f = \frac{1}{\tau} + \end{Equation} + 带宽与脉宽成反比。 + + 对于一般的周期信号,将幅度下降为$0.1|F_n|_{\mathrm{max}}$的频率区间定义为频带宽度。 + + 系统的通频带$>$信号的带宽时,信号才不失真。 +\end{BoxDefinition} diff --git a/Chapter04D.tex b/Chapter04D.tex new file mode 100644 index 0000000..dd24e35 --- /dev/null +++ b/Chapter04D.tex @@ -0,0 +1,201 @@ +\section{非周期信号的频谱} + +\subsection{傅里叶变换} + +\begin{BoxDefinition}[傅里叶变换] + $f(t)$的傅里叶变换 + \begin{Equation} + F(\mathrm{j}\omega)=\int_{-\infty}^{\infty}f(t)e^{-\mathrm{j}\omega t}dt + \end{Equation} + 简记为 + \begin{Equation} + F(\mathrm{j}\omega)=\mathscr{F}\left[f(t)\right] + \end{Equation} + + $F(\mathrm{j}\omega)$的傅里叶逆变换 + \begin{Equation} + f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\mathrm{j}\omega)e^{\mathrm{j}\omega t}d\omega + \end{Equation} + 简记为 + \begin{Equation} + f(t)=\mathscr{F}^{-1}\left[F(\mathrm{j}\omega)\right] + \end{Equation} + $F(\mathrm{j}\omega)$一般是复函数,写为 + \begin{Equation} + F(\mathrm{j}\omega) = |F(\mathrm{j}\omega)|e^{\mathrm{j}\varphi(\omega)}=R(\omega)+\mathrm{j}X(\omega) + \end{Equation} + 傅里叶变换存在的充分条件 + \begin{Equation} + \int_{-\infty}^{\infty}|f(t)|dt<\infty + \end{Equation} + 运用下列关系可以方便计算一些积分 + \begin{Equation} + f(0) = \frac{1}{2\pi}\int_{-\infty}^{\infty}F(\mathrm{j}\omega)d\omega + \end{Equation} + \begin{Equation} + F(0) = \int_{-\infty}^{\infty}f(t)dt + \end{Equation} +\end{BoxDefinition} + +\subsection{常用函数的傅里叶变换} + +\begin{BoxFormula}[门函数的傅里叶变换] + 门函数记为$g_{\tau}(t)$ + \begin{Figure}[门函数] + \includegraphics[width=15mm]{visio/4.7.pdf} + \end{Figure} + 其傅里叶变换为 + \begin{Equation} + F(\mathrm{j}\omega) = \int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}e^{-\mathrm{j}\omega t}dt = \tau\cdot\mathrm{Sa}(\frac{\omega\tau}{2}) + \end{Equation} +\end{BoxFormula} + +当频谱函数为实函数或虚函数时,只需要幅度频谱函数$|F(\mathrm{j}\omega)|$即可表示整个频谱函数,否则还需要相位频谱函数$\varphi(\omega)$。 +\begin{BoxFormula}[单边指数函数的傅里叶变换] + 单边指数函数 + \begin{Equation} + f(t)=e^{-\alpha t}\varepsilon(t) \ (\alpha > 0) + \end{Equation} + 傅里叶变换 + \begin{Equation} + \begin{aligned} + F(\mathrm{j}\omega) & = \int_{-\infty}^{\infty}f(t)e^{-\mathrm{j}\omega t}dt \\ + & = \int_{0}^{\infty} e^{-(\alpha+\mathrm{j}\omega)t}dt \\ + & = \frac{1}{\alpha+\mathrm{j}\omega} \\ + & = \frac{1}{\sqrt{\alpha^2+\omega^2}}e^{-\arctan\frac{\omega}{\alpha}} + \end{aligned} + \end{Equation} + 即 + \begin{Figure}[单边指数函数的频谱函数图像] + \begin{FigureSub}[单边指数函数的幅度频谱] + \quad\quad + \includegraphics[width=30mm]{visio/4.8-a.pdf} + \quad\quad + \end{FigureSub} + \begin{FigureSub}[单边指数函数的相位频谱] + \quad\quad + \includegraphics[width=30mm]{visio/4.8-a.pdf} + \quad\quad + \end{FigureSub} + \end{Figure} + \begin{Equation} + |F(\mathrm{j}\omega)| = \frac{1}{\sqrt{\alpha^2+\omega^2}},\ \varphi(t) = -\arctan\frac{\omega}{\alpha} + \end{Equation} +\end{BoxFormula} + +\begin{BoxFormula}[双边指数函数的傅里叶变换] + 双边指数函数 + \begin{Equation} + f(t) = e^{-\alpha|t|} \ (\alpha > 0) + \end{Equation} + 傅里叶变换 + \begin{Equation} + \begin{aligned} + F(\mathrm{j}\omega) & = \int_{-\infty}^{\infty}f(t)e^{-\mathrm{j}\omega t}dt \\ + & = \int_{-\infty}^{0}f(t)e^{-\mathrm{j}\omega t}dt + \int_{0}^{\infty}f(t)e^{-\mathrm{j}\omega t}dt \\ + & = \frac{1}{\alpha-\mathrm{j}\omega} + \frac{1}{\alpha+\mathrm{j}\omega} \\ + & = \frac{2\alpha}{\alpha^2+\omega^2} + \end{aligned} + \end{Equation} + \begin{Figure}[双边指数函数的频谱函数图像] + \includegraphics[width=30mm]{visio/4.9.pdf} + \end{Figure} + \ +\end{BoxFormula} + +\begin{BoxFormula}[冲激函数的傅里叶变换] + 对于冲激函数$\delta(t)$,其傅里叶变换为 + \begin{Equation} + F(\mathrm{j}\omega) = \int_{-\infty}^{\infty}\delta(t)e^{-\mathrm{j}\omega t }dt = 1 + \end{Equation} + 对于冲激偶$\delta'(t)$,其傅里叶变换为 + \begin{Equation} + F(\mathrm{j}\omega) = \int_{-\infty}^{\infty}\delta'(t)e^{-\mathrm{j}\omega t }dt = \mathrm{j}\omega + \end{Equation} +\end{BoxFormula} + +\begin{BoxDefinition}[广义傅里叶变换] + 由于一些信号使用定义方法求解傅里叶变换不满足绝对可积条件,故使用构造函数序列的方法求解。 + + 构造一个函数序列$\left\{f_\alpha(t)\right\}$满足$\lim\limits_{\alpha\rightarrow?}f_{\alpha}(t) = f(t)$ + + 此时对应的频谱函数序列$\left\{F_{\alpha}(\mathrm{j}\omega)\right\}$也满足$\lim\limits_{\alpha\rightarrow?}F_{\alpha}(\mathrm{j}\omega) = F(\mathrm{j}\omega)$ + + 这种方法即广义傅里叶变换。 +\end{BoxDefinition} + +\begin{BoxFormula}[直流信号的傅里叶变换] + 以$f(t)=1$为例,构造函数序列$f_{\alpha}(t) = e^{-\alpha|t|}$,满足 + \begin{Equation} + \lim\limits_{\alpha\rightarrow 0}f_{\alpha}(t) = 1 + \end{Equation} + 故其频谱函数 + \begin{Equation} + F(\mathrm{j}\omega) = \lim\limits_{\alpha\rightarrow 0}F_{\alpha}(\mathrm{j}\omega) = \lim\limits_{\alpha\rightarrow 0} \frac{2\alpha}{\alpha^2+\omega^2} + \end{Equation} + 当$\omega \neq 0$时,$F(\mathrm{j}\omega) = 0$ + + 当$\omega = 0$时 + \begin{Equation} + F(\mathrm{j}\omega) = \lim\limits_{\alpha\rightarrow 0}\int_{-\infty}^{\infty} \frac{2\alpha}{\alpha^2+\omega^2} d\omega = \lim\limits_{\alpha\rightarrow 0}\int_{-\infty}^{\infty} \frac{2}{1+(\frac{\omega}{\alpha})^2} d\frac{\omega}{\alpha} = \lim\limits_{\alpha\rightarrow 0} 2 \left.\arctan\frac{\omega}{\alpha}\right|_{-\infty}^{\infty} = 2\pi + \end{Equation} + 故频谱函数为 + \begin{Equation} + F(\mathrm{j}\omega) = 2\pi \delta(t) + \end{Equation} +\end{BoxFormula} + +\begin{BoxFormula}[符号函数的傅里叶变换] + 符号函数 + \begin{Equation} + \mathrm{sgn}(t) = + \left\{ + \begin{aligned} + 1 & , & t>0 \\ + 0 & , & t=0 \\ + -1 & , & t<0 + \end{aligned} + \right. + \end{Equation} + 构造一个函数序列 + \begin{Equation} + f_{\alpha}(t) = \left\{ + \begin{aligned} + e^{-\alpha t} & , & t>0 \\ + -e^{\alpha t} & , & t<0 + \end{aligned} + \right. \ (\alpha>0) + \end{Equation} + 满足$\lim\limits_{\alpha\rightarrow 0} f_{\alpha}(t) = \mathrm{sgn}(t)$ + 此时$F_{\alpha}(\mathrm{j}\omega)$ + \begin{Equation} + F_{\alpha}(\mathrm{j}\omega) = \frac{1}{\alpha+\mathrm{j}\omega} - \frac{1}{\alpha - \mathrm{j}\omega} = - \frac{\mathrm{j}2\omega}{\alpha^2+\omega^2} + \end{Equation} + 故符号函数的频谱函数为 + \begin{Equation} + F(\mathrm{j}\omega) = \lim\limits_{\alpha\rightarrow 0} F_{\alpha}(\mathrm{j}\omega) = \frac{2}{\mathrm{j}\omega} + \end{Equation} + \begin{Figure}[符号函数的频谱函数图像] + \begin{FigureSub}[符号函数的幅度频谱] + \quad + \includegraphics[width=30mm]{visio/4.10-a.pdf} + \quad + \end{FigureSub} + \begin{FigureSub}[符号函数的相位频谱] + \quad + \includegraphics[width=30mm]{visio/4.10-b.pdf} + \quad + \end{FigureSub} + \end{Figure} +\end{BoxFormula} + +\begin{BoxFormula}[阶跃函数的傅里叶变换] + 阶跃函数满足 + \begin{Equation} + \varepsilon(t) = \frac{1}{2} + \frac{1}{2}\mathrm{sgn}(t) + \end{Equation} + 故其傅里叶变换为 + \begin{Equation} + F(\mathrm{j}\omega) = \pi \delta(\omega) + \frac{1}{\mathrm{j}\omega} + \end{Equation} +\end{BoxFormula} \ No newline at end of file diff --git a/Chapter04E.tex b/Chapter04E.tex new file mode 100644 index 0000000..d2756d6 --- /dev/null +++ b/Chapter04E.tex @@ -0,0 +1,188 @@ +\section{傅里叶变换的性质} +\subsection{傅里叶变换的线性性质} + +\begin{BoxProperty}[傅里叶变换的线性性质] + 如果$f_1(t)\longleftrightarrow F_1(\mathrm{j} \omega),f_2(t)\longleftrightarrow F_2(\mathrm{j} \omega)$,那么 + \begin{Equation} + \left[af_1(t)+bf_2(t)\right]\longleftrightarrow\left[aF_1(t)+bF_2(t)\right] + \end{Equation} +\end{BoxProperty} + +\subsection{奇偶虚实性} + +\begin{BoxProperty}[傅里叶变换的奇偶虚实性] + 如果$f_1(t)$是实函数,且$f(t)\longleftrightarrow F(\mathrm{j} \omega)=\left|F(\mathrm{j} \omega)\right|e^{\mathrm{j} \varphi (\omega)}=R(\omega)+\mathrm{j} X(\omega)$ + + 其中$\left|F(\mathrm{j} \omega)\right|=\sqrt{R^2(\omega)+X^2(\omega)} \quad \varphi(\omega)=\arctan(\frac{X(\omega)}{\omega})$,那么 + \begin{Equation} + \begin{array}{l} + R(\omega)=R(-\omega), \quad X(\omega)=-X(-\omega) \\ + |F(\mathrm{j} \omega)|=|F(-\mathrm{j} \omega)|, \quad \varphi(\omega)=-\varphi(-\omega) \\ + f(-t) \longleftrightarrow F(-\mathrm{j} \omega)=F^{*}(\mathrm{j} \omega)\footnote[2]{*表示共轭} \\ + If \quad f(t)=f(-t) \quad then \quad X(\omega)=0, \quad F(\mathrm{j} \omega)=R(\omega) \\ + If \quad f(t)=-f(-t) \quad then \quad R(\omega)=0, \quad F(\mathrm{j} \omega)=\mathrm{j} X(\omega) + \end{array} + \end{Equation} + 简记为实部和幅度为偶函数,虚部和相位为奇函数。 + + 自变量取反时频谱函数自变量取反,即与原频谱函数共轭。 + + 原函数为偶函数时频谱函数为实函数,原函数为奇函数时频谱函数为虚函数。 +\end{BoxProperty} + +\subsection{对称性} + +\begin{BoxProperty}[傅里叶变换的对称性] + 如果$f(t)\longleftrightarrow F(\omega)$,那么 + \begin{Equation} + F(t) \longleftrightarrow 2\pi f(-\omega) + \end{Equation} + 证明: + \begin{Equation} + f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\mathrm{j}\omega) e^{\mathrm{j}\omega t}d\omega + \end{Equation} + $t$换为$-t$得 + \begin{Equation} + f(-t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\mathrm{j}\omega) e^{-\mathrm{j}\omega t}d\omega + \end{Equation} + $t$换元为$\omega$,原有$\omega$换元为$t$ + \begin{Equation} + f(-\omega) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\mathrm{j}t) e^{-\mathrm{j}\omega t}dt + \end{Equation} + 即 + \begin{Equation} + 2\pi f(-\omega) = \int_{-\infty}^{\infty} F(\mathrm{j}t) e^{-\mathrm{j}\omega t}dt\footnote{$\mathrm{j}$表示函数为复函数,因此$F(t)$和$F(\mathrm{j}t)$等价} + \end{Equation} +\end{BoxProperty} + +\subsection{尺度变换性质} + +\begin{BoxProperty}[傅里叶变换的尺度变换性质] + 如果$f(t)\longleftrightarrow F(\mathrm{j}\omega)$,那么 + \begin{Equation} + f(at)\longleftrightarrow\frac{1}{|a|}F\left(\mathrm{j}\frac{\omega}{a}\right) + \end{Equation} + $a$为非零实数。 +\end{BoxProperty} + +\subsection{傅里叶变换的时移特性} + +\begin{BoxProperty}[傅里叶变换的时移特性] + 如果$f(t)\longleftrightarrow F(\mathrm{j}\omega)$,那么 + \begin{Equation} + f(t-t_0)\longleftrightarrow e^{-\mathrm{j}\omega t_0}F(\mathrm{j}\omega) + \end{Equation} + $t_0$为实数。 +\end{BoxProperty} + +\subsection{频移性质} + +\begin{BoxProperty}[傅里叶变换的频移性质] + 如果$f(t)\longleftrightarrow F(\mathrm{j}\omega)$,那么 + \begin{Equation} + F\left[\mathrm{j}(\omega - \omega_0)\right]\longleftrightarrow e^{\mathrm{j}\omega_0 t}f(t) + \end{Equation} + $\omega_0$为实数。 +\end{BoxProperty} + +频移性质常用于信号的调制,信号调制的作用是\uwave{频谱搬移}和\uwave{频谱复用}。 + +\subsection{卷积性质} + +\begin{BoxTheorem}[时域卷积定理] + 如果$f_1(t)\longleftrightarrow F_1(\mathrm{j}\omega)$,$f_2(t)\longleftrightarrow F_2(\mathrm{j}\omega)$,那么 + \begin{Equation} + f_1(t)*f_2(t) \longleftrightarrow F_1(\mathrm{j}\omega)F_2(\mathrm{j}\omega) + \end{Equation} + \begin{Equation} + f_1(t)f_2(t) \longleftrightarrow \frac{1}{2\pi}F_1(\mathrm{j}\omega)*F_2(\mathrm{j}\omega) + \end{Equation} +\end{BoxTheorem} + +\subsection{时域的微分和积分} + +\begin{BoxTheorem}[傅里叶变换的时域微分定理] + 如果$f(t)\longleftrightarrow F(\mathrm{j}\omega)$,那么 + \begin{Equation} + f^{(n)}(t)\longleftrightarrow (\mathrm{j}\omega)^{n}F(\mathrm{j}\omega) + \end{Equation} + 如果$f^{(n)}(t)\longleftrightarrow F_n(\mathrm{j}\omega)$且$f(\infty)+f(-\infty)=0$,那么 + \begin{Equation} + F(\mathrm{j}\omega) = \frac{F_n(\mathrm{j}\omega)}{(\mathrm{j}\omega)^n} + \end{Equation} +\end{BoxTheorem} + +\begin{BoxTheorem}[傅里叶变换的时域积分定理] + 如果$f(t)\longleftrightarrow F(\mathrm{j}\omega)$,那么 + \begin{Equation} + \int_{-\infty}^{t} f(t) dt \longleftrightarrow \pi f(0) \delta(t) + \frac{F(\mathrm{j}\omega)}{\mathrm{j}\omega} + \end{Equation} +\end{BoxTheorem} + +\subsection{频域的微分和积分} + +\begin{BoxTheorem}[傅里叶变换的频域微分定理] + 如果$f(t)\longleftrightarrow F(\mathrm{j}\omega)$,那么 + \begin{Equation} + (-\mathrm{j}t)^nf(t)\longleftrightarrow F^{(n)}(\mathrm{j}\omega) + \end{Equation} +\end{BoxTheorem} + +\begin{BoxTheorem}[傅里叶变换的频域积分定理] + 如果$f(t)\longleftrightarrow F(\mathrm{j}\omega)$,那么 + \begin{Equation} + \pi f(0) \delta(t) + \frac{f(t)}{-\mathrm{j}t}\longleftrightarrow \int_{-\infty}^{\omega} F(\mathrm{j}x) dx + \end{Equation} + 其中 + \begin{Equation} + f(0) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\mathrm{j}\omega) d\omega + \end{Equation} +\end{BoxTheorem} + +\subsection{相关定理} + +\begin{BoxDefinition}[相关函数] + 相关函数即两函数间的相关性函数。 + 对于函数$f(t)$,其自相关函数为 + \begin{Equation} + R(\tau) = \int_{-\infty}^{\infty} f(t)f(t-\tau) dt + \end{Equation} + 根据卷积的定义,可以写为 + \begin{Equation} + R(\tau) = f(t)*f(-t) + \end{Equation} + 证明: + + 令$y(t)=f(-t)$ + \begin{Equation} + \begin{aligned} + f(t)*y(t) & = \int_{-\infty}^{\infty} f(\tau)y(t-\tau) d\tau \\ + & = \int_{-\infty}^{\infty} f(\tau)f(\tau-t) d\tau \\ + & = \int_{-\infty}^{\infty} f(t)f(t-\tau) dt + \end{aligned} + \end{Equation} + 证毕。 + + 同理,设$f_1(t)$,$f_2(t)$,其互相关函数为 + \begin{Equation} + R_{12}(\tau) = f_1(t)*f_2(-t) + \end{Equation} + \begin{Equation} + R_{21}(\tau) = f_2(t)*f_1(-t) + \end{Equation} +\end{BoxDefinition} + + +\begin{BoxTheorem}[相关定理] + 如果$f_1(t)\longleftrightarrow F_1(\mathrm{j}\omega)$,$f_2(t)\longleftrightarrow F_2(\mathrm{j}\omega)$,$f(t)\longleftrightarrow F(\mathrm{j}\omega)$,那么 + \begin{Equation} + \mathscr{F}[R_{12}(\tau)] = F_1(\mathrm{j}\omega) F_2^{*}(\mathrm{j}\omega) + \end{Equation} + \begin{Equation} + \mathscr{F}[R_{21}(\tau)] = F_2(\mathrm{j}\omega) F_1^{*}(\mathrm{j}\omega) + \end{Equation} + \begin{Equation} + \mathscr{F}[R(\tau)] = |F(\mathrm{j}\omega)|^2 + \end{Equation} + 其中*表示共轭,$F^{*}(\mathrm{j}\omega) = F(-\mathrm{j}\omega)$。 +\end{BoxTheorem} diff --git a/Chapter04F.tex b/Chapter04F.tex new file mode 100644 index 0000000..b87cc0d --- /dev/null +++ b/Chapter04F.tex @@ -0,0 +1,51 @@ +\section{能量谱和功率谱} + +\subsection{帕斯瓦尔关系} + +\begin{BoxDefinition}[帕斯瓦尔关系] + 帕斯瓦尔关系 + \begin{Equation} + E = \int_{-\infty}^{\infty} |f(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |F(\mathrm{j}\omega)|^2 d\omega + \end{Equation} +\end{BoxDefinition} + +\subsection{能量谱密度} + +\begin{BoxDefinition}[能量谱密度] + 能量谱指单位频率内信号的能量,记为$E(\omega)$。 + + 在频带$df$内信号的能量为$E(\omega)df$,因而信号在整个频率范围内的总能量 + + \begin{Equation} + E = \int_{-\infty}^{\infty} E(\omega) df = \frac{1}{2\pi} \int_{-\infty}^{\infty} E(\omega) d\omega + \end{Equation} + 由帕斯瓦尔关系可得 + \begin{Equation} + E(\omega) = |F(\mathrm{j}\omega)|^2 + \end{Equation} + 即 + \begin{Equation} + R(\tau) \longleftrightarrow E(\omega) + \end{Equation} + 能量谱函数与自相关函数是一对傅里叶变换对。 +\end{BoxDefinition} + +\subsection{功率谱} + +\begin{BoxDefinition}[功率谱] + 功率谱指单位频率的信号功率,记为$P(\mathrm{j}\omega)$。 + + 在频带$df$内信号的总功率为$P(\omega)df$,因而信号在整个频率范围的总功率 + \begin{Equation} + P = \int_{-\infty}^{\infty} P(\omega) df = \frac{1}{2\pi} \int_{-\infty}^{\infty} P(\omega) d\omega + \end{Equation} + 因此 + \begin{Equation} + P(\omega) = \lim\limits_{T\rightarrow\infty}\frac{|F_T(\mathrm{j}\omega)|^2}{T} + \end{Equation} + 即 + \begin{Equation} + R(\tau) \longleftrightarrow P(\omega) + \end{Equation} + 功率有限信号的功率谱与自相关函数是一对傅里叶变换对。 +\end{BoxDefinition} diff --git a/Chapter04G.tex b/Chapter04G.tex new file mode 100644 index 0000000..7a4e91a --- /dev/null +++ b/Chapter04G.tex @@ -0,0 +1,65 @@ +\section{周期信号的傅里叶变换} +\subsection{正、余弦的傅里叶变换} + +\begin{BoxFormula}[正弦函数的傅里叶变换] + 由\xref{def:傅里叶级数的指数形式}可知,正弦函数满足 + \begin{Equation} + \sin \omega_0 t = \frac{1}{2\mathrm{j}}(e^{\mathrm{j}\omega_0 t}-e^{-\mathrm{j}\omega_0 t}) + \end{Equation} + 由\xref{fml:直流信号的傅里叶变换}、\xref{ppt:傅里叶变换的频移性质}可得 + \begin{Equation} + \sin \omega_0 t \longleftrightarrow - \mathrm{j}\pi \delta(\omega - \omega_0) + \mathrm{j}\pi \delta(\omega+\omega_0) + \end{Equation} +\end{BoxFormula} + +\begin{BoxFormula}[余弦函数的傅里叶变换] + 由\xref{def:傅里叶级数的指数形式}可知,余弦函数满足 + \begin{Equation} + \cos \omega_0 t = \frac{1}{2} (e^{\mathrm{j}\omega_0 t}+e^{-\mathrm{j}\omega_0 t}) + \end{Equation} + 由\xref{fml:直流信号的傅里叶变换}、\xref{ppt:傅里叶变换的频移性质}可得 + \begin{Equation} + \cos \omega_0 t \longleftrightarrow \pi \delta(\omega + \omega_0) + \pi \delta(\omega-\omega_0) + \end{Equation} +\end{BoxFormula} + +\subsection{一般周期信号的傅里叶变换} + +\begin{BoxProperty}[一般周期信号的傅里叶变换] + 对于一般的周期信号,由\xref{ppt:傅里叶变换的频移性质},满足 + \begin{Equation} + f_T(t) = \sum\limits_{n=-\infty}^{\infty} F_n e^{\mathrm{j}n\omega t} \longleftrightarrow F_T(\mathrm{j}\omega) = 2\pi \sum\limits_{n=-\infty}^{\infty} F_n \delta(\omega-n\Omega) + \end{Equation} + 可见周期信号的傅里叶变换是冲激序列,即离散谱\footnote{冲激仅存在于谐波频率处,谱线幅度不是有限值,是冲激函数,换言之无限大}。易知非周期信号的频谱为连续谱。 + + 对于周期为$T$的冲激序列$\delta_T(t) = \sum\limits_{m=-\infty}^{\infty}\delta(t-mT)$,其傅里叶系数 + \begin{Equation} + F_n = \frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}\delta_T(t)e^{-\mathrm{j}n\Omega t}dt = \frac{1}{T} + \end{Equation} + 其傅里叶变换为 + \begin{Equation} + \mathscr{F}\left[\delta_T(t)\right] = \frac{2\pi}{T} \sum\limits_{n=-\infty}^{\infty} \delta(\omega-n\Omega) = \Omega \sum\limits_{n=-\infty}^{\infty} \delta(\omega-n\Omega) + \end{Equation} + 记$\delta_{\Omega}(\omega) = \sum\limits_{n=-\infty}^{\infty} \delta(\omega-n\Omega)$,则 + \begin{Equation} + \delta_T(t) \longleftrightarrow \Omega \delta_{\Omega}(\omega) + \end{Equation} + 对于任一周期信号$f(t)$,我们截取其中一个周期$(-\frac{T}{2},\frac{T}{2})$,即单脉冲信号$f_0(t)$,由\xref{ppt:与冲激函数卷积的筛选性}和\xref{ppt:卷积积分的时移特性},则 + \begin{Equation} + f_T(t) = \delta_T(t)*f_0(t) + \end{Equation} + 由\xref{thm:时域卷积定理} + \begin{Equation} + F(\mathrm{j}\omega) = \Omega \delta_{\Omega}(\omega)\cdot F_0(\mathrm{j}\omega) = \Omega \sum\limits_{n=-\infty}^{\infty}F_0(\mathrm{j}n\Omega)\delta(\omega - n\Omega) + \end{Equation} +\end{BoxProperty} + +\subsection{傅里叶系数与傅里叶变换关系} + +\begin{BoxFormula}[傅里叶系数与傅里叶变换关系] + 由\xref{ppt:一般周期信号的傅里叶变换}中$F(\mathrm{j}\omega)$的两种形式可知,$F_0(\mathrm{j}\omega)$和$F_n$关系满足 + \begin{Equation} + F_n = \left.\frac{1}{T} F_0(\mathrm{j}\omega)\right|_{\omega = n\Omega} + \end{Equation} + +\end{BoxFormula} \ No newline at end of file diff --git a/Chapter04H.tex b/Chapter04H.tex new file mode 100644 index 0000000..5807b78 --- /dev/null +++ b/Chapter04H.tex @@ -0,0 +1,187 @@ +\section{LTI系统的频域分析} + +\subsection[虚指数函数作用于LTI系统的响应]{虚指数函数$e^{j\omega t}$作用于LTI系统的响应} + +\begin{BoxDefinition}[频率响应函数] + 设LTI系统的冲激响应为$h(t)$,当激励为$e^{\mathrm{j}\omega t}$\footnote{幅度为1的虚指数函数}时,其零状态响应为 + \begin{Equation} + y(t) = h(t)*e^{\mathrm{j}\omega t} = \int_{-\infty}^{\infty} h(\tau)e^{\mathrm{j}\omega(t-\tau)}d\tau = \int_{-\infty}^{\infty} h(\tau) e^{-\mathrm{j}\omega \tau} d\tau \cdot e^{\mathrm{j}\omega t} + \end{Equation} + 其中$\int_{-\infty}^{\infty} h(\tau) e^{-\mathrm{j}\omega \tau} d\tau$正好为$\mathscr{F}\left[h(t)\right]$,记为$H(\mathrm{j}\omega)$,称为系统的频率响应函数,即 + \begin{Equation} + y(t) = H(\mathrm{j}\omega) e^{\mathrm{j}\omega t} + \end{Equation} + $H(\mathrm{j}\omega)$反应了$y(t)$的幅度和相位随频率的变化。 +\end{BoxDefinition} + +\subsection{一般信号\texorpdfstring{$f(t)$}作用于LTI系统的响应} + +\begin{BoxProperty}[傅里叶变换频域分析法] + 由LTI系统的齐次性和可加性可知,一般信号$f(t)$作用于系统的时候,响应满足 + \begin{Equation} + \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\mathrm{j}\omega) e^{\mathrm{j}\omega t}d\omega \longleftrightarrow \frac{1}{2\pi} \int_{-\infty}^{\infty} H(\mathrm{j}\omega)F(\mathrm{j}\omega) e^{\mathrm{j}\omega t} d\omega + \end{Equation} + 即 + \begin{Equation} + f(t) \longleftrightarrow y(t) = \mathscr{F}^{-1} \left[F(\mathrm{j}\omega)H(\mathrm{j}\omega)\right] + \end{Equation} + \begin{Equation} + Y(\mathrm{j}\omega) = F(\mathrm{j}\omega) H(\mathrm{j}\omega) + \end{Equation} + 因此频率响应可以定义为 + \begin{Equation} + H(\mathrm{j}\omega) = \frac{Y(\mathrm{j}\omega)}{F(\mathrm{j}\omega)} + \end{Equation} + \begin{Equation} + H(\mathrm{j}\omega) = \left|H(\mathrm{j}\omega)\right|e^{\mathrm{j}\theta(\omega)} = \frac{|Y(\mathrm{j}\omega)|}{|F(\mathrm{j}\omega)|}e^{\mathrm{j}\left[\varphi_y(\omega) - \varphi_f(\omega)\right]} + \end{Equation} + $|H(\mathrm{j}\omega)|$称为幅频响应(幅频特性),$\theta(\omega)$称为相频响应(相频特性)。 + + $|H(\mathrm{j}\omega)|$是$\omega$的偶函数,$\theta(\omega)$是$\omega$的奇函数。 +\end{BoxProperty} + +\begin{BoxProperty}[指数形式傅里叶级数频域分析法] + 设LTI系统的激励为$f_T(t) = \sum\limits_{n=-\infty}^{\infty} F_ne^{\mathrm{j}n\Omega t}$,则响应 + \begin{Equation} + y(t) = h(t) * f_T(t) = \sum\limits_{n=-\infty}^{\infty} F_n\left[h(t) * e^{\mathrm{j}n\Omega t}\right] = \sum\limits_{n=-\infty}^{\infty} F_n H(\mathrm{j}\omega)e^{\mathrm{j}n\Omega t} + \end{Equation} + 因此 + \begin{Equation} + Y_n = F_n H(\mathrm{j}n\Omega) + \end{Equation} + 即 + \begin{Equation} + y(t) = \sum\limits_{n=-\infty}^{\infty} F_nH(\mathrm{j}n\Omega) e^{\mathrm{j}n\Omega t} + \end{Equation} +\end{BoxProperty} + + +\begin{BoxProperty}[三角函数形式傅里叶级数频域分析法] + 设LTI系统的激励为$f_T(t) = \frac{A_0}{2} + \sum\limits_{n=1}^{\infty} A_n\cos(n\Omega t + \varphi_n)$,频率响应函数为$H(\mathrm{j}\omega) = |H(\mathrm{j}\omega)|e^{\mathrm{j}\theta(\omega)}$则响应 + \begin{Equation} + y(t) = \frac{A_0}{2}H(0) + \sum\limits_{n=1}^{\infty} A_n|H(\mathrm{j}n\Omega)| \cos\left[n\Omega t + \varphi_n + \theta(\omega)\right] + \end{Equation} +\end{BoxProperty} + +\subsection[频率响应的求法]{频率响应$H(j\omega)$的求法} + +\begin{BoxProperty}[已知微分方程求解频率响应] + 方程两边同时取傅里叶变换,可得$H(\mathrm{j}\omega)$关于$Y(\mathrm{j}\omega)$和$F(\mathrm{j}\omega)$的方程。 + + 移项即可得$H(\mathrm{j}\omega) = \frac{Y(\mathrm{j}\omega)}{F(\mathrm{j}\omega)}$ + + 根据$Y(\mathrm{j}\omega) = H(\mathrm{j}\omega)F(\mathrm{j}\omega)$可以进一步求响应$y(t)$。 +\end{BoxProperty} + +\begin{BoxProperty}[已知电路求解频率响应] + 根据电路列出$H(\mathrm{j}\omega) = \frac{Y(\mathrm{j}\omega)}{F(\mathrm{j}\omega)}$,标出电路各元件的阻抗即可根据各回路列方程解得所需比值。 + + 电感元件阻抗$\mathrm{j}\omega L$,电容元件阻抗$\frac{1}{\mathrm{j}\omega C}$,电阻元件阻抗$R$。 +\end{BoxProperty} + +\subsection{无失真传输与滤波} + +\begin{BoxDefinition}[无失真传输] + 无失真传输是指输出信号和输入信号相比,信号只有幅度的大小和出现的时间先后不同,没有波形上的变化。设输入信号为$f(t)$,即满足 + \begin{Equation} + y(t) = Kf(t-t_d) + \end{Equation} + 频谱关系满足 + \begin{Equation} + Y(\mathrm{j}\omega) = K e^{-\mathrm{j}\omega t_d}F(\mathrm{j}\omega) + \end{Equation} + 无失真传输满足以下条件 + \begin{Equation} + h(t) = K\delta(t-t_d) + \end{Equation} + \begin{Equation} + H(\mathrm{j}\omega) = \frac{Y(\mathrm{j}\omega)}{F(\mathrm{j}\omega)} = Ke^{-\mathrm{j}\omega t_d} + \end{Equation} + 即 + \begin{Equation} + |H(\mathrm{j}\omega)| = K,\quad \theta(\omega) = -\omega t_d + \end{Equation} + + \begin{Figure}[无失真传输幅频相频特性] + \includegraphics[width=30mm]{visio/4.11.pdf} + \end{Figure} + + 根据输入信号判断输出信号是否失真只需要看输入信号的各次谐波频率对应的$|H(\mathrm{j}\omega)|$是否相等,$\theta(\omega)$的值是否在同一条过原点的直线上(即保证$t_d$相同)即可。 + + 系统的群时延为 + \begin{Equation} + \tau = -\frac{d\theta(\omega)}{d\omega} + \end{Equation} + +\end{BoxDefinition} + +\begin{BoxDefinition}[失真相关概念] + 幅度失真是指各频率分量幅度产生不同程度衰减。 + + 相位失真是指各频率分量产生的相移不与频率成正比,使各频率分量在时间轴上的相对位置发生变化。 + + 线性失真是指只存在幅度或相位失真,不产生新的频率成分。 + + 非线性失真是指产生新的频率成分的失真。 +\end{BoxDefinition} + +\begin{BoxDefinition}[理想低通滤波器] + 具有如下图所示的幅频相频特性的系统称为理想低通滤波器。 + \begin{Figure}[理想低通滤波器幅频相频特性] + \includegraphics[width=30mm]{visio/4.12.pdf} + \end{Figure} + + $\omega_c$称为截止角频率,频率响应可以写为 + + \begin{Equation} + H(\mathrm{j}\omega) = + \left\{ + \begin{aligned} + e^{-\mathrm{j}\omega t_d} & , & |\omega| < \omega_c \\ + 0 & , & |\omega| > \omega_c + \end{aligned} + \right. = g_{2\omega_c}(\omega)e^{-\mathrm{j}\omega t_d} + \end{Equation} + 理想低通滤波器的冲激响应为 + \begin{Equation} + h(t) = \mathscr{F}^{-1}\left[H(\mathrm{j}\omega)\right] = \mathscr{F}^{-1}\left[g_{2\omega_c}(\omega)e^{-\mathrm{j}\omega t_d}\right] = \frac{\omega_c}{\pi} \mathrm{Sa}\left[\omega_c(t-t_d)\right] + \end{Equation} + 显然不是因果系统,因此理想低通滤波器不存在。 + + 理想低通滤波器的阶跃响应 + \begin{Equation} + g(t) = h(t) * \varepsilon(t) = \int_{-\infty}^{t} \frac{\omega_c}{\pi} \frac{\sin\left[\omega_c(\tau-t_d)\right]}{\omega_c(\tau-t_d)}d\tau = \frac{1}{2} + \frac{1}{\pi}\int_{0}^{\omega(t-t_d)}\frac{\sin x}{x} dx + \end{Equation} + 定义正弦积分为 + \begin{Equation} + Si(x) = \int_{0}^{y} \frac{\sin x}{x} dx + \end{Equation} + 则阶跃响应可以写为 + \begin{Equation} + g(t) = \frac{1}{2} + \frac{1}{\pi}Si\left[\omega_c(t-t_d)\right] + \end{Equation} + \begin{Figure}[理想低通滤波器的阶跃响应] + \includegraphics[width=60mm]{visio/4.13.pdf} + \end{Figure} + 其中最小值位置为$t_d-\omega$,最大值位置为$t_d+\omega$,上升时间$t_r = 2\cdot\frac{\pi}{\omega_c}$。 + + $g_{max}(t) = \frac{1}{2} + \frac{\mathrm{Si}(\pi)}{\pi} = 1.0895$ + + 显然,只要$\omega_c<\infty$,必有振荡,该由频率截断效应引起的振荡称为吉布斯现象。 +\end{BoxDefinition} + +\begin{BoxProperty}[物理可实现系统条件] + 从时域特性上来说,物理可实现系统的冲激响应满足 + \begin{Equation} + h(t)=0,\quad t<0 + \end{Equation} + 从频域特性来说,满足佩利-维纳准则(必要条件) + \begin{Equation} + \int_{-\infty}^{\infty} |H(\mathrm{j}\omega)|^2d\omega<\infty + \end{Equation} + 且 + \begin{Equation} + \int_{-\infty}^{\infty}\frac{\left|\ln|H(\mathrm{j}\omega)|\right|}{1+\omega^2}d\omega<\infty + \end{Equation} + 从该准则可看出,对于物理可实现系统,其幅频特性可在某些孤立频率点上为$0$,但不能在某个有限频带内为$0$。 +\end{BoxProperty} \ No newline at end of file diff --git a/Chapter04I.tex b/Chapter04I.tex new file mode 100644 index 0000000..68cf097 --- /dev/null +++ b/Chapter04I.tex @@ -0,0 +1,85 @@ +\section{取样定理} + +\subsection{信号的取样} + +\begin{BoxDefinition}[理想取样] + 理想取样,又叫周期单位冲激取样,设取样脉冲为$s(t) = \delta_{T_s}(t)$,则 + \begin{Equation} + s(t) = \delta_{T_s}(t) = \sum\limits_{n=-\infty}^{\infty} \delta(t-nT_s) \longleftrightarrow S(\mathrm{j}\omega) = \omega_s \sum\limits_{n=-\infty}^{\infty} \delta(\omega - n\omega_s) + \end{Equation} + 则 + \begin{Equation} + f_s(t) = f(t)\delta_{T_s}(t) = \sum\limits_{n=-\infty}^{\infty}f(nT_s)\delta(t-nT_s) + \end{Equation} + \begin{Equation} + F_s(\mathrm{j}\omega) = \mathscr{F}\left[f(t)\delta_{T_s}(t)\right] = \frac{1}{2\pi}F(\mathrm{j}\omega)*\omega_s\delta_{\omega_s}(\omega) = \frac{1}{T_s}\sum\limits_{n=-\infty}^{\infty}F\left[\mathrm{j}(\omega-n\omega_s)\right] + \end{Equation} +\end{BoxDefinition} + + +\begin{BoxProperty}[冲激取样信号的频谱] + 设$T_s$为取样间隔,$\omega_s$为取样角频率。当$\omega_s$满足下式时频谱不发生混叠,可以从$F_s(\mathrm{j}\omega)$中取出$F(\mathrm{j}\omega)$,即从$f_s(t)$中恢复原信号$f(t)$。 + \begin{Equation} + \omega_s \geq 2\omega_m + \end{Equation} +\end{BoxProperty} + +\subsection{时域取样定理} + +\begin{BoxTheorem}[时域取样定理] + 一个频谱在区间$(-\omega_m,\omega_m)$以外为$0$的带限信号$f(t)$,可唯一地由其在均匀间隔$T_s\left[T_s\leq\frac{1}{2f_m}\right]$上的样点值$f(kT_s)$确定。 +\end{BoxTheorem} + +\begin{BoxProperty}[取样信号恢复原信号] + 理想低通滤波器 + \begin{Equation} + H(\mathrm{j}\omega) = \left\{\begin{aligned} + T_s & , & |\omega| < \omega_c \\ + 0 & , & |\omega| > \omega_c + \end{aligned} + \right. + \end{Equation} + 则将取样信号通过理想低通滤波器即可恢复原信号 + \begin{Equation} + F(\mathrm{j}\omega) = F_s(\mathrm{j}\omega)H(\mathrm{j}\omega) \longleftrightarrow f(t) = f_s(t)*h(t) + \end{Equation} + 其中$\omega_c$满足$\omega_m\leq\omega_c\leq\omega_s-\omega_m$ + \begin{Figure}[取样信号恢复原信号] + \begin{FigureSub}[原始取样信号] + \includegraphics[width=50mm]{visio/4.14.pdf} + \end{FigureSub} + \begin{FigureSub}[理想低通滤波器] + \includegraphics[width=40mm]{visio/4.14-b.pdf} + \end{FigureSub} + \begin{FigureSub}[原始信号频谱] + \includegraphics[width=30mm]{visio/4.14-c.pdf} + \end{FigureSub} + \end{Figure} + 即 + \begin{Equation} + \begin{aligned} + f(t) = f_s(t)*h(t) & = \left[\sum\limits_{n=-\infty}^{\infty}f(nT_s)\delta(t-nT_s)\right]*\left[T_s\frac{\omega_s}{\pi}\mathrm{Sa}(\omega_c t)\right] \\ + & = T_s\frac{\omega_s}{\pi}\sum\limits_{-\infty}^{\infty}f(nT_s)\mathrm{Sa}\left[\omega_c(t-nT_s)\right] + \end{aligned} + \end{Equation} + 当$\omega_s=2\omega_m$,则$\omega_c = \omega_m$,$T_s = \frac{2\pi}{\omega_s} = \frac{\pi}{\omega_c}$,此时 + + \begin{Equation} + f(t) = \sum\limits_{n=-\infty}^{\infty}f(nT_s)\mathrm{Sa}\left[\omega_c(t-nT_s)\right] + \end{Equation} +\end{BoxProperty} + +\begin{BoxDefinition}[奈奎斯特频率与间隔] + 恢复原信号必须满足两个条件,一是$f(t)$必须是带限信号,二是取样频率不能太低,满足 + \begin{Equation} + f_s\geq2f_m + \end{Equation} + 或者取样间隔满足 + \begin{Equation} + T_s\leq\frac{1}{2f_m} + \end{Equation} + 否则会产生混叠。 + + 其中$f_s = 2f_m$称为奈奎斯特频率,$T_s = \frac{1}{2f_m}$称为奈奎斯特间隔。 +\end{BoxDefinition} + diff --git a/Chapter05.tex b/Chapter05.tex new file mode 100644 index 0000000..3658434 --- /dev/null +++ b/Chapter05.tex @@ -0,0 +1,6 @@ +\chapter{连续系统的s域分析} + +\input{Chapter05A.tex} +\input{Chapter05B.tex} +\input{Chapter05C.tex} +\input{Chapter05D.tex} \ No newline at end of file diff --git a/Chapter05A.tex b/Chapter05A.tex new file mode 100644 index 0000000..ad2cc3a --- /dev/null +++ b/Chapter05A.tex @@ -0,0 +1,226 @@ +\section{拉普拉斯变换} + +\subsection{从傅里叶变换到拉普拉斯变换} + +\begin{BoxDefinition}[拉普拉斯变换] + 为满足函数的绝对可积条件求解傅里叶变换,可用一个衰减因子$e^{-\sigma t}$($\sigma$为实数)乘信号$f(t)$使得$f(t)e^{-\sigma}$在$t\rightarrow\infty$时信号幅度趋于$0$,此时$f(t)e^{-\sigma}$的傅里叶变换存在,即 + \begin{Equation} + F_b(\sigma+\mathrm{j}\omega) = \mathscr{F}\left[f(t)e^{-\sigma t}\right] = \int_{-\infty}^{\infty} f(t)e^{-(\sigma+\mathrm{j}\omega)t}dt + \end{Equation} + 其傅里叶逆变换为 + \begin{Equation} + f(t)e^{-\sigma} = \frac{1}{2\pi} \int_{-\infty}^{\infty} F_b(\sigma + \mathrm{j}\omega)e^{\mathrm{j}\omega t}d\omega + \end{Equation} + 即 + \begin{Equation} + f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty}F_b(\sigma+\mathrm{j}\omega)e^{(\sigma+\mathrm{j}\omega)t}d\omega + \end{Equation} + 令$s=\sigma+\mathrm{j}\omega$,$d\omega = \frac{ds}{\mathrm{j}}$,则有 + \begin{Equation} + F_b(s) = \int_{-\infty}^{\infty}f(t)e^{-st}dt + \end{Equation} + \begin{Equation} + f(t) = \frac{1}{2\pi \mathrm{j}}\int_{\sigma-\mathrm{j}\infty}^{\sigma+\mathrm{j}\infty} F_b(s)e^{st}ds + \end{Equation} + $F_b(s)$称为$f(t)$的双边拉普拉斯变换,又称为象函数; + + $f(t)$称为$F_b(s)$的双边拉普拉斯逆变换,又称为原函数。 +\end{BoxDefinition} + +\subsection{收敛域} + +\begin{BoxDefinition}[收敛域] + 只有适当的$\sigma$值才能使得积分收敛,信号$f(t)$的拉普拉斯变换存在,其中$\sigma$的取值范围称为$F_b(s)$的收敛域。 +\end{BoxDefinition} + +\begin{BoxFormula}[指数函数因果信号的拉普拉斯变换] + 指数函数因果信号 + \begin{Equation} + f(t) = e^{\alpha t}\varepsilon(t) + \end{Equation} + 拉普拉斯变换 + \begin{Equation} + \begin{aligned} F_b(s) = \int_{0}^{\infty} e^{\alpha t}e^{-st}dt = \left.\frac{e^{-(s-\alpha)t}}{-(s-\alpha)}\right|_{0}^{\infty} &= \frac{1}{s-\alpha}\left[1-\lim\limits_{t\rightarrow\infty} e^{-(\sigma-\alpha)t}e^{-\mathrm{j}\omega t}\right] \\ + &= \left\{\begin{aligned} + \frac{1}{s-\alpha} & , & \mathrm{Re}\left[s\right] = \sigma > \alpha \\ + \text{不定} & , & \sigma = \alpha \\ + \text{无界} & , & \sigma < \alpha + \end{aligned} + \right. + \end{aligned} + \end{Equation} + 故收敛域如下图 + \begin{Figure}[指数函数因果信号的拉氏变换收敛域] + \includegraphics[width=40mm]{visio/5.1.pdf} + \end{Figure} +\end{BoxFormula} + +\begin{BoxFormula}[指数函数反因果信号的拉普拉斯变换] + 指数函数反因果信号 + \begin{Equation} + f(t) = e^{\beta t}\varepsilon(-t) + \end{Equation} + 拉普拉斯变换 + \begin{Equation} + \begin{aligned} F_b(s) = \int_{-\infty}^{0} e^{\beta t}e^{-st}dt = \left.\frac{e^{-(s-\beta)t}}{-(s-\beta)}\right|_{-\infty}^{0} &= \frac{1}{-(s-\beta)}\left[1-\lim\limits_{t\rightarrow-\infty} e^{-(\sigma-\beta)t}e^{-\mathrm{j}\omega t}\right] \\ + &= \left\{\begin{aligned} + \text{无界} & , & \mathrm{Re}\left[s\right] = \sigma > \beta \\ + \text{不定} & , & \sigma = \beta \\ + \frac{1}{-(s-\beta)} & , & \sigma < \beta + \end{aligned} + \right. + \end{aligned} + \end{Equation} + 故收敛域如下图 + \begin{Figure}[指数函数反因果信号的拉氏变换收敛域] + \includegraphics[width=40mm]{visio/5.2.pdf} + \end{Figure} +\end{BoxFormula} + +\begin{BoxFormula}[双边指数函数信号的拉普拉斯变换] + 双边指数函数信号 + \begin{Equation} + f(t) = \left\{\begin{aligned} + e^{\beta t} & , & t<0\\ + e^{\alpha t} & , & t>0 + \end{aligned} + \right. + \end{Equation} + 由\xref{fml:指数函数因果信号的拉普拉斯变换}和\xref{fml:指数函数反因果信号的拉普拉斯变换}可知,其拉普拉斯变换为 + \begin{Equation} + F_b(s) = \frac{1}{s-\alpha} + \frac{1}{-(s-\beta)} \quad , \quad \alpha < \mathrm{Re}\left[s\right] = \sigma < \beta + \end{Equation} + 故收敛域如下图 + \begin{Figure}[双边指数函数信号的拉氏变换收敛域] + \includegraphics[width=40mm]{visio/5.3.pdf} + \end{Figure} +\end{BoxFormula} + +双边拉氏变换必须写出收敛域,因为不同原函数对应的象函数可能相同,但收敛域不同。 + +\subsection{单边拉氏变换} + +\begin{BoxDefinition}[单边拉普拉斯变换] + 设信号的初始时刻为坐标原点,此时在$t<0$时$f(t)=0$,拉普拉斯变换为 + \begin{Equation} + F(s) = \int_{0_{-}}^{\infty} f(t)e^{-st}dt + \end{Equation} + \begin{Equation} + f(t) = \left[\frac{1}{2\pi\mathrm{j}}\int_{\sigma-\mathrm{j}\infty}^{\sigma+\mathrm{j}\infty}F(s)e^{st}ds\right]\varepsilon(t) + \end{Equation} + 简记为 + \begin{Equation} + F(s) = \mathscr{L}\left[f(t)\right] + \end{Equation} + 或 + \begin{Equation} + f(t) \longleftrightarrow F(s) + \end{Equation} + 其收敛域一定是$\mathrm{Re}\left[s\right]>\alpha$,可以省略。 +\end{BoxDefinition} + +\subsection{常见函数的拉普拉斯变换} + +\begin{BoxFormula}[冲激函数的拉普拉斯变换] + 冲激函数的拉普拉斯变换\footnote{本课程主要讨论单边拉普拉斯变换,若无特殊说明均为单边}为 + \begin{Equation} + \delta(t) \longleftrightarrow \int_{0_{-}}^{\infty} \delta(t) e^{-st}dt = 1 + \end{Equation} +\end{BoxFormula} + +\begin{BoxFormula}[指数函数的拉普拉斯变换] + 指数函数的拉普拉斯变换为 + \begin{Equation} + e^{s_0t} \longleftrightarrow \frac{1}{s-s_0} \quad , \quad \sigma > \mathrm{Re}\left[s_0\right] + \end{Equation} +\end{BoxFormula} + +\begin{BoxFormula}[阶跃函数的拉普拉斯变换] + 阶跃函数的拉普拉斯变换为 + \begin{Equation} + \varepsilon(t) \longleftrightarrow \frac{1}{s} \quad , \quad \sigma > 0 + \end{Equation} + $1$的拉普拉斯变换也为$\frac{1}{s}$。 +\end{BoxFormula} + +\begin{BoxFormula}[余弦函数的拉普拉斯变换] + 余弦函数的拉普拉斯变换为 + \begin{Equation} + \cos \omega_0 t = \frac{1}{2}(\e^{\mathrm{j}\omega_0 t}+e^{-\mathrm{j}\omega_0 t}) \longleftrightarrow \frac{s}{s^2+\omega_0^2} + \end{Equation} +\end{BoxFormula} + +\begin{BoxFormula}[正弦函数的拉普拉斯变换] + 正弦函数的拉普拉斯变换为 + \begin{Equation} + \sin \omega_0 t = \frac{1}{\mathrm{j}2}(\e^{\mathrm{j}\omega_0 t}-e^{-\mathrm{j}\omega_0 t}) \longleftrightarrow \frac{\omega_0}{s^2+\omega_0^2} + \end{Equation} +\end{BoxFormula} + +\begin{BoxFormula}[周期信号的拉普拉斯变换] + 周期信号的拉普拉斯变换为 + \begin{Equation} + F_T(s) = \int_{0}^{\infty} f_T(t)e^{-st}dt = \sum\limits_{n=0}^{\infty} \int_{nT}^{(n+1)T} f_T(t)e^{-st}dt + \end{Equation} + 令$t=t+nT$,则 + \begin{Equation} + \sum\limits_{n=0}^{\infty}e^{-nsT}\int_{0}^{T}f_T(t)e^{-st}dt = \frac{1}{1-e^{-sT}}\int_{0}^{T} f_T(t)e^{-st}dt + \end{Equation} + 当$f_T(t)=\delta_T(t)$时,有 + \begin{Equation} + \delta_T(t) \longleftrightarrow \frac{1}{1-e^{-sT}} + \end{Equation} +\end{BoxFormula} + +\subsection{单边拉氏变换与傅里叶变换的关系} + +\begin{BoxProperty}[单边拉氏变换与傅里叶变换的关系] + 为保证积分限一致,讨论该关系的前提是$f(t)$是因果信号。 + + 单边拉氏变换 + \begin{Equation} + F(s) = \int_{0}^{\infty} f(t)e^{-st} dt \quad , \quad \mathrm{Re}\left[s\right]>\sigma_0 + \end{Equation} + 傅里叶变换 + \begin{Equation} + F(\mathrm{j}\omega) = \int_{-\infty}^{\infty} f(t)e^{-\mathrm{j}\omega t} dt + \end{Equation} + + 当$\sigma_0<0$时,收敛域包含$s=\mathrm{j}\omega$,因此在$s=\mathrm{j}\omega$处满足 + \begin{Equation} + F(s) = \int_{0}^{\infty} f(t)e^{-\mathrm{j}\omega t} dt = F(\mathrm{j}\omega) + \end{Equation} + 当$\sigma_0=0$时,收敛域边界为$\mathrm{j}\omega$轴,此时 + \begin{Equation} + F(\mathrm{j}\omega) = \lim\limits_{\sigma\rightarrow 0} F(s) + \end{Equation} + + 当$\sigma_0>0$时,$F(\mathrm{j}\omega)$不存在 +\end{BoxProperty} + +对于$\sigma_0 = 0$的情况给出一个例子,例如$f(t)=\varepsilon(t)\longleftrightarrow \frac{1}{s}\quad(\sigma>0)$,此时 +\begin{Equation} + F(\mathrm{j}\omega) = \lim\limits_{\sigma\rightarrow 0} = \lim\limits_{\sigma\rightarrow 0}\frac{1}{\sigma + \mathrm{j}\omega} = \lim\limits_{\sigma\rightarrow 0}\frac{\sigma}{\sigma^2 + \omega^2} + \lim\limits_{\sigma\rightarrow 0}\frac{-\mathrm{j}\omega}{\sigma^2 + \omega^2} +\end{Equation} +其中易知$\lim\limits_{\sigma\rightarrow 0}\frac{-\mathrm{j}\omega}{\sigma^2 + \omega^2}=\frac{1}{\mathrm{j}\omega}$,而$\lim\limits_{\sigma\rightarrow 0}\frac{\sigma}{\sigma^2 + \omega^2}$需要讨论$\omega$ + +$\omega\neq 0$时,$\lim\limits_{\sigma\rightarrow 0}\frac{\sigma}{\sigma^2 + \omega^2}=0$; + +$\omega=0$时,根据狄拉克定义,$\delta(t)$可以等价为 + +\begin{Equation} + \delta(t)= \lim\limits_{a\rightarrow 0_{+}} \frac{1}{\pi}\frac{a}{a^2+x^2} +\end{Equation} + +因此 + +\begin{Equation} + \lim\limits_{\sigma\rightarrow 0}\frac{\sigma}{\sigma^2 + \omega^2} = \pi\delta(t) +\end{Equation} + +即 + +\begin{Equation} + F(\mathrm{j}\omega) = \pi \delta(\omega) + \frac{1}{\mathrm{j}\omega} +\end{Equation} + diff --git a/Chapter05B.tex b/Chapter05B.tex new file mode 100644 index 0000000..e485c9e --- /dev/null +++ b/Chapter05B.tex @@ -0,0 +1,119 @@ +\section{拉普拉斯变换性质} + +\subsection{拉氏变换的线性性质} + +\begin{BoxProperty}[拉普拉斯变换的线性性质] + 若$f_1(t)\longleftrightarrow F_1(s)\quad\mathrm{Re}\left[s\right]>\sigma_1$,$f_2(t)\longleftrightarrow F_2(s)\quad \mathrm{Re}\left[s\right]>\sigma_2$,则 + \begin{Equation} + a_1f_1(t)+a_2f_2(t)\longleftrightarrow a_1F_1(s)+a_2F_2(s) \quad \mathrm{Re}\left[s\right]>\max(\sigma_1,\sigma_2) + \end{Equation} +\end{BoxProperty} + +\subsection{拉氏变换的尺度变换} +\begin{BoxProperty}[拉普拉斯变换的尺度变换] + 若$f(t)\longleftrightarrow F(s)\quad\mathrm{Re}\left[s\right]>\sigma_0$,且$a>0$,则 + \begin{Equation} + f(at)\longleftrightarrow\frac{1}{a}F(\frac{s}{a}) + \end{Equation} +\end{BoxProperty} + +\subsection{拉氏变换的时移特性} + +\begin{BoxProperty}[拉普拉斯变换的时移特性] + 若$f(t)\longleftrightarrow F(s)\quad\mathrm{Re}\left[s\right]>\sigma_0$,且$t_0>0$,则 + \begin{Equation} + f(t-t_0)\varepsilon(t-t_0)\longleftrightarrow e^{-st_0}F(s)\quad\mathrm{Re}\left[s\right]>\sigma_0 + \end{Equation} + 与尺度变换结合 + \begin{Equation} + f(at-t_0)\varepsilon(at-t_0)\longleftrightarrow \frac{1}{a}e^{-\frac{t_0}{a}s}F(\frac{s}{a}) + \end{Equation} +\end{BoxProperty} + +\subsection{拉氏变换的复频移特性} + +\begin{BoxProperty}[拉普拉斯变换的复频移特性] + 若$f(t)\longleftrightarrow F(s)\quad\mathrm{Re}\left[s\right]>\sigma_0$,且$s_a = \sigma_a + \mathrm{j}\omega_a$,则 + \begin{Equation} + f(t)e^{s_a t}\longleftrightarrow F(s-s_a)\quad\mathrm{Re}\left[s\right]>\sigma_0+\sigma_a + \end{Equation} +\end{BoxProperty} + +\subsection{拉氏变换的时域的微分特性} + +\begin{BoxTheorem}[拉普拉斯变换的微分定理] + 若$f(t)\longleftrightarrow F(s)\quad\mathrm{Re}\left[s\right]>\sigma_0$,则 + \begin{Equation} + f'(t)\longleftrightarrow sF(s)-f(0_{-}) + \end{Equation} + 进一步推导可得 + \begin{Equation} + f^{(n)}(t) \longleftrightarrow s^nF(s)-\sum\limits_{m=0}^{n-1}s^{n-1-m}f^{(m)}(0_{-}) + \end{Equation} + 若$f(t)$为因果函数 + \begin{Equation} + f^{(n)}(t) \longleftrightarrow s^nF(s) \quad \mathrm{Re}\left[s\right] > \sigma_0 + \end{Equation} +\end{BoxTheorem} + +\subsection{拉氏变换的时域积分特性} + +\begin{BoxTheorem}[拉普拉斯变换的积分定理] + 若$\mathscr{L}\left[f(t)\right]= F(s)$,则 + \begin{Equation} + \mathscr{L}\left[\int_{-\infty}^{t}f(\tau)d\tau\right] = \frac{F(s)}{s}+\frac{f^{(-1)}(0_{-})}{s} + \end{Equation} + 若$f(t)$为因果信号,且$f^{(n)}(0_{-})=0$,则 + \begin{Equation} + \left(\int_{0_{-}}^{t}\right)^{n}f(x)dx \longleftrightarrow \frac{F(s)}{s^{n}} + \end{Equation} + 同理,若$f(t)$为因果信号,若$f^{(n)}(t)\longleftrightarrow F_{n}(s)$,则 + \begin{Equation} + f(t) \longleftrightarrow \frac{F_n(s)}{s^n} + \end{Equation} +\end{BoxTheorem} + +\subsection{拉氏变换的卷积定理} + +\begin{BoxTheorem}[拉普拉斯变换的时域卷积定理] + 若因果函数$f_1(t)\longleftrightarrow F_1(s)\quad \mathrm{Re}\left[s\right]>\sigma_1$, $f_2(t)\longleftrightarrow F_2(s) \quad \mathrm{Re}\left[s\right]>\sigma_2$,则 + \begin{Equation} + f_1(t)*f_2(t) \longleftrightarrow F_1(s)F_2(s) + \end{Equation} +\end{BoxTheorem} + +\begin{BoxTheorem}[拉普拉斯变换的复频域卷积定理] + 