## 2010-11-27

### Filter design (1)

This is about filter design of Hamming's digital filter. The chapter 3.9 is outstanding, so I would like to make a note.

While ago there was a soccer game. How to suppress the Vuvuzela sound (but other sounds should pass) was a popular topic at that time. In this case, you analyze the frequency of Vuvuzela sound and design a filter which doesn't pass that frequency. The design of digital filter is: which frequency of the signal will be passed and which frequency of the signal will not be passed.

See the following article for instance, http://blogs.mathworks.com/loren/2010/06/30/vuvuzela-denoising-with-parametric-equalizers/

In chapter 3.9, a simple non recursive filter design is described as an example.

The filter looks like the following.
This filter uses three samples. The design of filter is to determine the a,b to fit to your desire. First, we will calculate the transfer function of this filter. As I explained in a former blog post, a transfer function is an eigenvalue of the filter function. Thus, it shows the output of the input of filter F. Here the input signal x (this is a function of time t, x(t)) and the filter function is F, then the eigenvalue λ is F x = λ x. If you are familiar with linear algebra, you can consider the F as a matrix A, a scalar λ, then you will see that is the same as A x = λ x. Now we compute the transfer function:
Therefore, transfer function is
If you want to make your filter passing the π/3 frequency (=1) and you want to cut 2π/3 frequency (=0), we set this transfer function under that condition.
By solving the system, we get

The filter is:

We design a digital filter, but, it is still not clear what this is. So, next time, I will input the actual signal into this filter and see how this filter works.