本文为信号与线性系统上课笔记。
文章目录
1. 信号与系统
1.2 信号
信号自变量变换
- 平移 x ( t ) → x ( t + t 0 ) / x ( t − t 0 ) x\left(t\right)\rightarrow x\left(t+t_0\right)/x\left(t-t_0\right) x(t)→x(t+t0)/x(t−t0)
- 反转 x ( t ) → x ( − t ) x\left(t\right)\rightarrow x\left(-t\right) x(t)→x(−t)
- 连续信号尺度 x ( t ) → x ( a t ) x\left(t\right)\rightarrow x\left(at\right) x(t)→x(at)
- 离散信号尺度 x ( n ) → x ( N n ) / x ( n / N ) x\left(n\right)\rightarrow x\left(Nn\right)/x\left(n/N\right) x(n)→x(Nn)/x(n/N)
信号的特性
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奇信号和偶信号
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x o ( t ) = 1 2 [ x ( t ) − x ( − t ) ] x_o\left(t\right)=\frac{1}{2}\left[x\left(t\right)-x\left(-t\right)\right] xo(t)=21[x(t)−x(−t)]
x e ( t ) = 1 2 [ x ( t ) + x ( − t ) ] x_e\left(t\right)=\frac{1}{2}\left[x\left(t\right)+x\left(-t\right)\right] xe(t)=21[x(t)+x(−t)]
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周期信号和非周期信号
- 连续直流信号:基波周期无意义
- 离散直流信号:基波周期为1
基本常用信号
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连续时间正弦信号 x ( t ) = A cos ( Ω 0 t + φ ) x\left(t\right)=A\cos{\left({\Omega}_0t+\varphi\right)} x(t)=Acos(Ω0t+φ)
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离散时间正弦序列 x ( n ) = A cos ( ω 0 n + φ ) x\left(n\right)=A\cos{\left({\omega}_0n+\varphi\right)} x(n)=Acos(ω0n+φ)
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连续时间指数信号 x ( t ) = c e a t x\left(t\right)=c\mathrm{e}^{at} x(t)=ceat
- 单边指数信号 f ( t ) = { 0 t < 0 e − t τ t > 0 f\left(t\right)=\begin{cases}0&t<0\\e^{-\frac{t}{\tau}}&t>0\end{cases} f(t)={0e−τtt<0t>0
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离散时间指数信号 x ( n ) = c a n x\left(n\right)=ca^n x(n)=can
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单位阶跃信号
KaTeX parse error: Unknown column alignment: * at position 38: …{\begin{array}{*̲*lr**} 1&t>0\\0…
KaTeX parse error: Unknown column alignment: * at position 38: …{\begin{array}{*̲*lr**} 1&n\geq0…
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单位脉冲序列(离散)
KaTeX parse error: Unknown column alignment: * at position 43: …{\begin{array}{*̲*lr**} 1&n=0\\0…
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x ( n ) δ ( n ) = x ( 0 ) δ ( n ) x\left(n\right)\delta\left(n\right)=x\left(0\right)\delta\left(n\right) x(n)δ(n)=x(0)δ(n)
x ( n ) δ ( n − m ) = x ( m ) δ ( n − m ) x\left(n\right)\delta\left(n-m\right)=x\left(m\right)\delta\left(n-m\right) x(n)δ(n−m)=x(m)δ(n−m)
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δ ( n ) = u ( n ) − u ( n − 1 ) \delta\left(n\right)=u\left(n\right)-u\left(n-1\right) δ(n)=u(n)−u(n−1)
u ( n ) = ∑ k = 0 ∞ δ ( n − k ) u\left(n\right)=\sum\limits_{k=0}^{\infty}{\delta\left(n-k\right)} u(n)=k=0∑∞δ(n−k)
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单位冲击函数(连续)
KaTeX parse error: Unknown column alignment: * at position 22: …{\begin{array}{*̲*lr**} \int_{-\…
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δ ( t ) = d u ( t ) d t \delta\left(t\right)=\frac{\mathrm{d}u\left(t\right)}{\mathrm{d}t} δ(t)=dtdu(t)
∫ − ∞ t δ ( t ) d t = u ( t ) \int_{-\infty}^{t}{\delta\left(t\right)\mathrm{d}t}=u\left(t\right) ∫−∞tδ(t)dt=u(t)
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冲激函数及其性质
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定义: ∫ − ∞ t x ( t ) δ ( t − t 0 ) d t = x ( t 0 ) \int_{-\infty}^{t}{x\left(t\right)\delta\left(t-t_0\right)\mathrm{d}t}=x\left(t_0\right) ∫−∞tx(t)δ(t−t0)dt=x(t0)
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抽样性质: x ( t ) δ ( t − t 0 ) = x ( t 0 ) δ ( t − t 0 ) x\left(t\right)\delta\left(t-t_0\right)=x\left(t_0\right)\delta\left(t-t_0\right) x(t)δ(t−t0)=x(t0)δ(t−t0)
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奇偶性: δ ( t ) = δ ( − t ) \delta\left(t\right)=\delta\left(-t\right) δ(t)=δ(−t)
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尺度变换: δ ( a t ) = 1 ∣ a ∣ δ ( t ) \delta\left(at\right)=\frac{1}{\left|a\right|}\delta\left(t\right) δ(at)=∣a∣1δ(t)
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微分: ∫ − ∞ + ∞ x ( t ) δ ′ ( t ) d t = − x ′ ( 0 ) \int_{-\infty}^{+\infty}{x\left(t\right)\delta^{\prime}\left(t\right)\mathrm{d}t}=-x^{\prime}\left(0\right) ∫−∞+∞x(t)δ′(t)dt=−x′(0)
1.3 系统
时间系统基本单元
输入输出方程
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二阶系统
y ′ ′ + a 1 y ′ + a 0 y = b 1 x ′ + b 0 x y^{\prime\prime}+a_1y^{\prime}+a_0y=b_1x^{\prime}+b_0x y′′+a1y′+a0y=b1x′+b0x
q ′ ′ + a 1 q ′ + a 0 q = x q^{\prime\prime}+a_1q^{\prime}+a_0q=x q′′+a1q′+a0q=x
y = b 1 q ′ + b 0 q y=b_1q^{\prime}+b_0q y=b1q′+b0q
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n n n阶系统
y ( n ) + a n − 1 y n − 1 + ⋯ + a 1 y ′ + a 0 y = b n − 1 x n − 1 + ⋯ + b 1 x ′ + b 0 x y^{\left(n\right)}+a_{n-1}y^{n-1}+\cdots +a_1y^{\prime}+a_0y=b_{n-1}x^{n-1}+\cdots +b_1x^{\prime}+b_0x y(n)+an−1yn−1+⋯+a1y′+a0y=bn−1xn−1+⋯+b1x′+b0x
q ( n ) + a n − 1 q n − 1 + ⋯ + a 1 q ′ + a 0 q = x q^{\left(n\right)}+a_{n-1}q^{n-1}+\cdots +a_1q^{\prime}+a_0q=x q(n)+an−1qn−1+⋯+a1q′+a0q=x
y = b n − 1 q n − 1 + ⋯ + b 1 q ′ + b 0 q y=b_{n-1}q^{n-1}+\cdots +b_1q^{\prime}+b_0q y=bn−1qn−1+⋯+b1q′+b0q
系统性质
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及时系统/动态系统:在任何时刻的输入,只与当前时刻输入有关,则为即时系统
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可逆系统/不可逆系统:系统对不同的输入产生的输出都不同,即系统的输入与输出成一一对应关系,则称为可逆系统
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因果系统/非因果系统: t < t 0 , x ( t ) = 0 , y ( t ) = 0 t<t_0, x\left(t\right)=0, y\left(t\right)=0 t<t0,x(t)=0,y(t)=0
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稳定系统/不稳定系统:系统对任何有界输入产生的输出都是有界的,则称为稳定系统
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时变系统/时不变系统:输入信号在时间上有一个平移,则相应的输出信号也仅在时间上有一个同样的平移,而波形上没有任何变化,则为时变系统
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线性系统/非线性系统:既满足叠加性,同时又满足齐次性的系统,称为线性系统
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叠加性
x 1 ( t ) → y 1 ( t ) , x 2 ( t ) → y 2 ( t ) x_1\left(t\right)\rightarrow y_1\left(t\right),x_2\left(t\right)\rightarrow y_2\left(t\right) x1(t)→y1(t),x2(t)→y2(t)
x 1 ( t ) + x 2 ( t ) → y 1 ( t ) + y 2 ( t ) x_1\left(t\right)+x_2\left(t\right)\rightarrow y_1\left(t\right)+y_2\left(t\right) x1(t)+x2(t)→y1(t)+y2(t)
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齐次性
x ( t ) → y ( t ) x\left(t\right)\rightarrow y\left(t\right) x(t)→y(t)
k ⋅ x ( t ) → k ⋅ y ( t ) k\cdot x\left(t\right)\rightarrow k\cdot y\left(t\right) k⋅x(t)→k⋅y(t)
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增量线性系统:如果一个系统输出的增量与输入的增量之间成线性关系,则该系统为增量线性系统
2. 信号与系统的时域分析
2.1 连续LTI卷积
信号的时域分解
