# Wavelets

Wavelets for Audio Measurement Analysis

And the veil was lifted and the wavelet paradise appeared before our eyes...

Intuitive presentation how wavelet plot is generated

## Work Horse Impulse Responses

These impulse responses are used here to demonstrate different wavelets

Ideal impulse response:

download WAV file

Impulse response of consecutive ideal reflections in 1 ms intervals. Each reflection is half the amplitude of the previous one.

download WAV file

## Wavelet CSD (Cumulative Spectral Decay)

This wavelet CSD is similar to the conventional cumulative spectral decay or waterfall plot, but it uses a Gaussian envelope to maximise the resolution in the time-frequency domain.

Wavelet CSD on ideal impulse response:

Wavelet CSD on impulse response of ideal reflections:

## Constant Q Wavelet

Constant Q wavelet has the property of having constant ratio of center frequency to bandwidth, or 'Q'. It follows that the number of periods at every frequency is constant. This is the traditional wavelet.

Constant Q Wavelet on ideal impulse response:

Constant Q Wavelet on impulse response of ideal reflections:

## Bark Wavelet

Bark wavelet is based on the psychoacoustic Bark scale. It is relevant in the human psychoacoustics that it matches to the bandwidth (critical bandwidth) of the basilar membrane in cochlear in the inner ear.

Here the wavelet is defined by Gaussian envelope and the bandwidth equivalent to the critical bandwidth:

Bark Wavelet on ideal impulse response:

Bark Wavelet on impulse response of ideal reflections:

## Multiresolution Wavelets

Wavelets have tradable resolution in the time-frequency domain. Multiresolution wavelet gives multiple of resolutions at once. A sort of all-in-one.

Here is an example of time-frequency trading of wavelet CSD. Wouldn't it be nice to have all of these in one plot? That is what multiresolution does.

### Multiresolution Wavelet CSD

Multiresolution wavelet CSD on ideal impulse response:

Multiresolution wavelet CSD on impulse response of ideal reflections:

### Multiresolution Constant Q Wavelet

Multiresolution constant Q wavelet on ideal impulse response:

Multiresolution constant Q wavelet on impulse response of ideal reflections:

### Multiresolution Bark Wavelet

WTF? But Bark wavelet has only one resolution, that defined by the Bark scale.

Well, I took some liberties. Because I can...

Multiresolution Bark wavelet on ideal impulse response:

Multiresolution Bark wavelet on impulse response of ideal reflections:

Free software tools for wavelet analysis. Yes it is free.

Last modified 10.8.2013

© Elias Pekonen