本课程主要研究电信号随时间变化
连续时间信号:信号在连续时间内都有定义(只要求定义域连续)
离散信号:仅在一些离散的瞬间才有定义的信号
能量信号:能量有界,功率一定为0,离散变求和
功率信号:功率有界,E=∞,离散变求和
典型确定性信号:直流,单位斜坡,指数,正余弦,复指数,抽样
阶跃函数、冲激函数、冲击偶
ε ( t ) = ∫ − ∞ t δ ( τ ) d τ δ ( t ) = d ε ( t ) d t \varepsilon(t)=\int_{-\infty}^{t} \delta(\tau) d \tau \quad \delta(t)=\frac{d \varepsilon(t)}{d t} ε(t)=∫−∞tδ(τ)dτδ(t)=dtdε(t)
冲激函数的取样性
δ ( t ) f ( t ) = δ ( t ) f ( 0 ) ∫ − ∞ + ∞ δ ( t ) f ( t ) d t = f ( 0 ) \begin{array}{l} \delta(t) f(t)=\delta(t) f(0) \\ \int_{-\infty}^{+\infty} \delta(t) f(t) d t=f(0) \end{array} δ(t)f(t)=δ(t)f(0)∫−∞+∞δ(t)f(t)dt=f(0)
冲激偶(冲激函数的导数)
三个重要的性质
f ( t ) δ ′ ( t ) = f ( 0 ) δ ′ ( t ) − f ′ ( 0 ) δ ( t ) f(t) \delta^{\prime}(t)=f(0) \delta^{\prime}(t)-f^{\prime}(0) \delta(t) f(t)δ′(t)=f(0)δ′(t)−f′(0)δ(t)
∫ − ∞ ∞ δ ′ ( t ) f ( t ) d t = − f ′ ( 0 ) \int_{-\infty}^{\infty} \delta^{\prime}(t) f(t) d t=-f^{\prime}(0) ∫−∞∞δ′(t)f(t)dt=−f′(0)
∫ − ∞ t δ ′ ( t ) d t = δ ( t ) \int_{-\infty}^{t} \delta^{\prime}(t) d t=\delta(t) ∫−∞tδ′(t)dt=δ(t)
比例性 δ ( a t ) = 1 ∣ a ∣ δ ( t ) \delta(a t)=\frac{1}{|a|} \delta(t) δ(at)=∣a∣1δ(t)
1). x ( t ) = { e − t , t ≥ 0 0 , t < 0 x(\mathrm{t})=\left\{\begin{array}{ll}e^{-t}, & t \geq 0 \\ 0, & t<0\end{array}\right. x(t)={e−t,0,t≥0t<0
2). x ( n ) = { 1 , n ≥ 0 0 , n < 0 x(\mathrm{n})=\left\{\begin{array}{ll}1, & n \geq 0 \\ 0, & n<0\end{array}\right. x(n)={1,0,n≥0n<0
1). E = ∫ − ∞ ∞ ∣ x ( t ) ∣ 2 d t = ∫ 0 ∞ e − 2 t d t = 1 2 \quad E\\ =\int_{-\infty}^{\infty}|x(\mathrm{t})|^{2} d t\\ =\int_{0}^{\infty} e^{-2 t} d t\\ =\frac{1}{2} E=∫−∞∞∣x(t)∣2dt=∫0∞e−2tdt=21
能量信号
2). P = lim N → ∞ 1 2 N + 1 ∑ n = − N N ∣ x ( n ) ∣ 2 = lim N → ∞ 1 2 N + 1 ∑ n = 0 N 1 = lim N → ∞ N + 1 2 N + 1 = 1 2 \quad P\\ =\lim \limits_{N \rightarrow \infty} \frac{1}{2 N+1} \sum\limits_{n=-N}^{N}|x(\mathrm{n})|^{2}\\ =\lim \limits_{N \rightarrow \infty} \frac{1}{2 N+1} \sum\limits_{n=0}^{N} 1\\ =\lim \limits_{N \rightarrow \infty} \frac{N+1}{2 N+1}\\ =\frac{1}{2} \quad P=N→∞lim2N+11n=−N∑N∣x(n)∣2=N→∞lim2N+11n=0∑N1=N→∞lim2N+1N+1=21
功率信号
根据方程画框图
①左侧写激励,右侧写响应
②根据最高次数中间补对应数量的积分器,左侧补加法器
u C ′ ′ ( t ) + R L u C ′ ( t ) + 1 L C u C ( t ) = 1 L C u S ( t ) u_{C}^{\prime \prime}(t)+\frac{R}{L} u_{C}^{\prime}(t)+\frac{1}{L C} u_{C}(t)=\frac{1}{L C} u_{S}(t) uC′′(t)+LRuC′(t)+LC1uC(t)=LC1uS(t)
y ′ ′ ( t ) + a y ′ ( t ) + b y ( t ) = c f ( t ) y^{\prime \prime}(t)+a y^{\prime}(t)+b y(t)=c f(t) y′′(t)+ay′(t)+by(t)=cf(t)
a = R L b = 1 L C c = 1 L C a=\frac{R}{L} \quad b=\frac{1}{L C} \quad c=\frac{1}{L C} a=LRb=LC1c=LC1
y ′ ′ ( t ) = − a y ′ ( t ) − b y ( t ) + c f ( t ) y^{\prime \prime}(t)=-a y^{\prime}(t)-b y(t)+c f(t) y′′(t)=−ay′(t)−by(t)+cf(t)
根据框图画方程
①设置中间变量,左边高次,右边低次
②列写两个加法器
左边加法器
x ′ ′ ( t ) = − a 2 x ′ ( t ) − a 1 x ( t ) + f ( t ) x^{\prime \prime}(t)=-a_{2} x^{\prime}(t)-a_{1} x(t)+f(t) x′′(t)=−a2x′(t)−a1x(t)+f(t)
x ′ ′ ( t ) + a 2 x ′ ( t ) + a 1 x ( t ) = f ( t ) x^{\prime \prime}(t)+a_{2} x^{\prime}(t)+a_{1} x(t)=f(t) x′′(t)+a2x′(t)+a1x(t)=f(t)
右边加法器
y ( t ) = b 2 x ′ ′ ( t ) + b 1 x ′ ( t ) + x ( t ) a 1 y ( t ) = b 2 a 1 x ′ ′ ( t ) + b 1 a 1 x ′ ( t ) + a 1 x ( t ) a 2 y ′ ( t ) = b 2 ( a 2 x ′ ′ ( t ) ) ′ + b 1 ( a 2 x ′ ( t ) ) ′ + a 2 x ′ ( t ) y ′ ′ ( t ) = b 2 ( x ′ ′ ( t ) ) ′ ′ + b 1 ( x ′ ( t ) ) ′ ′ + x ′ ′ ( t ) y(t)=b_{2} x^{\prime \prime}(t)+b_{1} x^{\prime}(t)+x(t)\\ a_{1} y(t)=b_{2} a_{1} x^{\prime \prime}(t)+b_{1} a_{1} x^{\prime}(t)+a_{1} x(t) \\ a_{2} y^{\prime}(t) =b_{2}\left(a_{2} x^{\prime \prime}(t)\right)^{\prime}+b_{1}\left(a_{2} x^{\prime}(t)\right)^{\prime}+a_{2} x^{\prime}(t) \\ y^{\prime \prime}(t) =b_{2}\left(x^{\prime \prime}(t)\right)^{\prime \prime}+b_{1}\left(x^{\prime}(t)\right)^{\prime \prime}+x^{\prime \prime}(t) y(t)=b2x′′(t)+b1x′(t)+x(t)a1y(t)=b2a1x′′(t)+b1a1x′(t)+a1x(t)a2y′(t)=b2(a2x′′(t))′+b1(a2x′(t))′+a2x′(t)y′′(t)=b2(x′′(t))′′+b1(x′(t))′′+x′′(t)
