复变函数导数求解（包含矢量、矩阵形式）

文章目录

前言

一、复变函数导数

1.1 导数定义

1.2 求导法则

1.3 存在条件

二、常用求导结论

2.1 标量函数对标量的导数

2.2 标量函数对矢量的导数

2.3 标量函数对矩阵的导数

总结

前言

本文将从信号处理的角度简单阐明复变函数理论的重要性，并重点介绍能够用于信号处理领域的复变函数求导原理。

能够发射到空间中的信号只能是实信号。然而我们在处理接收信号时，往往是复信号形式的，这两者并不冲突。实信号频谱是左右对称的，也就是实信号有至少一半频谱携带的信息是冗余的，为了提高频谱利用率，IQ调制与解调技术被用于信号的发射与接收，接收的信号分为IQ两路信号。为了更好更高效的描述IQ信号，人们提出利用复变函数来描述IQ两路信号（复信号实际是不存在的，它只是描述IQ两路信号的最佳手段）。因此，基于实变函数的理论在复数中不再适用，包括估计理论中克拉美罗界的推导、匹配滤波理论的推导等需要从复变函数的角度重新开展，而其中一个重要的基础理论就是复变函数的求导理论。

一、复变函数导数

1.1 导数定义

设 $w=f\left ( z \right )$ 在点 $z_{0}$ 的某邻域 $N_{p}\left ( z_{0} \right )$ 内有定义，且 $\Delta z=z-z_{0}$ ，其中 $z\in N_{p}\left ( z_{0} \right )$ 。若下面极限存在

$\lim_{\Delta z\rightarrow 0}\frac{f\left ( z \right )-f\left ( z_{0} \right )}{z-z_{0}}=\lim_{\Delta z\rightarrow 0}\frac{\Delta w}{\Delta z}$

则称 $f\left ( z \right )$ 在点 $z_{0}$ 可导，其极限值称为 $f\left ( z \right )$ 在点 $z_{0}$ 的导数，记为 ${f}'\left ( z_{0} \right )$ 或 $\frac{dw}{dz}|_{z_{0}}$ 。

1.2 求导法则

复变函数 $f\left ( z \right )$ 和 $g\left ( z \right )$ 在复变量 $z$ 区域D内处处可导，则下列运算规则得到的复变函数的导数为

1）和差法则

${\left [f\left ( z \right )\pm g\left ( z \right ) \right ]}'={f}'\left ( z \right )\pm {g}'\left ( z \right )$

2) 积法则

${\left [f\left ( z \right ) g\left ( z \right ) \right ]}'={f}'\left ( z \right )g\left ( z \right )+f\left ( z \right ) {g}'\left ( z \right )$

3) 商法则

${\left [\frac{f\left ( z \right )}{g\left ( z \right )} \right ]}'=\frac{{f}'\left ( z \right )g\left ( z \right )-f\left ( z \right ) {g}'\left ( z \right )}{\left [ g^{2}\left ( z \right ) \right ]}$

4) 链式法则

若函数 $w=f\left ( \xi \right )$ 和 $\xi =\varphi \left ( z \right )$ 分别在区域G和D内可导，且 $\xi =\varphi \left ( z \right )$ 将D映射为 $D^{*}$ 使得 $D^{*}\subset G$ ，则复合函数 $w=f\left [ \varphi \left ( z \right ) \right ]$ 在D内可导，且

$\frac{dw}{dz}=f^{'}\left ( \xi \right )\varphi ^{'}\left ( z \right )\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \left ( \xi =\varphi \left ( z \right )\right )$

5) 反函数法则

若函数 $w=f\left (z \right )$ 在区域D内可导且将D一一映射到区域E。若在区域D内 ${f}'\left ( z \right )\neq 0$ 且反函数 $z=\varphi \left ( w \right )$ 在E连续，则 $\varphi \left ( w \right )$ 在E内可导，且

$\varphi ^{'}\left ( w \right )=\frac{1}{f ^{'}\left ( z \right )}\, \, \, \, \, \, \, \, \, \, \, \, \, \left ( z=\varphi \left ( w \right ) \right )$

1.3 存在条件

复变函数 $f\left ( z \right )=u\left ( x,y \right )+i v\left ( x,y \right )$ 在点 $z_{0}=x_{0}+i y_{0}$ 可导的充要条件是函数 $u\left ( x,y \right )$ ， $v\left ( x,y \right )$ 在点 $\left ( x_{0},y_{0} \right )$ 可微（四个一阶偏导数在该点存在且连续），其满足方程(Cauchy-Riemann方程)

