P. Moulin and M.K. Mihcak, "Theory and Design of SignalAdapted FIR Paraunitary Filter Banks," IEEE Trans. Signal Processing, vol. 46, no. 4, pp. 920--929, April 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....classes C. We then use the geometric approach described in [1] to show that this algorithm always produces a PCFB if one exists. Further we show that in absence of a PCFB, the algorithm will be suboptimal for a large number of objectives. Since this was previously shown only by numerical examples [4], it provides a new insight into the problem. Work supported in part by the National Science Foundation under Grant MIP 0703755. 2. REVIEW OF PCFB OPTIMALITY Following [1] we define the subband variance vector associated with the system of Fig. 1, as the vector v = oe M Gamma1 ) ....

....for the minima over the set E (as opposed to S or co(S) however it is not known to the authors at this time whether there are numerical search procedures that can exploit this fact. 4. THE SEQUENCE OF COMPACTION FILTERS ALGORITHM This is an algorithm that has sometimes been proposed [3, 4] to find a good FB in classes C that need not necessarily have PCFB s. We first state the algorithm in a precise way that holds for any general class C. This will show that it produces FB s for which the corresponding subband variance vector is a corner of co(S) The optimality of the algorithm ....

P.Moulin and M.K.Mihcak, "Theory and Design of Signal-Adapted FIR Paraunitary Filter Banks," IEEE Trans. SP, vol. 46, no. 4, pp. 920-929, April 1998.

....nature of both the concave objective at hand and the set E (which depends on the FB class C and input psd at hand) This explains why these optimizations are usually analytically intractable. 3. 3 The sequential compaction algorithm This is an algorithm that has sometimes been proposed [20] [13] to find a good FB in classes C that may not have PCFB s. We first state the algorithm in a precise manner that holds for any general class C. We This is because for any compact convex set D, the set of extreme points is the closure of the set of exposed points [16] which by definition are ....

....Find a family of input spectra for which there is no PCFB for some general FIR class, say that of all FB s with a given bound N on the McMillan degree or order of the polyphase matrix. At present, a few such results are known for specific low values of the bound N , for isolated input spectra [13], 11] Even in these cases, the proofs of PCFB nonexistence need numerical optimizations. Further one of these, from [11] is suspect due to the assumption that the FB maximizing its largest subband variance must contain a FIR compaction filter. Some insight may possibly be obtained by analytical ....

P.Moulin and M.K.Mihcak, "Theory and Design of Signal-Adapted FIR Paraunitary Filter Banks," IEEE Trans. Signal Processing, vol. 46, no. 4, pp. 920-929, April 1998.

....It includes the conjecture of [24] proved in [16] as a special case where h i # h for all i. It also includes the minimization objective of [13] as a special case when h i (x) f i (b i )x for all i. Filter bank design for quantization error minimization has also been studied by Moulin et al. [18], 19] The earlier stated form g = f i (b i )# i of the error requires modification for biorthogonal FB s. In an important preprint [19] Moulin et al. study the minimization of this modified objective over the class of all (unconstrained) biorthogonal FB s, for a broad class of f i (b i ) ....

....of all (unconstrained) biorthogonal FB s, for a broad class of f i (b i ) The authors examine the role of the properties of the PCFB for the unconstrained orthonormal FB class C in this problem. It is also claimed that pre and post filters around such a PCFB yield the optimal solution. In [18], an algorithm is proposed for PCFB design for a certain class of FIR orthonormal FB s. It involves a compaction filter design followed by a KLT matrix completion, and will produce the PCFB (which is known to maximize coding gain) if it exists. However, it is shown numerically that the designed ....

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P.Moulin and M.K.Mihcak, "Theory and Design of Signal-Adapted FIR Paraunitary Filter Banks," IEEE Trans. Signal Processing, vol. 46, no. 4, pp. 920-929, April 1998.

....and synthesis sections, and arises in a communications context related to orthogonal frequency division multiplexing. A problem of considerable interest is that of finding the best orthonormal transform for signal representation for a particular task. For example, it has been addressed in [1, 2] when the P i of Fig. 1 are quantizers for signal compression, under the high bit rate quantization noise models. In this case, the FB that minimizes the mean square reconstruction error is one that minimizes the product of the variances of its subband signals. Here it is well known that within ....

P.Moulin and M.K.Mihcak, "Theory and Design of Signal-Adapted FIR Paraunitary Filter Banks," IEEE Trans. SP, vol. 46, no. 4, pp. 920-929, April 1998.

....v j = v for all j = 1,2, J. To this end, since a0 v0 j, we have c0 = v. Hence v j T1. This in turn leads to ax v, and hence to a v and so on; until finally v v for all j = 1,2, J. When the class has a PCFB, all corners of co(S) cor respond to the PCFB. Hence the algorithm of Section 4.1 always produces the PCFB, and is thus optimal for many problems [11. The vector va of (3) here has an additional property: If its entries are arranged in increasing order, then in fact it becomes the least vector in S in the dictio nary ordering on M.2 On the other hand, if a PCFB does not ....