y ′ ′ ( t ) + a 2 y ′ ( t ) + a 1 y ( t ) = b 2 ( x ′ ′ ( t ) + a 2 x ′ ( t ) + a 1 x ( t ) ) ′ ′ + b 1 ( x ′ ′ ( t ) + a 2 x ′ ( t ) + a 1 x ( t ) ) ′ + x ′ ′ ( t ) + a 2 x ′ ( t ) + a 1 x ( t ) = b 2 f ′ ′ ( t ) + b 1 f ′ ( t ) + f ( t ) y^{\prime \prime}(t)+a_{2} y^{\prime}(t)+a_{1} y(t) \\ =b_{2}(x^{\prime \prime}(t)+a_{2} x^{\prime}(t)+a_{1} x(t))^{\prime \prime}+b_{1}\left(x^{\prime \prime}(t)+a_{2} x^{\prime}(t)+a_{1} x(t)\right)^{\prime}+x^{\prime \prime}(t)+a_{2} x^{\prime}(t)+a_{1} x(t)\\ =b_{2} f^{\prime \prime}(t)+b_{1} f^{\prime}(t)+f(t) y′′(t)+a2y′(t)+a1y(t)=b2(x′′(t)+a2x′(t)+a1x(t))′′+b1(x′′(t)+a2x′(t)+a1x(t))′+x′′(t)+a2x′(t)+a1x(t)=b2f′′(t)+b1f′(t)+f(t)
最终结果 y ′ ′ ( t ) + a 2 y ′ ( t ) + a 1 y ( t ) = b 2 f ′ ′ ( t ) + b 1 f ′ ( t ) + f ( t ) y^{\prime \prime}(t)+a_{2} y^{\prime}(t)+a_{1} y(t) =b_{2} f^{\prime \prime}(t)+b_{1} f^{\prime}(t)+f(t) y′′(t)+a2y′(t)+a1y(t)=b2f′′(t)+b1f′(t)+f(t)
y ( k ) − ( 1 + a − b ) y ( k − 1 ) = f ( k ) y(k)-(1+a-b) y(k-1)=f(k) y(k)−(1+a−b)y(k−1)=f(k)
左边加法器
x ( k ) = f ( k ) − a 2 x ( k − 1 ) − a 1 x ( k − 2 ) x(k)=f(k)-a_{2} x(k-1)-a_{1} x(k-2) x(k)=f(k)−a2x(k−1)−a1x(k−2)
f ( k ) = x ( k ) + a 2 x ( k − 1 ) + a 1 x ( k − 2 ) f(k)=x(k)+a_{2} x(k-1)+a_{1} x(k-2) f(k)=x(k)+a2x(k−1)+a1x(k−2)
右边加法器
y ( k ) = b 1 x ( k ) + x ( k − 2 ) \mathrm{y}(\mathrm{k})=b_{1} \mathrm{x}(\mathrm{k})+\mathrm{x}(\mathrm{k}-2) y(k)=b1x(k)+x(k−2)
a 2 y ( k − 1 ) = b 1 a 2 x ( k − 1 ) + a 2 x ( k − 3 ) a_{2} y(k-1)=b_{1} a_{2} x(k-1)+a_{2} x(k-3) a2y(k−1)=b1a2x(k−1)+a2x(k−3)
a 1 y ( k − 2 ) = b 1 a 1 x ( k − 2 ) + a 1 x ( k − 4 ) a_{1} y(k-2)=b_{1} a_{1} x(k-2)+a_{1} x(k-4) a1y(k−2)=b1a1x(k−2)+a1x(k−4)
y ( k ) + a 2 y ( k − 1 ) + a 1 y ( k − 2 ) = b 1 [ x ( k ) + a 2 x ( k − 1 ) + a 1 x ( k − 2 ) ] + [ x ( k − 2 ) + a 2 x ( k − 3 ) + a 1 x ( k − 4 ) ] = b 1 f ( k ) + f ( k − 2 ) y(k)+a_{2} y(k-1)+a_{1} y(k-2)\\ =b_{1}\left[x(k)+a_{2} x(k-1)+a_{1} x(k-2)\right]+\left[x(k-2)+a_{2} x(k-3)+a_{1} x(k-4)\right]\\ =b_{1} f(k)+f(k-2) y(k)+a2y(k−1)+a1y(k−2)=b1[x(k)+a2x(k−1)+a1x(k−2)]+[x(k−2)+a2x(k−3)+a1x(k−4)]=b1f(k)+f(k−2)
y ( k ) + a 2 y ( k − 1 ) + a 1 y ( k − 2 ) = b 1 f ( k ) + f ( k − 2 ) y(k)+a_{2} y(k-1)+a_{1} y(k-2)=b_{1} f(k)+f(k-2) y(k)+a2y(k−1)+a1y(k−2)=b1f(k)+f(k−2)
1.判断分解特性(写出两种输入相加)
2.判断零状态响应,零输入响应线性特征
( 1 ) y ( t ) = f ( t ) + 3 x ( 0 ) f ( t ) + 6 x ( 0 ) + 5 ( 2 ) y ( t ) = 5 ∣ f ( t ) ∣ + 6 x ( 0 ) \begin{array}{ll}(1) y(t)=f(t)+3 x(0) f(t)+6 x(0)+5 & (2) y(t)=5|f(t)|+6 x(0)\end{array} (1)y(t)=f(t)+3x(0)f(t)+6x(0)+5(2)y(t)=5∣f(t)∣+6x(0)
(3) y ( k ) = 7 f ( k ) + 2 x ( 0 ) 2 y(k)=7 f(k)+2 x(0)^{2} y(k)=7f(k)+2x(0)2
(1) y z i ( t ) = 6 x ( 0 ) + 5 y z s ( t ) = f ( t ) + 5 y ( t ) ≠ y z i ( t ) + y z s ( t ) y_{z i}(t)=6 x(0)+5 \quad y_{z s}(t)=f(t)+5 \quad y(t) \neq y_{z i}(t)+y_{z s}(t) \quad yzi(t)=6x(0)+5yzs(t)=f(t)+5y(t)=yzi(t)+yzs(t) 不满足分解特性,非线性系统
(2) y z i ( t ) = 6 x ( 0 ) y z s ( t ) = 5 ∣ f ( t ) ∣ y ( t ) = y z i ( t ) + y z s ( t ) \begin{array}{lll}\text {} y_{z i}(t)=6 x(0) & y_{z s}(t)=5|f(t)| & y(t)=y_{z i}(t)+y_{z s}(t) & \text { }\end{array} yzi(t)=6x(0)yzs(t)=5∣f(t)∣y(t)=yzi(t)+yzs(t) 满足分解特性
T [ { 0 } , { a f ( t ) } ] = 5 ∣ a f ( t ) ∣ ≠ a y z s ( t ) T[\{0\},\{a f(t)\}]=5|a f(t)| \neq a y_{z s}(t) \quad T[{0},{af(t)}]=5∣af(t)∣=ayzs(t) 不满足零状态响应线性特征, 非线性系统
(3) y z i ( t ) = 2 x ( 0 ) 2 y z s ( t ) = 7 f ( k ) y ( t ) = y z i ( t ) + y z s ( t ) \begin{array}{lll}\text { } y_{z i}(t)=2 x(0)^{2} & y_{z s}(t)=7 f(k) & y(t)=y_{z i}(t)+y_{z s}(t) & \text { }\end{array} yzi(t)=2x(0)2yzs(t)=7f(k)y(t)=yzi(t)+yzs(t) 满足分解特性
T [ { a x ( 0 ) } , { 0 } ] = 2 ( a x ( 0 ) ) 2 ≠ a y z i ( t ) T[\{a x(0)\},\{0\}]=2(a x(0))^{2} \neq a y_{z i}(t) \quad T[{ax(0)},{0}]=2(ax(0))2=ayzi(t) 不满足零输入响应线性特征, 非线性系统
若激励之前有变系数, 或反转、展缩变换, 则系统为时变系统
1.输入激励
2.从响应式子中做周期推移
( 1 ) y z s ( t ) = f ( t ) f ( t − 5 ) ( 2 ) y z s ( k ) = k f ( k − 2 ) ( 3 ) y z s ( k ) = f ( − k ) ( 4 ) y z s ( k ) = f ( 3 k ) (1) y_{z s}(t)=f(t) f(t-5)(2) y_{z s}(k)=k f(k-2)(3) y_{z s}(k)=f(-k)(4) y_{z s}(k)=f(3 k) (1)yzs(t)=f(t)f(t−5)(2)yzs(k)=kf(k−2)(3)yzs(k)=f(−k)(4)yzs(k)=f(3k)