$u_{x}^{'}=v_{y}^{'},u_{y}^{'}=-v_{x}^{'}$

当 $f\left ( z \right )$ 在点 $z=z_{0}$ 可导时，在该点有

$f^{'}\left ( z \right )=u_{x}^{'}\left ( x,y \right )+i v_{x}^{'}\left ( x,y \right )$

证明

假设 $w=f\left ( z \right )$ 在 $z=z_{0}$ 可导，则有

$\Delta w=A\Delta z+\rho \left ( \Delta z \right )$

进一步有

$\Delta u+i \Delta v=\left ( a+ib \right )\left (\Delta x+i \Delta y \right )+\rho_{1}+i\rho_{2}$

上式等价于

$\Delta u=a\Delta x-b\Delta y+\rho_{1}$

$\Delta v=b\Delta x+a\Delta y+\rho_{2}$

上式分别表示 $u\left ( x,y \right )$ ， $v\left ( x,y \right )$ 的微分形式，因此 $w=f\left ( z \right )$ 在 $z=z_{0}$ 可导等价于

$u_{x}^{'}=v_{y}^{'}=a,u_{y}^{'}=-v_{x}^{'}=b$

扩展1

利用 $u_{x}^{'}=v_{y}^{'}=a,u_{y}^{'}=-v_{x}^{'}=b$ ，导数可以表示为

$\frac{\partial f}{\partial z}=u_{x}^{'}\left ( x,y \right )+i v_{x}^{'}\left ( x,y \right )=\frac{1}{2}\left (\frac{\partial f}{\partial x} -i \frac{\partial f}{\partial y}\right )$

扩展2

$\Delta z=\Delta x-i\Delta y$ ，则 $\frac{\partial f}{\partial z^{*}}$ 存在的等效条件为

$u_{x}^{'}=-v_{y}^{'}=a,u_{y}^{'}=v_{x}^{'}=b$

对应导数为：

$\frac{\partial f}{\partial z^{*}}=u_{x}^{'}\left ( x,y \right )+i v_{x}^{'}\left ( x,y \right )=\frac{1}{2}\left (\frac{\partial f}{\partial x} +i \frac{\partial f}{\partial y}\right )$

推论

导数 $\frac{\partial f}{\partial z}=\frac{\partial f^{*}}{\partial z^{*}}$ 如果存在且不等于0，则导数 $\frac{\partial f}{\partial z^{*}}=\frac{\partial f^{*}}{\partial z}$ 必然不存在。（该结论是博主根据上述结论得到的新结论，没有细致调研，因此无法判断该结论是否有人证明过，感兴趣的可以去调研，也希望在评论区给出调研结果以及结论的证明过程）

二、常用求导结论

2.1 标量函数对标量的导数

标量 $f\left ( z \right )$ 是复变量 $z$ 的复变函数，下面为常见复变函数的导数：

${\left ( \ln z \right )}'=\frac{1}{z}\left ( z\neq 0\, \, \, \, \, \, \, -\pi<\arg\left ( z \right )<\pi \right )$

${\left ( z^{\alpha } \right )}'=\alpha z^{\alpha -1}\left (z\neq 0\, \, \, \, \, \, \, -\pi<\arg\left ( z \right )<\pi \right )$

${\left ( \sin z \right )}'=\cos z$

${\left ( \cos z \right )}'=-\sin z$

${\left ( \tan z \right )}'=\frac{1}{\cos^{2} z}$

${\left ( \sinh z \right )}'=\cosh z\left (\sinh z=\frac{1}{2}\left ( e^{z}-e^{-z} \right ) \, \, \, \, \, \cosh z=\frac{1}{2}\left ( e^{z}+e^{-z} \right ) \right )$

${\left ( \cosh z \right )}'=\sinh z$

$\frac{\partial z}{\partial z}=\frac{\partial z^{*}}{\partial z^{*}}=1$

$\frac{\partial z^{*}}{\partial z}=\frac{\partial z}{\partial z^{*}}=0\: \: \: \: \: \: \: (\text{Inexistence})$

