A impulse response for a LTI system is given by:
$$h[n]=\left(\frac{2}{3}\right)^n u[n]+2 \left(\frac{1}{5}\right)^n u [n]$$
and if the putput for the system is given by:
$$y[n]= \left(\frac{1}{3}\right)^n u [n] $$
- What is the input in the $z$-domain ($X(z)$)?
- And what is the output in the time-domain ?
I'm not sure what is the correct procedure for solving this type of problem. Can you just simply transform $y[n]$ into $Y(z)$ and then solve for $X(z)$?
$$\frac{Y(z)}{X(z)}=\frac{3-\frac{23}{12}z^{-1} }{\left(1-\frac 23z^{-1}\right)\left(1-\frac 15z^{-1}\right) }$$
Then I replace $ Y(z)$ with $\frac{1}{1-1/3z^{-1}}$ and then I isolate $ X(z)$.
$$\frac{\frac{1}{1-\frac 13z^{-1}}}{X(z)}=\frac{3-\frac{23}{12}z^{-1} }{\left(1-\frac 23z^{-1}\right)\left(1-\frac 15z^{-1}\right)}.$$
Then I get $X(z)=\frac{6z^-2-39z^{-1}+45}{23z^{-1}-11z^{-1}+135}$ and then I transform it into a sequence in the time domain.
Is it right? or is there a better method for solving this.