若因果函数$f_1(t)\longleftrightarrow F_1(s)\quad \mathrm{Re}\left[s\right]>\sigma_1$, $f_2(t)\longleftrightarrow F_2(s) \quad \mathrm{Re}\left[s\right]>\sigma_2$,则 + \begin{Equation} + f_1(t)f_2(t)\longleftrightarrow\frac{1}{2\pi\mathrm{j}}\int_{c-\mathrm{j}\infty}^{c+\mathrm{j}\infty} F_1(\eta)F_2(s-\eta)d\eta + \end{Equation} +\end{BoxTheorem} + +\subsection{s域微分和积分定理} + +\begin{BoxTheorem}[s域微分和积分定理] + 若$f(t)\longleftrightarrow F(s)$,$\mathrm{Re}\left[s\right]>\sigma_0$,则 + \begin{Equation} + (-t)^{n}f(t)\longleftrightarrow\frac{d^{n}F(s)}{ds} + \end{Equation} + 特别地 + \begin{Equation} + \frac{f(t)}{t}\longleftrightarrow\int_{s}^{\infty}F(\eta)d\eta + \end{Equation} +\end{BoxTheorem} + +\subsection{初值定理和终值定理} + +\begin{BoxTheorem}[初值定理] + 设函数$f(t)$不含$\delta(t)$及其各阶导数(即$F(s)$为真分式,若$F(s)$为假分式则化为真分式) + \begin{Equation} + f(0_{+}) = \lim\limits_{t\rightarrow 0_{+}}f(t) = \lim\limits_{s\rightarrow\infty}sF(s) + \end{Equation} +\end{BoxTheorem} + +\begin{BoxTheorem}[终值定理] + 若$f(t)$当$t\rightarrow \infty$时存在,并且$f(t)\longleftrightarrow F(s)$,$Re\left[s\right]>\sigma_0$,$\sigma_0<0$,则 + \begin{Equation} + f(\infty) = \lim\limits_{s\rightarrow 0 }sF(s) + \end{Equation} +\end{BoxTheorem} \ No newline at end of file diff --git a/Chapter05C.tex b/Chapter05C.tex new file mode 100644 index 0000000..bed799e --- /dev/null +++ b/Chapter05C.tex @@ -0,0 +1,121 @@ +\section{拉普拉斯逆变换} + +\subsection{零、极点的概念} + +\begin{BoxDefinition}[真分式和假分式] + 若象函数为$s$的有理分式,则可写成 + \begin{Equation} + F(s) = \frac{b_m s^m+b_{m-1}s^{m-1}+\dots+b_1 s+b_0}{s^n+a_{n-1}s^{n-1}+\dots+a_1 s+a_0} + \end{Equation} + 若$m\geq n$,则$F(s)$为假分式,可以用多项式除法化为 + \begin{Equation} + F(s) = P(s) + \frac{B_0(s)}{A(s)} + \end{Equation} + 若$m\sigma_0$ + + 离散因果系统的充分必要条件是单位响应满足 + \begin{Equation} + h(k)=0 \quad k<0 + \end{Equation} + 或者系统函数$H(z)$的收敛域为$|z|>\rho_0$ +\end{BoxDefinition} + +\subsection[系统函数与时域响应]{系统函数$H(\cdot)$与时域响应$h(\cdot)$} + +$H(s)$的极点在$s$平面上的位置可分为极点在左半开平面、虚轴和右半开平面三类。 + +\begin{BoxProperty}[极点在左半开平面的连续因果系统的时域响应] + 极点在左半开平面时,若系统函数有负实单极点$p=-\alpha(\alpha>0)$,则$A(s)$中有因子$(s+\alpha)$,其所对应的响应函数为 + \begin{Equation} + h_1(t) = Ke^{-\alpha t}\varepsilon(t) + \end{Equation} + 若有一对共轭复极点$p_{12}=-\alpha\pm\mathrm{j}\beta$,则$A(s)$中有因子$\left[(s+\alpha)^2+\beta^2\right]$,其所对应的响应函数为 + \begin{Equation} + h_2(t) = Ke^{-\alpha t}\cos(\beta t+\theta)\varepsilon(t) + \end{Equation} + 若有$r$重极点,则$A(s)$中有因子$(s+\alpha)^r$或$\left[(s+\alpha)^2+\beta^2\right]^r$,其所对应的响应函数为 + \begin{Equation} + h_3(t) = K_it_ie^{-\alpha t}\varepsilon(t) \quad (i=0,1,2,\dots,r-1) + \end{Equation} + 或 + \begin{Equation} + h_3(t) = K_it^ie^{-\alpha t}\cos(\beta t+\theta)\varepsilon(t) \quad (i=0,1,2,\dots,r-1) + \end{Equation} + 以上三种情况在$t\rightarrow \infty$时,响应均趋于$0$,为暂态分量,该系统稳定。 +\end{BoxProperty} + +\begin{BoxProperty}[极点在虚轴上的连续因果系统的时域响应] + 极点在虚轴上时,若系统函数有负实单极点$p=0$或$p_{12}=\pm\mathrm{j}\beta$,则响应函数为 + \begin{Equation} + h_1(t) = K\varepsilon(t) + \end{Equation} + 或 + \begin{Equation} + h_2(t) = K\cos(\beta t+ \theta)\varepsilon(t) + \end{Equation} + 此时响应为稳态分量,对应系统稳定。 + + 若系统函数有$r$重极点,相应$A(s)$中有$s^r$或$(s^2+\beta^2)^r$,其响应函数为 + \begin{Equation} + h_2(t) = K_i t^i \varepsilon(t) \quad (i=0,1,2,\dots,r-1) + \end{Equation} + 或 + \begin{Equation} + h_3(t) = K_i t^i \cos(\beta t+ \theta)\varepsilon(t) \quad (i=0,1,2,\dots,r-1) + \end{Equation} + 此时响应为递增函数,对应系统不稳定。 +\end{BoxProperty} + +\begin{BoxProperty}[极点在右半开平面的连续因果系统的时域响应] + 极点在右半开平面时,其对应的响应函数都是递增的,系统不稳定。 +\end{BoxProperty} + +由以上性质可知,LTI连续因果系统的$h(t)$函数形式由$H(t)$的极点确定。 + +\subsection{系统函数与频率响应} + +\begin{BoxDefinition}[全通函数] + 若系统的幅频响应$|H(\mathrm{j}\omega)|$为常数,则称为全通系统,其相应的$H(s)$称为全通函数。 + + 凡极点位于左半开平面,零点位于右半开平面,并且所有零点与极点相对于虚轴一一镜像对称的系统函数$H(s)$即为全通函数。 +\end{BoxDefinition} + +\begin{BoxDefinition}[最小相移函数] + 对于有相同幅频特性的系统函数而言,右半开平面没有零点的系统函数称为最小相移函数。 +\end{BoxDefinition} \ No newline at end of file diff --git a/Chapter07B.tex b/Chapter07B.tex new file mode 100644 index 0000000..2be79b9 --- /dev/null +++ b/Chapter07B.tex @@ -0,0 +1,21 @@ +\section{系统的稳定性} + +\subsection{稳定系统的的定义} + +\begin{BoxDefinition}[系统的稳定性] + 一个系统若对任意的有界输入,其零状态响应也是有界的,则称该系统为有界输入有界输出(BIBO\footnote{Bound Input Bound Output})稳定的系统,简称为稳定系统。 + + 即若系统对所有激励$|f(\cdot)|\leq M_f$,其零状态响应$|y_{zs}(\cdot)|\leq M_y$($M$为有限常数),则称系统稳定。 +\end{BoxDefinition} + +\subsection{连续稳定系统的充分必要条件} + +\begin{BoxTheorem}[连续稳定系统的充分必要条件] + 连续稳定系统的充分必要条件中,时域条件为 + \begin{Equation} + \int_{-\infty}^{\infty} |h(t)|dt\leq M + \end{Equation} + s域条件为若$H(s)$的收敛域包含虚轴,则该系统必是稳定系统。 + + 对于因果系统,若$H(s)$的极点均在左半开平面,则该系统必是稳定系统。 +\end{BoxTheorem} \ No newline at end of file diff --git a/Chapter07C.tex b/Chapter07C.tex new file mode 100644 index 0000000..fbe6138 --- /dev/null +++ b/Chapter07C.tex @@ -0,0 +1,118 @@ +\section{信号流图} + +\subsection{信号流图} + +\begin{BoxDefinition}[信号流图] + 信号流图是由结点和有向线段组成的几何图形。可以简化系统的表示并便于计算系统函数。 +\end{BoxDefinition} + +\begin{BoxDefinition}[信号流图的常用术语] + 结点表示系统中的变量或信号的点。 + + 连接两个结点之间的有向线段称为支路,支路上的权值又称支路增益,是两结点间的系统函数(转移函数)。 + + 源点是只有出支路的结点,又称输入结点;汇点是只有入支路的结点,又称输出结点;有入有出的结点称为混合结点。 + + 沿箭头方向从一个结点到其他结点的路径(多个支路)称为路径。 + + 如果通路与任一结点相遇不多于一次,则称为开通路;闭合的路径称为闭通路,又称回路或环;相互没有公共结点的回路,称为不接触回路;只有一个结点和一条支路的回路称为自回路。 + + 前向通路是从源点到汇点的开通路。 + + 前向通路增益和回路增益是通路中各支路增益的乘积。 +\end{BoxDefinition} + +\begin{BoxProperty}[信号流图的基本性质] + 信号只能沿支路箭头方向传输,且支路的输出=该支路的输入与支路增益的乘积。 + + 当结点有多个输入时,该结点将所有的输入支路的信号相加,并将和信号传输给所有与该结点相连的输出支路。 + + 混合结点可通过增加一个增益为$1$的出支路而变为汇点。 +\end{BoxProperty} + +系统框图与信号流图的转换如下 + +\begin{Figure}[系统框图与流图的转换] + \begin{FigureSub}[系统框图] + \includegraphics[width=75mm]{img/7.2.png} + \end{FigureSub} + \begin{FigureSub}[信号流图] + \includegraphics[width=75mm]{img/7.2-bn.png} + \end{FigureSub} +\end{Figure} + +注意加法器前引入了增益为$1$的支路。 + +\begin{BoxProperty}[信号流图的简化] + \begin{Figure}[支路串联] + \includegraphics[width=60mm]{img/7.3-a.png} + \end{Figure} + 支路串联 + \begin{Equation} + X_2 = H_2X_3 = H_1H_2X_1 + \end{Equation} + \begin{Figure}[支路并联] + \includegraphics[width=60mm]{img/7.4.png} + \end{Figure} + 支路并联 + \begin{Equation} + X_2 = H_1X_1 + H_2X_1 = (H_1+H_2)X_1 + \end{Equation} + \begin{Figure}[混联] + \begin{FigureSub}[多入支路单出支路混联] + \includegraphics[width=60mm]{img/7.5-a.png} + \end{FigureSub} + \begin{FigureSub}[多出支路单入支路混联] + \includegraphics[width=60mm]{img/7.5-b.png} + \end{FigureSub} + \end{Figure} + 多入支路单出支路混联 + \begin{Equation} + H_4=H_3X_3 = H_3(H_1X_1+H_2X_2) = H_1H_3X_1 + H_2H_3X_2 + \end{Equation} + 多出支路单入支路混联 + \begin{Equation} + X_3 = H_2X_2 = H_1H_2X_1 + \end{Equation} + \begin{Equation} + X_4 = H_3X_2 = H_1H_3X_1 + \end{Equation} + \begin{Figure}[自环的消除] + \includegraphics[width=60mm]{img/7.6.png} + \end{Figure} + 自环的消除 + \begin{Equation} + H_3 = H_1X_1 + H_2X_2 + H_3X_3 = \frac{H1}{1-H_3}X_1 + \frac{H_2}{1-H_3}X_2 + \end{Equation} +\end{BoxProperty} + +信号流图的化简过程不一定相同,但结果一定相同。 + +\subsection{梅森公式} + +\begin{BoxFormula}[梅森公式] + 系统函数$H(\cdot)$记为$H$,梅森公式为 + \begin{Equation} + H = \frac{1}{\Delta} \sum\limits_{i} p_i \Delta_i + \end{Equation} + 其中$\Delta$称为特征行列式,其表达式 + \begin{Equation} + \Delta = 1 - \sum\limits_{j} L_j + \sum\limits_{m,n}L_mL_n - \sum\limits_{p,q,r}L_pL_qL_r + \dots + \end{Equation} + 其中$\sum\limits_{j} L_j$为不同回路增益之和。 + + $\sum\limits_{m,n}L_mL_n$为两两不接触\footnote{即在所有回路里两回路没有公共结点}的回路的乘积增益之和。 + + $\sum\limits_{p,q,r}L_pL_qL_r$为三三不接触的回路的乘积增益之和。 + + $p_i$是第$i$条前向通路的增益,$\Delta_i$是与第$i$条前向通路不接触的子图的特征行列式,又叫前向通路的余因子。 +\end{BoxFormula} + +使用梅森公式根据信号流图求解系统函数步骤: + +\begin{itemize} + \item 列出所有回路的增益,符号$L_j$ + \item 求特征行列式 + \item 列出所有前向通路的增益,符号$p_i$ + \item 求各前向通路的余因子$\Delta_i$ +\end{itemize} \ No newline at end of file diff --git a/Chapter07D.tex b/Chapter07D.tex new file mode 100644 index 0000000..db88c13 --- /dev/null +++ b/Chapter07D.tex @@ -0,0 +1,45 @@ +\section{系统的结构} + +\subsection{直接实现} + +\begin{BoxProperty}[通过梅森公式直接实现构造信号流图] + 将系统函数化为以下形式 + \begin{Equation} + H(s) = \frac{b_ms^{-(n-m)}+b_{m-1}s^{-(n-m+1)}+\dots+b_0s^{-n}}{1-(-a_{n-1}s^{-1}-a_{n-2}s^{-2}-\dots-a_0s^{-n})} + \end{Equation} + 其中分子每项为一条前向通路,分母除$1$外每项为一条回路。 + 构造可得 + \begin{Figure}[梅森公式直接构造] + \includegraphics[width=80mm]{img/7.7.png} + \end{Figure} +\end{BoxProperty} + +\subsection{级联实现} + +\begin{BoxProperty}[级联实现构造信号流图] + 将$H$分解为若干一阶或二阶的系统函数的乘积,即 + \begin{Equation} + H = H_1H_2\dots H_n + \end{Equation} + 将分解后的系统级联即可。 + + 其中一阶系统函数形式为 + \begin{Equation} + H_i(s) = \frac{s+b_0}{s+a_0} + \end{Equation} + 其中二阶系统函数形式为 + \begin{Equation} + H_i(s) = \frac{s^2+b_{1i}s+b_{0i}}{s^2+a_{1i}s+a_{0i}} + \end{Equation} +\end{BoxProperty} + +\subsection{并联实现} + +\begin{BoxProperty}[并联实现构造信号流图] + 将$H$展开为部分分式,将每个分式分别作为支路增益,再将各支路并联。 + + 即将$H$化为以下形式再构造 + \begin{Equation} + H(s) = \frac{b_0}{s}+\frac{b_1}{s+a_1}+\frac{b_2}{s+a_2}+\dots+\frac{b_n}{s+a_n} + \end{Equation} +\end{BoxProperty} \ No newline at end of file diff --git a/Chapter08.tex b/Chapter08.tex new file mode 100644 index 0000000..5684cb6 --- /dev/null +++ b/Chapter08.tex @@ -0,0 +1,3 @@ +\chapter{系统的状态变量分析} + +%\input{Chapter08A.tex} \ No newline at end of file diff --git a/Local.sty b/Local.sty new file mode 100644 index 0000000..e9d44b6 --- /dev/null +++ b/Local.sty @@ -0,0 +1,7 @@ +\ProvidesPackage{Local} + +\DeclareMathOperator{\Grad}{grad} +\DeclareMathOperator{\Div}{div} +\DeclareMathOperator{\Curl}{curl} + +\newcommand{\Emf}{\mathscr{E}} \ No newline at end of file diff --git a/Makefile b/Makefile new file mode 100644 index 0000000..1c3b275 --- /dev/null +++ b/Makefile @@ -0,0 +1,42 @@ +PROJECT:=ElectromagneticField + +BUILD_DIR:=build + +OUTPUT:=${BUILD_DIR}/${PROJECT}.pdf + +FLAGS:=-xelatex -synctex=1 -interaction=nonstopmode -file-line-error -output-directory=${BUILD_DIR} + +STYS:=$(wildcard *.sty) +CLSS:=$(wildcard *.cls) +TEXS:=$(filter-out .fig.tex,$(wildcard *.tex)) +FIGS:=$(wildcard *.fig.tex) +OCTS:=$(wildcard *.fig.m) +WLSS:=$(wildcard *.fig.wls) + +FIGS_PDF:=$(addprefix ${BUILD_DIR}/,$(FIGS:.tex=.pdf)) +OCTS_PDF:=$(addprefix ${BUILD_DIR}/,$(OCTS:.m=.pdf)) +WLSS_PDF:=$(addprefix ${BUILD_DIR}/,$(WLSS:.wls=.pdf)) + +default: ${BUILD_DIR} ${OUTPUT} + @echo Makefile: Done! + +${OUTPUT}: ${TEXS} ${FIGS_PDF} ${OCTS_PDF} ${WLSS_PDF} ${STYS} ${CLSS} + latexmk ${FLAGS} ${PROJECT}.tex + +${BUILD_DIR}/%.fig.pdf:%.fig.tex ${STYS} ${CLSS} + latexmk ${FLAGS} $< + +${BUILD_DIR}/%.fig.pdf:%.fig.m + octave-cli $< + +${BUILD_DIR}/%.fig.pdf:%.fig.wls + wolframscript.exe -script $< + +${BUILD_DIR}: + mkdir -p ${BUILD_DIR} + +run:default + start ${OUTPUT} + +output: + @echo ${OUTPUT} \ No newline at end of file diff --git a/Reference.tex b/Reference.tex new file mode 100644 index 0000000..c5db8fd --- /dev/null +++ b/Reference.tex @@ -0,0 +1,4 @@ +\begin{thebibliography}{99}% + \bibitem{1} 吴大正,杨林耀,张永瑞,王松林,郭宝龙. 信号与线性系统分析[M]. 5版. 北京: 高等教育出版社, 2019. + \bibitem{2} MR\_Promethus. 信号流图、梅森公式、系统结构[OL]. https://blog.csdn.net/qq\_44431690/article/details/107121049, 2020-07-04/2024-06-06. +\end{thebibliography} diff --git a/img/7.1.png b/img/7.1.png new file mode 100644 index 0000000000000000000000000000000000000000..4f9163e4cef0c0a3b605687792cbad764a9a826d GIT binary patch literal 29398 zcmd?RcT`kc_a$0Jf(R-oAWBk@oIyYdl9Z%mCC8#bGDwahpdgBpL6DpTL^2YUsDR{P z5idzHNRS+Qox=N#{=L!f^?2QHjQ-;>?)6@*Qzz`b*IIMUIrj-vx_|d9;blS;3UyZY zo|Fm-b;1;dIzEMe8vZ6P;LmON4}tAHO$QW;xE}fUSOhCE6$*87$z1KxlShgQ!p1gM z9EK(~MraNfD_b}kg}N=~Vryt@fqp`3gvOX#i_kAu)X>wKn~2aq zN2|KsS2K3AFcvbQ7Zatu?IH{(utGmEq;;{fw00175uwMOR~Y^u`D;#kTHGN|EJWz< zAQz;4q^LwIX=9J3y~)ANZp_8SMSD|-!%*-hFSn5p8!ZnP4>u>55d6c=#U;#pQk(a^}o@rejM-0IOYSlRyPwAK!P#|cJ^)5Xx1 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+++ b/xCommon.sty @@ -0,0 +1,189 @@ +\ProvidesPackage{xCommon} + +%引入下划线和波浪线等修饰 +\RequirePackage{ulem} +%引入一个补丁使得ulem可以正确适用于中文 +\RequirePackage{xeCJKfntef} +%创建一个适用于中文的着重点命令 +\let\udotcn=\CJKunderdot + +%解决引入xeCJKfntef后的ulem命令不适用于混有数学公式的中文的问题 +%https://github.com/CTeX-org/ctex-kit/issues/614 +\ExplSyntaxOn +\cs_gset_protected:Npn \__xeCJK_ulem_CJK_and_Boundary:w + { + \xeCJK_if_ulem_patch:TF + { + \xeCJK_peek_catcode_ignore_spaces:NTF \c_math_toggle_token + { + \xeCJK_class_group_end: %\UL@stop %% remove + \CJKecglue + %\UL@start %% remove + } + { + \bool_if:NTF \l__xeCJK_peek_ignore_spaces_bool + { + \xeCJK_class_group_end: \UL@stop + \UL@start + { \xeCJK_make_node:n { CJK-space } } + } + { + \xeCJK_class_group_end: \UL@stop + \UL@start { \xeCJK_make_node:n { CJK } } + } + \xeCJK_make_group_tag: + } + } + { \__xeCJK_ulem_CJK_and_Boundary:w } + } +\cs_gset_protected:Npn \__xeCJK_peek_catcode_ignore_spaces_branches:w + { + \if_meaning:w \l_peek_token \c_space_token + \bool_set_true:N \l__xeCJK_peek_ignore_spaces_bool + \exp_after:wN \peek_after:Nw + \exp_after:wN \__xeCJK_peek_catcode_ignore_spaces_branches:w + \exp:w \exp_end_continue_f:w %% add + \tex_romannumeral:D 0 + \else: + \if_catcode:w + \exp_not:N \l_peek_token \exp_not:N \l__xeCJK_peek_search_token + \exp_after:wN \exp_after:wN + \exp_after:wN \__xeCJK_peek_catcode_true:w + \else: + \exp_after:wN \exp_after:wN + \exp_after:wN \__xeCJK_peek_catcode_false:w + \fi: + \fi: + } +\ExplSyntaxOff + +%定义一个强调环境 +\newcommand{\empx}[1]{\textbf{\uline{#1}}} + +%引入基于Unicode的表情符号 +\RequirePackage{pifont} + +%定义带圈数字 +%参数范围是1-10 +\newcommand{\circnum}[1]{\ding{\numexpr 171+#1}} +\newcommand{\circnumdark}[1]{\ding{\numexpr 181+#1}} + +%定义罗马数字 +\newcommand{\romnum}[1]{\romannumeral #1} +\newcommand{\Romnum}[1]{\expandafter\@slowromancap\romannumeral #1@} + +%引入更多颜色 +\PassOptionsToPackage{dvipsnames,svgnames,table}{xcolor} +\RequirePackage{xcolor} + +%引入向文档整页插入PDF的功能 +\RequirePackage{pdfpages} + +%引入可以跨页的文本框 +\RequirePackage{framed} + +%引入表达式解析fpeval +\RequirePackage{xfp} + +%引入基于LaTeX-3的新型命令和环境定义方式 +\RequirePackage{xparse} + +%引入循环命令foreach +\RequirePackage{pgffor} + +%引入判断命令ifthenelse +\RequirePackage{ifthen} +\RequirePackage{xifthen} + +%引入超链接 +\RequirePackage{xurl} %解决超链接无法正常换行的问题 +\RequirePackage[hidelinks]{hyperref} + +%创建一个自定义的标签解析对 +%用法:\xrefcreate{<键>}{<值>} +%调用该命令将会创建以下的命令 +% \xrefname<键> +% \xreftext<键> +% \xref<键> +% \fancyref<键> +\NewDocumentCommand{\xrefcreate}{mm}% +{% + %定义"\xrefname<键>"为"<值>" + \expandafter\def\csname xrefname#1\endcsname{#2}% + %判断"\xreftext<键>"是否已经被定义,这代表引用文字格式是否已经被给出 + %如果未被定义,则将"\xreftext<键>{<标签名>}"定义为默认的"<值>\ref*{<键>:<标签名>}",即形如"图1.1" + %如果已经定义,则不做定义,例如在DocumentClass中可能就章节已经规定了"第一章"和"第1.1节" + \@ifundefined{xreftext#1}% + {\expandafter\def\csname xreftext#1\endcsname##1{#2\hspace{0.10em}\ref*{#1:##1}\CJKsetecglue{}}}{}% + %定义"\xref<键>{<标签名>}"为指向"\label{<键>:<标签名>}"带有文字"\xreftext<键>{<标签名>}"的超链接 + \expandafter\def\csname xref#1\endcsname##1% + {% + \hyperref[#1:##1]% + {% + \expandafter\csname xreftext#1\endcsname{##1}% + }% + }% + %定义"\fancyref<键>{<标签名>}",其形如"[图1.1 图像的标题名称]" + \expandafter\def\csname fancyref#1\endcsname##1% + {% + \CJKsetecglue{ }% 整体和前续汉字保留一个空格的间距 + {\bfseries% 设置为粗体 + [% + \hyperref[#1:##1]% 中括号内部为指向"\label{<键>:<标签名>}"的超链接 + {% + \hspace{0.10em}% 左中括号和内部文字保留的距离 + %文字"\xreftext<键>{<标签名>}",即"图1.1" + %文字"<标签名>",即"图像的标题名称" + %两者间添加有一定的水平距离 + \expandafter\csname xreftext#1\endcsname{##1}\hspace{0.5em}\textnormal{##1}% + \hspace{0.08em}% 右中括号和内部文字保留的距离 + \CJKsetecglue{}% 右中括号和内部汉字间的默认空格应被取消 + }% + ]% + }% + \CJKsetecglue{ }% 整体和后续汉字汉字保留一个空格的间距 + }% +}% + +%定义文档层次标题的名称 +\xrefcreate{part}{部分} +\xrefcreate{chap}{章} +\xrefcreate{sec}{节} +\xrefcreate{subsec}{小节} + +%创建一个解析命令,将“<键>:<标签名>”拆分为两部分 +\def\decodetempA{} +\def\decodetempB{} + +\NewDocumentCommand{\decoderef}{>{\SplitArgument{1}{:}}m}% +{% + \decodesave #1% +}% + +\NewDocumentCommand{\decodesave}{mm}% +{% + \def\decodetempA{#1}% + \def\decodetempB{#2}% +}% + +%\fancyref{<键>:<标签名>} ==> \fancyref<键>{标签名} +\NewDocumentCommand{\fancyref}{m}% +{% + \decoderef{#1}% + \expandafter\csname fancyref\decodetempA\endcsname{\decodetempB}% +}% + +%\xref{<键>:<标签名>} ==> \xref<键>{标签名} +\NewDocumentCommand{\xref}{m}% +{% + \decoderef{#1}% + \expandafter\csname xref\decodetempA\endcsname{\decodetempB}% +}% + +%定义一个页面引用 +%如果页面引用的格式没有给出,那么使用这里的默认格式 +\ProvideDocumentCommand{\xreftextpage}{m}{页\CJKsetecglue{}\pageref*{#1}} +\NewDocumentCommand{\xrefpage}{m}% +{% + \hyperref[#1]{\xreftextpage{#1}}% +}% \ No newline at end of file diff --git a/xFloat.sty b/xFloat.sty new file mode 100644 index 0000000..0719358 --- /dev/null +++ b/xFloat.sty @@ -0,0 +1,257 @@ +\ProvidesPackage{xFloat} + +\RequirePackage{xCommon} + +%引入浮动选项[H]以固定浮动体 +\RequirePackage{float} + +%引入配置图表标题格式的宏包 +\RequirePackage{caption} +\captionsetup[table]{skip=8pt} %设定表标题与其后表格的间距 +\captionsetup[figure]{skip=10pt,labelsep=space} %设定图标题与其前图像的间距 + +%创建图的标签解析对 +\xrefcreate{fig}{图} + +%创建表的标签解析对 +\xrefcreate{tab}{表} + +%引入在表格单元格内输入多行文字的命令 +\RequirePackage{makecell} + +%引入在表格中创建跨越多行的单元格的命令 +\RequirePackage{multirow} + +%引入定义复杂表格列格式的功能 +\RequirePackage{array} + +%引入跨页长表格的命令 +\RequirePackage{longtable} + +%引入绘制粗表现的环境 +\RequirePackage{boldline} + +%引入对表头进行斜线分隔的命令 +\RequirePackage{diagbox} + +%定义可调粗细倍率的完整水平线\hlinex +\def\hlinex#1{\hlineB{#1}} +\def\hlinetop{\hlinex{2}} %顶线 +\def\hlinemid{\hlinex{1}} %中线 +\def\hlinebot{\hlinex{2}} %底线 +\def\hlinelig{\hlinex{0.5}} %细线 + +%定义可调粗细倍率的完整水平线\clinex +\def\clinex#1#2{\clineB{#1}{#2}} +\def\clinetop#1{\clinex{#1}{2}} %顶线 +\def\clinemid#1{\clinex{#1}{1}} %中线 +\def\clinebot#1{\clinex{#1}{2}} %底线 +\def\clinelig#1{\clinex{#1}{0.5}} %细线 + +%处理Table和Figure收到的标题名称参数 +%存在五种情况 +% 1. "<标题>" +% \caption{<标题>}\label{...:<标题>} +% 2. "<标题>;<标签>" +% \caption{<标题>}\label{...:<标签>} +% 3. "<标题>;" +% \caption{<标题>} +% 4. ";<标签>" +% \caption{}\label{...:<标签>} +% 5. ";" 或 "" +% \caption{} + +\def\captiontempA{}% +\def\captiontempB{}% + +%用法示例:\captionmake{tab}{<标题>;<标签>} +%用法示例:\captionmake{fig}{<标题>;<标签>} +%使用\captionmake*可以使得原先的"\caption{...}"变为"...",适用于\subcaptionbox命令 +\NewDocumentCommand{\captionmake}{sm>{\SplitArgument{1}{;}}m}% +{% + \captionsave #3% + \IfBooleanTF{#1} + {\captiontempA}% + {\ifthenelse{\equal{\captiontempA}{}}{\caption{}}{\caption{\captiontempA}}}% + \ifthenelse{\equal{\captiontempB}{}}{}{\label{#2:\captiontempB}}% +}% + +\NewDocumentCommand{\captionsave}{mm}% +{% + \def\captiontempA{#1}% + \IfValueTF{#2}% + {\def\captiontempB{#2}}% + {\def\captiontempB{#1}}% +}% + + +%标准表格环境和长表格环境现在具有完全相同的参数接口 +% #1 [表格名称] +% #2 {表格列格式} +% #3 [表格字体] 默认值为\small +% +% #4 布尔参数* 是否取消自动绘线 +% +% #5 <首处表头> 默认值为空 +% #6 (其余表头) 默认值为<首处表头> +% #7 <末处表尾> 默认值为空 +% #8 (其余表尾) 默认值为空 +% +% 其中"(其余表头)"和"(其余表尾)"对Table环境无效,仅作用于TableLong环境 +% +% Table环境的流程 +% 绘制表头线\hlinetop +% 判断<首处表头>是否为空 +% - 若为空,则不输出任何内容 +% - 若非空,则输出<首处表头>并绘制表中线\hlinemid +% 输出表格环境体内容 +% 判断<末处表位>是否为空 +% - 若为空,则绘制表尾线\hlinebot +% - 若非空,则输出<末处表尾> +% +% TableLong的流程基本相似,但在一般表头和一般表尾处以(其余表头)和(其余表尾)替之 +% TableLong中的默认值思路是 +% 1. 如果设置了<首处表头>,则会默认将其作为每一页开头的表头,除非另行设置(其余表头) +% 2. 