注：该式的导数是不存在的（不满足C-R方程），为了方便分析，一般认为 $z$ 和 $z^{*}$ 是相互独立的两个变量

$z$ 表示实变量，下面为常见实变函数的导数：

${\left ( \ln z \right )}'=\frac{1}{z}\left ( z\neq 0 \right )$

${\left ( z^{\alpha } \right )}'=\alpha z^{\alpha -1}\left (z\neq 0 \right )$

${\left ( \sin z \right )}'=\cos z$

${\left ( \cos z \right )}'=-\sin z$

${\left ( \tan z \right )}'=\frac{1}{\cos^{2} z}$

${\left ( \sinh z \right )}'=\cosh z\left (\sinh z=\frac{1}{2}\left ( e^{z}-e^{-z} \right ) \, \, \, \, \, \cosh z=\frac{1}{2}\left ( e^{z}+e^{-z} \right ) \right )$

${\left ( \cosh z \right )}'=\sinh z$

2.2 标量函数对矢量的导数

$\vec{b} =\left [ b_{1},b_{2},\cdots ,b_{M} \right ]^{T}$ 为复矢量， $f\left ( \vec{z} \right )$ 为复矢量 $\vec{z} =\left [ z_{1},z_{2},\cdots ,z_{M} \right ]^{T}$ 的标量函数， $\vec{f}\left ( \vec{z} \right )=\left [f_{1}\left ( \vec{z} \right ),f_{2}\left ( \vec{z} \right ),\cdots ,f_{N}\left ( \vec{z} \right ) \right ]^{T}$ 和 $\vec{g}\left ( \vec{z} \right )=\left [g_{1}\left ( \vec{z} \right ),g_{2}\left ( \vec{z} \right ),\cdots ,g_{N}\left ( \vec{z} \right ) \right ]^{T}$ 都为复矢量 $\vec{z} =\left [ z_{1},z_{2},\cdots ,z_{M} \right ]^{T}$ 的矢量函数，并作出如下导数定义

$\frac{\partial f\left ( \vec{z} \right )}{\partial \vec{z}}=\left [\frac{\partial f\left ( \vec{z} \right )}{\partial z_{1}},\frac{\partial f\left ( \vec{z} \right )}{\partial z_{2}},\cdots ,\frac{\partial f\left ( \vec{z} \right )}{\partial z_{M}} \right ]^{T}$ , $\frac{\partial f\left ( \vec{z}\right )}{\partial \vec{z}^{*} }=\left [\frac{\partial f\left ( \vec{z} \right )}{\partial z_{1}^{*} },\frac{\partial f\left ( \vec{z} \right )}{\partial z_{2}^{*} },\cdots ,\frac{\partial f\left ( \vec{z} \right )}{\partial z_{M}^{*} } \right ]^{T}$

$\frac{\partial \vec{f}^{T}\left ( \vec{z} \right )}{\partial \vec{z}}=\left [\frac{\partial {f}_{1}\left ( \vec{z} \right )}{\partial \vec{z}},\frac{\partial {f}_{2}\left ( \vec{z} \right )}{\partial \vec{z}},\cdots ,\frac{\partial {f}_{N}\left ( \vec{z} \right )}{\partial \vec{z}} \right ]=\begin{bmatrix} \frac{\partial {f}_{1}\left ( \vec{z} \right )}{\partial {z}_{1}}& \frac{\partial {f}_{2}\left ( \vec{z} \right )}{\partial {z}_{1}} & \cdots & \frac{\partial {f}_{N}\left ( \vec{z} \right )}{\partial {z}_{1}}\\ \frac{\partial {f}_{1}\left ( \vec{z} \right )}{\partial {z}_{2}}& \frac{\partial {f}_{2}\left ( \vec{z} \right )}{\partial {z}_{2}} & \cdots & \frac{\partial {f}_{N}\left ( \vec{z} \right )}{\partial {z}_{2}}\\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial {f}_{1}\left ( \vec{z} \right )}{\partial {z}_{M}} & \frac{\partial {f}_{2}\left ( \vec{z} \right )}{\partial {z}_{M}} & \cdots & \frac{\partial {f}_{N}\left ( \vec{z} \right )}{\partial {z}_{M}} \end{bmatrix}$