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矩形脉冲信号
KaTeX parse error: Unknown column alignment: * at position 52: …{\begin{array}{*̲*lr**} \frac{1}…
卷积
f 1 ( t ) ∗ f 2 ( t ) = ∫ − ∞ ∞ f 1 ( τ ) f 2 ( t − τ ) d τ f_1(t)*f_2(t)=\int_{-\infin}^{\infin}{f_1(\tau)f_2(t-\tau)\mathrm{d}\tau} f1(t)∗f2(t)=∫−∞∞f1(τ)f2(t−τ)dτ
y ( t ) = x ( t ) ∗ h ( t ) = ∫ 0 t x ( τ ) h ( t − τ ) d τ y(t)=x(t)*h(t)=\int_{0}^{t}{x(\tau)h(t-\tau)\mathrm{d}\tau} y(t)=x(t)∗h(t)=∫0tx(τ)h(t−τ)dτ
图解卷积
- 变换:改变图形中的横坐标,自变量由 t t t 变为 τ \tau τ
- 反转:将其中一个信号反转
- 平移:反转后的信号随参变量 t t t 平移,得到 h ( t − τ ) h(t-\tau) h(t−τ)。若 t > 0 t>0 t>0 则右向平移,若 t < 0 t<0 t<0 则左向平移
- 相乘:将 x ( τ ) x(\tau) x(τ) 与 h ( t − τ ) h(t-\tau) h(t−τ) 相乘
- 积分: x ( τ ) x(\tau) x(τ) 与 h ( t − τ ) h(t-\tau) h(t−τ) 乘积曲线下的面积即为 t t t 时刻的卷积值 (注意积分区域)
卷积性质
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交换律 x ( t ) ∗ h ( t ) = h ( t ) ∗ x ( t ) x(t)*h(t)=h(t)*x(t) x(t)∗h(t)=h(t)∗x(t)
结合律 [ x ( t ) ∗ h 1 ( t ) ] ∗ h 2 ( t ) = x ( t ) ∗ [ h 1 ( t ) ∗ h 2 ( t ) ] \left[x(t)*h_1(t)\right]*h_2(t)=x(t)*\left[h_1(t)*h_2(t)\right] [x(t)∗h1(t)]∗h2(t)=x(t)∗[h1(t)∗h2(t)]
- 串联系统的冲击响应,等于各子系统冲击响应之卷积
- 串联系统与子系统次序无关
分配律 x ( t ) ∗ [ h 1 ( t ) + h 2 ( t ) ] = x ( t ) ∗ h 1 ( t ) + x ( t ) ∗ h 2 ( t ) x(t)*\left[h_1(t)+h_2(t)\right]=x(t)*h_1(t)+x(t)*h_2(t) x(t)∗[h1(t)+h2(t)]=x(t)∗h1(t)+x(t)∗h2(t)
- 一个并联系统的冲激响应等于各个子系统冲激响应之和
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卷积微分 [ x ( t ) ∗ h ( t ) ] ′ = x ′ ( t ) ∗ h ( t ) = x ( t ) ∗ h ′ ( t ) [x(t)*h(t)]'=x'(t)*h(t)=x(t)*h'(t) [x(t)∗h(t)]′=x′(t)∗h(t)=x(t)∗h′(t)
卷积积分 ∫ − ∞ t [ x ( λ ) ∗ h ( λ ) ] d λ = [ ∫ − ∞ t x ( λ ) d λ ] ∗ h ( t ) = x ( t ) ∗ [ ∫ − ∞ t h ( λ ) d λ ] \int_{-\infin}^{t}{[x(\lambda)*h(\lambda)]\mathrm{d}\lambda}=\left[\int_{-\infin}^{t}{x(\lambda)\mathrm{d}\lambda}\right]*h(t)=x(t)*\left[\int_{-\infin}^{t}{h(\lambda)\mathrm{d}\lambda}\right] ∫−∞t[x(λ)∗h(λ)]dλ=[∫−∞tx(λ)dλ]∗h(t)=x(t)∗[∫−∞th(λ)dλ]
- 推论 y ( t ) = x ( t ) ∗ h ( t ) = x ′ ( t ) ∗ [ ∫ − ∞ t h ( λ ) d λ ] = [ ∫ − ∞ t x ( λ ) d λ ] ∗ h ′ ( t ) y(t)=x(t)*h(t)=x'(t)*\left[\int_{-\infin}^{t}{h(\lambda)\mathrm{d}\lambda}\right]=\left[\int_{-\infin}^{t}{x(\lambda)\mathrm{d}\lambda}\right]*h'(t) y(t)=x(t)∗h(t)=x′(t)∗[∫−∞th(λ)dλ]=[∫−∞tx(λ)dλ]∗h′(t)
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冲激函数卷积 x ( t − t 1 ) ∗ δ ( t − t 2 ) = x ( t − t 1 − t 2 ) x(t-t_1)*\delta(t-t_2)=x(t-t_1-t_2) x(t−t1)∗δ(t−t2)=x(t−t1−t2)
- 推论 x ( t − t 1 ) ∗ h ( t − t 2 ) = y ( t − t 1 − t 2 ) x(t-t_1)*h(t-t_2)=y(t-t_1-t_2) x(t−t1)∗h(t−t2)=y(t−t1−t2)
阶跃函数卷积 x ( t ) ∗ u ( t ) = ∫ − ∞ t x ( τ ) d τ x(t)*u(t)=\int_{-\infin}^{t}x(\tau)\mathrm{d}\tau x(t)∗u(t)=∫−∞tx(τ)dτ
- 推论 u ( t ) ∗ u ( t ) = t u ( t ) u(t)*u(t)=tu(t) u(t)∗u(t)=tu(t)
2.2 连续LTI单位冲激响应
冲激响应:系统对单位冲激信号的零状态响应
微分方程描述
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二阶通式 y ′ ′ + a 1 y ′ + a 0 y = b 1 x ′ + b 0 x y''+a_1 y'+a_0y=b_1x'+b_0x y′′+a1y′+a0y=b1x′+b0x
N N N 阶通式 y ( n ) + a n − 1 y ( n − 1 ) + ⋯ + a 1 y ′ + a 0 y = b m x ( m ) + ⋯ + b 1 x ′ + b 0 x y^{(n)}+a_{n-1}y^{(n-1)}+\cdots+a_1y'+a_0y=b_mx^{(m)}+\cdots+b_1x'+b_0x y(n)+an−1y(n−1)+⋯+a1y′+a0y=bmx(m)+⋯+b1x′+b0x
求和形式 ∑ k = 0 n a k y ( k ) ( t ) = ∑ k = 0 m b k x ( k ) ( t ) \sum\limits_{k=0}^{n}{a_k y^{(k)}(t)}=\sum\limits_{k=0}^{m}{b_k x^{(k)}(t)} k=0∑naky(k)(t)=k=0∑mbkx(k)(t)
单位冲激响应求解
y ( t ) = y 1 ( t ) y(t)=y_1(t) y(t)=y1(t)(齐次方程通解) + y 2 ( t ) +y_2(t) +y2(t)(非齐次方程特解)
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齐次方程 ∑ k = 0 n a k y ( k ) ( t ) = 0 \sum\limits_{k=0}^{n}{a_k y^{(k)}(t)}=0 k=0∑naky(k)(t)=0
y 1 ( t ) = ∑ k = 0 n C k e λ k t ( t ) y_1(t)=\sum\limits_{k=0}^{n}{C_k \mathrm{e}^{\lambda_kt}(t)} y1(t)=k=0∑nCkeλkt(t)
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微分算子
d n x d t n = p n x , ∫ − ∞ t x d τ = 1 p x \frac{d^nx}{dt^n}=p^nx, \int_{-\infin}^{t}{x\mathrm{d}\tau}=\frac{1}{p}x dtndnx=pnx,∫−∞txdτ=p1x
m p + n p = ( m + n ) p mp+np=(m+n)p mp+np=(m+n)p
p m p n = p m + n p^mp^n=p^{m+n} pmpn=pm+n( m , n m,n m,n同正负)
p 1 p ≠ 1 p p p\frac{1}{p}\neq\frac{1}{p}p pp1=p1p
p x ( t ) ↛ x ( t ) = y ( t ) px(t)\not\rightarrow x(t)=y(t) px(t)→x(t)=y(t)
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记 N N N 阶通式为 D ( p ) y ( t ) = N ( p ) x ( t ) D(p)y(t)=N(p)x(t) D(p)y(t)=N(p)x(t)
H ( p ) = N ( p ) D ( p ) H(p)=\frac{N(p)}{D(p)} H(p)=D(p)N(p)
y ( t ) = H ( p ) x ( t ) y(t)=H(p)x(t) y(t)=H(p)x(t)
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n > m n>m n>m
y ( t ) = H ( p ) x ( t ) ⇒ h ( t ) = H ( p ) δ ( t ) y(t)=H(p)x(t)\Rightarrow h(t)=H(p)\delta(t) y(t)=H(p)x(t)⇒h(t)=H(p)δ(t)
h ( t ) = H ( p ) δ ( t ) = ( k 1 p − λ 1 + k 2 p − λ 2 + ⋯ + k n p − λ n ) δ ( t ) h(t)=H(p)\delta(t)=\left(\frac{k_1}{p-\lambda_1}+\frac{k_2}{p-\lambda_2}+\cdots+\frac{k_n}{p-\lambda_n}\right)\delta(t) h(t)=H(p)δ(t)=(p−λ1k1+p−λ2k2+⋯+p−λnkn)δ(t)
特征方程的特征根为 λ k \lambda_k λk
令 h i ( t ) = k i p − λ i δ ( t ) h_i(t)=\frac{k_i}{p-\lambda_i}\delta(t) hi(t)=p−λikiδ(t)
h ( t ) = ∑ i = 1 n h i ( t ) = ∑ i = 1 n k i e λ i t u ( t ) h(t)=\sum\limits_{i=1}^{n}{h_i(t)}=\sum\limits_{i=1}^{n}{k_i\mathrm{e}^{\lambda_it}u(t)} h(t)=i=1∑nhi(t)=i=1∑nkieλitu(t)
若 λ k \lambda_k λk 均为 k k k 阶重根, h k ( t ) = ( A 1 + A 2 t + ⋯ + A k t k − 1 ) e λ 1 t u ( t ) h_k(t)=\left(A_1+A_2t+\cdots+A_kt^{k-1}\right)\mathrm{e}^{\lambda_1t}u(t) hk(t)=(A1+A2t+⋯+Aktk−1)eλ1tu(t)
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n = m n=m n=m
h ( t ) = ∑ i = 1 n k i e λ i t u ( t ) + b m δ ( t ) h(t)=\sum\limits_{i=1}^{n}{k_i\mathrm{e}^{\lambda_it}u(t)}+b_m\delta(t) h(t)=i=1∑nkieλitu(t)+bmδ(t)
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n < m n<m n<m
h ( t ) = H ( p ) δ ( t ) h(t)=H(p)\delta(t) h(t)=H(p)δ(t)
= ( A 0 p m − n + ⋯ + A m − n + 1 p + A m − n + k 1 p − λ 1 + k 2 p − λ 2 + ⋯ + k n p − λ n ) δ ( t ) =\left(A_0p^{m-n}+\cdots+A_{m-n+1}p+A_{m-n}+\frac{k_1}{p-\lambda_1}+\frac{k_2}{p-\lambda_2}+\cdots+\frac{k_n}{p-\lambda_n}\right)\delta(t) =(A0pm−n+⋯+Am−n+1p+Am−n+p−λ1k1+p−λ2k2+⋯+p−λnkn)δ(t)
h ( t ) = ∑ i = 1 n k i e λ i t u ( t ) + A 0 δ ( m − n ) ( t ) + ⋯ + A m − n δ ( n ) h(t)=\sum\limits_{i=1}^{n}{k_i\mathrm{e}^{\lambda_it}u(t)}+A_0\delta^{(m-n)}(t)+\cdots+A_{m-n}\delta(n) h(t)=i=1∑nkieλitu(t)+A0δ(m−n)(t)+⋯+Am−nδ(n)
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2.3 离散LTI卷积
卷积和
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y ( n ) = x ( n ) ∗ h ( n ) = ∑ k = − ∞ + ∞ x ( k ) h ( n − k ) y(n)=x(n)*h(n)=\sum\limits_{k=-\infin}^{+\infin}{x(k)h(n-k)} y(n)=x(n)∗h(n)=k=−∞∑+∞x(k)h(n−k)