如果设置了<末处表位>,则并不会影响(其余表尾),这两者是独立的 + +%创建Table环境 +\NewDocumentEnvironment{Table}{O{}G{lllllllll}O{\small}sD<>{}D(){#5}D<>{}D(){}b} +{ +\par +\begin{table}[H] + \centering + \captionmake{tab}{#1} + #3 + \begin{tabular}{#2} + \IfBooleanF{#4}{\hlinetop} + \if\relax\detokenize{#5}\relax + %... + \else + #5 + \IfBooleanF{#4}{\hlinemid} + \fi + #9 + \if\relax\detokenize{#7}\relax + \IfBooleanF{#4}{\hlinebot} + \else + #7 + \fi + \end{tabular} +\end{table} +}{} + +%创建TableLong环境 +%由于一些原因,在longtable中使用ifthenelse等语句会出现错误 +%因此这里我们手动处理名称参数,放弃对3,4,5情况的支持 +\NewDocumentEnvironment{TableLong}{>{\SplitArgument{1}{;}}O{}G{lllllllll}O{\small}sD<>{}D(){#5}D<>{}D(){}b} +{ +\par\captionsave #1 +#3 +\begin{longtable}{#2} + \caption*{续表} \\ + \IfBooleanF{#4}{\hlinetop} + \if\relax\detokenize{#6}\relax + %... + \else + #6 + \IfBooleanF{#4}{\hlinemid} + \fi + \endhead%---------------------------------------- + \caption{\captiontempA}\label{tab:\captiontempB} \\ + \IfBooleanF{#4}{\hlinetop} + \if\relax\detokenize{#5}\relax + %... + \else + #5 + \IfBooleanF{#4}{\hlinemid} + \fi + \endfirsthead%----------------------------------- + \if\relax\detokenize{#8}\relax + \IfBooleanF{#4}{\hlinebot} + \else + #8 + \fi + \endfoot%---------------------------------------- + \if\relax\detokenize{#7}\relax + \IfBooleanF{#4}{\hlinebot} + \else + #7 + \fi + \endlastfoot%------------------------------------ + #9 +\end{longtable} +}{} + +%使得单元格具有更大的上下间距 +%用法:\xgp[上间距][下间距]{内容} +%备注:若仅给出[上间距]而未给出[下间距],后者会默认定为前者 +\NewDocumentCommand{\xgp}{O{0ex}O{#1}m} +{\Gape[#1][#2]{#3}} + +%使得单元格内部可以换行 +%用法:\xcell<对齐模式>[上间距][下间距]{内容} +%<对齐模式>模式的默认值为左对齐,即l +\NewDocumentCommand{\xcell}{D<>{l}O{0ex}O{#2}m} +{\Gape[#2][#3]{\makecell[#1]{#4}}} + +%使得单元格跨行 +%用法:\mr{行数}[纵向位置调整]{内容} +%调整内容向上定[纵向位置调整]为正值 +%调整内容向下定[纵向位置调整]为负值 +\NewDocumentCommand{\mr}{mO{0ex}m} +{\multirow{#1}{*}[#2]{#3}} + +%使得单元格跨行的同时嵌套makecell +%用法:\mrx<对齐模式>{行数}[纵向位置调整]{内容} +%<对齐模式>模式的默认值为左对齐,即l +\NewDocumentCommand{\mrx}{D<>{l}mO{0ex}m} +{\multirow{#2}{*}[#3]{\xcell<#1>{#4}}} + +%使得单元格跨列 +%用法:\mc{列数}[上间距][下间距](列格式){内容} +%对于表格的首个单元格,此处的列格式会覆盖其左右两根纵线 +%对于表格的其余但约个,此处的列格式仅会对右侧的纵线有效 +%(列格式)的默认值是居中,即l +\NewExpandableDocumentCommand{\mc}{mO{0ex}O{#2}D(){l}m} +{\multicolumn{#1}{#4}{\xgp[#2][#3]{#5}}} + +%使得单元格跨列的同时嵌套makecell +%用法:\mcx<对齐模式>{列数}[上间距][下间距](列格式){内容} +%注意在\mcx中(列格式)中的l,c,r是无效的,对齐模式将被<对齐模式>接管控制 +%因而在\mcx中(列格式)设置的意义仅在于控制纵线 +%<对齐模式>模式的默认值为左对齐,即l +\NewExpandableDocumentCommand{\mcx}{D<>{c}mO{0ex}O{#3}D(){l}m} +{\multicolumn{#2}{#5}{\xcell<#1>[#3][#4]{#6}}} + +%引入插图的功能 +\RequirePackage{graphicx} + +%引入绘制子图表的宏包 +%list=true 在插图目录中显示子图 +%labelformat=simple 在子图的caption中显示a而不是(a) +\RequirePackage[labelformat=simple]{subcaption} %这是子图标题对\thesubfigure的修饰 + +%将\thesubcaption的显示格式由a改为(a) +\renewcommand\thesubfigure{(\alph{subfigure})} %这是计数器\thesubfigure本身的格式 + +%经过以上两项修改,现在子图的caption和ref中就同时能以(a)的形式显示了 + +%创建Figure环境 +\NewDocumentEnvironment{Figure}{O{}b} +{ + \begin{figure}[H] + \centering + #2 + \captionmake{fig}{#1} + \vspace{-12pt} + \end{figure} +}{} + +%创建FigureSub环境 +\NewDocumentEnvironment{FigureSub}{O{}b} +{% + \subcaptionbox{\captionmake*{fig}{#1}}% + {% + #2 + }% +}{}% \ No newline at end of file diff --git a/xMath.sty b/xMath.sty new file mode 100644 index 0000000..1990cd9 --- /dev/null +++ b/xMath.sty @@ -0,0 +1,434 @@ +\ProvidesPackage{xMath} + +\RequirePackage{xCommon} + +%AMS 数学宏包 +\RequirePackage{amsmath} + +%AMS 扩展的数学符号 +\RequirePackage{amssymb} %包括\mathbb在内的特殊数学字体 + +%AMS 定理和证明环境 +\RequirePackage{amsthm} + +%引入直立希腊字母 +\RequirePackage{upgreek} +\def\i{\mathrm{i}} %定义正体虚数单位i +\def\e{\mathrm{e}} %定义正体自然常数e +\def\j{\mathrm{j}} %适用于电工的虚数单位j +\let\dipi=\pi %令斜体的pi从原有的\pi转移\dipi +\let\pi=\uppi %令正体的pi使用默认的命令\pi + +\newcommand{\N}{\mathbb{N}} %自然数集 +\newcommand{\Z}{\mathbb{Z}} %整数集 +\newcommand{\Q}{\mathbb{Q}} %有理数集 +\newcommand{\R}{\mathbb{R}} %实数集 +\newcommand{\C}{\mathbb{C}} %复数集 + +\DeclareMathOperator{\sgn}{sgn} %取整函数 + +%三角函数 +%其余的三角函数已经由physics宏包引入 +\DeclareMathOperator{\arsinh}{arsinh} %反双曲正弦 +\DeclareMathOperator{\arcosh}{arcosh} %反双曲余弦 +\DeclareMathOperator{\artanh}{artanh} %反双曲正切 +\DeclareMathOperator{\arcoth}{arcoth} %反双曲余切 +\DeclareMathOperator{\arsech}{arsech} %反双曲正割 +\DeclareMathOperator{\arcsch}{arcsch} %反双曲余割 + +%定义增量的简写 +\newcommand{\delt}{\Delta} + +%定义增量的比值 +\newcommand{\detv}[2]{\frac{\delt #1}{\delt #2}} + +%定义偏微记号的简写 +\newcommand{\pdd}{\partial} + +%定义泛函微分记号的简写 +\newcommand{\fdd}{\delta} + +%重新定义上极限和下极限的记号,使两者可以对齐 +\DeclareMathOperator*{\xlimsup}{lim\hspace{0.20em}sup} +\DeclareMathOperator*{\xliminf}{lim\hspace{0.35em}i\hspace{0.025em}n\hspace{0.025em}f\hspace{0.165em}} + +%巨型运算符 +\NewDocumentCommand{\Lim}{O{}o}{\lim_{#1\IfValueT{#2}{\to #2}}} %极限符号 +\NewDocumentCommand{\Limsup}{sO{}o} %上极限符号 +{\IfBooleanTF{#1}{\xlimsup}{\limsup}_{#2\IfValueT{#3}{\to #3}}} +\NewDocumentCommand{\Liminf}{sO{}o} %下极限符号 +{\IfBooleanTF{#1}{\xliminf}{\liminf}_{#2\IfValueT{#3}{\to #3}}} + +\NewDocumentCommand{\Sum}{O{}O{}}{\sum_{#1}^{#2}} %连续求和符号 +\NewDocumentCommand{\Prod}{O{}O{}}{\prod_{#1}^{#2}} %连续求积符号 + +%引入更多的积分符号 +\RequirePackage{esint} +\NewDocumentCommand{\Int}{O{}O{}}{\int_{#1}^{#2}} %定积分 +\NewDocumentCommand{\Idnt}{O{}}{\iint\nolimits_{#1}} %二重积分 d-double +\NewDocumentCommand{\Itnt}{O{}}{\iiint\nolimits_{#1}} %三重积分 t-triple +\NewDocumentCommand{\Ilnt}{O{}}{\int\nolimits_{#1}} %曲线积分 l-line +\NewDocumentCommand{\Ilot}{O{}}{\oint\nolimits_{#1}} %环路曲线积分 +\NewDocumentCommand{\Isnt}{O{}}{\iint\nolimits_{#1}} %曲面积分 s-surface +\NewDocumentCommand{\Isot}{O{}}{\oiint\nolimits_{#1}} %环路曲面积分 + +\newcommand{\xvec}[1]{\overrightarrow{#1}} %向量AB的简写 \vec只能得到短箭头 +\newcommand{\xbar}[1]{\overline{#1}} %模长AB的简写 \bar只能得到短横线 + +%傅里叶变换 +\NewDocumentCommand{\F}{sg} +{\mathcal{F}\IfBooleanT{#1}{^{-1}}\IfValueT{#2}{\qty{#2}}} + +%拉普拉斯变换 +\RenewDocumentCommand{\L}{sg} +{\mathcal{L}\IfBooleanT{#1}{^{-1}}\IfValueT{#2}{\qty{#2}}} + +%特殊函数 +\newcommand{\Beta}{\mathrm{B}} %贝塔函数 +\newcommand{\PolyGamma}{\uppsi} %双对数函数 +\newcommand{\Jbessel}{\mathrm{J}} %第一类贝塞尔函数 +\newcommand{\Nbessel}{\mathrm{N}} %第二类贝塞尔函数 +\newcommand{\Hbessel}{\mathrm{H}} %第三类贝塞尔函数 +\newcommand{\Ibessel}{\mathrm{I}} %第一类修正贝塞尔函数 +\newcommand{\Kbessel}{\mathrm{K}} %第二类修正贝塞尔函数 +\newcommand{\Plegendre}{\mathrm{P}} %第一类勒让德函数 +\newcommand{\Qlegendre}{\mathrm{Q}} %第二类勒让德函数 +\newcommand{\Hermite}{\text{H}} %厄米多项式 + +%狄拉克函数 +\newcommand{\dirac}{\updelta} + +%哈密顿算符 +\newcommand{\Hami}{\mathcal{H}} + +%霍奇星算子 +\newcommand{\hodge}{{\star}} + +%空集 +\let\EMPTY=\empty %备份\empty命令 +\renewcommand{\empty}{\varnothing} + +%玻尔兹曼常数 +\newcommand{\kB}{k_\text{B}} + +%阿伏伽德罗常数 +\newcommand{\NA}{N_\text{A}} + +%尖括号 +\def\<#1>{\langle#1\rangle} + +%引入为符号左侧添加上下标的命令\prescript +%用法:\prescript{<上标>}{<下标>}{<符号>} +\RequirePackage{mathtools} + +%引入更多的分式命令 +\RequirePackage{xfrac} + +%引入副对角线方向的省略号\iddots并使\vdots和\ddots适用于任意字号 +\RequirePackage{mathdots} + +%引入可以绘制带有复杂格式的矩阵的环境 +\RequirePackage{nicematrix} + +%引入一些常用数学符号的简便记法 +\RequirePackage{physics} + +\newcommand{\vi}{\vb*{i}} %单位向量i +\newcommand{\vj}{\vb*{j}} %单位向量j +\newcommand{\vk}{\vb*{k}} %单位向量k + +\newcommand{\dx}{\dd{x}} %微分dx +\newcommand{\dy}{\dd{y}} %微分dy +\newcommand{\dz}{\dd{z}} %微分dz + +%引入可以简便控制数学公式形式的功能 +\RequirePackage{relsize} +\let\mal=\mathlarger %使得行内公式以行间形式显示 +\let\mas=\mathsmaller %使得行间公式以行内形式显示 + +%引入带字的等号和箭头 +\RequirePackage{extarrows} + +%带字等号 +\NewDocumentCommand{\xleq}{O{}O{}}{\xlongequal[\text{#2}]{\text{#1}}} + +%带字箭头 +\NewDocumentCommand{\xlarr}{O{}O{}}{\xlongrightarrow[\text{#2}]{\text{#1}}} + +\NewDocumentCommand{\xlarrL}{O{}O{}}{\xlongleftarrow[\text{#2}]{\text{#1}}} %单线左箭头 +\NewDocumentCommand{\xlarrR}{O{}O{}}{\xlongrightarrow[\text{#2}]{\text{#1}}} %单线右箭头 +\NewDocumentCommand{\xlarrB}{O{}O{}}{\xlongleftrightarrow[\text{#2}]{\text{#1}}} %单线双箭头 + +\NewDocumentCommand{\xLarrL}{O{}O{}}{\xLongleftarrow[\text{#2}]{\text{#1}}} %双线左箭头 +\NewDocumentCommand{\xLarrR}{O{}O{}}{\xLongrightarrow[\text{#2}]{\text{#1}}} %双线右箭头 +\NewDocumentCommand{\xLarrB}{O{}O{}}{\xLongleftrightarrow[\text{#2}]{\text{#1}}} %双线双箭头 + +%引入弧的记号 +\RequirePackage{yhmath} +\let\xpar=\wideparen %弧AB的简写 +\let\xhat=\widehat %尖AB的简写 \hat只能得到单个符号的尖 +\let\xwav=\widetilde %波AB的简写 + +%引入数学花体mathscr +\RequirePackage{mathrsfs} + +%引入多项式长除 +\RequirePackage{polynom} + +%引入物理单位的命令\si{} +\RequirePackage{siunitx} +\sisetup +{ + inter-unit-product = \ensuremath{{}\cdot{}} %将形如\si{N.m}以N\cdot m的形式显示 +} + +%引入有机化学的命令 +\RequirePackage{chemfig} + +%引入无机化学的命令 +\RequirePackage{mhchem} +\newcommand{\xce}[1]{\hspace{0.3em}\ce{#1}} %使用\xce{}可以在化学式和前序中文间添加空格 + +%引入彩色文本盒的环境 +\RequirePackage{tcolorbox} +\tcbuselibrary{breakable} + +%生成一个标签盒计数器的命令 +\NewDocumentCommand{\labelboxcnt}{m} +{ + \newcounter{#1}[section] %计数器的上级为节 + %定义\the<计数器名称>的格式为<章>.<节>.<计数器名称> + \expandafter\def\csname the#1\endcsname{\thesection.