Consider an IIR filter with impulse response $h[n] = (\frac{1}{3})^n u[n]$. The input signal $x[n] = 3^n u[-n - 1]$ is entered into the IIR filter to generate the output signal $y[n]$.
(a) (11%) Determine the transfer function $H(z)$ of the IIR filter. Remember to state the ROC of $H(z)$. Draw the pole-zero plot of $H(z)$, and explain whether the IIR filter is stable or not.
(b) (6%) Determine the z-transform $X(z)$ of the input signal $x[n]$. Remember to state the ROC of $X(z)$.
(c) (15%) Determine the output signal $y[n]$, i.e., give an expression for $y[n]$.
Useful z-transform results:
$\cdot a^n u[n] \leftrightarrow \frac{z}{1 - az^{-1}}$, ROC = $\{|z| > |a|\}$
$\cdot -a^n u[-n - 1] \leftrightarrow \frac{z}{1 - az^{-1}}$, ROC = $\{|z| < |a|\}$