因果系统 y ( n ) = = ∑ k = 0 n x ( k ) h ( n − k ) y(n)==\sum\limits_{k=0}^{n}{x(k)h(n-k)} y(n)==k=0∑nx(k)h(n−k)
图解法
- 反转:将 h ( k ) h(k) h(k) 以纵轴为对称轴反转得到 h ( − k ) h(-k) h(−k)
- 平移:将 h ( − k ) h(-k) h(−k) 随参变量平移得到 h ( n − k ) h(n-k) h(n−k)
- 相乘:将 x ( n ) x(n) x(n) 与 h ( n − k ) h(n-k) h(n−k) 各对应点相乘
- 求和:将相乘后的各点值相加
性质
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交换律、结合律、分配律
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长度有限性 l y = l x + l h − 1 l_y=l_x+l_h-1 ly=lx+lh−1
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x ( n − n 1 ) ∗ δ ( n − n 2 ) = x ( n − n 1 − n 2 ) x(n-n_1)*\delta(n-n_2)=x(n-n_1-n_2) x(n−n1)∗δ(n−n2)=x(n−n1−n2)
x ( n ) ∗ u ( n ) = ∑ k = − ∞ n x ( k ) x(n)*u(n)=\sum\limits_{k=-\infin}^{n}{x(k)} x(n)∗u(n)=k=−∞∑nx(k)
u ( n ) ∗ h ( n ) = ∑ k = − ∞ n h ( k ) = s ( n ) u(n)*h(n)=\sum\limits_{k=-\infin}^{n}{h(k)}=s(n) u(n)∗h(n)=k=−∞∑nh(k)=s(n)
h ( n ) = s ( n ) − s ( n − 1 ) h(n)=s(n)-s(n-1) h(n)=s(n)−s(n−1)
2.4 离散LTI单位脉冲响应
差分方程描述
∑ k = 0 N a k y ( n + k ) = ∑ k = 0 M b k x ( n + k ) \sum\limits_{k=0}^{N}{a_ky(n+k)}=\sum\limits_{k=0}^{M}{b_kx(n+k)} k=0∑Naky(n+k)=k=0∑Mbkx(n+k)
差分方程阶数:差分方程的阶定义为响应最大移序与最小移序之差
单位脉冲响应求解
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移位算子: S ⋅ y ( k ) = y ( k + 1 ) S\cdot y(k)=y(k+1) S⋅y(k)=y(k+1)
差分方程变为 ( S N + ⋯ + a 1 S 1 + a 0 ) y ( n ) = ( b M S M + ⋯ + b 1 S 1 + b 0 ) x ( n ) \left(S^N+\cdots+a_1S_1+a_0\right)y(n)=\left(b_MS_M+\cdots+b_1S_1+b_0\right)x(n) (SN+⋯+a1S1+a0)y(n)=(bMSM+⋯+b1S1+b0)x(n)
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y ( n ) = H ( S ) x ( n ) y(n)=H(S)x(n) y(n)=H(S)x(n)
H i ( S ) = A i S − v i H_i(S)=\frac{A_i}{S-v_i} Hi(S)=S−viAi
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m < n m<n m<n
h ( n ) = ∑ r = 1 N A r v n − 1 u ( n − 1 ) h(n)=\sum\limits_{r=1}^{N}{A_rv^{n-1}u(n-1)} h(n)=r=1∑NArvn−1u(n−1)
若 v r v_r vr 为 l l l 阶重根, h r ( n ) = A ( n − 1 ) ! ( l − 1 ) ! ( n − 1 ) ! v r n − l u ( n − 1 ) h_r(n)=\frac{A(n-1)!}{(l-1)!(n-1)!}v_r^{n-l}u(n-1) hr(n)=(l−1)!(n−1)!A(n−1)!vrn−lu(n−1)
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m = n m=n m=n
H ( S ) = A 0 + H 1 ( S ) + ⋯ + H N ( S ) H(S)=A_0+H_1(S)+\cdots+H_N(S) H(S)=A0+H1(S)+⋯+HN(S)
h ( n ) = A 0 δ ( n ) + ∑ r = 1 N A r v n − 1 u ( n − 1 ) h(n)=A_0\delta(n)+\sum\limits_{r=1}^{N}{A_rv^{n-1}u(n-1)} h(n)=A0δ(n)+r=1∑NArvn−1u(n−1)
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m > n m>n m>n:非因果系统,不考虑
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2.4 系统性质分析
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及时系统: h ( n ) = a δ ( n ) , h ( t ) = a δ ( t ) h(n)=a\delta(n),h(t)=a\delta(t) h(n)=aδ(n),h(t)=aδ(t)
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恒等系统: h ( n ) = δ ( n ) , h ( t ) = δ ( t ) h(n)=\delta(n),h(t)=\delta(t) h(n)=δ(n),h(t)=δ(t)
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可逆系统: h ( n ) ∗ h I ( n ) = δ ( n ) , h ( t ) ∗ h I ( t ) = δ ( t ) h(n)*h_I(n)=\delta(n),h(t)*h_I(t)=\delta(t) h(n)∗hI(n)=δ(n),h(t)∗hI(t)=δ(t)
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因果系统: n < 0 ⇒ h ( n ) = 0 , t < 0 ⇒ h ( t ) = 0 n<0\Rightarrow h(n)=0,t<0\Rightarrow h(t)=0 n<0⇒h(n)=0,t<0⇒h(t)=0
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稳定性:对于任何有界的输入,其输出有界
∑ k = − ∞ + ∞ ∣ h ( k ) ∣ < ∞ , ∫ − ∞ + ∞ ∣ h ( t ) ∣ d t < ∞ \sum\limits_{k=-\infin}^{+\infin}{\left|h(k)\right|}<\infin,\int_{-\infin}^{+\infin}{\left|h(t)\right|\mathrm{d}t}<\infin k=−∞∑+∞∣h(k)∣<∞,∫−∞+∞∣h(t)∣dt<∞
2.5 离散LTI系统方框图
∑ k = 0 N a k y ( n − k ) = ∑ k = 0 M b k x ( n − k ) \sum\limits_{k=0}^{N}{a_ky(n-k)}=\sum\limits_{k=0}^{M}{b_kx(n-k)} k=0∑Naky(n−k)=k=0∑Mbkx(n−k)
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解法1
∑ k = 0 N a k y ( n + k ) = ∑ k = 0 M b k x ( n + k ) \sum\limits_{k=0}^{N}{a_ky(n+k)}=\sum\limits_{k=0}^{M}{b_kx(n+k)} k=0∑Naky(n+k)=k=0∑Mbkx(n+k)
y ( n ) = b M S M + ⋯ + b 1 S + b 0 a N S N + ⋯ + a 1 S + a 0 x ( n ) y(n)=\frac{b_MS^M+\cdots+b_1S+b_0}{a_NS^N+\cdots+a_1S+a_0}x(n) y(n)=aNSN+⋯+a1S+a0bMSM+⋯+b1S+b0x(n)
令 q ( n ) = 1 a N S N + ⋯ + a 1 S + a 0 x ( n ) q(n)=\frac{1}{a_NS^N+\cdots+a_1S+a_0}x(n) q(n)=aNSN+⋯+a1S+a01x(n)
则 a N q ( n ) = x ( n ) − ⋯ a_Nq(n)=x(n)-\cdots aNq(n)=x(n)−⋯
y ( n ) = ( b M S M + ⋯ + b 1 S + b 0 ) q ( n ) y(n)=\left(b_MS^M+\cdots+b_1S+b_0\right)q(n) y(n)=(bMSM+⋯+b1S+b0)q(n)
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解法2
w ( n ) = ∑ k = 0 M b k x ( n − k ) w(n)=\sum\limits_{k=0}^{M}{b_kx(n-k)} w(n)=k=0∑Mbkx(n−k)
y ( n ) = 1 a 0 [ w ( n ) − ∑ k = 1 N a k y ( n − k ) ] y(n)=\frac{1}{a_0}\left[w(n)-\sum\limits_{k=1}^{N}{a_ky(n-k)}\right] y(n)=a01[w(n)−k=1∑Naky(n−k)]
2.6 连续LTI系统方框图
∑ k = 0 N a k y ( N − k ) ( t ) = ∑ k = 0 M b k x ( N − k ) ( t ) \sum\limits_{k=0}^{N}{a_k y^{(N-k)}(t)}=\sum\limits_{k=0}^{M}{b_k x^{(N-k)}(t)} k=0∑Naky(N−k)(t)=k=0∑Mbkx(N−k)(t)
w ( n ) = ∑ k = 0 M b k x ( N − k ) ( t ) w(n)=\sum\limits_{k=0}^{M}{b_kx^{(N-k)}}(t) w(n)=k=0∑Mbkx(N−k)(t)
y ( n ) = 1 a N [ w ( n ) − ∑ k = 1 N − 1 a k y ( N − k ) ( t ) ] y(n)=\frac{1}{a_N}\left[w(n)-\sum\limits_{k=1}^{N-1}{a_ky^{(N-k)}(t)}\right] y(n)=aN1[w(n)−k=1∑N−1aky(N−k)(t)]
3 连续时间信号与系统的频域分析
3.1 信号分解
复指数信号 x ( t ) = e s t x(t)=e^{st} x(t)=est
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复频域分析 s = σ + h Ω s=\sigma+h\Omega s=σ+hΩ
频域分析 σ = 0 , s = j Ω \sigma=0,s=j\Omega σ=0,s=jΩ
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欧拉公式 e j Ω 0 t = cos Ω 0 t + j sin Ω 0 t e^{j\Omega_0t}=\cos{\Omega_0t}+j\sin{\Omega_0t} ejΩ0t=cosΩ0t+jsinΩ0t
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令 x ( t ) = e s t , y ( t ) = e s t ∫ − ∞ ∞ h ( τ ) e − s t d τ = H ( s ) e s t x(t)=e^{st},y(t)=e^{st}\int_{-\infin}^{\infin}{h(\tau)e^{-st}\mathrm{d}\tau}=H(s)e^{st} x(t)=est,y(t)=est∫−∞∞h(τ)e−stdτ=H(s)est
s s t s^{st} sst 为特征函数, H ( s ) H(s) H(s) 为特征值
x ( t ) = ∑ k a k e s k t → y ( t ) = ∑ k a k H ( s k ) e s k t x(t)=\sum\limits_k{a_ke^{s_kt}}\rightarrow y(t)=\sum\limits_k{a_kH(s_k)e^{s_kt}} x(t)=k∑akeskt→y(t)=k∑akH(sk)eskt
3.2 周期信号傅立叶级数
x ( t ) = x ( x + T 0 ) x(t)=x(x+T_0) x(t)=x(x+T0)
周期信号 e j Ω 0 t e^{j\Omega_0t} ejΩ0t:基波周期 T 0 = 2 π Ω 0 T_0=\frac{2\pi}{\Omega_0} T0=Ω02π,基波频率 Ω 0 = 2 π T 0 \Omega_0=\frac{2\pi}{T_0} Ω0=T02π
x ( t ) = ∑ k = − ∞ ∞ A k ˙ e j k Ω 0 t x(t)=\sum\limits_{k=-\infin}^{\infin}{\dot{A_k}e^{jk\Omega_0t}} x(t)=k=−∞∑∞Ak˙ejkΩ0t
A k ˙ \dot{A_k} Ak˙ 为傅立叶系数, k = ± N k=\pm N k=±N 称为 N N N 次谐波分量
A k ˙ = 1 T 0 ∫ 0 T 0 x ( t ) e − j k Ω 0 t d t \dot{A_k}=\frac{1}{T_0}\int_{0}^{T_0}{x(t)e^{-jk\Omega_0t}\mathrm{d}t} Ak˙=T01∫0T0x(t)e−jkΩ0tdt
性质
