Abstract
The theory of orthogonal multiwavelets offers enhanced flexibility for signal processing applications and analysis by employing multiple waveforms simultaneously, rather than a single one. When implementing them with polyphase filter banks, it has been recognized that balanced vanishing moments are needed to prevent undesirable artifacts to occur, which otherwise compromise the interpretation and usefulness of the multiwavelet analysis. In the literature, several such balanced orthogonal multiwavelets have been constructed and published; but however useful, their choice is still limited. In this work we present a full parameterization of the space of all orthogonal multiwavelets with two balanced vanishing moments (of orders 0 and 1), for arbitrary given multiplicity and degree of the polyphase filter. This allows one to search for matching multiwavelets for a given application, by optimizing a suitable design criterion. We present such a criterion, which is sparsity-based and useful for detection purposes, which we illustrate with an example from electrocardiographic signal analysis. We also present explicit conditions to build in a third balanced vanishing moment (of order 2), which can be used as a constraint together with the earlier parameterization. This is demonstrated by constructing a balanced orthogonal multiwavelet of multiplicity three, having three balanced vanishing moments, but this approach can easily be employed for arbitrary multiplicity.
Highlights
Wavelets [1, 2] are a popular signal processing tool, able to provide a time-frequency representation of signals
Compact support translates into finite impulse response (FIR) filters, whereas orthogonality is captured conveniently when switching from a filter bank
By inspecting what happens for balanced orthogonal multiwavelets generated from scalar orthogonal wavelets with vanishing moments through shifting, just as explained earlier for the Haar multiwavelet, we find that the value of λ varies between multiwavelets
Summary
Wavelets [1, 2] are a popular signal processing tool, able to provide a time-frequency representation of signals. Though there are various viewpoints on wavelets, here we will restrict ourselves to wavelets from filter banks Within this class, various desirable properties are possible to achieve, e.g. orthogonality, linear phase, compact support, symmetry and vanishing moments. In this paper we will develop such a parameterization for orthogonal multiwavelets with compact support based on lossless systems, with an arbitrary multiplicity r We will show in this paper how all balanced orthogonal multiwavelets of orders 0 and 1 can be obtained for arbitrary multiplicity r and any given polyphase filter order (McMillan degree), which we consider a major step forward in making multiwavelets applicable
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