Computing impulse response (help)

[steve910]

I am trying to find the impulse response h[n] of a LSI system: y[n]+ay[n-1]=bx[n]-cx[n-1]. The way that I think to solve this problem is compute the z-transform, and from this H(z)=Y(z)/X(z). Then take the inverse z-transform to find h[n]. Is there another way to compute this? Also, for H(z) I get (b-cz^(-1))/(1-az^(-1)). I am having trouble taking the inverse z-transform of this... I can separate the numerator into 2 fractions but I'm not sure if the inverse z-transform of (cz^(-1))/(1-az^(-1)) = c*a^n*u[n-1]?

 

[Valkarie]

Any help is appreciated!A:You can directly find the impulse response $h[n]$ from the difference equation, by using the fact that$$h[n]=y[n]\quad \textrm{for } n=0$$and$$h[n]=y[n]-ay[n-1]\quad \textrm{for } n\ge 1$$For the particular difference equation given by you, this results in$$h[n]=\begin{cases}\;b & \textrm{for } n=0\\ -c & \textrm{for } n=1\\ 0 & \textrm{else}\end{cases}$$