....in increasing order, then in fact it becomes the least vector in S in the dictio nary ordering on M.2 On the other hand, if a PCFB does not exist, then there will be at least two corners that are not equivalent, i.e. whose coordinates are not permutations of each other. The algorithm of Section 4.1 produces one corner, but the minima could easily be at other non equivalent corners. Thus the algorithm could be suboptimum. To illustrate this point, consider the following hypo thetical example with M 3 channels: Let co(S) co(E) where the set E consists of vectors vx = 3,2,1)W,ve = ....

[Article contains additional citation context not shown here]

P. Moulin and M.K.Mihcak, "Theory and Design of Signal-Adapted FIR Paraunitary Filter Banks," IEEE Trans. $P, vol. 46, no. 4, pp. 920-929, April 1998.

....specific nature of both the concave objective at hand and the set (which depends on the FB class and input psd at hand) This explains why these optimizations are usually analytically intractable. C. The Sequential Compaction Algorithm This is an algorithm that has sometimes been proposed [20] [13] to find a good FB in classes that may not have PCFBs. We first state the algorithm in a precise manner that holds for any general class . We then show that it produces FBs for which the corresponding subband variance vector is an extreme point of co . We examine the optimality of the algorithm ....

....problem. Find a family of input spectra for which there is no PCFB for some general FIR class, say that of all FBs with a given bound on the McMillan degree or order of the polyphase matrix. At present, a few such results are known for specific low values of the bound , for isolated input spectra [13], 11] Even in these cases, the proofs of PCFB nonexistence need numerical optimizations. Further, one of these, from [11] is suspect due to the assumption that the FB maximizing its largest subband variance must contain an FIR compaction filter. Some insight may possibly be obtained by ....

P. Moulin and M. K. Mihcak, "Theory and design of signal-adapted FIR paraunitary filter banks," IEEE Trans. Signal Processing, vol. 46, pp. 920--929, Apr. 1998.

.... brickwall filters) was given by Rao and Pearlman [50] for the pyramid structure, and further results along those lines have been reported by Fischer [19] and de Queiroz and Malvar [16] The design of optimal signal adapted filter banks for FIR and IIR cases has also been addressed by Moulin et al. [42, 43] who also show how the results extend for the biorthogonal case. Several important results in this direction can be found in [44] 1.5. Scope and Outline Most of this paper is restricted to the case of uniform orthonormal filter banks. In Section 2 we give an overview of situations where ....

P. Moulin and M. K. Mihcak, Theory and design of signal adapted FIR paraunitary filter banks, IEEE Trans. Signal Process. 46 (1998), 920--929.

....of applications, as shown in Section VI. It includes the conjecture of [25] proved in [17] as a special case where for all . It also includes the minimization objective of [14] as a special case when for all . FB design for quantization error minimization has also been studied by Moulin et al. [19], 20] The earlier stated form of the error requires modification for biorthogonal FBs. In an important paper [20] Moulin et al. study the minimization of this modified objective over the class of all (unconstrained) biorthogonal FBs for a broad class of . The authors examine the role of the ....

....over the class of all (unconstrained) biorthogonal FBs for a broad class of . The authors examine the role of the properties of the PCFB for the unconstrained orthonormal FB class in this problem. It is also claimed that pre and post filters around such a PCFB yield the optimal solution. In [19], an algorithm is proposed for PCFB design for a certain class of FIR orthonormal FBs. It involves a compaction filter design followed by a KLT matrix completion and will produce the PCFB (which is known to maximize coding gain) if it exists. However, it is shown numerically that the designed ....

[Article contains additional citation context not shown here]

P. Moulin and M. K. Mihcak, "Theory and design of signal-adapted FIR paraunitary filter banks," IEEE Trans. Signal Processing, vol. 46, pp. 920--929, Apr. 1998.

....The reduction of the execution time is achieved by reducing the original data set by a factor of four, and by placing a larger portion (if not all the data) of the reduced data set in the internal memory. Some previous work in this area has been done using different approaches. For example in [5] a similar problem is formulated with respect to the product filter in the frequency domain. In this formulation every locally optimal solution is a global one. However, this optimization problem has a linear objective function with infinitely many linear constraints. The optimization technique ....

P. Moulin and M. Mihcak, "Theory and Design of Signal-Adapted FIR Paraunitary Filter Banks", IEEE Transactions on Signal Processing, VOL. 46, NO. 4, April 1998.

....of FIR orthonormal filter banks, it has recently been shown that there does not always exist a PCFB for a given input. This case is addressed in this paper with some new insights. 1. INTRODUCTION Optimization of filter banks for subband coding (SBC) of signals has been an active area of research [1, 2, 10, 17, 18]. Subband coding involves a linear transform part and a nonlinear quantization part. Block transform coding, overlapped transform coding, and wavelet coding are special cases. In the block transform coding case, the optimal orthogonal transform matrix is well known to be the Karhunen Loeve ....

