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Wavelets Signal Representations With Important Advantages

Mark O. Freeman

The goal, of course, is to get the most for the least work. In the signal analysis and processing world, this is strongly influenced by how one chooses to represent information. In 1822, Jean-Baptiste Joseph Fourier devised an excellent way to do this: represent a signal as the sum of its frequencies. Power spectra, carrier frequencies and bandwidths, clock frequencies, brain activity frequency bands—these global characterizations provide much information in a concise manner. Some of the power of this representation begins to fade, though, when one attempts to represent information that changes its character unpredictably during the course of the signal. The standard analogy is a musical score. A global representation like the Fourier transform will tell what notes were played somewhere within the piece of music, but the timing of the notes is buried in a complicated frequency representation. A better representation, like the musical score, tells what notes were played when.