Discrete-time Convolution Convergence Issue I'm trying to solve a problem on convolution from Alan V.Oppenheim:
Find the convolution output $y[n]$ for the following signals:
$$x[n]= u[n]\quad\text{and}\quad h[n]=a^{-n}u[-n-1], \ a&gt;1 $$
I started the evaluation:
$$y[n]=\sum_{k=-\infty}^{+\infty} u[k]a^{-n+k}u[-n+k-1]$$
considering that $u[k]=1$ for $k\ge0$ and $u[-n+k-1]=1$ for $k\ge n+1$
which I evaluated to
$$y[n]=a^{-n}\sum_{k=n+1}^{+\infty} a^k$$ where $n$ could be $&lt;0$ or $&gt;0$
and I tried to evaluate for $n&gt;0$:
where I faced an issue:
I subsittuted $k-1$ to $m$ and reframed the equation as:
$$y[n]=a^{-n+1}\sum_{m=n}^{+\infty} a^m=a^{-n+1}\left[\left(\sum_{m=0}^{+\infty} a^m\right)-\left(\sum_{m=0}^{n-1} a^m\right)\right]$$
now for $a&gt;1, \ \displaystyle \sum_{m=0}^{+\infty} a^m$ will not converge. How will I evaluate this?
In the third edition of Oppenheim and Schafer's Discrete-Time Signal Processing, the corresponding exercise is given with $h[n]=a^nu[-n-1]$, $a&gt;1$ (note the positive sign in the power of $a$). In that case everything works out fine. So either there's a typo in your edition or you made a mistake copying the exercise.