(1) 令 g ( t ) = f ( t − t d ) , 令g(t)=f\left(t-t_{d}\right), 令g(t)=f(t−td), 则 T [ { 0 } , { g ( t ) } ] = g ( t ) g ( t − 5 ) = f ( t − t d ) f ( t − t d − 5 ) T[\{0\},\{g(t)\}]=g(t) g(t-5)=f\left(t-t_{d}\right) f\left(t-t_{d}-5\right) T[{0},{g(t)}]=g(t)g(t−5)=f(t−td)f(t−td−5)
y z s ( t − t d ) = f ( t − t d ) f ( t − t d − 5 ) T [ { 0 } , { f ( t − t d ) } ] = y z s ( t − t d ) y_{z s}\left(t-t_{d}\right)=f\left(t-t_{d}\right) f\left(t-t_{d}-5\right) T\left[\{0\},\left\{f\left(t-t_{d}\right)\right\}\right]=y_{z s}\left(t-t_{d}\right) yzs(t−td)=f(t−td)f(t−td−5)T[{0},{f(t−td)}]=yzs(t−td) 所以是时不变系统
(2) 令 g ( k ) = f ( k − k d ) , {\text { 令}} g(k)=f\left(k-k_{d}\right), 令g(k)=f(k−kd), 则 T [ { 0 } , { g ( k ) } ] = k g ( k − 2 ) = k f ( k − k d − 2 ) T[\{0\},\{g(k)\}]=k g(k-2)=k f\left(k-k_{d}-2\right) T[{0},{g(k)}]=kg(k−2)=kf(k−kd−2)
y z s ( k − k d ) = ( k − k d ) f ( k − k d − 2 ) T [ { 0 } , { f ( k − k d ) } ] ≠ y z s ( k − k d ) y_{z s}\left(k-k_{d}\right)=\left(k-k_{d}\right) f\left(k-k_{d}-2\right) T\left[\{0\},\left\{f\left(k-k_{d}\right)\right\}\right] \neq y_{z s}\left(k-k_{d}\right) yzs(k−kd)=(k−kd)f(k−kd−2)T[{0},{f(k−kd)}]=yzs(k−kd) 所以是时变系统
(3) 令 g ( k ) = f ( k − k d ) , 令 g(k)=f\left(k-k_{d}\right), \quad 令g(k)=f(k−kd), 则 T [ { 0 } , { g ( k ) } ] = g ( − k ) = f ( − k − k d ) T[\{0\},\{g(k)\}]=g(-k)=f\left(-k-k_{d}\right) T[{0},{g(k)}]=g(−k)=f(−k−kd)
y z s ( k − k d ) = f ( − ( k − k d ) ) T [ { 0 } , { f ( k − k d ) } ] ≠ y z s ( k − k d ) y_{z s}\left(k-k_{d}\right)=f\left(-\left(k-k_{d}\right)\right) \quad T\left[\{0\},\left\{f\left(k-k_{d}\right)\right\}\right] \neq y_{z s}\left(k-k_{d}\right) yzs(k−kd)=f(−(k−kd))T[{0},{f(k−kd)}]=yzs(k−kd)
所以是时变系统
(4) 令 g ( k ) = f ( k − k d ) , 令g(k)=f\left(k-k_{d}\right), 令g(k)=f(k−kd), 则 T [ { 0 } , { g ( k ) } ] = g ( 3 k ) = f ( 3 k − k d ) T[\{0\},\{g(k)\}]=g(3 k)=f\left(3 k-k_{d}\right) T[{0},{g(k)}]=g(3k)=f(3k−kd)
y z s ( k − k d ) = f ( 3 ( k − k d ) ) T [ { 0 } , { f ( k − k d ) } ] ≠ y z s ( k − k d ) y_{z s}\left(k-k_{d}\right)=f\left(3\left(k-k_{d}\right)\right) \quad T\left[\{0\},\left\{f\left(k-k_{d}\right)\right\}\right] \neq y_{z s}\left(k-k_{d}\right) \quad yzs(k−kd)=f(3(k−kd))T[{0},{f(k−kd)}]=yzs(k−kd) 所以是时变系统
1.令 t < t 0 时 f ( t ) = 0 t<t_{0}时f{(t})=0 t<t0时f(t)=0
2.将 t 0 t_0 t0代入式中
若 t < t 0 ( t<t_{0}\left(\right. t<t0( 或 k < k 0 ) \left.k<k_{0}\right) k<k0) 时, \quad 激励 f ( t ) ( f(\mathrm{t}) \quad( f(t)( 或 f ( k ) ) = 0 , f(\mathrm{k}))=0, f(k))=0, 则当 t < t 0 ( t<t_{0} \quad\left(\right. t<t0( 或 k < k 0 ) \left.k<k_{0}\right) k<k0) 时, y z s ( t ) = 0 ( \mathrm{y}_{\mathrm{zs}}(\mathrm{t})=0\left(\right. yzs(t)=0( 或 y z s ( k ) = 0 ) \left.\mathrm{y}_{\mathrm{zs}}(\mathrm{k})=0\right) yzs(k)=0)
(1) y z s ( t ) = f ( t − 1 ) y_{z s}(t)=f(t-1) yzs(t)=f(t−1)
(2) y z s ( t ) = f ( t + 1 ) y_{z s}(t)=f(t+1) yzs(t)=f(t+1)
(3) y z s ( t ) = f ( 2 t ) y_{z s}(t)=f(2 t) yzs(t)=f(2t)
(1) 若 t < t 0 时, f ( t ) = 0 , 有 y z s ( t 0 ) = f ( t 0 − 1 ) = 0 , 是因果系统 \begin{array}{ll}\text { (1) 若 } t<t_{0} \text { 时, } f(\mathrm{t})=0, & \text { 有 } \mathrm{y}_{\mathrm{zs}}\left(\mathrm{t}_{0}\right)=f\left(\mathrm{t}_{0}-1\right)=0, \text { 是因果系统 }\end{array} (1) 若 t<t0 时, f(t)=0, 有 yzs(t0)=f(t0−1)=0, 是因果系统
(2) 若 t < t 0 时, f ( t ) = 0 , 有 y z s ( t 0 ) = f ( t 0 + 1 ) ≠ 0 , 是非因果系统 \begin{array}{ll}\text { (2) 若 } t<t_{0} \text { 时, } f(\mathrm{t})=0, & \text { 有 } \mathrm{y}_{\mathrm{zs}}\left(\mathrm{t}_{0}\right)=f\left(\mathrm{t}_{0}+1\right) \neq 0, \text { 是非因果系统 }\end{array} (2) 若 t<t0 时, f(t)=0, 有 yzs(t0)=f(t0+1)=0, 是非因果系统
(3) 若 t < t 0 时, f ( t ) = 0 , 有y z s ( t 0 ) = f ( 2 t 0 ) ≠ 0 , 是非因果系统 \begin{array}{ll}\text { (3) 若 } t<t_{0} \text { 时, } f(\mathrm{t})=0, & \text { 有y }_{\mathrm{zs}}\left(\mathrm{t}_{0}\right)=f\left(2\mathrm{t}_{0}\right) \neq 0, \text { 是非因果系统 }\end{array} (3) 若 t<t0 时, f(t)=0, 有y zs(t0)=f(2t0)=0, 是非因果系统 (因为是 1 2 t 0 到 t 0 之 间 \frac{1}{2}t_0到t_0之间 21t0到t0之间?)
信号是信息的物理表现和传偷載体,它一般是一种随时间变化而变化的物理量. 根据物理属性,信号可以分为电信号和非电信号。
电信号:随时间变化的电压或电流.
