WAVELETS

THE WAVELET HISTORY

Pierre Simon Laplace imagined a single formula that would describe the motions of every object in the universe, for all time. He said “ An intelligence that knew, for a given instant, all the forces that animate nature and respective states of everything that composes it, and that in addition was vast enough to submit these facts to analysis, would encompass in the same formula all the movements of the largest bodies in the universe and those of the lightest atom. Nothing would be uncertain of it, and future, like past, would be present before our eyes. ”

The optimism foundered on the shoals of reality. Solving differential equation is not easy. So a scientist Jean Baptiste Joseph Fourier provided a practical way to extract the truth from a whole class of such equations; linear partial differential equations.

There are two parts to fouriers contribution; mathematical statement and an explanation why the statement is useful. The mathematical statement is that any periodic function can be represented as a sum of sines and cosines- Fourier series.

The trick is to multiply sines and cosines by coefficient to change their amplitude and to shift them so they either add or cancel.

Fourier series or transform can be used to study natural phenomena such as heat, which was numerically impossible till then.

Fourier analysis translates the tantalizing curves of a signal, which contain all the information of the signal and hide it from our comprehension, into forms that makes sense, transforming a signal that varies with time into a new function, the fourier transform of the signal, which tells us how much of each frequency the signal contains.

Sampling Theorem

The basis for digital technology is given by the sampling theorem, proved by J. Whittaker and applied by Claude Shannon.

The theorem states that if the range of frequencies of a signal is measured in cycles per second is n, then the signal can be represented with complete accuracy by measuring its amplitude 2n times per second.

This is a remarkable theorem, Ordinarily continuous curve can be only approximately characterized by stating any finite number of points through which it passes, and an infinite number would in general be required to complete information about the curve. However, if the curve is band limited- composed of limited range of frequencies -it can be produced accurately from finite number of samples.

The sampling theorem has opened the door to digital technology: a sampled signal could be expressed in a series of digits.

Fueling the revolution was the fast fourier transform, a mathematical trick which catapulted the calculation of fourier transform from the horse buggy into supersonic jet.

Fourier analysis works with linear equations. A nonlinear problem tends to be much harder and is much less predictable: a small change in the input can bring about a big change in the output.

“One can say the great discovery of 19^th century is that the equations of nature were linear and the great discovery of 20^th century is they weren’t.”

Fourier analysis had great limits.

· The results obtained were not easy to interpret.

· It is not helpful when the frequencies are changing.

· It hides the information about time. The information of time is not destroyed but is buried deep in it.

· It is terribly vulnerable to errors.

In order to analyze a signal both in time and in frequency, we use Windowed Fourier Transform (WFT). The idea is to study the frequencies in the signal, segment by segment. The window that defines the size of the segment to be analyzed-and, which remains, constant-is a little piece of curve and this curve is filled with oscillating functions of different frequencies.

While in classical FT compares the entire signal successively to infinite sines and cosines of different frequencies, to see how much of each frequency it contains, WFT compares a segment of the signal to bits of oscillating curves, first of one frequency then the other and so on. When one segment is analyzed, the window slides along the signal, to analyze another segment.

This method imposes compromises. The smaller the window, the better you can locate sudden changes, but blinder to low frequency component of signal. These lower frequencies just don’t fit into the little window. If we choose a bigger window, we can see more of low frequencies, but we would do worse at “localizing in time.”

To overcome this Morlet the father of wavelets, instead of keeping the size of the window fixed and filling with oscillations of different frequencies, he did the reverse: he kept the number of oscillations in the window constant and varied the width of the window, stretching and compressing it. When the wavelet was stretched, the oscillation inside the window was stretched, decreasing the frequency, when he squeezed the wavelet; the oscillations inside were squeezed, producing higher frequencies.

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