\arabic{#1}} + %定义\add<计数器名称>为计数器自增后显示计数器数值 + \expandafter\def\csname add#1\endcsname{\refstepcounter{#1}\expandafter\csname the#1\endcsname} +} + +\labelboxcnt{cntdef} %创建定义族标签盒的计数器cntdef +\labelboxcnt{cntthm} %创建定理族标签盒的计数器cntthm +\labelboxcnt{cntexp} %创建示例族标签盒的计数器cntexp + +%生成一个带有特定格式的标签盒环境族 +%用法:\labelboxstyle{<族环境名>}{<边框颜色>} +%用法: +%\begin{<族环境名>}{<键>}{<值>}{<族计数器>}{<标题>} +% ... +%\end{<族环境名>} +%注意这里传入的<族计数器>应是对应计数器带有自增并显示的命令"\add<计数器名称>" +\NewDocumentCommand{\labelboxstyle}{mm}% +{% + %定义一个新的tcolorbox并命名为<族环境名> + %该tcolorbox的边框颜色设定为<边框颜色> + %该tcolorbox的标题格式取决于该环境调用时的四个参数 + %左侧:<值><族计数器>\label{<键>:<标题>} + %右侧:<标题> + \newtcolorbox{#1}[4]{ + arc=0.1mm, + colframe=#2, + colback=black!3!white, + fonttitle=\bfseries, + left=0.2cm, + right=0.2cm, + toptitle=0.05cm, + bottomtitle=0.05cm, + title={##2\textnormal{##3\label{##1:##4}}\hfill ##4}} + %定义一个名为“m<族环境名>”作为“<族环境名>”的可跨页版 + \newtcolorbox{m#1}[4]{ + breakable, + arc=0.1mm, + colframe=#2, + colback=black!3!white, + fonttitle=\bfseries, + left=0.2cm, + right=0.2cm, + toptitle=0.05cm, + bottomtitle=0.05cm, + title={##2\textnormal{##3\label{##1:##4}}\hfill ##4}} +}% + +\labelboxstyle{boxdef}{gray!80!green!80!black} %定义族 绿色 +\labelboxstyle{boxthm}{gray!70!blue} %定理族 蓝色 +\labelboxstyle{boxexp}{black!50!cyan} %示例族 青色 + +\renewcommand{\thempfootnote}{\fnsymbol{mpfootnote}} + +%记录上一个使用的标签盒环境的名称,作为公式标签的自动前缀 +\gdef\prefixeq{} + +%创建一个修改\prefixeq的命令 +\newcommand{\setpeq}[1]{\gdef\prefixeq{#1}} + +%记录上一个使用的标签盒的类型,这会在证明环境的标题中使用 +\gdef\proplast{} + +%生成一个标签盒环境的实例 +%用法:\labelboxgen{<键>}{<值>}{<族环境名>}{<族计数器>}{<环境实例名>} +\NewDocumentCommand{\labelboxgen}{mmmmm} +{% + \xrefcreate{#1}{#2}% 创建<键>:<值>标签解析对 + \NewDocumentEnvironment{#5}{O{}s}% 创建<环境实例名>,带有可选参数[<标题>]和布尔参数<是否断页> + {% + \gdef\prefixeq{##1}% 将该环境的<标题>作为公式标签的自动前缀 + \gdef\proplast{#1}% 将该环境的<键>记录在宏中 + %调用族环境\begin{<环境名>}{<键>}{<值>}{<族计数器>}{<标题>} + \begin{\IfBooleanT{##2}{m}#3}{#1}{#2}{#4}{##1}% + \setlength{\parskip}{6pt}% + }% %... + {% %... + \end{\IfBooleanT{##2}{m}#3}% + }% +}% + +%"定义族"的环境实例 +\labelboxgen{def}{定义}{boxdef}{\addcntdef}{BoxDefinition} + +%"定理族"的环境实例 +\labelboxgen{lem}{引理}{boxthm}{\addcntthm}{BoxLemma} +\labelboxgen{thm}{定理}{boxthm}{\addcntthm}{BoxTheorem} +\labelboxgen{col}{推论}{boxthm}{\addcntthm}{BoxCorollary} + +\labelboxgen{pps}{命题}{boxthm}{\addcntthm}{BoxProposition} +\labelboxgen{ppt}{性质}{boxthm}{\addcntthm}{BoxProperty} + +\labelboxgen{eqt}{方程}{boxthm}{\addcntthm}{BoxEquation} +\labelboxgen{fml}{公式}{boxthm}{\addcntthm}{BoxFormula} + +\labelboxgen{law}{定律}{boxthm}{\addcntthm}{BoxLaw} +\labelboxgen{pri}{原理}{boxthm}{\addcntthm}{BoxPrinciple} + +%"示例族"的环境实例 +\labelboxgen{exp}{例}{boxexp}{\addcntexp}{BoxExample} + +%证明环境 +\newenvironment{Proof}[1][\xref{\proplast:\prefixeq}]% +{% + \begin{proof}[\indent\bf 证明]% + \quad\dotfill\hspace{0.1em} #1\par% +}% +{% + \end{proof}\nopagebreak[4]% + \hrule\par\goodbreak\vspace{15pt plus 10pt minus 10pt}% +}% + +%求解环境 +\newenvironment{Solution}[1][\xref{\proplast:\prefixeq}]% +{% + \begin{proof}[\indent\bf 解]% + \quad\dotfill\hspace{0.1em} #1\par% +}% +{% + \end{proof}\nopagebreak[4]% + \hrule\par\goodbreak\vspace{15pt plus 10pt minus 10pt}% +}% + +%设定数学公式的编号的上级为节 +\numberwithin{equation}{section} + +%创建数学公式的标签解析对 +\xrefcreate{eq}{式} + +%定义\xlabeleq{...}作为\label{eq:...}的简写 +\NewDocumentCommand{\xlabeleq}{m}% +{% + \label{eq:#1}% +}% + +%创建带有\prefixeq前缀的标签 +\NewDocumentCommand{\xlabelpeq}{om}% +{% + \IfValueTF{#1}% + {\xlabeleq{#1#2}}% + {\xlabeleq{\prefixeq#2}}% +}% + +%创建带有\prefixeq前缀的引用 +\NewDocumentCommand{\xrefpeq}{om}% +{% + \IfValueTF{#1}% + {\xrefeq{#1#2}}% + {\xrefeq{\prefixeq#2}}% +}% + +%Equation环境 +\NewDocumentEnvironment{Equation}{st&o} +{% + \IfBooleanTF{#1}% + {\begin{equation*}}% + {% + \begin{equation}% + \IfValueTF{#3}% + {% + \IfBooleanTF{#2}% + {\xlabeleq{\prefixeq #3}}% + {\xlabeleq{#3}}% + }{}% + }% +}% +{% + \IfBooleanTF{#1}% + {\end{equation*}}% + {\end{equation}}% + \par% +}% + +%Split环境 +\NewDocumentEnvironment{Split}{st&ob} +{% + \IfBooleanTF{#1}% + {% + \begin{equation*}% + \begin{split}% + #4 + \end{split}% + \end{equation*}% + }% + {% + \begin{equation}% + \IfValueTF{#3}% + {% + \IfBooleanTF{#2}% + {\xlabeleq{\prefixeq #3}}% + {\xlabeleq{#3}}% + }{}% + \begin{split}% + #4 + \end{split}% + \end{equation}% + }% + \par% +}{} + +%Gather环境 +\NewDocumentEnvironment{Gather}{sO{\jot}b}% +{% + \setlength\jot{#2}% + \IfBooleanTF{#1}% + {% + \begin{gather*}% + #3 + \end{gather*}% + }% + {% + \begin{gather}% + #3 + \end{gather}% + }% + \par% +}{}% + +%Align环境 +\NewDocumentEnvironment{Align}{sO{\jot}b}% +{% + \setlength\jot{#2}% + \IfBooleanTF{#1}% + {% + \begin{align*}% + #3 + \end{align*}% + }% + {% + \begin{align}% + #3 + \end{align}% + }% + \par% +}{}% \ No newline at end of file diff --git a/xNoteBook.cls b/xNoteBook.cls new file mode 100644 index 0000000..39d6846 --- /dev/null +++ b/xNoteBook.cls @@ -0,0 +1,198 @@ +\ProvidesClass{xNoteBook} + +\let\xtitle=\@title +\let\xauthor=\@author +\let\xdate=\@date + +%基于ctexbook改造的文档类型 +\LoadClass[a4paper,openany,oneside,sub4section,heading=true,draft]{ctexbook} +\RequirePackage{xeCJKfntef} + +%引入超链接 +\PassOptionsToPackage{hyphens}{url} +\RequirePackage[hidelinks]{hyperref} +\hypersetup{bookmarksnumbered=true} +\hypersetup{pdftitle=\xtitle} +\hypersetup{pdfauthor=\xauthor} + +\RequirePackage{tocbibind} + +%设置文档页面布局 +\RequirePackage{geometry} + +%设置文档页眉页脚 +\RequirePackage{fancyhdr} + +%定制文档字体 +\RequirePackage{fontspec} + +%定制文档标题 +\RequirePackage{titlesec} + +%处理附录事宜 +\RequirePackage{apptools} + +%定制文档脚注 +\RequirePackage[multiple,perpage]{footmisc} + +%--------------------------------% + +%设定页面边距 +\geometry{left=3cm,right=3cm,top=3cm,bottom=3cm} + +%设定西文字体 +\setmonofont{Consolas} + +%设定中文字体 +\setCJKmainfont[AutoFakeSlant=0.25]{NotoSerifCJKsc-Regular.otf} %谷歌衬线字体 +\setCJKsansfont[AutoFakeSlant=0.25]{NotoSansSC-Regular.ttf} %谷歌无衬线字体 +\setCJKmonofont[AutoFakeBold,AutoFakeSlant=0.25]{FangSong} %仿宋字体 + +%设定页眉页脚格式 +\pagestyle{fancy} +\fancyhf{} +\lhead{\sffamily\leftmark} %页眉左侧显示章名 +\rhead{\sffamily\rightmark} %页眉右侧显示节名 +\lfoot{\small\LaTeXnotebook} %页脚左侧显示徽标 +\rfoot{--\thepage--} %页脚右侧显示页码 +\renewcommand{\headrulewidth}{0.4pt} %设定页眉横线的粗细 +\setlength{\headheight}{13pt} %设定页眉区域的高度 + +\fancypagestyle{plain} %特殊页面的页眉页脚设置 +{ %适用于目录和章首第一页等位置 + \fancyhf{} + \renewcommand{\headrulewidth}{0.0pt} + \cfoot{--\thepage--} +} + +%设定取消段落缩进 +\setlength{\parindent}{0em} + +%设定取消脚注缩进 +\setlength{\footnotemargin}{0em} + +%设定脚注间的间距 +\addtolength{\footnotesep}{0.65mm} + +%设定页面底部对齐 +\flushbottom + +%设定前言部分格式 +\let\frontmatterold=\frontmatter +\renewcommand\frontmatter +{ + \frontmatterold + \setlength{\parskip}{1pt} %段落间距为1pt + \pagenumbering{Roman} %使用罗马数字表示页码 +} + +\let\mainmatterold=\mainmatter +\renewcommand\mainmatter +{ + \mainmatterold + \setlength{\parskip}{6pt} %段落间距为6pt + \pagenumbering{arabic} %使用阿拉伯数字表示页码 +} + +%设定标题编号至subparagraph +\setcounter{secnumdepth}{5} + +%设定目录读取至subsubsection +\setcounter{tocdepth}{3} + +%设定part格式 +\titleformat{\part}[display] +{\raggedright\bfseries\fontsize{40}{60}} +{\Huge 第\chinese{part}部分\mdseries\hfill\dag\arabic{part}} +{0em}{\vspace{1ex}\titlerule\vspace{2ex}} +\let\partold=\part +\renewcommand{\part}[1]{\partold{#1}\label{part:#1}} + +%设定chapter格式 +%这里需要使用宏包apptools的IfAppendix特殊处理附录的情况 +\titleformat{\chapter}[block] +{\centering\bfseries\Huge} +{\IfAppendix{\appendixname\thechapter}{第\chinese{chapter}章}} +{1em}{} +\let\chapterold=\chapter +\RenewDocumentCommand{\chapter}{sO{#3}m}% +{% + \IfBooleanTF{#1}% + {\chapterold*{#3}}% + {\chapterold[#2]{#3}}% + \label{chap:#2}% +}% + +%设定section格式 +\titleformat{\section}[block] +{\centering\bfseries\LARGE} +{\thesection} +{1em}{} +\let\sectionold=\section +\RenewDocumentCommand{\section}{sO{#3}m}% +{% + \IfBooleanTF{#1}% + {\sectionold*{#3}}% + {\sectionold[#2]{#3}}% + \label{sec:#2}% +}% + +%设定subsection格式 +\titleformat{\subsection}[hang] +{\bfseries\Large} +{\thesubsection} +{1em}{} +\let\subsectionold=\subsection +\RenewDocumentCommand{\subsection}{sO{#3}m}% +{% + \IfBooleanTF{#1}% + {\subsectionold*{#3}}% + {\subsectionold[#2]{#3}}% + \label{subsec:#2}% +}% + +%设定subsubsection格式 +\titleformat{\subsubsection}[hang] +{\bfseries\large} +{\thesubsubsection} +{1em}{} + +%设定paragraph格式 +\renewcommand\theparagraph{\arabic{paragraph}} +\titleformat{\paragraph}[hang] +{\bfseries} +{\large\theparagraph.\normalsize} +{0.2em}{} + +%设定subparagraph格式 +\renewcommand\thesubparagraph{\theparagraph.\arabic{subparagraph}} +\titleformat{\subparagraph}[runin] +{\bfseries} +{} +{0.0em}{}[] +\titlespacing*{\subparagraph}{0pt}{0pt}{1em} + +%预先重定义章节层次标签在引用时的格式 +\newcommand{\xreftextpart}[1]{第\ref*{part:#1}部分} +\newcommand{\xreftextchap}[1]{第\ref*{chap:#1}章} +\newcommand{\xreftextsec}[1]{第\CJKsetecglue{}\ref*{sec:#1}节} +\newcommand{\xreftextsubsec}[1]{第\CJKsetecglue{}\ref*{subsec:#1}小节} + +%预先重定义页码引用的格式 +\newcommand{\xreftextpage}[1]{第\CJKsetecglue{}\pageref*{#1}页} + +%定义"LaTeX NoteBook"的徽标和显示徽标页的命令 +\newcommand{\LaTeXnotebook}{\textsc{\xtitle}} +%\newcommand{\LaTeXnotebook}{\LaTeX~\textsc{NoteBook}} +\newcommand{\makesymb} +{ + \thispagestyle{empty} + \vspace*{\stretch{0.6}} + \begin{center} + \Huge + \LaTeXnotebook + \end{center} + \vspace*{\stretch{1.0}} + \newpage + \addtocounter{page}{-1} +} diff --git a/xStandalone.cls b/xStandalone.cls new file mode 100644 index 0000000..84912cc --- /dev/null +++ b/xStandalone.cls @@ -0,0 +1,13 @@ +\ProvidesClass{xStandalone} + +\LoadClass{standalone} + +\RequirePackage{ctex} + +\RequirePackage{xCommon} +\RequirePackage{xMath} +\RequirePackage{xFloat} +\RequirePackage{xTikZ} + + +\RequirePackage{Local} \ No newline at end of file diff --git a/xTikZ.sty b/xTikZ.sty new file mode 100644 index 0000000..fcede0f --- /dev/null +++ b/xTikZ.sty @@ -0,0 +1,19 @@ +\ProvidesPackage{xTikZ} + +\RequirePackage{tikz} + +\usetikzlibrary{intersections} +\usetikzlibrary{decorations.markings} + +\usetikzlibrary{quotes} +\usetikzlibrary{angles} + +\tikzset{point/.style={circle,draw=#1,solid,thin,radius=0.03,fill=white,inner sep=0.8pt},point/.default=black} + +\RequirePackage[european,americaninductors,americanvoltages,americancurrents,RPvoltages]{circuitikz} + +\ctikzset{logic ports=ieee} +\ctikzset{voltage/distance from node=0.85} + +\RequirePackage{pgfplots} +\pgfplotsset{trig format plots=rad} \ No newline at end of file

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