Using an impulse response and an output to find an input using Z-transform

[ztjust]

Homework Statement

An LTI digital system with impulse response h[n] = 2(1/4)^n u[n-1] produces an output y[n] = (-3)^n u[n-1]. Determine the corresponding input x[n].

Homework Equations

I believe that Y[z] = X[z]H[z]

The Attempt at a Solution

So I started by finding the z-transform of h[n]. Which I think is (0.25z)/(z-0.25)^2.

I then went on the find to the z-transform of y[n]. Which I think is z/(z+3).

So then I divided them and got 4((z-0.25)^2) / (z+3)..

I can't figure out how to take the inverse z-transform to get x[n]. I think my z-transforms might be wrong, or I'm just dividing incorrectly.

 

[KataKoniK]

Please help!

Your approach is correct, but there are a few small errors in your calculations. The z-transform of h[n] is actually (0.5z)/(z-0.25), and the z-transform of y[n] is (z-3)/(z+3). When you divide them, you should get 0.5(z+3)/(z-0.25)^2. To take the inverse z-transform, you can use the partial fraction decomposition method, which involves breaking the z-transform into simpler fractions and then finding the inverse z-transform of each term. Alternatively, you can use the inverse z-transform table to find the corresponding time-domain signal. I hope this helps. Good luck with your calculations!