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共轭性 A k ˙ = A − k ˙ ∗ \dot{A_k}=\dot{A_{-k}}^{*} Ak˙=A−k˙∗
x ( t ) = A 0 ˙ + 2 ∑ k = 1 ∞ R e { A k ˙ e j k Ω 0 t } x(t)=\dot{A_0}+2\sum\limits_{k=1}^{\infin}{\mathrm{Re}\left\{\dot{A_k}e^{jk\Omega_0t}\right\}} x(t)=A0˙+2k=1∑∞Re{Ak˙ejkΩ0t}
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三角函数形式 x ( t ) = A 0 ˙ + 2 ∑ k = 1 ∞ A k ˙ cos ( k Ω 0 t + θ k ) x(t)=\dot{A_0}+2\sum\limits_{k=1}^{\infin}{\dot{A_k}\cos{(k\Omega_0t+\theta_k)}} x(t)=A0˙+2k=1∑∞Ak˙cos(kΩ0t+θk)
= a 0 + 2 ∑ k = 1 ∞ [ a k cos k Ω 0 t − b k sin k Ω 0 t ] = a 0 + ∑ n = 1 ∞ [ a n ′ cos n Ω 0 t − b n ′ sin n Ω 0 t ] =a_0+2\sum\limits_{k=1}^{\infin}{\left[a_k\cos{k\Omega_0t}-b_k\sin{k\Omega_0t}\right]}=a_0+\sum\limits_{n=1}^{\infin}{\left[a'_n\cos{n\Omega_0t}-b'_n\sin{n\Omega_0t}\right]} =a0+2k=1∑∞[akcoskΩ0t−bksinkΩ0t]=a0+n=1∑∞[an′cosnΩ0t−bn′sinnΩ0t]
A 0 = a 0 = 1 T 0 ∫ 0 T 0 x ( t ) d t A_0=a_0=\frac{1}{T_0}\int_{0}^{T_0}{x(t)\mathrm{d}t} A0=a0=T01∫0T0x(t)dt
a k = 1 2 ( A k ˙ + A − k ˙ ) = 1 T 0 ∫ 0 T 0 x ( t ) ⋅ cos k Ω 0 t d t a_k=\frac{1}{2}\left(\dot{A_k}+\dot{A_{-k}}\right)=\frac{1}{T_0}\int_{0}^{T_0}{x(t)\cdot \cos{k\Omega_0t}\mathrm{d}t} ak=21(Ak˙+A−k˙)=T01∫0T0x(t)⋅coskΩ0tdt
b k = 1 2 j ( A k ˙ − A − k ˙ ) = 1 T 0 ∫ 0 T 0 x ( t ) ⋅ sin k Ω 0 t d t b_k=\frac{1}{2j}\left(\dot{A_k}-\dot{A_{-k}}\right)=\frac{1}{T_0}\int_{0}^{T_0}{x(t)\cdot \sin{k\Omega_0t}\mathrm{d}t} bk=2j1(Ak˙−A−k˙)=T01∫0T0x(t)⋅sinkΩ0tdt
a 1 ′ cos n Ω 0 t − b 1 ′ sin n Ω 0 t a'_1\cos{n\Omega_0t}-b'_1\sin{n\Omega_0t} a1′cosnΩ0t−b1′sinnΩ0t 为基波分量,其余为谐波分量
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a k a_k ak 为偶信号 x e ( t ) x_e(t) xe(t) 的傅立叶系数, j b k jb_k jbk 为奇信号 x o ( t ) x_o(t) xo(t) 的傅立叶系数
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奇谐函数:周期为 T T T 的函数,任意半个周期的波形可由将前半周期波形沿x轴反转得到 a 2 k = b 2 k = 0 a_{2k}=b_{2k}=0 a2k=b2k=0
偶谐函数:将奇谐函数的负半周沿 x x x 轴反转为正半周,此时的函数为偶谐函数 a 2 k + 1 = b 2 k + 1 = 0 a_{2k+1}=b_{2k+1}=0 a2k+1=b2k+1=0
3.3 傅立叶变换
频谱
所有谐波分量的复振幅随频率的分布称为信号的频谱
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振幅频谱: A k A_k Ak
相位频谱: θ k \theta_k θk
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特点
- 离散性:它由不连续的线条组成;
- 谐波性:线条只出现在基波频率的整数倍点上;
- 收敛性:实际信号的幅频特性总是随频率趋向无穷大而趋向于零
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S a ( x ) = sin x x Sa(x)=\frac{\sin{x}}{x} Sa(x)=xsinx
x ( t ) = A τ T ∑ k = − ∞ ∞ S a ( n Ω 0 τ 2 ) e j k Ω 0 t x(t)=\frac{A\tau}{T}\sum\limits_{k=-\infin}^{\infin}{Sa\left(\frac{n\Omega_0\tau}{2}\right)e^{jk\Omega_0t}} x(t)=TAτk=−∞∑∞Sa(2nΩ0τ)ejkΩ0t
X ( Ω ) = A τ S a ( τ Ω 2 ) X(\Omega)=A\tau Sa(\frac{\tau\Omega}{2}) X(Ω)=AτSa(2τΩ)
- X ( Ω ) = T ⋅ A n ˙ ∣ n Ω 0 = Ω X(\Omega)=T\cdot \dot{A_n}|_{n\Omega_0=\Omega} X(Ω)=T⋅An˙∣nΩ0=Ω
- 时域非周期则频域连续,时域周期则频域离散
非周期信号傅立叶变换
傅立叶变换 X ( Ω ) = ∫ − ∞ + ∞ x ( t ) e − j Ω t d t X(\Omega)=\int_{-\infin}^{+\infin}{x(t)e^{-j\Omega t}\mathrm{d}t} X(Ω)=∫−∞+∞x(t)e−jΩtdt ( X ( Ω ) X(\Omega) X(Ω) 为频谱密度函数,简称频谱)
傅立叶反变换 x ( t ) = 1 2 π ∫ − ∞ ∞ X ( Ω ) e j Ω t d Ω x(t)=\frac{1}{2\pi}\int_{-\infin}^{\infin}{X(\Omega)e^{j\Omega t}\mathrm{d}\Omega} x(t)=2π1∫−∞∞X(Ω)ejΩtdΩ
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傅立叶变换存在条件
- ∫ − ∞ ∞ ∣ x ( t ) ∣ d t < ∞ \int_{-\infin}^{\infin}{\left|x(t)\right|\mathrm{d}t}<\infin ∫−∞∞∣x(t)∣dt<∞
- 在任何有限区间内只有有限个极值点,且极值有限
- 在任何有限区间内只有有限个间断点,且不连续值有限
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x ( t ) = 1 2 π ∫ − ∞ ∞ ∣ X ( Ω ) ∣ e j ( Ω t + ϕ ) d Ω x(t)=\frac{1}{2\pi}\int_{-\infin}^{\infin}{\left|X(\Omega)\right|e^{j(\Omega t+\phi)}\mathrm{d}\Omega} x(t)=2π1∫−∞∞∣X(Ω)∣ej(Ωt+ϕ)dΩ
x ( t ) = 1 π ∫ 0 ∞ ∣ X ( Ω ) ∣ cos ( Ω t + ϕ ) d Ω x(t)=\frac{1}{\pi}\int_{0}^{\infin}{\left|X(\Omega)\right|\cos{(\Omega t+\phi)}\mathrm{d}\Omega} x(t)=π1∫0∞∣X(Ω)∣cos(Ωt+ϕ)dΩ
∣ X ( Ω ) ∣ \left|X(\Omega)\right| ∣X(Ω)∣ 为幅度频谱, ϕ ( Ω ) \phi(\Omega) ϕ(Ω) 为相位频谱
常用傅立叶变换
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单边指数信号: x ( t ) = e − α t u ( t ) , α > 0 x(t)=e^{-\alpha t}u(t),\alpha>0 x(t)=e−αtu(t),α>0, X ( Ω ) = 1 α + j Ω X(\Omega)=\frac{1}{\alpha+j\Omega} X(Ω)=α+jΩ1
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单位冲激信号: X ( Ω ) = 1 X(\Omega)=1 X(Ω)=1
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单位阶跃信号: X ( Ω ) = π δ ( Ω ) + 1 j Ω X(\Omega)=\pi\delta(\Omega)+\frac{1}{j\Omega} X(Ω)=πδ(Ω)+jΩ1
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复指数信号
周期信号傅立叶变换
x ( t ) ↔ 2 π ∑ n = − ∞ ∞ A n ˙ δ ( Ω − n Ω 0 ) x(t)\leftrightarrow 2\pi \sum\limits_{n=-\infin}^{\infin}{\dot{A_n}\delta(\Omega-n\Omega_0)} x(t)↔2πn=−∞∑∞An˙δ(Ω−nΩ0)
3.4 傅立叶变换性质
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线性特性: x 1 ( t ) ↔ X 1 ( Ω ) , x 2 ( t ) ↔ X 2 ( Ω ) x_1(t)\leftrightarrow X_1(\Omega),x_2(t)\leftrightarrow X_2(\Omega) x1(t)↔X1(Ω),x2(t)↔X2(Ω)
a ⋅ x 1 ( t ) + b ⋅ x 2 ( t ) ↔ a ⋅ X 1 ( Ω ) + b ⋅ X 2 ( Ω ) a\cdot x_1(t)+b\cdot x_2(t)\leftrightarrow a\cdot X_1(\Omega)+b\cdot X_2(\Omega) a⋅x1(t)+b⋅x2(t)↔a⋅X1(Ω)+b⋅X2(Ω)
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共轭对称性: X ∗ ( Ω ) = X ( − Ω ) X^*(\Omega)=X(-\Omega) X∗(Ω)=X(−Ω)( x x x 为实信号)
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时移特性: x ( t − t 0 ) ↔ X ( Ω ) e − j Ω t 0 x(t-t_0)\leftrightarrow X(\Omega)e^{-j\Omega t_0} x(t−t0)↔X(Ω)e−jΩt0
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移频特性: x ( t ) e j Ω 0 t ↔ X ( Ω − Ω 0 ) x(t)e^{j\Omega_0 t}\leftrightarrow X(\Omega-\Omega_0) x(t)ejΩ0t↔X(Ω−Ω0)
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尺度变换: x ( a t ) ↔ 1 ∣ a ∣ X ( Ω a ) x(at)\leftrightarrow \frac{1}{|a|}X\left(\frac{\Omega}{a}\right) x(at)↔∣a∣1X(aΩ)
- x ( − t ) ↔ X ( − Ω ) x(-t)\leftrightarrow X(-\Omega) x(−t)↔X(−Ω)
- u ( − t ) ↔ π δ ( Ω ) − 1 j Ω u(-t)\leftrightarrow \pi\delta(\Omega)-\frac{1}{j\Omega} u(−t)↔πδ(Ω)−jΩ1
- 1 = u ( t ) + u ( − t ) ↔ 2 π δ ( Ω ) 1=u(t)+u(-t)\leftrightarrow 2\pi\delta(\Omega) 1=u(t)+u(−t)↔2πδ(Ω)
- s g n ( t ) = u ( t ) − u ( − t ) ↔ 2 j Ω \mathrm{sgn}(t)=u(t)-u(-t)\leftrightarrow \frac{2}{j\Omega} sgn(t)=u(t)−u(−t)↔jΩ2
- e − a ∣ t ∣ = e − a t u ( t ) + e a t u ( − t ) ↔ 2 a a 2 + Ω 2 e^{-a|t|}=e^{-at}u(t)+e^{at}u(-t)\leftrightarrow \frac{2a}{a^2+\Omega^2} e−a∣t∣=e−atu(t)+eatu(−t)↔a2+Ω22a
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对偶特性: X ( t ) ↔ 2 π X ( − Ω ) X(t)\leftrightarrow 2\pi X(-\Omega) X(t)↔2πX(−Ω)
- 若 x ( t ) x(t) x(t) 为实偶函数,则 X ( Ω ) X(\Omega) X(Ω) 为实偶函数, X ( t ) ↔ 2 π x ( Ω ) X(t)\leftrightarrow 2\pi x(\Omega) X(t)↔2πx(Ω)
- 若 x ( t ) x(t) x(t) 为实奇函数,则 X ( Ω ) X(\Omega) X(Ω) 为虚奇函数, X ( t ) ↔ − 2 π x ( Ω ) X(t)\leftrightarrow -2\pi x(\Omega) X(t)↔−2πx(Ω)