$\frac{\partial \vec{f}^{T}\left ( \vec{z} \right )}{\partial \vec{z}^{*}}=\left [\frac{\partial {f}_{1}\left ( \vec{z} \right )}{\partial \vec{z}^{*}},\frac{\partial {f}_{2}\left ( \vec{z} \right )}{\partial \vec{z}^{*}},\cdots ,\frac{\partial {f}_{N}\left ( \vec{z} \right )}{\partial \vec{z}^{*}} \right ]=\begin{bmatrix} \frac{\partial {f}_{1}\left ( \vec{z} \right )}{\partial {z}_{1}^{*}}& \frac{\partial {f}_{2}\left ( \vec{z} \right )}{\partial {z}_{1}^{*}} & \cdots & \frac{\partial {f}_{N}\left ( \vec{z} \right )}{\partial {z}_{1}^{*}}\\ \frac{\partial {f}_{1}\left ( \vec{z} \right )}{\partial {z}_{2}^{*}}& \frac{\partial {f}_{2}\left ( \vec{z} \right )}{\partial {z}_{2}^{*}} & \cdots & \frac{\partial {f}_{N}\left ( \vec{z} \right )}{\partial {z}_{2}^{*}}\\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial {f}_{1}\left ( \vec{z} \right )}{\partial {z}_{M}^{*}} & \frac{\partial {f}_{2}\left ( \vec{z} \right )}{\partial {z}_{M}^{*}} & \cdots & \frac{\partial {f}_{N}\left ( \vec{z} \right )}{\partial {z}_{M}^{*}} \end{bmatrix}$

则

$\frac{\partial \vec{z}^{T}}{\partial \vec{z}}=\begin{bmatrix} \frac{\partial {z}_{1}}{\partial {z}_{1}}& \frac{\partial {z}_{2}}{\partial {z}_{1}} & \cdots & \frac{\partial {z}_{M}}{\partial {z}_{1}}\\ \frac{\partial {z}_{1}}{\partial {z}_{2}}& \frac{\partial {z}_{2}}{\partial {z}_{2}} & \cdots & \frac{\partial {z}_{M}}{\partial {z}_{2}}\\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial {z}_{1}}{\partial {z}_{M}} & \frac{\partial {z}_{2}}{\partial {z}_{M}} & \cdots & \frac{\partial {z}_{M}}{\partial {z}_{M}} \end{bmatrix}=I=\frac{\partial \vec{z}^{H}}{\partial \vec{z}^{*}}$

$\frac{\partial \vec{z}^{H}}{\partial \vec{z}}=\begin{bmatrix} \frac{\partial {z}_{1}^{*}}{\partial {z}_{1}}& \frac{\partial {z}_{2}^{*}}{\partial {z}_{1}} & \cdots & \frac{\partial {z}_{M}^{*}}{\partial {z}_{1}}\\ \frac{\partial {z}_{1}^{*}}{\partial {z}_{2}}& \frac{\partial {z}_{2}^{*}}{\partial {z}_{2}} & \cdots & \frac{\partial {z}_{M}^{*}}{\partial {z}_{2}}\\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial {z}_{1}^{*}}{\partial {z}_{M}} & \frac{\partial {z}_{2}^{*}}{\partial {z}_{M}} & \cdots & \frac{\partial {z}_{M}^{*}}{\partial {z}_{M}} \end{bmatrix}=O=\frac{\partial \vec{z}^{T}}{\partial \vec{z}^{*}}\: \: \: \: \: \: \: (\text{Inexistence})$

$\frac{\partial \vec{b}^{H}\vec{z}}{\partial \vec{z}}=\begin{bmatrix} \frac{\partial \sum_{m=1}^{M} b_{m}^{*}z_{m}}{\partial z_{1}}\\ \frac{\partial \sum_{m=1}^{M} b_{m}^{*}z_{m}}{\partial z_{2}}\\ \vdots \\ \frac{\partial \sum_{m=1}^{M} b_{m}^{*}z_{m}}{\partial z_{M}} \end{bmatrix}=\vec{b}^{*}=\frac{\partial \vec{z}^{T}\vec{b}^{*}}{\partial \vec{z}}=\left (\frac{\partial \vec{z}^{H}\vec{b}}{\partial \vec{z}^{*}} \right )^{*}$

$\frac{\partial \vec{b}^{H}\vec{z}}{\partial \vec{z}^{*}}=\begin{bmatrix} \frac{\partial \sum_{m=1}^{M} b_{m}^{*}z_{m}}{\partial z_{1}^{*}}\\ \frac{\partial \sum_{m=1}^{M} b_{m}^{*}z_{m}}{\partial z_{2}^{*}}\\ \vdots \\ \frac{\partial \sum_{m=1}^{M} b_{m}^{*}z_{m}}{\partial z_{M}^{*}} \end{bmatrix}=O=\frac{\partial \vec{z}^{T}\vec{b}^{*}}{\partial \vec{z}^{*}}=\left (\frac{\partial \vec{z}^{H}\vec{b}}{\partial \vec{z}} \right )^{*}\: \: \: \: \: \: \: (\text{Inexistence})$

