# 欧拉公式

ejx=cos(x)+jsin(x)ejx=cos(x)jsin(x)\begin{aligned} e^{jx} & = \cos(x)+j\sin(x)\\ e^{-jx} & = \cos(x)-j\sin(x) \end{aligned}

# 傅里叶级数

# 连续信号

x(t)=k=+akejkw0tan=1TTx(t)ejnw0tw0=2πT\begin{aligned} x(t) & = \sum_{k=-\infty}^{+\infty}a_ke^{jkw_0t}\\ a_n & = \frac{1}{T} \int_T x(t)e^{-jnw_0t}\\ w_0 & = \frac{2\pi}{T} \end{aligned}

# 离散信号

x[n]=k=<N>akejkw0nak=1Nn=<N>x[n]ejkw0nw0=2πN\begin{aligned} x[n] & = \sum_{k=<N>} a_ke^{jkw_0n}\\ a_k & = \frac{1}{N}\sum_{n = <N>}x[n]e^{-jkw_0n}\\ w_0 & = \frac{2\pi}{N} \end{aligned}

# 特殊例子

方波信号傅里叶级数

ak=sin(kw0T)πka_k = \frac{\sin(kw_0T)}{\pi k}

# 傅里叶级数的性质

# 傅里叶变换

# 连续傅里叶变换对

x(t)=12π+X(jw)ejwtdwX(jw)=+x(t)ejwtdt\begin{aligned} x(t) & = \frac{1}{2\pi} \int_{-\infty}^{+\infty}X(jw)e^{jwt} dw \\ X(jw) & = \int_{-\infty}^{+\infty}x(t)e^{-jwt}dt \end{aligned}

# 周期信号的连续傅里叶变换

X(jw)=k=+2πakδ(wkw0)\begin{aligned} X(jw) = \sum_{k=-\infty}^{+\infty}2 \pi a_k\delta(w-kw_0) \end{aligned}

# 连续傅里叶变换的系统函数

H(jw)=Y(jw)X(jw)=k=0Mbk(jw)kk=0Nak(jw)kH(jw) = \frac{Y(jw)}{X(jw)} = \frac{\sum_{k=0}^{M}b_k(jw)^k}{\sum_{k=0}^{N}a_k(jw)^k}

# 连续傅里叶变换的性质

# 常见的连续傅里叶变换对

# 离散傅里叶变换对

x[n]=12π2πX(ejw)ejwndwX(ejw)=k=+x[n]ejwn\begin{aligned} x[n] & = \frac{1}{2\pi} \int_{2\pi} X(e^{jw})e^{jwn} dw\\ X(e^{jw}) & = \sum_{k=-\infty}^{+\infty} x[n]e^{-jwn} \end{aligned}

# 周期信号的离散傅里叶变换

X(ejw)=k=+2πakδ(w2πkN)\begin{aligned} X(e^{jw}) = \sum_{k=-\infty}^{+\infty}2\pi a_k \delta(w-\frac{2\pi k}{N}) \end{aligned}

# 离散傅里叶变换系统函数

H(jw)=Y(jw)X(jw)=k=0Mbkejkwk=0NakejkwH(jw) = \frac{Y(jw)}{X(jw)} = \frac{\sum_{k=0}^{M}b_ke^{-jkw}}{\sum_{k=0}^{N}a_ke^{-jkw}}

# 离散傅里叶变换的性质

# 常见的离散傅里叶变换对

# 信号与系统的时域和频域特性

# 连续

X(jw)=X(jw)ejX(jw)X(jw) = |X(jw)|e^{j\sphericalangle X(jw)}

# 离散

X(ejw)=X(ejw)ejX(ejw)X(e^{jw}) = |X(e^{jw})|e^{j\sphericalangle X(e^{jw})}

# 频域响应

Y(jw)=H(jw)X(jw)Y(jw)=H(jw)X(jw)Y(jw)=H(jw)+X(jw)\begin{aligned} Y(jw) & = H(jw)*X(jw)\\ \Rightarrow |Y(jw)| & = |H(jw)|*|X(jw)|\\ \sphericalangle Y(jw) & = \sphericalangle H(jw) + \sphericalangle X(jw) \end{aligned}