电信号容易产生,便于控制,易于处理。本课程主要讨论电信号,简称为信号。
信号的描述方法:
(1)数学函数表达式
(2)图形表达形式
按实际用途划分
电视信号
雷达信号
控制信号
通信信号
广播信号
按时间特性分为:
一维信号和多维信号
确定信号和随机信号
连续信号和离散信号
庄期信号和非周期信号
实信号和复信号
能量信号和功率信号
一维信号: 只由一个自变量描述的信号, 如语音信号
多维信号: 由多个自变量描述的信号, 如图徽信号
确定性信号
可用确定的时间函数表示的信号。
随机信号
取值具有未可预知的不确定性信号。
伪随机信号
看似随机但却道循一定规律的信号 ( 如伪随机码 )
连续时间信号 : 信号在连续时间内都有定义
用t表示连续时间变量
这里的 "连续” 指函数的定义域 - 时间是连续的,并不要求值域也连续,即信号可含间断点。
仅在一些离散的瞬间才有定义的信号,简称离散信号。
周期信号
定义在 ( − ∞ , ∞ ) (-\infty, \infty) (−∞,∞) 区间 , , , 每隔一定时间T (或整数 N \mathrm{N} N ),按相同规律重复变化的信号。
f ( n ) = f ( t + n T ) , n = 0 , ± 1 , ± 2 , … f ( k ) = f ( k + n N ) , n = 0 , ± 1 , ± 2 , … f(n)=f(t+n T), \quad n=0,\pm 1,\pm 2, \ldots\\ f(k)=f(k+n N), \quad n=0,\pm 1,\pm 2, \ldots f(n)=f(t+nT),n=0,±1,±2,…f(k)=f(k+nN),n=0,±1,±2,…
满足上述关系的最小T(或整数N)称为该信号的周期。
非周期信号
不具有周期性的信号称为非周期信号。
给定信号 f ( t ) , \mathrm{f}(\mathrm{t}), f(t), 若其施加于 1 Ω 1 \Omega 1Ω 电阻上,它所消耗的瞬时功率为 ∣ f ( t ) ∣ 2 , |f(\mathrm{t})|^{2}, ∣f(t)∣2, 则该信号在区间 ( − ∞ , ∞ ) (-\infty, \infty) (−∞,∞) 的能量和平均功率定义如下
信号的能量 E = ∫ − ∞ ∞ ∣ f ( t ) ∣ 2 d t \quad E=\int_{-\infty}^{\infty}|f(\mathrm{t})|^{2} d t E=∫−∞∞∣f(t)∣2dt
信号的功率 P = lim T → + ∞ 1 T ∫ − T / 2 T / 2 ∣ f ( t ) ∣ 2 d t \mathrm{P}=\lim\limits_{T \rightarrow+\infty} \frac{1}{T} \int_{-T / 2}^{T / 2}|f(\mathrm{t})|^{2} d t P=T→+∞limT1∫−T/2T/2∣f(t)∣2dt
能量信号
若 E < ∞ , E<\infty, E<∞, 即信号的能量有界,则称信号 f ( t ) f(t) f(t) 为能量信号。此时 p = 0 p=0 p=0
功率信号
若 P < ∞ P<\infty P<∞, 即信号的功率有界,则称信号 f ( t ) f(t) f(t) 为功率信号。此时 E = ∞ E=\infty E=∞
能量 E = ∑ ∣ f ( k ) ∣ 2 E=\sum|f(\mathrm{k})|^{2} E=∑∣f(k)∣2
功率 P = lim N → + ∞ 1 N ∑ k = − N / 2 N / 2 ∣ f ( k ) ∣ 2 \mathrm{P}=\lim \limits_{N \rightarrow+\infty} \frac{1}{N} \sum_{k=-N / 2}^{N / 2}|f(\mathrm{k})|^{2} P=N→+∞limN1∑k=−N/2N/2∣f(k)∣2
能量信号: 若 E < ∞ , E<\infty, E<∞, 则为能量信号
功率信号: 若 P < ∞ , P<\infty, P<∞, 则为功率信号
确定性信号:可用确定的时间函数表示的信号。
1.直流信号
x ( t ) = C , − ∞ < t < ∞ x(t)=C, \quad-\infty<t<\infty x(t)=C,−∞<t<∞
2.单位斜坡信号
单位指的是斜率为1
R ( t ) = { t t > 0 0 t < 0 R(t)=\left\{\begin{array}{ll}t & t>0 \\ 0 & t<0\end{array}\right. R(t)={t0t>0t<0
3.指数信号
双边指数信号
x ( t ) = A e − a t x(\mathrm{t})=A e^{-\mathrm{at}} x(t)=Ae−at
单边指数信号
x ( t ) = { A e − a t , t > 0 0 , t < 0 x(\mathrm{t})=\left\{\begin{array}{ll}A e^{-\mathrm{at}}, & t>0 \\ 0, & t<0\end{array}\right. x(t)={Ae−at,0,t>0t<0
4.正余弦信号
x ( t ) = A cos ( w t + θ ) = A cos [ w ( t − t 0 ) ] = A sin w t + θ + π 2 \quad x(\mathrm{t})\\ =A \cos (w t+\theta)\\ =A \cos [w(t-t_{0})] \\ =A \sin w t+\theta+\frac{\pi}{2} x(t)=Acos(wt+θ)=Acos[w(t−t0)]=Asinwt+θ+2π
振幅: A A A
周期 : T = 2 π w = 1 f : T=\frac{2 \pi}{w}=\frac{1}{f} :T=w2π=f1
频率: f f f
角频率 : w = 2 π f : w=2 \pi f :w=2πf
初相: θ \theta θ
5.复指数信号
x ( t ) = A e s t ; s = σ + j w = A e σ t e j w t = A e σ t ( cos ( w t ) + j sin ( w t ) ) 讨论 : [ σ = 0 , w = 0 , 直流 σ > 0 , w = 0 升指数信号 σ < 0 , w = 0 衰减指数信号 σ = 0 , w ≠ 0 , 等幅振荡 σ > 0 , w ≠ 0 , 增幅振荡 σ < 0 , w ≠ 0 , 衰减振荡 \begin{aligned} x(\mathrm{t})=& A \mathrm{e}^{s t} ; \quad s=\sigma+j w \\=& A e^{\sigma t} e^{j w t} \\=& A e^{\sigma t}(\cos (w t)+j \sin (w t)) \\ \text { 讨论 }: &\left[\begin{array}{ll}\sigma=0, w=0, & \text { 直流 } \\ \sigma>0, w=0 & \text { 升指数信号 } \\ \sigma<0, w=0 & \text { 衰减指数信号 } \\ \sigma=0, w \neq 0, & \text { 等幅振荡 } \\ \sigma>0, w \neq 0, & \text { 增幅振荡 } \\ \sigma<0, w \neq 0, & \text { 衰减振荡 }\end{array}\right.\end{aligned} x(t)=== 讨论 :Aest;s=σ+jwAeσtejwtAeσt(cos(wt)+jsin(wt))⎣⎢⎢⎢⎢⎢⎢⎡σ=0,w=0,σ>0,w=0σ<0,w=0σ=0,w=0,σ>0,w=0,σ<0,w=0, 直流 升指数信号 衰减指数信号 等幅振荡 增幅振荡 衰减振荡
6.抽样信号
x ( t ) = S a ( t ) = sin t t x(t)=S a(\mathrm{t})=\frac{\sin t}{t} x(t)=Sa(t)=tsint
7.阶跃函数(冲激函数的积分)
单位阶跃函数 ε ( t ) = { 0 , t < 0 0.5 , t = 0 1 , t > 0 \varepsilon(t)=\left\{\begin{array}{c}0, t<0 \\ 0.5, t=0 \\ 1, t>0\end{array}\right. ε(t)=⎩⎨⎧0,t<00.5,t=01,t>0
对于t=0点不同学校定义可能不同,但无关紧要,阶跃函数可以变化出很多变种
8.冲激函数/狄拉克δ函数(阶跃函数的导数)
δ ( t ) = 0 , t ≠ 0 ∫ − ∞ + ∞ δ ( t ) d t = ∫ 0 − 0 + δ ( t ) d t = 1 \delta(t)=0, t \neq 0 \\ \int_{-\infty}^{+\infty} \delta(t) d t=\int_{0_{-}}^{0_{+}} \delta(t) d t=1 δ(t)=0,t=0∫−∞+∞δ(t)dt=∫0−0+δ(t)dt=1
函数值只在 t = 0 t=0 t=0 时不为零
积分面积为1
t=0时 , δ ( t ) → ∞ , , \delta(t) \rightarrow \infty, ,δ(t)→∞, 为无界函数。
ε ( t ) = ∫ − ∞ t δ ( τ ) d τ δ ( t ) = d ε ( t ) d t \varepsilon(t)=\int_{-\infty}^{t} \delta(\tau) d \tau \quad \delta(t)=\frac{d \varepsilon(t)}{d t} ε(t)=∫−∞tδ(τ)dτδ(t)=dtdε(t)
阶跃函数组成的函数求导,只在台阶处有值,且正负与台阶上升还是下降有关
如果 f ( t ) f(t) f(t) 在 t = 0 t=0 t=0 处连续,且处处有界,则有