$\frac{\partial \vec{f}^{H}\left (\vec{z} \right )\vec{g}\left (\vec{z} \right )}{\partial \vec{z}}=\left [\frac{\partial \vec{f}^{H}\left (\vec{z} \right )}{\partial \vec{z}} \right ]\vec{g}\left (\vec{z} \right )+\left [\frac{\partial \vec{g}^{T}\left (\vec{z} \right )}{\partial \vec{z}} \right ]\vec{f}^{*}\left (\vec{z} \right )$

$\frac{\partial \vec{z}^{H} R \vec{z}}{\partial \vec{z}}=\left [\frac{\partial \vec{z}^{H}}{\partial \vec{z}} \right ]R \vec{z} +\left [\frac{\partial \vec{z}^{T}}{\partial \vec{z}} \right ]R^{T}\vec{z}^{*}=R^{T}\vec{z}^{*}$

$\frac{\partial \vec{z}^{H} R \vec{z}}{\partial \vec{z}^{*}}=\left [\frac{\partial \vec{z}^{H}}{\partial \vec{z}^{*}} \right ]R \vec{z} +\left [\frac{\partial \vec{z}^{T}}{\partial \vec{z}^{*}} \right ]R^{T}\vec{z}^{*}=R \vec{z}$

$\frac{\partial \vec{z}^{H} R \vec{z}^{*}}{\partial \vec{z}^{*}}=\left [\frac{\partial \vec{z}^{H}}{\partial \vec{z}^{*}} \right ]R \vec{z}^{*} +\left [\frac{\partial \vec{z}^{H}}{\partial \vec{z}^{*}} \right ]R^{T}\vec{z}^{*}=\left (R+R^{T} \right ) \vec{z}^{*}$

$\vec{b} =\left [ b_{1},b_{2},\cdots ,b_{M} \right ]^{T}$ 为实矢量， $f\left ( \vec{z} \right )$ 为实矢量 $\vec{z} =\left [ z_{1},z_{2},\cdots ,z_{M} \right ]^{T}$ 的实函数， $\vec{f}\left ( \vec{z} \right )=\left [f_{1}\left ( \vec{z} \right ),f_{2}\left ( \vec{z} \right ),\cdots ,f_{N}\left ( \vec{z} \right ) \right ]^{T}$ 和 $\vec{g}\left ( \vec{z} \right )=\left [g_{1}\left ( \vec{z} \right ),g_{2}\left ( \vec{z} \right ),\cdots ,g_{N}\left ( \vec{z} \right ) \right ]^{T}$ 都为实矢量 $\vec{z} =\left [ z_{1},z_{2},\cdots ,z_{M} \right ]^{T}$ 的矢量函数，则

$\frac{\partial \vec{b}^{T}\vec{z}}{\partial \vec{z}}=\begin{bmatrix} \frac{\partial \sum_{m=1}^{M} b_{m}z_{m}}{\partial z_{1}}\\ \frac{\partial \sum_{m=1}^{M} b_{m}z_{m}}{\partial z_{2}}\\ \vdots \\ \frac{\partial \sum_{m=1}^{M} b_{m}z_{m}}{\partial z_{M}} \end{bmatrix}=\vec{b}$

$\frac{\partial \vec{f}^{T}\left (\vec{z} \right )\vec{g}\left (\vec{z} \right )}{\partial \vec{z}}=\left [\frac{\partial \vec{f}^{T}\left (\vec{z} \right )}{\partial \vec{z}} \right ]\vec{g}\left (\vec{z} \right )+\left [\frac{\partial \vec{g}^{T}\left (\vec{z} \right )}{\partial \vec{z}} \right ]\vec{f}\left (\vec{z} \right )$

$\frac{\partial \vec{z}^{T} R \vec{z}}{\partial \vec{z}}=\left [\frac{\partial \vec{z}^{T}}{\partial \vec{z}} \right ]R \vec{z} +\left [\frac{\partial \vec{z}^{T}}{\partial \vec{z}} \right ]R^{T}\vec{z}=\left (R+R^{T} \right ) \vec{z}$

2.3 标量函数对矩阵的导数

$A$ 和 $B$ 分别是 $m\times n$ 和 $n\times m$ 的矩阵，则

$\frac{\partial Tr\left ( AB \right ) }{\partial A}=B^{T}$ , $\frac{\partial Tr\left ( AB \right ) }{\partial B}=A^{T}$ , $\frac{\partial Tr\left ( A^{H}B \right ) }{\partial A}=O_{m\times n}$ , $\frac{\partial Tr\left ( AB^{H} \right ) }{\partial B}=O_{n\times m}$