- δ ( t ) ↔ 1 ⟹ 1 ↔ 2 π δ ( Ω ) \delta(t)\leftrightarrow 1\Longrightarrow 1\leftrightarrow 2\pi\delta(\Omega) δ(t)↔1⟹1↔2πδ(Ω)
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时域微分特性: x ′ ( t ) ↔ j Ω X ( Ω ) x'(t)\leftrightarrow j\Omega X(\Omega) x′(t)↔jΩX(Ω)
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时域积分特性: ∫ − ∞ t x ( τ ) d τ ↔ X ( Ω ) j Ω + π δ ( Ω ) X ( 0 ) \int_{-\infin}^{t}{x(\tau)\mathrm{d}\tau}\leftrightarrow \frac{X(\Omega)}{j\Omega}+\pi\delta(\Omega)X(0) ∫−∞tx(τ)dτ↔jΩX(Ω)+πδ(Ω)X(0)
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频域微积分特性: − j t x ( t ) ↔ X ′ ( Ω ) -jtx(t)\leftrightarrow X'(\Omega) −jtx(t)↔X′(Ω)
− x ( t ) j t + π x ( 0 ) δ ( t ) ↔ ∫ − ∞ Ω X ( Ω ) d Ω -\frac{x(t)}{jt}+\pi x(0)\delta(t)\leftrightarrow \int_{-\infin}^{\Omega}{X(\Omega)\mathrm{d}\Omega} −jtx(t)+πx(0)δ(t)↔∫−∞ΩX(Ω)dΩ
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卷积特性
x 1 ( t ) ∗ x 2 ( t ) ↔ X 1 ( Ω ) ⋅ X 2 ( Ω ) x_1(t)*x_2(t)\leftrightarrow X_1(\Omega)\cdot X_2(\Omega) x1(t)∗x2(t)↔X1(Ω)⋅X2(Ω)
x 1 ( t ) ⋅ x 2 ( t ) ↔ 1 2 π X 1 ( Ω ) ∗ X 2 ( Ω ) x_1(t)\cdot x_2(t)\leftrightarrow \frac{1}{2\pi}X_1(\Omega)* X_2(\Omega) x1(t)⋅x2(t)↔2π1X1(Ω)∗X2(Ω)
3.5 连续时间系统频域分析
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H ( Ω ) = Y ( Ω ) X ( Ω ) = ∣ H ( Ω ∣ e j ϕ ( Ω ) H(\Omega)=\frac{Y(\Omega)}{X(\Omega)}=\left|H(\Omega\right|e^{j\phi(\Omega)} H(Ω)=X(Ω)Y(Ω)=∣H(Ω∣ejϕ(Ω)
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分析方法
- 将时域激励信号分解为频域信号 x ( t ) → X ( Ω ) x(t)\rightarrow X(\Omega) x(t)→X(Ω)
- 确定系统频率响应函数 H ( Ω ) H(\Omega) H(Ω)
- 求取激励信号的频域响应 Y ( Ω ) = X ( Ω ) ⋅ H ( Ω ) Y(\Omega)=X(\Omega)\cdot H(\Omega) Y(Ω)=X(Ω)⋅H(Ω)
- 对频域响应函数求傅立叶反变换得到系统的时域响应函数 Y ( Ω ) → y ( t ) Y(\Omega)\rightarrow y(t) Y(Ω)→y(t)
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系统函数的确定
∑ k = 0 n a k y ( k ) ( t ) = ∑ k = 0 m b k x ( k ) ( t ) \sum\limits_{k=0}^{n}{a_k y^{(k)}(t)}=\sum\limits_{k=0}^{m}{b_k x^{(k)}(t)} k=0∑naky(k)(t)=k=0∑mbkx(k)(t)
H ( Ω ) = ∑ k = 0 m b k ( j Ω ) k ∑ k = 0 n a k ( j Ω ) k H(\Omega)=\frac{\sum\limits_{k=0}^{m}{b_k (j\Omega)^k}}{\sum\limits_{k=0}^{n}{a_k (j\Omega)^k}} H(Ω)=k=0∑nak(jΩ)kk=0∑mbk(jΩ)k
理想低通滤波器
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系统不失真条件
y ( t ) = K x ( t − t 0 ) y(t)=Kx(t-t_0) y(t)=Kx(t−t0)
H ( Ω ) = K e − j Ω t 0 H(\Omega)=Ke^{-j\Omega t_0} H(Ω)=Ke−jΩt0
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频率特征: H ( Ω ) = { K e − j Ω t 0 ∣ Ω ∣ < ω c 0 0 其 它 H(\Omega)=\begin{cases}Ke^{-j\Omega t_0}&|\Omega|<\omega_{c0}\\0&其它\end{cases} H(Ω)={Ke−jΩt00∣Ω∣<ωc0其它
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单位冲激响应 h ( t ) = K ω c 0 π S a [ ω c 0 ( t − t 0 ) ] h(t)=\frac{K\omega_{c0}}{\pi}Sa\left[\omega_{c0}(t-t_0)\right] h(t)=πKωc0Sa[ωc0(t−t0)]
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单位阶跃响应 y ( t ) = K 2 + K π S i [ ω c 0 ( t − t 0 ) ] y(t)=\frac{K}{2}+\frac{K}{\pi}Si\left[\omega_{c0}(t-t_0)\right] y(t)=2K+πKSi[ωc0(t−t0)]
S i ( x ) = ∫ 0 x sin y y d y Si(x)=\int_0^x{\frac{\sin{y}}{y}\mathrm{d}y} Si(x)=∫0xysinydy
调制与解调
连续信号的时域抽样
4 连续时间信号与系统的频域分析
4.1 信号分解
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令 x ( n ) = z n , y ( n ) = z n ∑ k = − ∞ ∞ h ( k ) z − k = H ( s ) ⋅ z n x(n)=z^{n},y(n)=z^{n}\sum\limits_{k=-\infin}^{\infin}{h(k)z^{-k}}=H(s)\cdot z^{n} x(n)=zn,y(n)=znk=−∞∑∞h(k)z−k=H(s)⋅zn
z n z^{n} zn 为特征函数, H ( z ) H(z) H(z) 为特征值
x ( n ) = ∑ k a k z k n → y ( t ) = ∑ k a k H ( z k ) z k n x(n)=\sum\limits_k{a_kz_{k}^n}\rightarrow y(t)=\sum\limits_k{a_kH(z_k)z_k^{n}} x(n)=k∑akzkn→y(t)=k∑akH(zk)zkn
4.2 离散时间周期信号傅立叶级数
x ( n ) = x ( n + N ) x(n)=x(n+N) x(n)=x(n+N)
周期信号 e j 2 π N n e^{j\frac{2\pi}{N}n} ejN2πn
成谐波关系的复指数信号集 ϕ k ( n ) = { e j 2 π N k n } , ϕ k ( n ) = ϕ k + N ( n ) , k = 0 , ± 1 , ⋯ \phi_k(n)=\left\{e^{j\frac{2\pi}{N}kn}\right\},\phi_k(n)=\phi_{k+N}(n),k=0,\pm1,\cdots ϕk(n)={ejN2πkn},ϕk(n)=ϕk+N(n),k=0,±1,⋯
x ( n ) = ∑ k = < N > A k ˙ e j 2 π N k n x(n)=\sum\limits_{k=<N>}^{}{\dot{A_k}e^{j\frac{2\pi}{N}kn}} x(n)=k=<N>∑Ak˙ejN2πkn
A k ˙ = 1 N ∑ n = < N > x ( n ) e − j 2 π N k n \dot{A_k}=\frac{1}{N}\sum\limits_{n=<N>}{x(n)e^{-j\frac{2\pi}{N}kn}} Ak˙=N1n=<N>∑x(n)e−jN2πkn
4.3 傅立叶变换
非周期信号傅立叶变换
ω = 2 π N k \omega=\frac{2\pi}{N}k ω=N2πk
X ( e j ω ) = ∑ n = − ∞ ∞ x ( n ) e − j ω n X(e^{j\omega})=\sum\limits_{n=-\infin}^{\infin}{x(n)e^{-j\omega n}} X(ejω)=n=−∞∑∞x(n)e−jωn
x ( n ) = 1 2 π ∫ 2 π X ( e j ω ) e j ω n d ω x(n)=\frac{1}{2\pi}\int_{2\pi}{X(e^{j\omega})e^{j\omega n}\mathrm{d}\omega} x(n)=2π1∫2πX(ejω)ejωndω
- 收敛条件:平方可和 ∑ n = − ∞ ∞ ∣ x ( n ) ∣ 2 < ∞ \sum\limits_{n=-\infin}^{\infin}{|x(n)|^2}<\infin n=−∞∑∞∣x(n)∣2<∞
常用序列傅立叶变换
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单边指数序列: x ( n ) = a n u ( n ) , ∣ a ∣ < 1 x(n)=a^nu(n),|a|<1 x(n)=anu(n),∣a∣<1
X ( e j ω ) = 1 1 − a e − j ω = ∣ X ( e j ω ) ∣ e j φ ( ω ) X(e^{j\omega})=\frac{1}{1-ae^{-j\omega}}=|X(e^{j\omega})|e^{j\varphi(\omega)} X(ejω)=1−ae−jω1=∣X(ejω)∣ejφ(ω)
- 幅度频谱 ∣ X ( e j ω ) ∣ |X(e^{j\omega})| ∣X(ejω)∣ 偶对称
- 相位频谱 φ ( ω ) \varphi(\omega) φ(ω) 奇对称
- X ( e j ω ) X(e^{j\omega}) X(ejω) 以 2 π 2\pi 2π 为周期
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双边指数序列: x ( n ) = a ∣ n ∣ , ∣ a ∣ < 1 x(n)=a^{|n|},|a|<1 x(n)=a∣n∣,∣a∣<1
X ( e j ω ) = 1 1 − a e − j ω + a e j ω 1 − a e j ω = 1 − a 2 1 − 2 a cos ω + a 2 X(e^{j\omega})=\frac{1}{1-ae^{-j\omega}}+\frac{ae^{j\omega}}{1-ae^{j\omega}}=\frac{1-a^2}{1-2a\cos{\omega}+a^2} X(ejω)=1−ae−jω1+1−aejωaejω=1−2acosω+a21−a2
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单位脉冲序列: x ( n ) = δ ( n ) x(n)=\delta(n) x(n)=δ(n)
X ( e j ω ) = 1 X(e^{j\omega})=1 X(ejω)=1
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常数序列: x ( n ) = 1 x(n)=1 x(n)=1
X ( e j ω ) = 2 π ∑ k = − ∞ ∞ δ ( ω − 2 π k ) X(e^{j\omega})=2\pi\sum\limits_{k=-\infin}^{\infin}{\delta(\omega-2\pi k)} X(ejω)=2πk=−∞∑∞δ(ω−2πk)
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符号函数序列
X ( e j ω ) = − j sin ω 1 − cos ω X(e^{j\omega})=\frac{-j\sin{\omega}}{1-\cos{\omega}} X(ejω)=1−cosω−jsinω
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单位阶跃函数序列: x ( n ) = u ( n ) x(n)=u(n) x(n)=u(n)
X ( e j ω ) = 1 ( 1 − e − j ω ) + π ∑ k = − ∞ ∞ δ ( ω − 2 π k ) X(e^{j\omega})=\frac{1}{(1-e^{-j\omega})}+\pi\sum\limits_{k=-\infin}^{\infin}{\delta(\omega-2\pi k)} X(ejω)=(1−e−jω)1+πk=−∞∑∞δ(ω−2πk)
周期信号傅立叶变换
X ( e j ω ) = 2 π ∑ k = − ∞ ∞ A k ˙ δ ( ω − 2 π N k ) X(e^{j\omega})=2\pi\sum\limits_{k=-\infin}^{\infin}{\dot{A_k}\delta\left(\omega-\frac{2\pi}{N}k\right)} X(ejω)=2πk=−∞∑∞Ak˙δ(ω−N2πk)
