Finding linear convolution of two time series

Question

PROBLEM

Two real, causal time series

$$f_k = a^k \quad \text{and} \quad g_k = b^k$$

where $a \neq b$, $|a| < 1$, and $|b| < 1$, are given for $k = 0,1,2,...$

Find the linear convolution $h_n$ of the time series $f_k$ and $g_k$.

ATTEMPTED SOLUTION

I think perhaps my attempted solution here can be improved upon. We have from the definition:

$$h_n = \sum_{k=0}^{n} f_k g_{n-k} = \sum_{k=0}^{n} a^k b^{n-k} = b^n \sum_{k=0}^{n} \left(\frac{a}{b} \right)^k$$

If we let $R = \left(\frac{a}{b} \right)$, the last term can be written as:

$$h_n = b^n \cdot \frac{1 - R^{n+1}}{1 - R}$$

or

$$h_n = \frac{b^n - \frac{a^{n+1}}{b}}{1 - \frac{a}{b}}$$

$$h_n = \frac{b^{n+1} - a^{n+1}}{b-a}$$

I don't see how I can improve upon this answer though, but since we are given certain conditions in the problem, I have a feeling that it is possible to get a more elegant solution. I see that as $n \to \infty$, $h_n \to 0$, but is there any way I can "improve" on the answer for any $n$?

Answer 1

There is a small mistake in your solution. Here $f_k$ and $g_k$ are infinite duration signal, ie. $n \to \infty$. You made a mistake that you took n in $h_n$ and number of coefficients in $f_k$ or $g_k$ as same, but it is different.

$$h_n = \sum_{k=0}^{\infty} f_k g_{n-k} = \sum_{k=0}^{\infty} a^k b^{n-k} = b^n \sum_{k=0}^{\infty} \left(\frac{a}{b} \right)^k$$

$$h_n = b^n \cdot \frac{1}{1 - \frac{a}{b}}$$

$$h_n = \frac{b^{n+1}}{b-a}$$

If you want to finite the number of coefficients in $f_k$ and $g_k$, use other variables other than n.

Then n in the RHS of your equation will replace with the new variable.