数学知识：其他

目录

• 求导
  • 向量求导
  • 矩阵求导
• 卷积
  • 线性卷积
  • 周期卷积
  • 循环卷积
  • 连续函数和离散函数的卷积
• 傅里叶变换
  • FT
  • DFT
  • FFT
• 内积、外积、点乘、叉乘
  • 结果

求导

• 链式法则适用(

$$\frac{\partial f(x)g(x)}{\partial x}=f'(x)g(x)+f(x)g'(x)$$

• 偏导结果与变量形状相同

$$\frac{\partial f(x)}{\partial x_{m\times n}}=y_{m\times n}$$

• 相乘时要满足可乘条件

$$a_{m\times n}b_{n\times p}c_{p\times q}=d_{m\times q}$$

向量求导

$$\dfrac{\partial x^Ta}{\partial x}=\dfrac{\partial a^Tx}{\partial x}=a$$

$$\dfrac{\partial x^Tx}{\partial x}=2x$$

$$\dfrac{\partial x^TAx}{\partial x}=Ax+A^Tx$$

$$\dfrac{\partial a^Txx^Tb}{\partial X}=ab^Tx+ba^Tx$$

矩阵求导

$$\dfrac{\partial a^TXb}{\partial X}=ab^T$$

$$\dfrac{\partial a^TX^Tb}{\partial X}=ba^T$$

$$\dfrac{\partial a^TXX^Tb}{\partial X}=ab^TX+ba^TX$$

$$\dfrac{\partial a^TX^TXb}{\partial X}=Xba^T+Xab^T$$

卷积

线性卷积

周期卷积

循环卷积

连续函数和离散函数的卷积

1. 举例
   连续函数

$$f(t) = sin(2\pi t)\rm{rect}(\it{t})$$

离散序列

$$a = (1, 1, 1, 1, 1, 1)$$

离散函数

$$g(t) = \sum\limits_{i=0}^{5}a_i\delta(t-i)$$

卷积

$$\begin{align*} f(t)*g(t)&=\sin(2\pi t)rect(t)*\sum\limits_{i=0}^{5}a_i\delta(t-i) \\ &=\sum\limits_{i=0}^{5}a_i\sin[2\pi*(t-i)]rect(t-i) \\ &=\sum\limits_{i=0}^{5}\sin[2\pi*(t-i)]rect(t-i) \end{align*}$$

仿真补充：！！！！！！！！！！！！！！！！

2. 举例
   两个“离散的连续函数”卷积

$$f(t)=\cos(2\pi t)rect(t)*\sum\limits_{i=0}^{5}a_i\delta(t-i) \\ g(t)=f(-t)$$

卷积

$$\begin{align*} f(t)*g(t) = &[\cos(2\pi t)rect(t)*\sum\limits_{i=0}^{5}a_i\delta(t-i)] \\ &*[\cos(2\pi t)rect(t)*\sum\limits_{j=0}^{5}a_j\delta(t+j)] \\ =&[\cos(2\pi t)rect(t)*\cos(2\pi t)rect(t)] \\ &*[\sum\limits_{i=0}^{5}a_i\delta(t-i)*\sum\limits_{j=0}^{5}a_j\delta(t+j)] \\ =&\dfrac{1}{2}\cos(2\pi t)rect(\dfrac{t}{2})*\sum\limits_{i=0}^{5}\sum\limits_{j=0}^{5}a_ia_j\delta(t-i+j) \\ =&\dfrac{1}{2}\sum\limits_{i=0}^{5}\sum\limits_{j=0}^{5}a_ia_j\cos[2\pi(t-i+j)]rect(\dfrac{t-i+j}{2}) \end{align*}$$

傅里叶变换

FT

$$f(t)=\sum\limits_{i=0}^{N-1}f(x_i)e^{-j\pi t}$$

DFT

FFT

内积、外积、点乘、叉乘

结果

向量（同维）内积后得数值，外积（不同维）得矩阵，点乘（同维）得数值，叉乘（同维）得同维向量。

参考：https://blog.csdn.net/Dust_Evc/article/details/127502272