For decades engineers used techniques to analyse a signal of interest by transforming it into time domain or frequency (FT) domain. Short Term (windowing) Fourier Transform (STFT) attempts to combine both domains, but due to:

· Heisenberg's Uncertainty Principle: the transformation ends up with either poorer frequency resolution or poorer time resolution.

· The windowing dilemma: the difficulty to decide the window size ends up with a messy framework.

· There is no algorithm to reconstruct the original signal!

STFT proved to be unrealistic solution to multiresolution transformation.

Wavelets (WT) allow us to study the signal in time (space) and frequency (scale) domains combined, therefore it is an ideal multiresolution transform for non-periodic or non-stationary signals.It is the only technique that able to extract and show any number of fundamental frequencies in a signal over a period of time.

It offers excellent features for quite few applications. For example in image, speech, and music applications, WT already has proved to be the natural signal compression technique. Other applications that implement WT are the astronomy, acoustics, nuclear engineering, sub-band coding, neurophysiology, magnetic resonance imaging, optics, fractals, turbulence, earthquake-prediction, radar, and human vision.

Next section provides a brief description to the required mathematical formulas for WT. These formulas have been programmed using the assembly language for the SHARC processor as described in "The Program on ADSP-21065L" section. The "The required Hardware" section outlines the main components to implement WT in any application. The performance in terms of the number of cycles to execute this program and the memory size required for WT has been given in the last section. We will publish these details gradually, so please revisit this site and send your constructive comments.

For obvious reasons, WT has been explained in Textbooks, different articles and some web sites proceeded by the FT discussion. Here, we are not going to discuss the FT, DFT & FFT, simply because:

· Fourier Transforms (FT, DFT and FFT) are subsets of WT. Therefore, if you are new to the FT, and interested to implement WT, then you are not going to miss much from the FT!

· If the signal of interest is non-stationary, the WT is the efficient transform, and

· The aim of this article is not to compete with FFT.

The required mathematical formulas to implement WT are listed here.

1) The following function:

is called Continues Wavelets Transform (CWT), where:

W(a,b) is the output wavelets transform,

f(t) is the input function,

Y* is the Mother wavelet function & the (*) indicates the complex conjugate,

a is the scale (dilation) value,

b is the time (translation) value, and

is the energy normalising factor (to insure that the energy stays the same for all a and b).

2) The Continues Wavelet Transform (CWT) of a discrete sequence xb is defined as the convolution of xb with a scaled and translated version of :

where dt is the equal time spacing,

Y* is the Mother wavelet function & the (*) indicates the complex conjugate,

a is the scale (dilation) value,

b is the time (translation) value, and

is the energy normalising factor (to insure that the energy stays the same for all a and b).

More details on the Subband Coding and the Multiresolution Analysis will be published soon...

Will be published soon!

Will be published soon!

Will be published soon!

http://www.public.iastate.edu/~rpolikar/WAVELETS/waveletindex.html

http://www.amara.com/IEEEwave/IEEEwavelet.html

http://www.wavelet.org/wavelet/index.html

http://www.ee.umn.edu/groups/msp/index.html