δ ( t ) f ( t ) = δ ( t ) f ( 0 ) ∫ − ∞ + ∞ δ ( t ) f ( t ) d t = f ( 0 ) \begin{array}{l} \delta(t) f(t)=\delta(t) f(0) \\ \int_{-\infty}^{+\infty} \delta(t) f(t) d t=f(0) \end{array} δ(t)f(t)=δ(t)f(0)∫−∞+∞δ(t)f(t)dt=f(0)
同理可得:
f ( t ) δ ( t − t 0 ) = f ( t 0 ) δ ( t − t 0 ) f(t) \delta\left(t-t_{0}\right)=f\left(t_{0}\right) \delta\left(t-t_{0}\right) f(t)δ(t−t0)=f(t0)δ(t−t0)
∫ − ∞ + ∞ δ ( t − t 0 ) f ( t ) d t = f ( t 0 ) \int_{-\infty}^{+\infty} \delta\left(t-t_{0}\right) f(t) d t=f\left(t_{0}\right) ∫−∞+∞δ(t−t0)f(t)dt=f(t0)
f ( t ) δ ′ ( t ) = f ( 0 ) δ ′ ( t ) − f ′ ( 0 ) δ ( t ) f(t) \delta^{\prime}(t)=f(0) \delta^{\prime}(t)-f^{\prime}(0) \delta(t) f(t)δ′(t)=f(0)δ′(t)−f′(0)δ(t)
∫ − ∞ ∞ δ ′ ( t ) f ( t ) d t = − f ′ ( 0 ) \int_{-\infty}^{\infty} \delta^{\prime}(t) f(t) d t=-f^{\prime}(0) ∫−∞∞δ′(t)f(t)dt=−f′(0)
∫ − ∞ t δ ′ ( t ) d t = δ ( t ) \int_{-\infty}^{t} \delta^{\prime}(t) d t=\delta(t) ∫−∞tδ′(t)dt=δ(t)
推导如下
f ( t ) δ ′ ( t ) = f ( 0 ) δ ′ ( t ) − f ′ ( 0 ) δ ( t ) [ f ( t ) δ ( t ) ] ′ = f ( t ) δ ′ ( t ) + f ′ ( t ) δ ( t ) f ( t ) δ ′ ( t ) = [ f ( t ) δ ( t ) ] ′ − f ′ ( t ) δ ( t ) = f ( 0 ) δ ′ ( t ) − f ′ ( 0 ) δ ( t ) ∫ − ∞ ∞ δ ′ ( t ) f ( t ) d t = − f ′ ( 0 ) ∫ − ∞ ∞ δ ′ ( t ) f ( t ) d t = f ( t ) δ ( t ) ∣ − ∞ ∞ − ∫ − ∞ ∞ f ′ ( t ) δ ( t ) d t = − f ′ ( 0 ) ∫ − ∞ t δ ′ ( t ) d t = δ ( t ) \begin{array}{c} f(t) \delta^{\prime}(t)=f(0) \delta^{\prime}(t)-f^{\prime}(0) \delta(t) \\ {[f(t) \delta(t)]^{\prime}=f(t) \delta^{\prime}(t)+f^{\prime}(t) \delta(t) \quad f(t) \delta^{\prime}(t)=[f(t) \delta(t)]^{\prime}-f^{\prime}(t) \delta(t)=f(0) \delta^{\prime}(t)-f^{\prime}(0) \delta(t)} \\ \int_{-\infty}^{\infty} \delta^{\prime}(t) f(t) d t=-f^{\prime}(0) \\ \int_{-\infty}^{\infty} \delta^{\prime}(t) f(t) d t=\left.f(t) \delta(t)\right|_{-\infty} ^{\infty}-\int_{-\infty}^{\infty} f^{\prime}(t) \delta(t) d t=-f^{\prime}(0) \\ \int_{-\infty}^{t} \delta^{\prime}(t) d t=\delta(t) \end{array} f(t)δ′(t)=f(0)δ′(t)−f′(0)δ(t)[f(t)δ(t)]′=f(t)δ′(t)+f′(t)δ(t)f(t)δ′(t)=[f(t)δ(t)]′−f′(t)δ(t)=f(0)δ′(t)−f′(0)δ(t)∫−∞∞δ′(t)f(t)dt=−f′(0)∫−∞∞δ′(t)f(t)dt=f(t)δ(t)∣−∞∞−∫−∞∞f′(t)δ(t)dt=−f′(0)∫−∞tδ′(t)dt=δ(t)
δ ( a t ) = 1 ∣ a ∣ δ ( t ) \delta(a t)=\frac{1}{|a|} \delta(t) δ(at)=∣a∣1δ(t)
证明过程如下
p ( t ) \mathrm{p}(\mathrm{t}) p(t) 面积为1 , δ ( t ) , \quad \delta(t) ,δ(t) 强度为1
p ( a t ) \mathrm{p}(\mathrm{at}) p(at) 面积为 1 ∣ a ∣ , δ ( a t ) \frac{1}{|a|}, \quad \delta(a t) ∣a∣1,δ(at) 强度为 1 ∣ a ∣ \frac{1}{|a|} ∣a∣1
τ → 0 \tau \rightarrow 0 τ→0 时 , p ( t ) → δ ( t ) , p ( a t ) → 1 ∣ a ∣ δ ( t ) , p(t) \rightarrow \delta(t), p(a t) \rightarrow \frac{1}{|a|} \delta(t) ,p(t)→δ(t),p(at)→∣a∣1δ(t)
ε ( k ) = { 1 , k ≥ 0 0 , k < 0 \varepsilon(k)=\left\{\begin{array}{l}1, k \geq 0 \\ 0, k<0\end{array}\right. ε(k)={1,k≥00,k<0
δ ( k ) = ε ( k ) − ε ( k − 1 ) \delta(k)=\varepsilon(k)-\varepsilon(k-1) δ(k)=ε(k)−ε(k−1)
ε ( k ) = ∑ i = − ∞ k δ ( i ) \varepsilon(k)=\sum_{i=-\infty}^{k} \delta(i) ε(k)=∑i=−∞kδ(i)
δ ( k ) = { 1 , k = 0 0 , k ≠ 0 \delta(k)=\left\{\begin{array}{l}1, k=0 \\ 0, k \neq 0\end{array}\right. δ(k)={1,k=00,k=0
δ ( k ) f ( k ) = δ ( k ) f ( 0 ) \delta(k) f(k)=\delta(k) f(0) δ(k)f(k)=δ(k)f(0)
f ( k ) δ ( k − k 0 ) = f ( k 0 ) δ ( k − k 0 ) f(k) \delta\left(k-k_{0}\right)=f\left(k_{0}\right) \delta\left(k-k_{0}\right) f(k)δ(k−k0)=f(k0)δ(k−k0)
∑ k = − ∞ ∞ f ( k ) δ ( k ) = f ( 0 ) \sum\limits_{k=-\infty}^{\infty} f(k) \delta(k)=f(0) k=−∞∑∞f(k)δ(k)=f(0)
相加和相乘:同一瞬间/同一序列对应值相加/相乘。
平移:左加右减
反转:对折
没有对应的器件,但是数字信号处理有相应的概念,如退栈的先进后出
尺度变换:a大压缩,a小扩展(大小以1为对应)
对于离散信号,由于 f (ak) 仅在ak为整数时才有意义,进行尺度变换时可能会使部分信号丢失。因此一般不作波形的尺度变换。
一般建议先平移,再尺度变换,再反转
只需要记住所有的变换都是对应自变量而言即可,可以取值验证
由相互作用、相互联系的事物按一定规律组成的具有特定功能的整体,称为系统。手机电视都是系统。
连续系统:系统的激励是连续信号,响应也是连续信号。
离散系统:系统的激励是离散信号,响应也是离散信号。
混合系统:系统的激励是连续信号,响应是离散信号;或反之。
分析系统时,需要建立描述该系统的数学模型,求解,并对结果赋予实际意义。
依据基尔霍夫电压定律:
u L ( t ) + u R ( t ) + u C ( t ) = u S ( t ) u_{L}(t)+u_{R}(t)+u_{C}(t)=u_{S}(t) uL(t)+uR(t)+uC(t)=uS(t)
依据各元件的电压与电流关系:
u L ( t ) = L i ′ ( t ) u R ( t ) = R i ( t ) i ( t ) = C u C ′ ( t ) u_{L}(t)=L i^{\prime}(t) u_{R}(t)=\mathrm{R} i(t) i(t)=C u_{C}^{\prime}(t) uL(t)=Li′(t)uR(t)=Ri(t)i(t)=CuC′(t)
整理可得,
u C ′ ′ ( t ) + R L u C ′ ( t ) + 1 L C u C ( t ) = 1 L C u S ( t ) u_{C}^{\prime \prime}(t)+\frac{R}{L} u_{C}^{\prime}(t)+\frac{1}{L C} u_{C}(t)=\frac{1}{L C} u_{S}(t) uC′′(t)+LRuC′(t)+LC1uC(t)=LC1uS(t)
上式即为描述该电路的微分方程。
响应一般写左侧,激励写右侧
根据方程画框图
①左侧写激励,右侧写响应
②根据最高次数中间补对应数量的积分器,左侧补加法器
