How to find out the transfer function of a FIR filter? $$h[n]=\begin{cases}a^n &amp; \text{if } 0 \le n &lt; N \\
0 &amp; \text{otherwise}\end{cases}$$
And for which values of $a$ the filter is stable 

I know that the transfer function will be 

$$H(z)=\frac{z}{z-a}~,~|z|&gt;a$$ 

how to find out the values of $a$ for it's stability ?
\begin{align}
y[n] &amp;= h[n]  * x[n]\\
&amp;= h[0]x[n] + h[1]x[n-1] + \ldots + h[N-2]x[n-(N-2)] + h[N-1]x[n-(N-1)]\\
&amp;= a^{0}x[n] +a^{1}x[n-1] + \ldots + a^{N-2} x[n-(N-2)] + a^{N-1} x[n-(N-1)]
\end{align}
So

$$\mathcal Z\{y[n]\} = Y(z) = X(z)\left(1 + az^{-1} + ... + a^{N-2}z^{-(N-2)} + a^{N-1}z^{-(N-1)}\right)$$

and

$$H(z) = \dfrac{Y(z)}{X(z)} = 1 + az^{-1} + ... + a^{N-2}z^{-(N-2)} + a^{N-1}z^{-(N-1)}$$

For $a$ finite, $H(z)$ is finite and thus stable for finite input values.