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It has been shown that the thresholding of wavelet coefficients has near optimal noise reduction property for many classes of signals [link] . The thresholding scheme used in the approximation in the proposed FAFT algorithm is exactly the hard thresholding schemeused to denoise the data. Soft thresholding can also be easily embedded in the FAFT. Thus the proposed algorithm also reduces the noise whiledoing approximation. If we need to compute the DFT of noisy signals, the proposed algorithm not only can reduce the numerical complexity but alsocan produce cleaner results.
In the past, the FFT has been used to calculate the DWT [link] , [link] , [link] , which leads to an efficient algorithm when filters are infinite impulse response (IIR). Inthis chapter, we did just the opposite – using DWT to calculate FFT. We have shown that when no intermediate coefficients are dropped and noapproximations are made, the proposed algorithm computes the exact result, and its computational complexity is on the same order of the FFT; i.e., $O\left(N{log}_{2}N\right)$ . The advantage of our algorithm is two fold. From the input data side, the signals are made sparse by the wavelet transform, thusapproximation can be made to speed up the algorithm by dropping the insignificant data. From the transform side, since the twiddle factors of the new algorithm have decreasingmagnitudes, approximation can be made to speed up the algorithm by pruning the section of the algorithm which corresponds to the insignificant twiddle factors. Since wavelets are an unconditionalbasis for many classes of signals [link] , [link] , [link] , the algorithm is very efficient and has built-in denoising capacity.An alternative approach has been developed by Shentov, Mitra, Heute, and Hossen [link] , [link] using subband filter banks.
Wavelets became known to most engineers and scientists with the publication of Daubechies' important paper [link] in 1988. Indeed, the work of Daubechies [link] , Mallat [link] , [link] , [link] , Meyer [link] , [link] , and others produced beautiful and interesting structures, but many engineers and applied scientist felt they had a“solution looking for a problem." With the recent work of Donoho and Johnstone together with ideas from Coifman, Beylkin and others, the fieldis moving into a second phase with a better understanding of why wavelets work. This new understanding combined with nonlinear processing not only solves currently important problems, but gives the potential offormulating and solving completely new problems. We now have a coherence of approach and a theoretical basis for the success of our methods thatshould be extraordinarily productive over the next several years. Some of the Donoho and Johnstone references are [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] and related ones are [link] , [link] , [link] , [link] , [link] . Ideas from Coifman are in [link] , [link] , [link] , [link] , [link] , [link] , [link] .
These methods are based on taking the discrete wavelet transform (DWT) of a signal, passing this transform through a threshold, which removes thecoefficients below a certain value, then taking the inverse DWT, as illustrated in [link] . They are able to remove noise and achieve high compression ratios because of the “concentrating" ability ofthe wavelet transform. If a signal has its energy concentrated in a small number of wavelet dimensions, its coefficients will be relatively largecompared to any other signal or noise that has its energy spread over a large number of coefficients. This means that thresholding or shrinkingthe wavelet transform will remove the low amplitude noise or undesired signal in the wavelet domain, and an inverse wavelet transform will thenretrieve the desired signal with little loss of detail. In traditional Fourier-based signal processing, we arrange our signals such that thesignals and any noise overlap as little as possible in the frequency domain and linear time-invariant filtering will approximately separatethem. Where their Fourier spectra overlap, they cannot be separated. Using linear wavelet or other time-frequency or time-scale methods, onecan try to choose basis systems such that in that coordinate system, the signals overlap as little as possible, and separation is possible.
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