4.4 傅立叶变换性质
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周期性: X ( e j ω ) = X ( e j ( ω + 2 π ) ) X(e^{j\omega})=X(e^{j(\omega+2\pi)}) X(ejω)=X(ej(ω+2π))
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线性特性: x 1 ( n ) ↔ X 1 ( e j ω ) , x 2 ( n ) ↔ X 2 ( e j ω ) x_1(n)\leftrightarrow X_1(e^{j\omega}),x_2(n)\leftrightarrow X_2(e^{j\omega}) x1(n)↔X1(ejω),x2(n)↔X2(ejω), a ⋅ x 1 ( n ) + b ⋅ x 2 ( n ) ↔ a ⋅ X 1 ( e j ω ) + b ⋅ X 2 ( e j ω ) a\cdot x_1(n)+b\cdot x_2(n)\leftrightarrow a\cdot X_1(e^{j\omega})+b\cdot X_2(e^{j\omega}) a⋅x1(n)+b⋅x2(n)↔a⋅X1(ejω)+b⋅X2(ejω)
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共轭对称性: x ∗ ( n ) ↔ X ∗ ( e j ω ) x^{*}(n)\leftrightarrow X^{*}(e^{j\omega}) x∗(n)↔X∗(ejω), X ( e j ω ) ↔ X ∗ ( e − j ω ) X(e^{j\omega})\leftrightarrow X^{*}(e^{-j\omega}) X(ejω)↔X∗(e−jω)
- 实偶函数变换为实偶函数,实奇函数变换为虚奇函数
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时延特性: x ( n − n 0 ) ↔ X ( e j ω ) e − j ω n 0 x(n-n_0)\leftrightarrow X(e^{j\omega})e^{-j\omega n_0} x(n−n0)↔X(ejω)e−jωn0
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频移特性: x ( n ) e j ω 0 n ↔ X ( e j ( ω − ω 0 ) ) x(n)e^{j\omega_0 n}\leftrightarrow X\left(e^{j(\omega-\omega_0)}\right) x(n)ejω0n↔X(ej(ω−ω0))
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尺度变换: x ( k ) ( n ) ↔ X ( e j k ω ) x_{(k)}(n)\leftrightarrow X(e^{jk\omega}) x(k)(n)↔X(ejkω), x ( − n ) ↔ X ( e − j ω ) x(-n)\leftrightarrow X(e^{-j\omega}) x(−n)↔X(e−jω)
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时域差分与求和: x ( n ) − x ( n − 1 ) ↔ ( 1 − e − j ω ) X ( e j ω ) x(n)-x(n-1)\leftrightarrow (1-e^{-j\omega})X(e^{j\omega}) x(n)−x(n−1)↔(1−e−jω)X(ejω)
∑ k = − ∞ n x ( k ) ↔ X ( e j ω ) 1 − e − j ω + π X ( e j 0 ) ∑ k = − ∞ ∞ δ ( ω − 2 π k ) \sum\limits_{k=-\infin}^{n}{x(k)}\leftrightarrow \frac{X(e^{j\omega})}{1-e^{-j\omega}}+\pi X(e^{j0})\sum\limits_{k=-\infin}^{\infin}{\delta(\omega-2\pi k)} k=−∞∑nx(k)↔1−e−jωX(ejω)+πX(ej0)k=−∞∑∞δ(ω−2πk)
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频域微分特性: n x ( n ) ↔ j d X ( e j ω ) d ω nx(n)\leftrightarrow j\frac{\mathrm{d}X(e^{j\omega})}{\mathrm{d}\omega} nx(n)↔jdωdX(ejω)
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时域卷积特性: x ( n ) ∗ h ( n ) ↔ X ( e j ω ) H ( e j ω ) x(n)*h(n)\leftrightarrow X(e^{j\omega})H(e^{j\omega}) x(n)∗h(n)↔X(ejω)H(ejω)
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频域卷积特性: x ( n ) y ( n ) ↔ 1 2 π X ( e j ω ) ⊗ Y ( e j ω ) = 1 2 π ∫ 2 π X ( e j θ ) Y ( e j ( ω − θ ) ) d θ x(n)y(n)\leftrightarrow \frac{1}{2\pi}X(e^{j\omega})\otimes Y(e^{j\omega})=\frac{1}{2\pi}\int_{2\pi}{X(e^{j\theta})Y\left(e^{j(\omega-\theta)}\right)\mathrm{d}\theta} x(n)y(n)↔2π1X(ejω)⊗Y(ejω)=2π1∫2πX(ejθ)Y(ej(ω−θ))dθ称为周期卷积
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对偶特性: X ( e j t ) ↔ x ( − n ) X(e^{jt})\leftrightarrow x(-n) X(ejt)↔x(−n)
4.5 离散时间系统频域分析
∑ k = 0 N a k y ( n − k ) = ∑ k = 0 M b k x ( n − k ) \sum\limits_{k=0}^{N}{a_ky(n-k)}=\sum\limits_{k=0}^{M}{b_kx(n-k)} k=0∑Naky(n−k)=k=0∑Mbkx(n−k)
两边同时傅立叶变换 ∑ k = 0 N a k e − j ω k Y ( e j ω ) = ∑ k = 0 M b k e − j ω k X ( e j ω ) \sum\limits_{k=0}^{N}{a_ke^{-j\omega k}Y(e^{j\omega})}=\sum\limits_{k=0}^{M}{b_ke^{-j\omega k}X(e^{j\omega})} k=0∑Nake−jωkY(ejω)=k=0∑Mbke−jωkX(ejω)
∑ k = 0 M b k e − j ω k \sum\limits_{k=0}^{M}{b_ke^{-j\omega k}} k=0∑Mbke−jωk
4.6 离散傅里叶变换
X ( k ) = ∑ n = 0 N − 1 x ( n ) W N k n X(k)=\sum\limits_{n=0}^{N-1}{x(n)W_N^{kn}} X(k)=n=0∑N−1x(n)WNkn, W N = e − j 2 π N W_N=e^{-j\frac{2\pi}{N}} WN=e−jN2π
性质
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圆周移位
x 1 ( n ) = x ( ( n − n 0 ) ) N R N ( n ) x_1(n)=x((n-n_0))_NR_N(n) x1(n)=x((n−n0))NRN(n)
X 1 ( k ) = W N k n 0 X ( k ) X_1(k)=W_N^{kn_0}X(k) X1(k)=WNkn0X(k)
5. 拉普拉斯变换
x ( t ) = e s t x(t)=e^{st} x(t)=est
y ( t ) = e s t ∫ − ∞ ∞ h ( τ ) e − s t d τ = H ( s ) e s t y(t)=e^{st}\int_{-\infin}^{\infin}{h(\tau)e^{-st}\mathrm{d}\tau}=H(s)e^{st} y(t)=est∫−∞∞h(τ)e−stdτ=H(s)est
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双边拉普拉斯变换
X ( s ) = ∫ − ∞ ∞ x ( t ) e − s t d t = ∫ − ∞ ∞ [ x ( t ) e − σ t ] e − j Ω t d t , s = σ + j ω X(s)=\int_{-\infin}^{\infin}{x(t)e^{-st}\mathrm{d}t}=\int_{-\infin}^{\infin}{\left[x(t) e^{-\sigma t}\right]e^{-j\Omega t}\mathrm{d}t},s=\sigma+j\omega X(s)=∫−∞∞x(t)e−stdt=∫−∞∞[x(t)e−σt]e−jΩtdt,s=σ+jω
L { x ( t ) } = F { x ( t ) e − σ t } \mathscr{L}\left\{x(t)\right\}=\mathscr{F}\left\{x(t)e^{-\sigma t}\right\} L{x(t)}=F{x(t)e−σt}
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双边拉普拉斯反变换
x ( t ) = 1 2 π j ∫ σ − j ∞ σ + j ∞ X ( s ) e s t d s x(t)=\frac{1}{2\pi j}\int_{\sigma-j\infin}^{\sigma+j\infin}{X(s)e^{st}\mathrm{d}s} x(t)=2πj1∫σ−j∞σ+j∞X(s)estds
5.1 收敛域
将 σ \sigma σ 允许的取值范围称为 x ( t ) x(t) x(t) 拉普拉斯变换的收敛域
- 拉普拉斯变换收敛域的几何表示:零极点图
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X ( s ) = E ( s ) D ( s ) X(s)=\frac{E(s)}{D(s)} X(s)=D(s)E(s),零点为 E ( s ) E(s) E(s) 的根 o o o ,极点为 D ( s ) D(s) D(s) 的根 × \times ×
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收敛域由平行于虚轴的带状区域构成;收敛域内不包含任何极点
右边信号,收敛域位于其最右边极点的右边;左边信号,收敛域位于其最左边极点的左边;双边信号,收敛域为一带状区域
如果信号为时限的,并且至少存在一个 s s s 值,使其拉斯变换存在,则收敛域为整个 s s s 平面
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5.2 常用拉普拉斯变换
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t t t 的指数类函数 e a t u ( t ) e^{at}u(t) eatu(t): L [ e a t u ( t ) ] = 1 s − a ( σ > a ) \mathscr{L}\left[e^{at}u(t)\right]=\frac{1}{s-a}(\sigma>a) L[eatu(t)]=s−a1(σ>a)
- L [ cos ( Ω t ) u ( t ) ] = s s 2 + Ω 2 ( σ > 0 ) \mathscr{L}\left[\cos{(\Omega t)}u(t)\right]=\frac{s}{s^2+\Omega^2}(\sigma>0) L[cos(Ωt)u(t)]=s2+Ω2s(σ>0)
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t t t 的幂函数类 t n u ( t ) , n ∈ Z + t^nu(t),n\in \mathbb{Z}^+ tnu(t),n∈Z+: L [ t n u ( t ) ] = n s L [ t n − 1 u ( t ) ] = { L [ t n u ( t ) ] = n ! s n + 1 L [ t u ( t ) ] = 1 s 2 ( σ > 0 ) \mathscr{L}\left[t^nu(t)\right]=\frac{n}{s}\mathscr{L}\left[t^{n-1}u(t)\right]=\begin{cases}\mathscr{L}\left[t^{n}u(t)\right]=\frac{n!}{s^{n+1}}\\\mathscr{L}\left[tu(t)\right]=\frac{1}{s^2}\end{cases}(\sigma>0) L[tnu(t)]=snL[tn−1u(t)]={L[tnu(t)]=sn+1n!L[tu(t)]=s21(σ>0)
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单位冲激函数: L [ δ ( t ) ] = 1 , \mathscr{L}\left[\delta(t)\right]=1, L[δ(t)]=1, 收敛域为整个平面
5.3 双边拉普拉斯变换性质
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线性: a ⋅ x 1 ( t ) + b ⋅ x 2 ( t ) ↔ a ⋅ X 1 ( s ) + b ⋅ X 2 ( s ) , R 1 ∩ R 2 ∈ R O C a\cdot x_1(t)+b\cdot x_2(t)\leftrightarrow a\cdot X_1(s)+b\cdot X_2(s),R_1\cap R_2\in \mathrm{ROC} a⋅x1(t)+b⋅x2(t)↔a⋅X1(s)+b⋅X2(s),R1∩R2∈ROC