u C ′ ′ ( t ) + R L u C ′ ( t ) + 1 L C u C ( t ) = 1 L C u S ( t ) u_{C}^{\prime \prime}(t)+\frac{R}{L} u_{C}^{\prime}(t)+\frac{1}{L C} u_{C}(t)=\frac{1}{L C} u_{S}(t) uC′′(t)+LRuC′(t)+LC1uC(t)=LC1uS(t)
y ′ ′ ( t ) + a y ′ ( t ) + b y ( t ) = c f ( t ) y^{\prime \prime}(t)+a y^{\prime}(t)+b y(t)=c f(t) y′′(t)+ay′(t)+by(t)=cf(t)
a = R L b = 1 L C c = 1 L C a=\frac{R}{L} \quad b=\frac{1}{L C} \quad c=\frac{1}{L C} a=LRb=LC1c=LC1
y ′ ′ ( t ) = − a y ′ ( t ) − b y ( t ) + c f ( t ) y^{\prime \prime}(t)=-a y^{\prime}(t)-b y(t)+c f(t) y′′(t)=−ay′(t)−by(t)+cf(t)
根据框图画方程
①设置中间变量,左边高次,右边低次
左边加法器
x ′ ′ ( t ) = − a 2 x ′ ( t ) − a 1 x ( t ) + f ( t ) x^{\prime \prime}(t)=-a_{2} x^{\prime}(t)-a_{1} x(t)+f(t) x′′(t)=−a2x′(t)−a1x(t)+f(t)
x ′ ′ ( t ) + a 2 x ′ ( t ) + a 1 x ( t ) = f ( t ) x^{\prime \prime}(t)+a_{2} x^{\prime}(t)+a_{1} x(t)=f(t) x′′(t)+a2x′(t)+a1x(t)=f(t)
右边加法器
y ( t ) = b 2 x ′ ′ ( t ) + b 1 x ′ ( t ) + x ( t ) a 1 y ( t ) = b 2 a 1 x ′ ′ ( t ) + b 1 a 1 x ′ ( t ) + a 1 x ( t ) a 2 y ′ ( t ) = b 2 ( a 2 x ′ ′ ( t ) ) ′ + b 1 ( a 2 x ′ ( t ) ) ′ + a 2 x ′ ( t ) y ′ ′ ( t ) = b 2 ( x ′ ′ ( t ) ) ′ ′ + b 1 ( x ′ ( t ) ) ′ ′ + x ′ ′ ( t ) y(t)=b_{2} x^{\prime \prime}(t)+b_{1} x^{\prime}(t)+x(t)\\ a_{1} y(t)=b_{2} a_{1} x^{\prime \prime}(t)+b_{1} a_{1} x^{\prime}(t)+a_{1} x(t) \\ a_{2} y^{\prime}(t) =b_{2}\left(a_{2} x^{\prime \prime}(t)\right)^{\prime}+b_{1}\left(a_{2} x^{\prime}(t)\right)^{\prime}+a_{2} x^{\prime}(t) \\ y^{\prime \prime}(t) =b_{2}\left(x^{\prime \prime}(t)\right)^{\prime \prime}+b_{1}\left(x^{\prime}(t)\right)^{\prime \prime}+x^{\prime \prime}(t) y(t)=b2x′′(t)+b1x′(t)+x(t)a1y(t)=b2a1x′′(t)+b1a1x′(t)+a1x(t)a2y′(t)=b2(a2x′′(t))′+b1(a2x′(t))′+a2x′(t)y′′(t)=b2(x′′(t))′′+b1(x′(t))′′+x′′(t)
y ′ ′ ( t ) + a 2 y ′ ( t ) + a 1 y ( t ) = b 2 ( x ′ ′ ( t ) + a 2 x ′ ( t ) + a 1 x ( t ) ) ′ ′ + b 1 ( x ′ ′ ( t ) + a 2 x ′ ( t ) + a 1 x ( t ) ) ′ + x ′ ′ ( t ) + a 2 x ′ ( t ) + a 1 x ( t ) = b 2 f ′ ′ ( t ) + b 1 f ′ ( t ) + f ( t ) y^{\prime \prime}(t)+a_{2} y^{\prime}(t)+a_{1} y(t) \\ =b_{2}(x^{\prime \prime}(t)+a_{2} x^{\prime}(t)+a_{1} x(t))^{\prime \prime}+b_{1}\left(x^{\prime \prime}(t)+a_{2} x^{\prime}(t)+a_{1} x(t)\right)^{\prime}+x^{\prime \prime}(t)+a_{2} x^{\prime}(t)+a_{1} x(t)\\ =b_{2} f^{\prime \prime}(t)+b_{1} f^{\prime}(t)+f(t) y′′(t)+a2y′(t)+a1y(t)=b2(x′′(t)+a2x′(t)+a1x(t))′′+b1(x′′(t)+a2x′(t)+a1x(t))′+x′′(t)+a2x′(t)+a1x(t)=b2f′′(t)+b1f′(t)+f(t)
最终结果 y ′ ′ ( t ) + a 2 y ′ ( t ) + a 1 y ( t ) = b 2 f ′ ′ ( t ) + b 1 f ′ ( t ) + f ( t ) y^{\prime \prime}(t)+a_{2} y^{\prime}(t)+a_{1} y(t) =b_{2} f^{\prime \prime}(t)+b_{1} f^{\prime}(t)+f(t) y′′(t)+a2y′(t)+a1y(t)=b2f′′(t)+b1f′(t)+f(t)
响应左边激励右边
y ( k ) − ( 1 + a − b ) y ( k − 1 ) = f ( k ) y(k)-(1+a-b) y(k-1)=f(k) y(k)−(1+a−b)y(k−1)=f(k)
左边加法器
x ( k ) = f ( k ) − a 2 x ( k − 1 ) − a 1 x ( k − 2 ) x(k)=f(k)-a_{2} x(k-1)-a_{1} x(k-2) x(k)=f(k)−a2x(k−1)−a1x(k−2)
f ( k ) = x ( k ) + a 2 x ( k − 1 ) + a 1 x ( k − 2 ) f(k)=x(k)+a_{2} x(k-1)+a_{1} x(k-2) f(k)=x(k)+a2x(k−1)+a1x(k−2)
右边加法器
y ( k ) = b 1 x ( k ) + x ( k − 2 ) \mathrm{y}(\mathrm{k})=b_{1} \mathrm{x}(\mathrm{k})+\mathrm{x}(\mathrm{k}-2) y(k)=b1x(k)+x(k−2)
a 2 y ( k − 1 ) = b 1 a 2 x ( k − 1 ) + a 2 x ( k − 3 ) a_{2} y(k-1)=b_{1} a_{2} x(k-1)+a_{2} x(k-3) a2y(k−1)=b1a2x(k−1)+a2x(k−3)
a 1 y ( k − 2 ) = b 1 a 1 x ( k − 2 ) + a 1 x ( k − 4 ) a_{1} y(k-2)=b_{1} a_{1} x(k-2)+a_{1} x(k-4) a1y(k−2)=b1a1x(k−2)+a1x(k−4)
y ( k ) + a 2 y ( k − 1 ) + a 1 y ( k − 2 ) = b 1 [ x ( k ) + a 2 x ( k − 1 ) + a 1 x ( k − 2 ) ] + [ x ( k − 2 ) + a 2 x ( k − 3 ) + a 1 x ( k − 4 ) ] = b 1 f ( k ) + f ( k − 2 ) y(k)+a_{2} y(k-1)+a_{1} y(k-2)\\ =b_{1}\left[x(k)+a_{2} x(k-1)+a_{1} x(k-2)\right]+\left[x(k-2)+a_{2} x(k-3)+a_{1} x(k-4)\right]\\ =b_{1} f(k)+f(k-2) y(k)+a2y(k−1)+a1y(k−2)=b1[x(k)+a2x(k−1)+a1x(k−2)]+[x(k−2)+a2x(k−3)+a1x(k−4)]=b1f(k)+f(k−2)
y ( k ) + a 2 y ( k − 1 ) + a 1 y ( k − 2 ) = b 1 f ( k ) + f ( k − 2 ) y(k)+a_{2} y(k-1)+a_{1} y(k-2)=b_{1} f(k)+f(k-2) y(k)+a2y(k−1)+a1y(k−2)=b1f(k)+f(k−2)
可从多角度观察和分析系统, 将系统分成多种类别。
连续与离散系统
线性与非线性系统
时变与时不变系统
因果与非因果系统
记忆与非记忆系统
稳定与发散系统
连续系统:系统的激励是连续信号,响应也是连续信号。 f ( t ) , y ( t ) f(t), y(t) f(t),y(t)
离散系统:系统的激励是离散信号,响应也是离散信号。 f ( k ) , y ( k ) f(k), y(k) f(k),y(k)
混合系统:系统的激励是连续信号, 响应是离散信号; 或反之。
满足线性性质(齐次性和可加性)和分解特性的系统为线性系统, 否则为非
线性系统。
f ( ⋅ ) → 系统 T → y ( ⋅ ) y ( ⋅ ) = T [ f ( ⋅ ) ] f ( ⋅ ) → y ( ⋅ ) \begin{array}{l|l|rl} f(\cdot) \rightarrow& \text { 系统 } \mathbf{T} & \rightarrow y(\cdot) & \begin{array}{l} y(\cdot)=T[f(\cdot)] \\ f(\cdot) \rightarrow y(\cdot) \end{array} \end{array} f(⋅)→ 系统 T→y(⋅)y(⋅)=T[f(⋅)]f(⋅)→y(⋅)
