Eigenfunction of LTI causal system Z-transform I am studying for my DSP final and I came across this question from the Oppenhiem and Schafer book 3rd edition. The question says


  3.18 A casual LTI system has the system function
  
  $$H(z)=\dfrac{1+2z^{-1}+z^{-2}}{\left(z+\frac{1}{2}z^{-1}
&gt; \right)\left(1-z^{-1} \right)} $$
  
  (a) Find the impulse response of the system, $h[n]$
  
  (b) Find the output of this system, $y[n]$, for the input
  
  $$x[n]=e^{j(\pi/2)n} $$


For the first part, I got
$$h[n] =  -2 \delta[n]+\frac{1}{3}  \left(\frac{-1}{2}\right)^2 u[n]+\frac{8}{3} u[n]$$   

which matches the answer at the back of the book, but the second part is difficult. We are asked to find the function $y[n]$ if $x[n] = 2^n$

I know that $2^n$ is an eigenfunction of $y[n]$, but I am not sure how to substitute it in $H(e^{j\omega})$. The answer at the back of the book is $y[n] = \frac{18}{5} 2^n$

Can anyone help me? Thanks in advance
Hint: $y[n] = H(e^{j (\pi / 2)}) x[n]$. 

Since $x[n]$ is a complex exponential, it is an eigenfunction of a LTI system with eigenvalue given by the frequency response at the frequency of $x[n]$ (which in this case is $H(e^{j (\pi/2)})$. In less fancy words, if you put a complex exponential into a LTI system, you get the same complex exponential out, except scaled by the transfer function evaluated at the frequency of the complex exponential.