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Multiscale difference equation signal models. I. Theory
Ali, M.; Tewfik, A.H.;
Signal Processing, IEEE Transactions on [see also Acoustics, Speech, and Signal Processing, IEEE Transactions on]
Volume 43,
Issue 10,
Oct. 1995
Page(s):2332
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2345
Abstract:
The paper studies multiscale difference equation models for l-D and M-D signals. In this modeling technique, the signal of interest is viewed as a solution to a multiscale difference equation (MSDE). The model completely characterizes the signal as well as a number of its higher derivatives. It provides a recursive signal interpolation scheme as a function of scale. It also leads naturally to multigrid signal filtering, detection and estimation algorithms. An MSDE model must be uniquely decodable, i.e., it must correspond to a unique signal. Therefore, one must guarantee that the modeling MSDE has a unique solution. The authors investigate the existence and uniqueness of L1 and L2 solutions-to multiscale difference equations. Using Fourier domain techniques, they derive conditions for the existence of L1 solutions to an MSDE. They provide conditions under which the L1 solution is unique (up to a multiplicative constant) and has compact support. They also derive sufficient, but not necessary, conditions for the existence of a unique L2 solution to a subclass of MSDEs. The results extend known facts about the solutions of two-scale difference equations. The paper concludes with several examples of MSDE signal models that highlight the modeling advantages of MSDEs over two-scale difference equation models
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