Wavelet technology offers designers alternative to Fourier analysis.
The primary difference between Fourier and wavelet signal processing lies in the wavelet technique's ability to analyze signals and images containing sharp changes and small or irregular details.
For continuously smooth signals, Fourier transform analyses are usually satisfactory. For piecewise smooth functions, however, a wavelet analysis is better, says Gilbert Strang, a researcher at the Massachusetts Institute of Technology, Cambridge, in his book Wavelets and Filter Banks.
The underlying assumption for Fourier transforms is that the signal being analyzed is infinite in length. A windowed Fourier analysis will lose resolution or blur, whereas a wavelet decomposition of these signals will provide good resolution at both high and low signal frequencies. "Its a nonstatistical method for analyzing signals that change over time," says Ken Karnofsky, a market-segment manager at The MathWorks, Natick, Mass. "Wavelet decompositions are particularly useful for signal compression and denoising."
While wavelet technologies have been around for more than 15 years, software implementations have only recently surfaced. One of these is the Wavelet Toolbox for use with The MathWorks' MATLAB numeric and visualization system.
Lance Martin, a principal at Martin Consulting, Ponce Inlet, Fla., uses the Toolbox to work on a classified high-resolution, wide-bandwidth radar system. "With a wide-bandwidth application, signal-processing designers have to deal with relativistic effects resulting from the compression or expansion of waveforms as they move forward or recede," he says. "Conventional fast Fourier transforms don't address this situation. As a result, you can obtain numerous approximations and resultant losses."
The inherent nature of the wavelet transforms, however, is to scale these waveform compressions and expansions in a time domain without the need for approximations. Martin cannot ensure that his wavelet designs will be totally successful, but "they will provide better performance with a higher bandwidth than a Fourier-based solution."
MIT graduate student Vance Bjorn has a more direct profit motive for his wavelet research. He's using the Wavelet Toolbox to extract components from stock-market data. "Wavelet techniques can be used to distinguish, in real time, the components pointing to short- and long-term stock traders," he says. Long-term traders include the likes of institutional traders, while short-term traders are generally floor traders or individuals.
"I plan to port the wavelet analysis to a neural network to obtain a continuous frequency spectrum of the stock-trader components," says Bjorn.
Wavelet techniques also work well in imaging applications. The FBI uses it to compress and efficiently store fingerprint data. The compressed image is almost indistinguishable from the original.
The easy-to-use Wavelet Toolbox can also be used to enhance edge detection in other image-processing applications, discover trends in noisy or faulty data, and study the fractal properties of signals and images. Other applications include speech and audio processing, image and video processing, geophysics, finance, and medicine.
The authors of the Wavelet Toolbox are members of the famed Laboratoire de Mathematique at the Univ. of France in Paris.
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|Title Annotation:||Software Review|
|Publication:||R & D|
|Date:||May 1, 1996|
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