....as the number of channels, we know that a PCFB exists and therefore optimal for subband coding. What happens in the intermediate case where the filter lengths are finite but larger than the number of channels This case turns out to be very difficult to analyze as confirmed by several researchers [10, 15, 19] who devised numerical techniques for suboptimal solutions. One approach for a suboptimal solution is to design an optimal energy compaction filter [7] that pushes most of the signal energy into the first channel, and then complete the filter bank in some optimal fashion [10] Although this ....

[Article contains additional citation context not shown here]

P. Moulin and M. K. Mihcak. Theory and design of signaladapted FIR paraunitary filter banks. IEEE Trans. on Signal Proc., 46:920--929, April 1998.

....a particular permutation is optimum M Gammachannel nonuniform orthonormal filter bank for subband coding. As in the uniform case, the results are valid at arbitrary bit rates. 1. INTRODUCTION Optimization of filter banks for subband coding (SBC) of signals has been an active area of research [1, 2, 7, 8, 13, 14, 15, 17]. Subband coding involves a linear transform part and a nonlinear quantization part. Block transform coding, overlapped transform coding, and wavelet based coding are special cases. In the implementation of wavelet based coders, one uses a dyadic tree structured filter bank. This is equivalent to ....

P. Moulin and M. K. Mihcak. Theory and design of signaladapted FIR paraunitary filter banks. IEEE Trans. on Signal Proc. Special Issue on Theory and Application of Filter Banks and Wavelet Transforms, 46:920--929, April 1998.

....(see (3) below) Our results represent a generalization of[6] where only a first order all pass filter A(z) is considered, whereas here we consider a general N th order all pass filter. The problem of optimal compaction FIR filter design received wide attention in several recent papers [1] [3], but optimal IIR filter design was presented only for a very restricted filter class in [6] The problem we discuss here, designing the IIR filter for optimum compaction, is clearly important for coding applications, since it enables to achieve the same compaction gains as in FIR filter case, but ....

P. Moulin and M.K. Mih¸cak. Theory and design of signal--adapted FIR paraunitary filter banks. IEEE Trans. SP, 46(4):920--929, April 1998.

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P. Moulin and M. K. Mhcak, "Theory and Design of Signal-Adapted FIR Paraunitary Filter Banks," IEEE Transactions on Signal Processing, Special Issue on Wavelets and Filter Banks, Apr. 1998. Postscript file available at: http://www.ifp.uiuc.edu/~mihcak/publications/sp98.ps .

....are challenging, due to the nonlinear nature of the constraints (filter class constraints) This is particularly true of problems that involve constrained length filters. See [1] 6] for optimization of two channel, Quadrature Mirror Filter (QMF) banks under the coding gain criterion, [7] for optimization of M channel paraunitary filter banks under an energy compaction criterion, 8] for optimization of QMF banks under a general rate distortion criterion, and [9] for optimization of two channel biorthogonal filter banks under the coding gain criterion again. The analysis is ....

....is that the problems of optimal filter design and optimal bit allocation to subbands are decoupled in the IIR biorthogonal case, but not in the FIR case. 6.2 Nonstationary Processes The cost function (1. 5) also arises in a deterministic formulation of the filter bank design problem, introduced in [7]. Assume that (A1) the input signal x(n) has finite length MP and is replicated by periodic extension, and (A2) the signals before and after decimation in Fig. 1 have equal energy. Then the spectral density matrix S(f) is the Discrete Time Fourier Transform of the empirical temporal covariance ....

P. Moulin and K. M. Mih¸cak, "Theory and Design of Signal-Adapted FIR Paraunitary Filter Banks," to appear in IEEE Trans. on Signal Processing, Special Issue on Wavelets and Filter Banks, Apr. 1998.

No context found.

P. Moulin and M.K. Mihcak, "Theory and Design of SignalAdapted FIR Paraunitary Filter Banks," IEEE Trans. Signal Processing, vol. 46, no. 4, pp. 920--929, April 1998.

No context found.

P. Moulin and M. K. Mihcak, \Theory and design of signal-adapted FIR paraunitary lter banks," IEEE Transactions on Signal Processing, vol. 46, no. 4, pp. 920-929, April 1998.

No context found.

P.Moulin and M.K.Mihcak, "Theory and Design of Signal-Adapted FIR Paraunitary Filter Banks," IEEE Trans. Signal Processing, vol. 46, no. 4, pp. 920-929, April 1998.

No context found.

P. Moulin and M.K. Mthcak, "Theory and design of signaladapted FIR paraunitary filter banks," IEEE Trans. Signal Processing, vol. 46, pp. 920-929, Apr. 1998.

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