信号与系统(Python) 学习笔记 (7) 电路与系统函数

求得: [ ∑ i = 0 n a i s i ] Y ( s ) − ∑ i = 0 n a i [ ∑ p = 0 i − 1 s i − 1 − p y ( p ) ( 0 − ) ] = [ ∑ j = 0 m b j s j ] F ( s ) \big[\sum^{n}_{i=0}a_i s^i\big]Y(s) - \sum^{n}_{i=0}a_i \big[\sum^{i-1}_{p=0}s^{i-1-p} y^{(p)}(0_-)\big] = \big[\sum^{m}_{j=0}b_j s^j\big]F(s) [i=0∑n​ai​si]Y(s)−i=0∑n​ai​[p=0∑i−1​si−1−py(p)(0−​)]=[j=0∑m​bj​sj]F(s)

Y ( s ) = ∑ i = 0 n a i [ ∑ p = 0 i − 1 s i − 1 − p y ( p ) ( 0 − ) ] ∑ i = 0 n a i s i Y z i ( s ) + ∑ j = 0 m b j s j ∑ i = 0 n a i s i F ( s ) Y z s ( s ) = M ( s ) A ( s ) + B ( s ) A ( s ) F ( s ) \begin{aligned}Y(s) & = \displaystyle \overset{{\color{blue}Y_{zi}(s)}}{\displaystyle\frac{\displaystyle\sum^{n}_{i=0}a_i \big[\displaystyle\sum^{i-1}_{p=0}s^{i-1-p} y^{(p)}(0_-)\big]}{\displaystyle\sum^{n}_{i=0}a_i s^i}} + \overset{{\color{blue}Y_{zs}(s)}}{\displaystyle\frac{\displaystyle\sum^{m}_{j=0}b_j s^j}{\displaystyle\sum^{n}_{i=0}a_i s^i}F(s)} \\ & = \displaystyle \frac{M(s)}{A(s)} + \frac{B(s)}{A(s)}F(s)\end{aligned} Y(s)​=i=0∑n​ai​sii=0∑n​ai​[p=0∑i−1​si−1−py(p)(0−​)]​Yzi​(s)​+i=0∑n​ai​sij=0∑m​bj​sj​F(s)Yzs​(s)​=A(s)M(s)​+A(s)B(s)​F(s)​

Y z i ( s ) + Y z s ( s ) = Y ( s ) → y ( t ) = y z i ( t ) + y z s ( t ) {\color{blue} Y_{zi}(s) + Y_{zs}(s) = Y(s) \to y(t) = y_{zi}(t) + y_{zs} (t) } Yzi​(s)+Yzs​(s)=Y(s)→y(t)=yzi​(t)+yzs​(t)