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时域平移: x ( t − t 0 ) ↔ X ( s ) e − s t 0 , x(t-t_0)\leftrightarrow X(s)e^{-st_0}, x(t−t0)↔X(s)e−st0, 收敛域不变
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复频域平移: x ( t ) e s 0 t ↔ X ( s − s 0 ) , x(t)e^{s_0t}\leftrightarrow X(s-s_0), x(t)es0t↔X(s−s0), 收敛域右移 R e { s 0 } \mathrm{Re}\left\{s_0\right\} Re{s0}
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尺度变换: x ( a t ) ↔ 1 ∣ a ∣ X ( s a ) , R 1 = a R x(at)\leftrightarrow\frac{1}{|a|}X\left(\frac{s}{a}\right),R_1=aR x(at)↔∣a∣1X(as),R1=aR
- x ( − t ) ↔ X ( − s ) , R 1 = − R x(-t)\leftrightarrow X(-s),R_1=-R x(−t)↔X(−s),R1=−R
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卷积定理
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时域卷积: x 1 ( t ) ∗ x 2 ( t ) ↔ X 1 ( s ) ⋅ X 2 ( s ) , R 1 ∩ R 2 ∈ R O C x_1(t)*x_2(t)\leftrightarrow X_1(s)\cdot X_2(s),R_1\cap R_2\in \mathrm{ROC} x1(t)∗x2(t)↔X1(s)⋅X2(s),R1∩R2∈ROC
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复频域卷积: x 1 ( t ) ⋅ x 2 ( t ) ↔ 1 2 π j [ X 1 ( s ) ∗ X 2 ( s ) ] x_1(t)\cdot x_2(t)\leftrightarrow \frac{1}{2\pi j}\left[X_1(s)*X_2(s) \right] x1(t)⋅x2(t)↔2πj1[X1(s)∗X2(s)]
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时域微分: x ′ ( t ) ↔ s X ( s ) x'(t)\leftrightarrow sX(s) x′(t)↔sX(s)
x ( n ) ( t ) ↔ s n X ( s ) , R ∈ R O C , x^{(n)}(t)\leftrightarrow s^nX(s),R\in \mathrm{ROC}, x(n)(t)↔snX(s),R∈ROC, 收敛域可能放大
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时域积分: ∫ − ∞ t x ( τ ) d τ ↔ X ( s ) s , \int_{-\infin}^{t}{x(\tau)\mathrm{d}\tau}\leftrightarrow \frac{X(s)}{s}, ∫−∞tx(τ)dτ↔sX(s), 收敛域为 R ∩ ( σ > 0 ) R\cap(\sigma>0) R∩(σ>0) 或 R R R( R R R 在 s = 0 s=0 s=0 处有 0 0 0 点)
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复频域微分: t x ( t ) ↔ − X ′ ( s ) , tx(t)\leftrightarrow -X'(s), tx(t)↔−X′(s), 收敛域不变
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复频域积分: x ( t ) t ↔ ∫ s ∞ X ( s ) d s , \frac{x(t)}{t}\leftrightarrow \int_{s}^{\infin}{X(s)\mathrm{d}s}, tx(t)↔∫s∞X(s)ds, 收敛域不变
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初值定理: x ( 0 + ) = lim t → 0 + x ( t ) = lim s → ∞ s X ( s ) x(0^+)=\lim\limits_{t\rightarrow 0^+}{x(t)}=\lim\limits_{s\rightarrow\infin}{sX(s)} x(0+)=t→0+limx(t)=s→∞limsX(s)
- 若极限不存在,则 X ( s ) = a 0 + a 1 s + ⋯ + a p s p + X p ( s ) X(s)=a_0+a_1s+\cdots+a_ps^p+X_p(s) X(s)=a0+a1s+⋯+apsp+Xp(s), x ( 0 + ) = lim s → ∞ s X p ( s ) x(0^+)=\lim\limits_{s\rightarrow\infin}{sX_p(s)} x(0+)=s→∞limsXp(s)
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终值定理
设右边函数 x ( t ) x(t) x(t) 及其导数存在并有拉普拉斯变换且的所有极点都位于 S S S 平面的左半边(包括在原点处的单极点),则 x ( ∞ ) = lim t → ∞ x ( t ) = lim s → 0 s X ( s ) x(\infin)=\lim\limits_{t\rightarrow \infin}{x(t)}=\lim\limits_{s\rightarrow 0}{sX(s)} x(∞)=t→∞limx(t)=s→0limsX(s)
- 如果有极点落在 S S S 平面右半边,则 x ( t ) → ∞ x(t)\rightarrow \infin x(t)→∞
- 如果有极点落在虚轴上,则 x ( t ) → x(t)\rightarrow x(t)→ 等幅振荡
- 如果原点处极点为重极点,则 x ( t ) → x(t)\rightarrow x(t)→ 随时间增长的函数
5.4 拉普拉斯反变换
x ( t ) = 1 2 π j ∫ σ − j ∞ σ + j ∞ X ( s ) e s t d s x(t)=\frac{1}{2\pi j}\int_{\sigma-j\infin}^{\sigma+j\infin}{X(s)e^{st}\mathrm{d}s} x(t)=2πj1∫σ−j∞σ+j∞X(s)estds
X ( s ) = N ( s ) D ( s ) = b m s m + ⋯ + b 1 s + b 0 s n + a n − 1 s n − 1 + ⋯ + a 1 s + a 0 X(s)=\frac{N(s)}{D(s)}=\frac{b_ms^m+\cdots+b_1s+b_0}{s^n+a_{n-1}s^{n-1}+\cdots+a_1s+a_0} X(s)=D(s)N(s)=sn+an−1sn−1+⋯+a1s+a0bmsm+⋯+b1s+b0
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m > n m>n m>n
X ( s ) = X(s)= X(s)= 多项式 + 有理真分式
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m < n m<n m<n 且 D ( s ) = 0 D(s)=0 D(s)=0 无重根
D ( s ) = ( s − s 1 ) ⋯ ( s − s n ) D(s)=(s-s_1)\cdots(s-s_n) D(s)=(s−s1)⋯(s−sn)
X ( s ) = K 1 s − s 1 + ⋯ + K n s − s n X(s)=\frac{K_1}{s-s_1}+\cdots+\frac{K_n}{s-s_n} X(s)=s−s1K1+⋯+s−snKn
K k = [ ( s − s k ) N ( s ) D ( s ) ] s = s k K_k=\left[(s-s_k)\frac{N(s)}{D(s)}\right]_{s=s_k} Kk=[(s−sk)D(s)N(s)]s=sk
K k s − s k ↔ { K k e s k t u ( t ) − K k e s k t u ( − t ) \frac{K_k}{s-s_k}\leftrightarrow\begin{cases}K_ke^{s_kt}u(t)\\-K_ke^{s_kt}u(-t) \end{cases} s−skKk↔{Kkesktu(t)−Kkesktu(−t)
- 极点位于收敛域左边或左边界:右边函数
- 极点位于收敛域右边或右边界:左边函数
- 极点位于收敛域两边或外边界:双边函数
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m < n m<n m<n 且 D ( s ) = 0 D(s)=0 D(s)=0 有重根
设 D ( s ) = 0 D(s)=0 D(s)=0 有 p p p 重根, D ( s ) = ( s − s 1 ) p ( s − s p + 1 ) ⋯ ( s − s n ) D(s)=(s-s_1)^p(s-s_{p+1})\cdots(s-s_n) D(s)=(s−s1)p(s−sp+1)⋯(s−sn)
X ( s ) = K 1 p ( s − s 1 ) p + K 1 ( p − 1 ) ( s − s 1 ) p − 1 + ⋯ + K 11 s − s 1 + K p + 1 s − s p + 1 + ⋯ + K n s − s n X(s)=\frac{K_{1p}}{(s-s_{1})^{p}}+\frac{K_{1(p-1)}}{(s-s_{1})^{p-1}}+\cdots+\frac{K_{11}}{s-s_{1}}+\frac{K_{p+1}}{s-s_{p+1}}+\cdots+\frac{K_{n}}{s-s_{n}} X(s)=(s−s1)pK1p+(s−s1)p−1K1(p−1)+⋯+s−s1K11+s−sp+1Kp+1+⋯+s−snKn
K 1 p = [ ( s − s 1 ) p N ( s ) D ( s ) ] s = s 1 K_{1p}=\left[(s-s_1)^p\frac{N(s)}{D(s)}\right]_{s=s_1} K1p=[(s−s1)pD(s)N(s)]s=s1
K 1 k = 1 ( p − k ) ! d p − k d s p − k [ ( s − s 1 ) p N ( s ) D ( s ) ] s = s 1 K_{1k}=\frac{1}{(p-k)!}\frac{\mathrm{d}^{p-k}}{\mathrm{d}s^{p-k}}\left[(s-s_1)^p\frac{N(s)}{D(s)}\right]_{s=s_1} K1k=(p−k)!1dsp−kdp−k[(s−s1)pD(s)N(s)]s=s1
L − 1 [ X ( s ) ] = [ K 1 p ( p − 1 ) ! t p − 1 + K 1 ( p − 1 ) ( p − 2 ) ! t p − 2 + ⋯ + K 12 t + K 11 ] e s 1 t u ( t ) + ∑ q = p + 1 n K k e s q t u ( t ) \mathscr{L^{-1}}\left[X(s)\right]=\left[\frac{K_{1p}}{(p-1)!}t^{p-1}+\frac{K_{1(p-1)}}{(p-2)!}t^{p-2}+\cdots+K_{12}t+K_{11} \right]e^{s_1t}u(t)+\sum\limits_{q=p+1}^{n}{K_ke^{s_qt}u(t)} L−1[X(s)]=[(p−1)!K1ptp−1+(p−2)!K1(p−1)tp−2+⋯+K12t+K11]es1tu(t)+q=p+1∑nKkesqtu(t)
5.5 连续时间系统复频域分析方法
- 将激励信号分解为 e s t e^{st} est 形式的指数分量(求拉氏变换) x ( t ) → X ( s ) x(t)\rightarrow X(s) x(t)→X(s)
- 确定复频域的系统函数 H ( s ) H(s) H(s)
- 求取每一分量的响应 Y ( s ) = X ( s ) ⋅ H ( s ) Y(s)=X(s)\cdot H(s) Y(s)=X(s)⋅H(s)
- 对响应复频谱函数求拉氏反变换得到系统的响应函数 Y ( s ) → y ( t ) Y(s)\rightarrow y(t) Y(s)→y(t)
∑ k = 0 N a k s k Y ( s ) = ∑ k = 0 M b k s k X ( s ) \sum\limits_{k=0}^{N}{a_ks^kY(s)}=\sum\limits_{k=0}^{M}{b_ks^kX(s)} k=0∑NakskY(s)=k=0∑MbkskX(s)
H ( s ) = ∑ k = 0 M b k s k ∑ k = 0 N a k s k = b M a N ∏ k = 1 M ( s − z k ) ∏ k = 1 N ( s − p k ) H(s)=\frac{\sum\limits_{k=0}^{M}{b_ks^k}}{\sum\limits_{k=0}^{N}{a_ks^k}}=\frac{b_M}{a_N}\frac{\prod\limits_{k=1}^{M}{(s-z_k)}}{\prod\limits_{k=1}^{N}{(s-p_k)}} H(s)=k=0∑Nakskk=0∑Mbksk=aNbMk=1∏N(s−pk)k=1∏M(s−zk), z k z_k zk 为零点, p k p_k pk 为极点
因果且稳定的 LTI 系统,系统函数的收敛域一定包含虚轴,且系统函数的全部极点一定位于 S S S 平面的左半平面
5.6 单边拉普拉斯变换