齐次性 f ( ⋅ ) → y ( ⋅ ) aff ⋅ ) → a y ( ⋅ ) f(\cdot) \rightarrow y(\cdot) \quad \text { aff } \cdot) \rightarrow a y(\cdot) f(⋅)→y(⋅) aff ⋅)→ay(⋅)
可加性 f 1 ( ⋅ ) → y 1 ( ⋅ ) f 2 ( ⋅ ) → y 2 ( ⋅ ) f 1 ( ⋅ ) + f 2 ( ⋅ ) → y 1 ( ⋅ ) + y 2 ( ⋅ ) \quad f_{1}(\cdot) \rightarrow y_{1}(\cdot) \quad f_{2}(\cdot) \rightarrow y_{2}(\cdot) \quad f_{1}(\cdot)+f_{2}(\cdot) \rightarrow \mathrm{y}_{1}(\cdot)+\mathrm{y}_{2}(\cdot) f1(⋅)→y1(⋅)f2(⋅)→y2(⋅)f1(⋅)+f2(⋅)→y1(⋅)+y2(⋅)
线性性质
a f 1 ( ⋅ ) + b f 2 ( ⋅ ) → a y 1 ( ⋅ ) + b y 2 ( ⋅ ) a f_{1}(\cdot)+b f_{2}(\cdot) \rightarrow a y_{1}(\cdot)+b y_{2}(\cdot) af1(⋅)+bf2(⋅)→ay1(⋅)+by2(⋅)
系统的响应取决于系统的激励 { f ( ⋅ ) } \{f(\cdot)\} {f(⋅)} 和初始状态 { x ( 0 ) } , \{\mathrm{x}(0)\}, {x(0)}, 系统的全响应为:
y ( ⋅ ) = T [ { x ( 0 ) } , { f ( ⋅ ) } ] \mathrm{y}(\cdot)=T[\{\mathrm{x}(0)\},\{f(\cdot)\}] y(⋅)=T[{x(0)},{f(⋅)}]
零输入响应 y z i ( ⋅ ) = T [ { x ( 0 ) } , { 0 } ] \quad y_{z i}(\cdot)=T[\{x(0)\},\{0\}] \quad yzi(⋅)=T[{x(0)},{0}]
零状态响应 y z s ( ⋅ ) = T [ { 0 } , { f ( ⋅ ) } ] \quad y_{z s}(\cdot)=T[\{0\},\{f(\cdot)\}] yzs(⋅)=T[{0},{f(⋅)}]
分解特性 y ( ⋅ ) = y z i ( ⋅ ) + y z s ( ⋅ ) \quad y(\cdot)=y_{z i}(\cdot)+y_{z s}(\cdot) y(⋅)=yzi(⋅)+yzs(⋅)
零输入响应线性
T [ { a x 1 ( 0 ) + b x 2 ( 0 ) } , { 0 } ] = a T [ { x 1 ( 0 ) } , { 0 } ] + b T [ { x 2 ( 0 ) } , { 0 } ] \quad T\left[\left\{a x_{1}(0)+b x_{2}(0)\right\},\{0\}\right]=a T\left[\left\{x_{1}(0)\right\},\{0\}\right]+b T\left[\left\{x_{2}(0)\right\},\{0\}\right] T[{ax1(0)+bx2(0)},{0}]=aT[{x1(0)},{0}]+bT[{x2(0)},{0}]
零状态响应线性
T [ { 0 } , { a f 1 ( t ) + b f 2 ( t ) } ] = a T [ { 0 } , { f 1 ( t ) } ] + b T [ { 0 } , { f 2 ( t ) } ] \quad T\left[\{0\},\left\{a f_{1}(t)+b f_{2}(t)\right\}\right]=a T\left[\{0\},\left\{f_{1}(t)\right\}\right]+b T\left[\{0\},\left\{f_{2}(t)\right\}\right] T[{0},{af1(t)+bf2(t)}]=aT[{0},{f1(t)}]+bT[{0},{f2(t)}]
例题
( 1 ) y ( t ) = f ( t ) + 3 x ( 0 ) f ( t ) + 6 x ( 0 ) + 5 ( 2 ) y ( t ) = 5 ∣ f ( t ) ∣ + 6 x ( 0 ) \begin{array}{ll}(1) y(t)=f(t)+3 x(0) f(t)+6 x(0)+5 & (2) y(t)=5|f(t)|+6 x(0)\end{array} (1)y(t)=f(t)+3x(0)f(t)+6x(0)+5(2)y(t)=5∣f(t)∣+6x(0)
(3) y ( k ) = 7 f ( k ) + 2 x ( 0 ) 2 y(k)=7 f(k)+2 x(0)^{2} y(k)=7f(k)+2x(0)2
(1)$ y_{z i}(t)=6 x(0)+5 \quad y_{z s}(t)=f(t)+5 \quad y(t) \neq y_{z i}(t)+y_{z s}(t) \quad$ 不满足分解特性,非线性系统
(2) y z i ( t ) = 6 x ( 0 ) y z s ( t ) = 5 ∣ f ( t ) ∣ y ( t ) = y z i ( t ) + y z s ( t ) \begin{array}{lll}\text {} y_{z i}(t)=6 x(0) & y_{z s}(t)=5|f(t)| & y(t)=y_{z i}(t)+y_{z s}(t) & \text { }\end{array} yzi(t)=6x(0)yzs(t)=5∣f(t)∣y(t)=yzi(t)+yzs(t) 满足分解特性
T [ { 0 } , { a f ( t ) } ] = 5 ∣ a f ( t ) ∣ ≠ a y z s ( t ) T[\{0\},\{a f(t)\}]=5|a f(t)| \neq a y_{z s}(t) \quad T[{0},{af(t)}]=5∣af(t)∣=ayzs(t) 不满足零状态响应线性特征, 非线性系统
(3) y z i ( t ) = 2 x ( 0 ) 2 y z s ( t ) = 7 f ( k ) y ( t ) = y z i ( t ) + y z s ( t ) \begin{array}{lll}\text { } y_{z i}(t)=2 x(0)^{2} & y_{z s}(t)=7 f(k) & y(t)=y_{z i}(t)+y_{z s}(t) & \text { }\end{array} yzi(t)=2x(0)2yzs(t)=7f(k)y(t)=yzi(t)+yzs(t) 满足分解特性
T [ { a x ( 0 ) } , { 0 } ] = 2 ( a x ( 0 ) ) 2 ≠ a y z i ( t ) T[\{a x(0)\},\{0\}]=2(a x(0))^{2} \neq a y_{z i}(t) \quad T[{ax(0)},{0}]=2(ax(0))2=ayzi(t) 不满足零输入响应线性特征, 非线性系统
如果激励 f ( ⋅ ) f(\cdot) f(⋅) 作用于系统的响应为 y z s ( ⋅ ) , y_{z s}(\cdot), yzs(⋅), 那么当激励延迟一定的时间 t d t_{d} td ( \left(\right. ( 或 k d ) \left.k_{d}\right) kd) 时, 其零状态响应也延迟同样的时间的系统为时不变系统, 反
之为时变系统。
T [ { 0 } , { f ( t − t d ) } ] = y z s ( t − t d ) T [ { 0 } , { f ( k − k d ) } ] = y z s ( k − k d ) T\left[\{0\},\left\{f\left(t-t_{d}\right)\right\}\right]=y_{z s}\left(t-t_{d}\right)\\ T\left[\{0\},\left\{f\left(k-k_{d}\right)\right\}\right]=y_{z s}\left(k-k_{d}\right) T[{0},{f(t−td)}]=yzs(t−td)T[{0},{f(k−kd)}]=yzs(k−kd)
( 1 ) y z s ( t ) = f ( t ) f ( t − 5 ) ( 2 ) y z s ( k ) = k f ( k − 2 ) ( 3 ) y z s ( k ) = f ( − k ) ( 4 ) y z s ( k ) = f ( 3 k ) (1) y_{z s}(t)=f(t) f(t-5)(2) y_{z s}(k)=k f(k-2)(3) y_{z s}(k)=f(-k)(4) y_{z s}(k)=f(3 k) (1)yzs(t)=f(t)f(t−5)(2)yzs(k)=kf(k−2)(3)yzs(k)=f(−k)(4)yzs(k)=f(3k)
(1) 令 g ( t ) = f ( t − t d ) , 令g(t)=f\left(t-t_{d}\right), 令g(t)=f(t−td), 则 T [ { 0 } , { g ( t ) } ] = g ( t ) g ( t − 5 ) = f ( t − t d ) f ( t − t d − 5 ) T[\{0\},\{g(t)\}]=g(t) g(t-5)=f\left(t-t_{d}\right) f\left(t-t_{d}-5\right) T[{0},{g(t)}]=g(t)g(t−5)=f(t−td)f(t−td−5)
