I am given the following signals :
input signal
$x_{[n]} = (\frac{1}{2})^nu_{[n]} + 2^nu_{[-n-1]}$
The signal apparetnly starts in $ n=-\infty$ and goes to $n=\infty$.
system output
$y_{[n]} = 6(\frac{1}{2})^nu_{[n]} - 6(\frac{3}{4})^nu_{[n]}$
The response starts at $n=0$.
The question is to state the impulse response $h_{[n]}$ of such system.
my question
How to understand a system that "tolerates" infinite input from $n=-\infty$ to $n=-1$ and only then, at $n=0$ starts to respond ? For me this seems somehow flawed.
analysis
I have tried to calculate z-transform, to obtain $H_{(z)}$, I believe I can get rid of the same roots and zeros and thus arriving at :
$H_{(z)} = \frac{Y_{(z)}}{X_{(z)}} = \frac{1-2z^{-1}}{1-(\frac{3}{4})z^{-1}}$
Which still somehow doesnt provide me with much choice for inverse z-transform.