could be fractal (since Tn0 f(t) converges only in L2). One way to prevent this is to impose K-regularity - one forces H0(z) to be of the form

H0(z) =

?1 + z?1 + : : : + z?(M?1)

M

?K

Q(z): (11)

for maximal possible K (and Q(z) a polynomial in the FIR

case). The autocorrelation (H0(z)H0(z?1)) of such scaling filters that also satisfy Eqn. 3 can be explicitly obtained [12, 27, 53, 19]. In this case, one obtains all wavelet filters from Eqn. 6. Also one can solve numerically for the vi's and V0 that satisfy Eqn. 11 to obtain K-regular WTFs. Regularity as defined here is a property of h0, not a statement that (t)

(and hence the wavelets) are in CK?1. Practically it means

that H0(!) is (K ? 1)th order flat at ! = 0, or equivalently the scaling function and its translates represent polynomials of degree (K ? 1) exactly.

WTFs have a 1-1 correspondence to unitary FBs with

H0(!) = pM . This suggests that one can impose this linear constraint on other types of unitary FBs to get corresponding WTFs. For unitary MFBs this demands that P
k ?l;k = ss4 + ss2M ( 2 ? l), f?l;kg is the set of lattice angle

parameters of the lth two-channel unitary FIR FB [23, 22]. WTFs corresponding to unitary FBs with symmetry restrictions (like linear-phase etc.) can also be obtained [52, 24].

4. FB TREES AND WAVELET PACKETS

M-band WTFs give a flexible tiling of the TF plane. They are associated with a particular tree-structured FB, where the lowpass channel at any depth is split into M bands. Clearly, an arbitrary tree-structured filter bank, where all channels are split into sub-channels (using FBs with potentially different number of bands), would give a very flexible signal decomposition. The wavelet analog of this is known as the wavelet packet decomposition (WPD) [10]. For a given signal (or class of signals), one can (for a fixed set of filters) obtain the best (in some sense) FB tree-topology. For a binary tree an efficient scheme using entropy as the criterion has been developed - the best wavelet packet basis algorithm [10].

5. TIME-VARYING FBs AND LOCAL BASES

FB trees (wavelet packets) afford flexible decompositions of the TF plane. However, all TF tiles at a particular frequency level have the same time-resolution. The frequency-axis in the TF plane is split arbitrarily (by the choice of tree-topology) while the time-axis split is governed by the sampling rate in each channel (which is constant in a particular channel). Timevarying FBs (wavelet analogs being local sine/cosine bases [1]) are useful when the time-axis is to be split arbitrarily (the frequency-axis split being constant bandwidth). Here the time-axis is split into bins, and in each bin a certain FB is used. By combining the two dual concepts of FB trees, and time-varying FBs one can obtain arbitrary tiling of the TF plane [28, 29, 30]. One can use Eqn. 5 to design and implement time-varying filter bank trees [21].

We now give a list of FB structures and their TF analogs. A single FB (many channels) is like the DSTFT, a lowpass recursive FB tree gives the DWT, an arbitrary FB tree gives the WPD, time-varying FBs give local bases and arbitrary time-varying FB trees give arbitrary TF decompositions. Orthonormality of all these decompositions can be obtained by allowing only unitary FBs.

6. GENERALIZATIONS

Just as unitary FBs give ON wavelet bases, so also general PR filter banks give rise to wavelet biorthogonal bases [9, 58, 20]. Multi-d unitary FBs give ON wavelet bases on IRn [34].

7. CONCLUSION

Unitary FBs with an additional linear constraint give rise to ON wavelet bases. Unitary FB trees give rise to wavelet packet bases. Time-Varying FBs give rise to local bases. TimeVarying FB trees give arbitrary tiling of the TF plane. The advantage of using these techniques for TF decompositions is that unitary FBs are easy to design and implement. Other techniques to tile the TF plane also do exist: for example, see the matching pursuits algorithm in [37].

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