y z s ( t − t d ) = f ( t − t d ) f ( t − t d − 5 ) T [ { 0 } , { f ( t − t d ) } ] = y z s ( t − t d ) y_{z s}\left(t-t_{d}\right)=f\left(t-t_{d}\right) f\left(t-t_{d}-5\right) T\left[\{0\},\left\{f\left(t-t_{d}\right)\right\}\right]=y_{z s}\left(t-t_{d}\right) yzs(t−td)=f(t−td)f(t−td−5)T[{0},{f(t−td)}]=yzs(t−td) 所以是时不变系统
(2) 令 g ( k ) = f ( k − k d ) , {\text { 令}} g(k)=f\left(k-k_{d}\right), 令g(k)=f(k−kd), 则 T [ { 0 } , { g ( k ) } ] = k g ( k − 2 ) = k f ( k − k d − 2 ) T[\{0\},\{g(k)\}]=k g(k-2)=k f\left(k-k_{d}-2\right) T[{0},{g(k)}]=kg(k−2)=kf(k−kd−2)
y z s ( k − k d ) = ( k − k d ) f ( k − k d − 2 ) T [ { 0 } , { f ( k − k d ) } ] ≠ y z s ( k − k d ) y_{z s}\left(k-k_{d}\right)=\left(k-k_{d}\right) f\left(k-k_{d}-2\right) T\left[\{0\},\left\{f\left(k-k_{d}\right)\right\}\right] \neq y_{z s}\left(k-k_{d}\right) yzs(k−kd)=(k−kd)f(k−kd−2)T[{0},{f(k−kd)}]=yzs(k−kd) 所以是时变系统
(3) 令 g ( k ) = f ( k − k d ) , 令 g(k)=f\left(k-k_{d}\right), \quad 令g(k)=f(k−kd), 则 T [ { 0 } , { g ( k ) } ] = g ( − k ) = f ( − k − k d ) T[\{0\},\{g(k)\}]=g(-k)=f\left(-k-k_{d}\right) T[{0},{g(k)}]=g(−k)=f(−k−kd)
y z s ( k − k d ) = f ( − ( k − k d ) ) T [ { 0 } , { f ( k − k d ) } ] ≠ y z s ( k − k d ) y_{z s}\left(k-k_{d}\right)=f\left(-\left(k-k_{d}\right)\right) \quad T\left[\{0\},\left\{f\left(k-k_{d}\right)\right\}\right] \neq y_{z s}\left(k-k_{d}\right) yzs(k−kd)=f(−(k−kd))T[{0},{f(k−kd)}]=yzs(k−kd)
所以是时变系统
(4) 令 g ( k ) = f ( k − k d ) , 令g(k)=f\left(k-k_{d}\right), 令g(k)=f(k−kd), 则 T [ { 0 } , { g ( k ) } ] = g ( 3 k ) = f ( 3 k − k d ) T[\{0\},\{g(k)\}]=g(3 k)=f\left(3 k-k_{d}\right) T[{0},{g(k)}]=g(3k)=f(3k−kd)
y z s ( k − k d ) = f ( 3 ( k − k d ) ) T [ { 0 } , { f ( k − k d ) } ] ≠ y z s ( k − k d ) y_{z s}\left(k-k_{d}\right)=f\left(3\left(k-k_{d}\right)\right) \quad T\left[\{0\},\left\{f\left(k-k_{d}\right)\right\}\right] \neq y_{z s}\left(k-k_{d}\right) \quad yzs(k−kd)=f(3(k−kd))T[{0},{f(k−kd)}]=yzs(k−kd) 所以是时变系统
若激励之前有变系数, 或反转、展缩变换, 则系统为时变系统
因果系统:当且仅当输入信号激励系统时, 系统才出现零状态响应输出的系统。即,系统的零状态响应不出现于激励之前。反之则为非因果系统。
1.令 t < t 0 时 f ( t ) = 0 t<t_{0}时f{(t})=0 t<t0时f(t)=0
2.将 t 0 t_0 t0代入式中
若 t < t 0 ( t<t_{0}\left(\right. t<t0( 或 k < k 0 ) \left.k<k_{0}\right) k<k0) 时, \quad 激励 f ( t ) ( f(\mathrm{t}) \quad( f(t)( 或 f ( k ) ) = 0 , f(\mathrm{k}))=0, f(k))=0, 则当 t < t 0 ( t<t_{0} \quad\left(\right. t<t0( 或 k < k 0 ) \left.k<k_{0}\right) k<k0) 时, y z s ( t ) = 0 ( \mathrm{y}_{\mathrm{zs}}(\mathrm{t})=0\left(\right. yzs(t)=0( 或 y z s ( k ) = 0 ) \left.\mathrm{y}_{\mathrm{zs}}(\mathrm{k})=0\right) yzs(k)=0)
(1) y z s ( t ) = f ( t − 1 ) y_{z s}(t)=f(t-1) yzs(t)=f(t−1)
(2) y z s ( t ) = f ( t + 1 ) y_{z s}(t)=f(t+1) yzs(t)=f(t+1)
(3) y z s ( t ) = f ( 2 t ) y_{z s}(t)=f(2 t) yzs(t)=f(2t)
(1) 若 t < t 0 时, f ( t ) = 0 , 有 y z s ( t 0 ) = f ( t 0 − 1 ) = 0 , 是因果系统 \begin{array}{ll}\text { (1) 若 } t<t_{0} \text { 时, } f(\mathrm{t})=0, & \text { 有 } \mathrm{y}_{\mathrm{zs}}\left(\mathrm{t}_{0}\right)=f\left(\mathrm{t}_{0}-1\right)=0, \text { 是因果系统 }\end{array} (1) 若 t<t0 时, f(t)=0, 有 yzs(t0)=f(t0−1)=0, 是因果系统
(2) 若 t < t 0 时, f ( t ) = 0 , 有 y z s ( t 0 ) = f ( t 0 + 1 ) ≠ 0 , 是非因果系统 \begin{array}{ll}\text { (2) 若 } t<t_{0} \text { 时, } f(\mathrm{t})=0, & \text { 有 } \mathrm{y}_{\mathrm{zs}}\left(\mathrm{t}_{0}\right)=f\left(\mathrm{t}_{0}+1\right) \neq 0, \text { 是非因果系统 }\end{array} (2) 若 t<t0 时, f(t)=0, 有 yzs(t0)=f(t0+1)=0, 是非因果系统
(3) 若 t < t 0 时, f ( t ) = 0 , 有y z s ( t 0 ) = f ( 2 t 0 ) ≠ 0 , 是非因果系统 \begin{array}{ll}\text { (3) 若 } t<t_{0} \text { 时, } f(\mathrm{t})=0, & \text { 有y }_{\mathrm{zs}}\left(\mathrm{t}_{0}\right)=f\left(2\mathrm{t}_{0}\right) \neq 0, \text { 是非因果系统 }\end{array} (3) 若 t<t0 时, f(t)=0, 有y zs(t0)=f(2t0)=0, 是非因果系统 (因为是 1 2 t 0 到 t 0 之 间 \frac{1}{2}t_0到t_0之间 21t0到t0之间?)
记忆系统又称为动态系统, 即系统的输出不仅与当前时刻的输入有关, 还
与过去/将来的输入相关。如, 含有电容、电感的系统
非记忆系统又称为即时系统, 即系统的输出仅与当前时刻的输入有关, 与
过去/将来的输入无关。如, 仅还有电阻的简单系统
某系统对于有界激励 f ( ⋅ ) f(\cdot) f(⋅) 产生的零状态响应 y z s ( ⋅ ) y_{z s}(\cdot) yzs(⋅) 也是有界时,称该系统 为有界输入有界输出系统, 简称为稳定系统。即若 ∣ f ( ⋅ ) ∣ < ∞ , |f(\cdot)|<\infty, ∣f(⋅)∣<∞, 有 ∣ y z s ( ⋅ ) ∣ < ∞ \left|y_{z s}(\cdot)\right|<\infty ∣yzs(⋅)∣<∞
的系统。否则,称为发散系统, 或不稳定系统。
y ( t ) = f ( t − 1 ) + f ( t + 2 ) y(t)=f(t-1)+f(t+2) \quad y(t)=f(t−1)+f(t+2) 稳定系统
y ( t ) = ∫ − ∞ t f ( x ) d x 发散系统 y(t)=\int_{-\infty}^{t} f(x) d x \quad \text { 发散系统 } y(t)=∫−∞tf(x)dx 发散系统
f ( t ) = ε ( t ) f(t)=\varepsilon(t) f(t)=ε(t) 有界, y ( t ) = t ε ( t ) y(t)=t \varepsilon(t) y(t)=tε(t) 无界