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If the redundancy measure $A$ in [link] and [link] is one, the matrices must be square and the system has an orthonormal basis.
Frames are over-complete versions of non-orthogonal bases and tight frames are over-complete versions of orthonormal bases. Tight frames areimportant in wavelet analysis because the restrictions on the scaling function coefficients discussed in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients guarantee not that the wavelets will be a basis, but a tight frame. In practice, however, theyare usually a basis.
An example of an infinite-dimensional tight frame is the generalized Shannon's sampling expansion for the over-sampled case [link] . If a function is over-sampled but the sinc functions remains consistentwith the upper spectral limit $W$ , the sampling theorem becomes
or using $R$ as the amount of over-sampling
we have
where the sinc functions are no longer orthogonal now. In fact, they are no longer a basis as they are not independent. They are, however, a tightframe and, therefore, act as though they were an orthogonal basis but now there is a “redundancy" factor $R$ as a multiplier in the formula.
Notice that as $R$ is increased from unity, [link] starts as [link] where each sample occurs where the sinc function is one or zero but becomes an expansion with the shifts still being $t=Tn$ , however, the sinc functions become wider so that the samples are no longer at thezeros. If the signal is over-sampled, either the expression [link] or [link] could be used. They both are over-sampled but [link] allows the spectrum of the signal to increase up to the limit without distortion while [link] does not. The generalized sampling theorem [link] has a built-in filtering action which may be an advantage or it may not.
The application of frames and tight frames to what is called a redundant discrete wavelet transform (RDWT) is discussed later in Section: Overcomplete Representations, Frames, Redundant Transforms, and Adaptive Bases and their use in Section: Nonlinear Filtering or Denoising with the DWT . They are also needed for certain adaptive descriptionsdiscussed at the end of Section: Overcomplete Representations, Frames, Redundant Transforms, and Adaptive Bases where an independent subset of theexpansion vectors in the frame are chosen according to some criterion to give an optimal basis.
A powerful point of view used by Donoho [link] gives an explanation of which basis systems are best for a particular class of signals and whythe wavelet system is good for a wide variety of signal classes.
Donoho defines an unconditional basis as follows. If we have a functionclass $F$ with a norm defined and denoted $\left|\right|\xb7{\left|\right|}_{F}$ and a basis set ${f}_{k}$ such that every function $g\in F$ has a unique representation $g={\sum}_{k}{a}_{k}\phantom{\rule{0.166667em}{0ex}}{f}_{k}$ with equality defined as a limit using the norm, we consider the infinite expansion
If for all $g\in F$ , the infinite sum converges for all $|{m}_{k}|\le 1$ , the basis is called an unconditional basis . This is very similar to unconditional or absolute convergence of a numerical series [link] , [link] , [link] . If the convergence depends on ${m}_{k}=1$ for some $g\left(t\right)$ , the basis is called a conditional basis .
An unconditional basis means all subsequences converge and all sequences of subsequences converge. It means convergence does not depend on the orderof the terms in the summation or on the sign of the coefficients. This implies a very robust basis where the coefficients drop off rapidly forall members of the function class. That is indeed the case for wavelets which are unconditional bases for a very wide set of function classes [link] , [link] , [link] .
Unconditional bases have a special property that makes them near-optimal for signal processing in several situations. This property has to do withthe geometry of the space of expansion coefficients of a class of functions in an unconditional basis. This is described in [link] .
The fundamental idea of bases or frames is representing a continuous function by a sequence of expansion coefficients. We have seen that theParseval's theorem relates the ${L}^{2}$ norm of the function to the ${\ell}^{2}$ norm of coefficients for orthogonal bases and tight frames [link] . Different function spaces are characterized by different norms on the continuous function. If we have an unconditional basis for thefunction space, the norm of the function in the space not only can be related to some norm of the coefficients in the basis expansion, but theabsolute values of the coefficients have the sufficient information to establish the relation. So there is no condition on the sign or phaseinformation of the expansion coefficients if we only care about the norm of the function, thus unconditional .
For this tutorial discussion, it is sufficient to know that there are theoretical reasons why wavelets are an excellent expansion system for awide set of signal processing problems. Being an unconditional basis also sets the stage for efficient and effective nonlinear processing of thewavelet transform of a signal for compression, denoising, and detection which are discussed in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients .
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