# 非失真传输

H(jw)=kejwtH(jw) = ke^{-jwt}

其中H(jw)|H(jw)| 为常数,H(jw)\sphericalangle H(jw) 过原点

# 采样

# 冲击串采样

xp(t)=x(t)p(t)p(t)=k=+δ(tnT)Xp(jw)=12π+X(jw)P(j(wθ))dθP(jw)=2πTk=+δ(wkws)XP(jw)=1Tk=+X(j(wkws))\begin{aligned} x_p(t) & = x(t)*p(t) \\ p(t) & = \sum_{k=-\infty}^{+\infty}\delta(t-nT) \\ X_p(jw) & = \frac{1}{2\pi}\int_{-\infty}^{+\infty} X(jw)P(j(w-\theta)){\,\,}d\theta \\ P(jw) & = \frac{2\pi}{T}\sum_{k=-\infty}^{+\infty}\delta(w-kw_s) \Rightarrow X_P(jw) & = \frac{1}{T} \sum_{k=-\infty}^{+\infty}X(j(w-kw_s)) \end{aligned}

# 采样定理

# 拉普拉斯变换

# 拉普拉斯变换

X(s)=+x(t)estdts=σ+jw\begin{aligned} X(s) & = \int_{-\infty}^{+\infty}x(t)e^{-st} dt \\ s & = \sigma + jw \end{aligned}

收敛域要保证X(s)X(s) 收敛,记为{\cal Re}\left\

# 拉普拉斯逆变换

x(t)=12π+X(s)estds\begin{aligned} x(t) & = \frac{1}{2\pi}\int_{-\infty}^{+\infty}X(s)e^{st} ds \end{aligned}

# 拉普拉斯变换性质

# 常见的拉普拉斯变换对

# 微分方程描述的因果线性时不变系统

andny(t)dtn+an1dn1y(t)dtn1+an2dn2y(t)dtn2++a1dy(t)dt+a0y(t)=bndnx(t)dtn+bn1dn1x(t)dtn1+bn2dn2x(t)dtn2++b1dx(t)dt+b0x(t)\begin{aligned} & a_n\frac{d^ny(t)}{dt^n}+a_{n-1}\frac{d^{n-1}y(t)}{dt^{n-1}}+a_{n-2}\frac{d^{n-2}y(t)}{dt^{n-2}}+ \dotsb + a_{1}\frac{dy(t)}{dt} + a_0y(t) = \\ & b_n\frac{d^nx(t)}{dt^n}+b_{n-1}\frac{d^{n-1}x(t)}{dt^{n-1}}+b_{n-2}\frac{d^{n-2}x(t)}{dt^{n-2}}+ \dotsb + b_{1}\frac{dx(t)}{dt} + b_0x(t) \end{aligned}

系统函数表示为

H(s)=k=0nbkskk=0naksk\begin{aligned} {\cal H}(s) = \frac{\sum_{k=0}^{n}b_ks^k}{\sum_{k=0}^{n}a_ks^k} \end{aligned}

# Z\cal Z 变换

# Z\cal Z 变换

X(z)=n=+x[n]znz=rejw\begin{aligned} & X({\cal z}) = \sum_{n=-\infty}^{+\infty}x[n] {\cal z}^{-n}\\ & z = re^{jw} \end{aligned}

收敛于要保证X(z)X({\cal z}) 收敛

# Z\cal Z 逆变换

x[n]=12πjX(z)zn1dz\begin{aligned} x[n] = \frac{1}{2\pi j}\oint X({\cal z})z^{n-1} dz \end{aligned}

# Z\cal Z 变换性质

# 常见的Z\cal Z 变换性质