Cascaded One-Zero Filter IR

Question

given the one-zero digital filter,

$$y[n] = x[n]\cdot \frac{1}{2} + x[n-1] \cdot \frac{1}{2}$$

Which has the transfer function:

$$H(z) = (1+z^{-1})\cdot \frac{1}{2}$$

and taking $M$ of these filters in series as a cascade, giving a filter $G$ with

$$G(z) = H(z)^{M}$$ Is there a known closed form of G's impulse response?

I have calculated one and am not sure if it's documented somewhere. I know we can take the inverse fourier transform of $G(z)$, but we arrive at an integral/sum. I wonder about a closed form.

Thank you very much.

Answer 1

This is called a binomial filter. Its transfer function can be written as

$$H(z)=\frac{1}{2^M}\sum_{n=0}^M{M\choose n}z^{-n}\tag{1}$$

which gives

$$h[n]=\frac{1}{2^M}{M\choose n},\qquad 0\le n\le M\tag{2}$$

for its impulse response.

As mentioned in a comment, such a filter can be used to approximate a Gaussian filter. Also take a look at this related question and its answers.