信号处理中的求导

本文对信号处理中一些常用的求导作以记录，如有表述不当之处欢迎批评指正。欢迎任何形式的转载，但请务必注明出处。

1. 实值函数相对于复变量的梯度

  a. 定义$f(w)$是复变量$w$的实函数；
  b. 其中$w$和$w^{\ast }$分别表示为：$$w=x+jy$$$$w^{\ast }=x-jy$$
  c. 那么：$$\frac{\partial f}{\partial w}=\frac{1}{2}(\frac{\partial f}{\partial x}-j\frac{\partial f}{\partial y})$$$$\frac{\partial f}{\partial w^{\ast }}=\frac{1}{2}(\frac{\partial f}{\partial x}+j\frac{\partial f}{\partial y})$$$$\frac{\partial w}{\partial w}=1$$$$\frac{\partial w}{\partial w^{\ast }}=0$$

2. 标量函数相对于复向量的梯度

  a. 定义$f(\vec{\mathbf{w}})$是复向量$\vec{\mathbf{w}}$的标量函数；
  b. 其中$$\vec{\mathbf{w}}=[w_0,...,w_{M-1}{]}^T$$$$w_n=x_n+j{y}_n$$
  c. 那么
$$\frac{\partial f}{\partial \vec{\mathbf{w}}}={\nabla }_{\vec{\mathbf{w}}}f(\vec{\mathbf{w}})=[\frac{\partial f}{\partial w_0},...,\frac{\partial f}{\partial w_{M-1}}{]}^T$$$$\frac{\partial f}{\partial \overrightarrow{{\mathbf{w}}^{\ast }}}={\nabla }_{\overrightarrow{{\mathbf{w}}^{\ast }}}f(\vec{\mathbf{w}})=[\frac{\partial f}{\partial w_0^{\ast }},...,\frac{\partial f}{\partial w_{M-1}^{\ast }}{]}^T$$

3. 向量函数相对于复向量的梯度

  a. 定义$\vec{\mathbf{f}}$是复向量$\vec{\mathbf{w}}$的向量函数
  b. 其中
$$\vec{\mathbf{f}}=[f_0(\vec{\mathbf{w}}),...,f_{M-1}(\vec{\mathbf{w}}){]}^T$$$$\vec{\mathbf{w}}=[w_0,...,w_{M-1}{]}^T$$$$w_n=x_n+j{y}_n$$
  c. 那么
$$\frac{\partial \vec{\mathbf{f}}}{\partial \vec{\mathbf{w}}}=\left[\begin{array}{c}\frac{\partial f_0}{\partial w_0},...,\frac{\partial f_0}{\partial w_{M-1}}\\ \frac{\partial f_1}{\partial w_0},...,\frac{\partial f_1}{\partial w_{M-1}}\\ .........\\ \frac{\partial f_{M-1}}{\partial w_0},...,\frac{\partial f_{M-1}}{\partial w_{M-1}}\end{array}\right]$$
  d. 结合上述概念可得到$\frac{\partial \vec{\mathbf{f}}}{\partial \overrightarrow{{\mathbf{w}}^{\ast }}}$

4. 关于复变量/向量的一些结论

  a. 定义$$\nabla =(\frac{\partial }{\partial z_0^{\ast }},...,\frac{\partial }{\partial z_{N-1}^{\ast }}{)}^T$$$$z_n=xn+j{y}_n$$$$n=0,...,N-1$$
  b. $$\frac{\partial }{\partial z_n^{\ast }}=\frac{1}{2}(\frac{\partial }{\partial x_n}+j\frac{\partial }{\partial y_n})$$$$z_n=xn+j{y}_n$$$$n=0,...,N-1$$
  c.
$$\{\begin{array}{l}\frac{\partial z}{\partial z^{\ast }}=0\\ \frac{\partial z}{\partial z}=1\\ \frac{\partial z^{\ast }}{\partial z^{\ast }}=1\end{array}$$
  d.
$$\{\begin{array}{l}\nabla ({\vec{\mathbf{a}}}^H\vec{\mathbf{z}})=0\\ \nabla ({\vec{\mathbf{z}}}^H\vec{\mathbf{a}})=\vec{\mathbf{a}}\\ \nabla ({\vec{\mathbf{z}}}^H\vec{\mathbf{R}}\vec{\mathbf{z}})=\vec{\mathbf{R}}\vec{\mathbf{z}}\end{array}$$
  其中$\vec{\mathbf{a}}$和$\vec{\mathbf{z}}$均为列向量，$\vec{\mathbf{R}}$为矩阵
  e.
$$f(\vec{\mathbf{w}})={\vec{\mathbf{p}}}^H\vec{\mathbf{w}}\Rightarrow \{\begin{array}{l}\frac{\partial f}{\partial \vec{\mathbf{w}}}={\vec{\mathbf{p}}}^{\ast }\\ \frac{\partial f}{\partial {\vec{\mathbf{w}}}^{\ast }}=0\end{array}$$
  其中$\vec{\mathbf{p}}$和$\vec{\mathbf{w}}$均为列向量，$f(\vec{\mathbf{w}})$为标量
  f.
$$f(\vec{\mathbf{w}})={\vec{\mathbf{w}}}^H\vec{\mathbf{p}}\Rightarrow \{\begin{array}{l}\frac{\partial f}{\partial \vec{\mathbf{w}}}=0\\ \frac{\partial f}{\partial {\vec{\mathbf{w}}}^{\ast }}=\vec{\mathbf{p}}\end{array}$$
  其中$\vec{\mathbf{p}}$和$\vec{\mathbf{w}}$均为列向量，$f(\vec{\mathbf{w}})$为标量
  g.
$$f(\vec{\mathbf{w}})={\vec{\mathbf{w}}}^H\vec{\mathbf{A}}\vec{\mathbf{w}}\Rightarrow \{\begin{array}{l}\frac{\partial f}{\partial \vec{\mathbf{w}}}={\vec{\mathbf{A}}}^T{\vec{\mathbf{w}}}^{\ast }\\ \frac{\partial f}{\partial {\vec{\mathbf{w}}}^{\ast }}=\vec{\mathbf{A}}\vec{\mathbf{w}}\end{array}$$
  其中$\vec{\mathbf{w}}$为列向量，$\vec{\mathbf{A}}$为矩阵，$f(\vec{\mathbf{w}})$为标量

5. 关于实变量/向量的一些结论

  a. $$\frac{\partial {\vec{\mathbf{a}}}^T\vec{\mathbf{b}}}{\partial \vec{\mathbf{b}}}=\frac{\partial {\vec{\mathbf{b}}}^T\vec{\mathbf{a}}}{\partial \vec{\mathbf{b}}}=\vec{\mathbf{a}}$$
  b. $$\frac{\partial {\vec{\mathbf{b}}}^T\vec{\mathbf{A}}\vec{\mathbf{b}}}{\partial \vec{\mathbf{b}}}=2\vec{\mathbf{A}}\vec{\mathbf{b}}=2{\vec{\mathbf{b}}}^T\vec{\mathbf{A}}$$
  其中$\vec{\mathbf{a}}$和$\vec{\mathbf{b}}$均为列向量，$\vec{\mathbf{A}}$为对称矩阵

6. 参考文献

[1] 张贤达．矩阵分析与应用[M]．北京：高等教育出版社，2004．