How do I take the complex convolution of this impulse response and input?

Question

I derived a an impulse response of $h[n] = (3/4) (-j3/4)^{n} u[n]$, where $u[n]$ is the unit step function. I have an input $x[n] = u[n-5]$. I can find a vector representation of the convolution of these two functions. But I this wouldn't be a closed form solution since the vector goes on infinitely. How can I find the closed form convolution of this function? I'm not sure how to graph it because it contains real and complex parts mixed in. thanks

Answer 1

A closed form solution is straightforward to obtain. You just have to actually do the computation.

Let $\theta = -i \frac{3}{4}$. (To avoid confusing myself, I will use $H$, as in Heaviside, to denote the unit step function.)

Then \begin{eqnarray} y_n = (h * x)_n &=& \sum_{k \in \mathbb{Z}} x_k h_{n-k} \\ &=& \sum_{k \in \mathbb{Z}} H_{k-5} \frac{3}{4} \theta^{n-k} H_{n-k} \\ &=& H_{n-5} \frac{3}{4} \sum_{k=5}^n \theta^{n-k} \\ &=& H_{n-5} \frac{3}{4} \sum_{k=0}^{n-5} \theta^k \\ &=& H_{n-5} \frac{3}{4} \frac{1-\theta^{n-4}}{1-\theta} \end{eqnarray}