Wavelet analysis
Any
engineer or mathematician will be familiar with fourier transforms which in engineering are used to
extract frequency information from an amplitude/time signal. The limitation of fourier is that it tells you
nothing about when in a signal the high or low frequency components are found.
Wavelet transformations provide information resolved in time as well as frequency.
My
understanding comes mainly from reading this site from
Rowan University’s electronics department.
A sample
waveform (mother wavelet ) is multiplied, integrated
and energy normalised across the input signal with its position along the input
signal time base being varied (time resolution) and with the width (scale) of
the mother wavelet being varied (frequency resolution).
The
transform is generally reversible so that the original signal can be recovered,
and like fourier a digital
version is available and actually requires fewer calculations than the fourier.
What I am
not clear on from a quick scan of the literature on the web is how you choose a
mother wavelet, clearly the function describing your wavelet has be tractable within
the maths of the transformation but there are several
possibilities. The abstract of Moortel et al suggests that a tighter wavelet (such as
the Paul)
will give better time resolution and a broader multipeaked
mother wavelet like the Morlet will give better
frequency resolution. This seems perfectly logical, but I wonder whether in
practice what you do is choose a mother wavelet whose shape most closely
matches that of the signal you are looking for in the data.