###
** R. Brigola
**

### Some Examples on Engineering Mathematics with Matlab

We discuss some examples of discrete denoising filters against the Vuvuzela noise
in recordings

of a soccer game during the World-Cup 2010 in South-Africa and listen to the corresponding denoising effects.

We implement a discrete 2nd order notch and a discrete 3rd order Butterworth lowpass filter.

The notch filter is repeatedly used to eliminate several noise frequencies.

The notches in the frequency response of the filters are chosen for example at the Vuvuzela frequencies

233 Hz, 466 Hz (Octave), 699 Hz (Quint) and 932 Hz (Octave).

With a Butterworth lowpass cutoff frequency at 2000 HZ
and dc-gain 2 (to get back signal energy lost by filtering),

that types of "Anti-Vuvuzela-Filtering" have results as in the following examples

Original Recording with Vuvuzela Noise

Filtered Version of that Recording

A Second Original Recording with Vuvuzela Noise for Tests

These recordings were made with the following m-files

A demo m-file that realizes the filtering difference equations for a recording with Vuvuzela noise

m-file, that computes the coefficients of a 2nd order discrete notch filter

m-file, that computes the coefficients of a 3rd order Butterworth lowpass filter

m-file, that computes the output of a notch biquad

m-file, that computes the output of a 3rd order IIR filter

A C++-file, which allows to generate the filter function "myfilter" with MEX

filter function "myfilter" - analogous to Matlab's filter function - generated with mex (platform dependent !)

m-file, that computes a vector of complex exponentials at sampling frequencies

The absolute amplitude distortions of the filters with the lowpass cutoff frequency at 2500 Hz is shown in the following figure

Students should proceed with a similar test by writing and applying a discrete parametric equalizer instead of a notch filter

or by writing for tests more sophisticated denoising algorithms like time-frequency block-thresholding or wavelet methods

with adaptive block attenuation. We will talk on those methods and their mathematical requisites in our lessons.