X ( s ) = ∫ 0 ∞ x ( t ) e − s t d t \mathscr{X}(s)=\int_{0}^{\infin}{x(t)e^{-st}\mathrm{d}t} X(s)=∫0∞x(t)e−stdt
存在冲激函数及其导数时, X ( s ) = ∫ 0 − ∞ x ( t ) e − s t d t \mathscr{X}(s)=\int_{0^-}^{\infin}{x(t)e^{-st}\mathrm{d}t} X(s)=∫0−∞x(t)e−stdt
反变换 x ( t ) u ( t ) = [ 1 2 π j ∫ σ − j ∞ σ + j ∞ X ( s ) e s t d t ] u ( t ) x(t)u(t)=\left[\frac{1}{2\pi j}\int_{\sigma-j\infin}^{\sigma+j\infin}{X(s)e^{st}\mathrm{d}t} \right]u(t) x(t)u(t)=[2πj1∫σ−j∞σ+j∞X(s)estdt]u(t)
- 右边信号:单边拉普拉斯变换与双边拉普拉斯变换相同
双边信号:单边拉普拉斯变换与双边拉普拉斯变换不同
性质
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时域微分: x ′ ( t ) ↔ s X ( s ) − x ( 0 − ) x'(t)\leftrightarrow s\mathscr{X}(s)-x(0^-) x′(t)↔sX(s)−x(0−)
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时域积分: ∫ − ∞ t x ( τ ) d τ ↔ 1 s X ( s ) + ∫ − ∞ 0 − x ( τ ) d τ s \int_{-\infin}^{t}{x(\tau)\mathrm{d}\tau}\leftrightarrow \frac{1}{s}\mathscr{X}(s)+\frac{\int_{-\infin}^{0^-}{x(\tau)\mathrm{d}\tau}}{s} ∫−∞tx(τ)dτ↔s1X(s)+s∫−∞0−x(τ)dτ
6. Z \mathscr{Z} Z变换
x ( n ) = z n x(n)=z^n x(n)=zn
y ( n ) = z n ∑ k = − ∞ ∞ h ( k ) z − k = z n H ( z ) y(n)=z^n\sum\limits_{k=-\infin}^{\infin}{h(k)z^{-k}}=z^nH(z) y(n)=znk=−∞∑∞h(k)z−k=znH(z)
6.1 Z \mathscr{Z} Z变换
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双边 Z \mathscr{Z} Z变换 Z [ x ( n ) ] = X ( z ) = ∑ n = − ∞ ∞ x ( n ) z − n \mathscr{Z}[x(n)]=X(z)=\sum\limits_{n=-\infin}^{\infin}{x(n)z^{-n}} Z[x(n)]=X(z)=n=−∞∑∞x(n)z−n
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单边 Z \mathscr{Z} Z变换 Z [ x ( n ) u ( n ) ] = X ( z ) = ∑ n = 0 ∞ x ( n ) z − n \mathscr{Z}[x(n)u(n)]=\mathscr{X}(z)=\sum\limits_{n=0}^{\infin}{x(n)z^{-n}} Z[x(n)u(n)]=X(z)=n=0∑∞x(n)z−n
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收敛域
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有限长序列 x ( n ) ( n 1 ≤ n ≤ n 2 ) x(n)(n_1\leq n\leq n_2) x(n)(n1≤n≤n2)
- n 2 > n 1 ≥ 0 n_2>n_1\geq 0 n2>n1≥0 或 n 2 ≥ n 1 > 0 ⇒ o < ∣ z ∣ ≤ ∞ n_2\geq n_1>0\Rightarrow o<|z|\leq\infin n2≥n1>0⇒o<∣z∣≤∞
- n 2 > 0 , n 1 < 0 ⇒ 0 < ∣ z ∣ < ∞ n_2>0,n_1<0\Rightarrow 0<|z|<\infin n2>0,n1<0⇒0<∣z∣<∞
- 0 ≥ n 2 > n 1 0\geq n_2>n_1 0≥n2>n1 或 1 > n 2 ≥ n 1 ⇒ 0 ≤ ∣ z ∣ < ∞ 1>n_2\geq n_1\Rightarrow 0\leq|z|<\infin 1>n2≥n1⇒0≤∣z∣<∞
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右边序列(因果序列): R r < ∣ z ∣ ≤ ∞ R_r<|z|\leq\infin Rr<∣z∣≤∞
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左边序列(反因果序列): ∣ z ∣ < R l |z|<R_l ∣z∣<Rl
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双边序列:若 R l > R r R_l>R_r Rl>Rr, R r < ∣ z ∣ < R l R_r<|z|<R_l Rr<∣z∣<Rl;若 R l > R r R_l>R_r Rl>Rr,没有收敛,没有 Z \mathscr{Z} Z变换
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Z \mathscr{Z} Z变换和拉普拉斯变换关系: z − e s T z-e^{sT} z−esT
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Z \mathscr{Z} Z变换和离散时间傅立叶变换: z = r e j ω z=re^{j\omega} z=rejω
6.2 常用 Z \mathscr{Z} Z变换
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单位冲激函数: δ ( n ) ↔ 1 ( o ≤ ∣ z ∣ ≤ ∞ ) \delta(n)\leftrightarrow1(o\leq|z|\leq\infin) δ(n)↔1(o≤∣z∣≤∞), Z [ δ ( n ) ] = 1 \mathscr{Z}[\delta(n)]=1 Z[δ(n)]=1
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单位阶跃序列: u ( n ) ↔ z z − 1 ( ∣ z ∣ > 1 ) 右 边 序 列 u(n)\leftrightarrow \frac{z}{z-1}(|z|>1)右边序列 u(n)↔z−1z(∣z∣>1)右边序列, Z [ u ( n ) ] = 1 1 − z − 1 \mathscr{Z}[u(n)]=\frac{1}{1-z^{-1}} Z[u(n)]=1−z−11
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单边指数序列: a n u ( n ) ↔ z z − a ( ∣ z ∣ > ∣ a ∣ ) 右 边 序 列 a^nu(n)\leftrightarrow \frac{z}{z-a}(|z|>|a|)右边序列 anu(n)↔z−az(∣z∣>∣a∣)右边序列, Z [ a n u ( n ) ] = z z − a , ∣ a z − 1 ∣ < 1 , ∣ z ∣ > a \mathscr{Z}[a^nu(n)]=\frac{z}{z-a},|az^{-1}|<1,|z|>a Z[anu(n)]=z−az,∣az−1∣<1,∣z∣>a
6.3 双边 Z \mathscr{Z} Z变换常用性质
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时域平移: x ( n − n 0 ) ↔ z − n 0 X ( z ) x(n-n_0)\leftrightarrow z^{-n_0}X(z) x(n−n0)↔z−n0X(z), R R R 在原点或无穷远处可能发生变化
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线性特征: a 1 x 1 ( n ) + a 2 x 2 ( n ) ↔ a 1 X 1 ( z ) + a 2 X 2 ( z ) , R 1 ∩ R 2 ∈ R a_1x_1(n)+a_2x_2(n)\leftrightarrow a_1X_1(z)+a_2X_2(z),R_1\cap R_2\in R a1x1(n)+a2x2(n)↔a1X1(z)+a2X2(z),R1∩R2∈R
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移频特性: e j ω 0 n x ( n ) ↔ X ( z e − j ω 0 ) , e^{j\omega_0 n}x(n)\leftrightarrow X(ze^{-j\omega_0}), ejω0nx(n)↔X(ze−jω0), 收敛域不变
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Z \mathscr{Z} Z域尺度变换特性: z 0 n x ( n ) ↔ X ( z z 0 ) , ∣ z 0 ∣ R z_0^nx(n)\leftrightarrow X\left(\frac{z}{z_0}\right),|z_0|R z0nx(n)↔X(z0z),∣z0∣R, z 0 = r 0 e j ω 0 z_0=r_0e^{j\omega_0} z0=r0ejω0
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时域反转特性: x ( − n ) ↔ X ( z − 1 ) , 1 R x(-n)\leftrightarrow X(z^{-1}),\frac{1}{R} x(−n)↔X(z−1),R1
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卷积定理: x 1 ( n ) ∗ x 2 ( n ) ↔ X 1 ( z ) ⋅ X 2 ( z ) x_1(n)*x_2(n)\leftrightarrow X_1(z)\cdot X_2(z) x1(n)∗x2(n)↔X1(z)⋅X2(z)
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Z \mathscr{Z} Z域微分特性: n x ( n ) ↔ − z X ′ ( z ) , R nx(n)\leftrightarrow -zX'(z),R nx(n)↔−zX′(z),R 不变
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时域求和性质: ∑ k = − ∞ n x ( k ) ↔ z z − 1 X ( z ) , R ∩ ( ∣ z ∣ > 1 ) \sum\limits_{k=-\infin}^{n}{x(k)}\leftrightarrow \frac{z}{z-1}X(z),R\cap(|z|>1) k=−∞∑nx(k)↔z−1zX(z),R∩(∣z∣>1)
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初值定理: x ( 0 ) = lim z → ∞ X ( z ) x(0)=\lim\limits_{z\rightarrow\infin}{X(z)} x(0)=z→∞limX(z)
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终值定理:除了单位圆上允许有一阶极点之外,其余极点都在单位圆之内
x ( ∞ ) = lim z → 1 [ ( z − 1 ) X ( z ) ] x(\infin)=\lim\limits_{z\rightarrow 1}{[(z-1)X(z)]} x(∞)=z→1lim[(z−1)X(z)]
6.4 Z \mathscr{Z} Z反变换
x ( n ) = 1 2 π j ∮ C X ( z ) z n − 1 d z x(n)=\frac{1}{2\pi j}\oint_{C}{X(z)z^{n-1}\mathrm{d}z} x(n)=2πj1∮CX(z)zn−1dz, C C C 是在收敛域内包围z平面原点的闭合积分路线
幂级数展开法
X ( z ) = ∑ n = − ∞ ∞ x ( n ) z − n , z = r e j ω X(z)=\sum\limits_{n=-\infin}^{\infin}{x(n)z^{-n}},z=re^{j\omega} X(z)=n=−∞∑∞x(n)z−n,z=rejω
X ( z ) = N ( z ) D ( z ) = ⋯ + x ( − 1 ) z + x ( 0 ) + x ( 1 ) z − 1 + x ( 2 ) z − 2 + ⋯ + x ( n ) z − n + ⋯ X(z)=\frac{N(z)}{D(z)}=\cdots+x(-1)z+x(0)+x(1)z^{-1}+x(2)z^{-2}+\cdots+x(n)z^{-n}+\cdots X(z)=D(z)N(z)=⋯+x(−1)z+x(0)+x(1)z−1+x(2)z−2+⋯+x(n)z−n+⋯
展开方法(长除法):对右边的序列按 z z z 的降幂的顺序排列;对左边的序列按 z z z 的升幂的顺序排列
部分式展开法
6.5 离散时间LTI的 Z \mathscr{Z} Z域分析方法
∑ k = 0 n a k y ( n − k ) = ∑ k = 0 m b k x ( n − k ) \sum\limits_{k=0}^{n}{a_ky(n-k)}=\sum\limits_{k=0}^{m}{b_kx(n-k)} k=0∑naky(n−k)=k=0∑mbkx(n−k), H ( z ) = ∑ k = 0 M b k z − k ∑ k = 0 N a k z − k H(z)=\frac{\sum\limits_{k=0}^{M}{b_kz^{-k}}}{\sum\limits_{k=0}^{N}{a_kz^{-k}}} H(z)=k=0∑Nakz−kk=0∑Mbkz−k