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数字信号处理数学基础

2019-07-13 17:06发布

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  1. 泰勒级数
    f(x)=f(x0)+f′(x0)(x−x0)+f′′(x0)2!(x−x0)2+⋯+f(n)(x0)n!(x−x0)n+Rn(x)fleft( x ight)=fleft( {{x}_{0}} ight)+{{f}^{'}}left( {{x}_{0}} ight)left( x-{{x}_{0}} ight)+frac{{{f}^{''}}left( {{x}_{0}} ight)}{2!}left( x-{{x}_{0}} ight)^2+cdots +frac{{{f}^{left( n ight)}}left( {{x}_{0}} ight)}{n!}{{left( x-{{x}_{0}} ight)}^{n}}+{{R}_{n}}left( x ight)
    其中Rn(x)=f(n+1)(ξ)(n+1)!(x−x0)n+1{{R}_{n}}left( x ight)=frac{{{f}^{left( n+1 ight)}}left( xi ight)}{left( n+1 ight)!}{{left( x-{{x}_{0}} ight)}^{n+1}}
  2. 复指数函数
    ez=lim⁡n→∞ (1+zn)n=lim⁡n→∞ ∑i=0n1i!zi{{e}^{z}}=underset{n o infty }{mathop{lim }},{{left( 1+frac{z}{n} ight)}^{n}}=underset{n o infty }{mathop{lim }},sumlimits_{i=0}^{n}{frac{1}{i!}}{{z}^{i}}
  3. 欧拉公式
    eix=cos⁡x+isin⁡x{{e}^{ix}}=cos x+isin x
  4. 卷积
    离散：y[n]=x[n]∗h[n]=∑k=−∞∞x[k]h[n−k]yleft[ n ight]=xleft[ n ight]*hleft[ n ight]=sumlimits_{k=-infty }^{infty }{xleft[ k ight]hleft[ n-k ight]}
    连续：y(t)=x(t)∗h(t)=∫τ=−∞∞x(τ)h(t−τ)yleft( t ight)=xleft( t ight)*hleft( t ight)=int olimits_{ au =-infty }^{infty }{xleft( au ight)hleft( t- au ight)}
  5. 连续傅里叶变换
  6. 正变换： X(w)=F[x(t)]=∫−∞∞x(t)e−jwtdtXleft( w ight)={mathcal{F}}left[ xleft( t ight) ight]=int_{-infty }^{infty }{xleft( t ight){{e}^{-jwt}}dt}
  7. 逆变换： x(t)=F−1[X(w)]=12π∫−∞∞X(w)ejwtdwxleft( t ight)={{mathcal{F}}^{-1}}left[ Xleft( w ight) ight]=frac{1}{2pi }int_{-infty }^{infty }{Xleft( w ight){{e}^{jwt}}dw}
  8. 典型函数的傅里叶变换
    门函数：
    x(t)={A,−τ/2≤t≤τ/20,other →FX(w)=AτSa⁡(wτ/2)=Aτsinc⁡(wτ/2π)xleft( t ight)=left{ egin{matrix}A,- au /2le tle au /2 \0, ext{other } \ end{matrix} ight.xrightarrow{mathcal{F}}Xleft( w ight)=A auoperatorname{Sa}left( w au /2 ight)=A auoperatorname{sinc}left( w au /2pi ight)
    冲激函数：
    x(t)=δ(t)→FX(w)=1xleft( t ight)=delta left( t ight)xrightarrow{mathcal{F}}Xleft( w ight)=1
    周期函数：
    x(t)=sin⁡(

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