optical orthogonal codes new bounds and an optimal construction a technique for constructing optimal oocs optical orthogonal codes is presented it provides the only known family of optimal with respect to family size oocs having lambda 2 the parameters n omega lambda are respectively p sup 2m 1 p sup m 1 2 where p is any prime and the family size is p sup m 2 three distinct upper bounds on the size of an ooc are presented that for many values of the parameter set n omega lambda improve upon the tightest previously known bound

perfect reconstruction versus mmse filter banks in source coding classically the filter banks fbs used in source coding schemes have been chosen to possess the perfect reconstruction pr property or to be maximally selective quadrature mirror filters qmfs this paper puts this choice back into question and solves the problem of minimizing the reconstruction distortion which in the most general case is the sum of two terms a first one due to the non pr property of the fb and the other being due to signal quantization in the subbands the resulting filter banks are called minimum mean square error mmse filter banks several quantization noise models are considered first under the classical white noise assumption the optimal positive bit rate allocation in any filter bank possibly nonorthogonal is expressed analytically and an efficient optimization method of the mmse filter banks is derived then it is shown that while in a pr fb the improvement brought by an accurate noise model over the classical white noise one is noticeable it is not the case for the mmse fb the optimization of the synthesis filters is also performed for two measures of the bit rate the classical one which is defined for uniform scalar quantization and the order one entropy measure finally the comparison of rate distortion curves where the distortion is minimized for a given bit rate budget enables us to quantify the snr improvement brought by mmse solutions

optimal subband filter banks for multiple description coding multiple description coding refers to the encoding of an information source into multiple bit streams in a manner that each bit stream independently represents a coarse description of the source while multiple descriptions jointly convey a higher fidelity source representation thus multiple description coding allows enhanced resilience to loss of individual descriptions and can be viewed as a form of source based diversity coding this paper addresses the design of subband filter banks to optimize the rate distortion performance of multiple description subband coding the problem is formulated in a lagrangian optimization framework and is solved directly in the frequency domain to produce the optimal filter spectral responses by varying the lagrangian parameter our design obtains all points on the redundancy distortion curve for gaussian wide sense stationary input processes we consider the cases of both orthonormal and more general biorthogonal filter banks we also analyze the connections between these two solutions and quantify the gap in their optimal performance application of the proposed methods to the popular first order autoregressive ar 1 model yields very interesting and insightful results in the two extreme cases of maximum and minimum redundancy our solutions degenerate to previously known results finally comparisons with popularly deployed filter banks reveal large gaps from the theoretical performance bound

allpass delay chain based iir pr filterbank and its application to multiple description subband coding a class of infinite impulse response iir perfect reconstruction pr filterbank is obtained with an allpass delay chain and finite impulse response fir matrices at the analysis side such a formulation leads to a filterbank that can be optimized to any desired response for a first order allpass the synthesis bank becomes fir design examples showing warping of the frequency scale and filters with unequal passband widths are presented the latter part of the work deals with designing a filterbank for a multiple description coding scenario this is achieved in two parts by obtaining the mmse synthesis bank for a given analysis bank and channel state and by optimizing the analysis response to minimize the average distortion in the presence of channel erasures and quantization the proposed iir filterbank as well as the well known fir paraunitary filterbank are optimized for various channels and their performances for the two channel case have been compared with the derived optimal performance

constructions for optimal spl upsi 4 1 optical orthogonal codes four direct constructions three of which are by way of skew starters are given in this correspondence for optimal spl upsi 4 1 optical orthogonal codes ooc s these improve the known existence results concerning optimal spl upsi 4 1 oocs in particular it is shown that an optimal spl upsi 4 1 ooc exists for all positive integers spl upsi spl equiv 6 mod 12 or spl upsi spl equiv 24 mod 48 it is also shown that an optimal 12 spl upsi 4 1 ooc exists for any positive integer spl upsi whose prime factors are all congruent to 1 modulo 4

a new class of decomposition for inverting asymmetric and indefinite matrices abstract an innovative decomposition for inverting a nonsingular asymmetric and indefinite matrix a of order n n is derived in this paper the inverse of a is written as a 1 l d u where l is a lower triangular matrix d is a diagonal matrix and u is an upper triangular matrix by the method the solution of a x b may be easily and efficiently computed by matrix vector multiplications as x l d u b this technique requires a minimal amount of computer memories and can be easily transformed into parallel procedures with high efficiencies performances in inverting asymmetric and indefinite matrices and in solving systems of linear equations on an alliant fx8 computer are reported

a sparsity for decomposing a symmetric matrix abstract this paper studies a sparse configuration for a new class of decomposition derived by the author based upon the sparsity the inverse of a sparse and symmetric matrix and the solution of a system of linear equations can be economically computed numerical examples are included

optimal 2 d n times m 3 2 1 optical orthogonal codes optical orthogonal codes are commonly used as signature codes for optical code division multiple access systems so far research on 2 d optical orthogonal codes has mainly concentrated on the same autocorrelation and cross correlation constraints in this paper we are concerned about optimal 2 d optical orthogonal codes with the autocorrelation a and the cross correlation 1 some combinatorial constructions for 2 d n m k a 1 optical orthogonal codes are presented when k 3 and a 2 the exact number of codewords of an optimal 2 d n m 3 2 1 optical orthogonal code is determined for any positive integers n 0 1 3 6 9 10 mod 12 and m 2 mod 4

bounds and constructions of optimal n 4 2 1 optical orthogonal codes in this paper a tight upper bound on the maximum possible code size of n 4 2 1 oocs and some direct and recursive constructions of optimal n 4 2 1 oocs attaining the upper bound are give

optical orthogonal codes design analysis and applications an optical orthogonal code ooc is a family of 0 1 sequences with good auto and cross correlation properties i e the autocorrelation of each sequence exhibits the thumbtack shape and the cross correlation between any two sequences remains low throughout the use of optical orthogonal codes enables a large number of asynchronous users to transmit information efficiently and reliably the thumbtack shaped autocorrelation facilitates the detection of the desired signal and low profile cross correlation reduces interference from unwanted signals theoretical upper and lower bounds on the maximum possible size of oocs are derived methods for the design and analysis of oocs using tools from projective geometry the greedy algorithm iterative constructions algebraic coding theory block design and various other combinational disciplines are discussed

extended concept of stair shape sparsity for the inverse of an asymmetric matrix abstract this paper extends the concept of stair shape sparsity introduced in a previous work of the author for analysis of a symmetric and sparse matrix to an asymmetric matrix the essential idea first considers an asymmetric matrix as a combination of an upper triangular matrix and a lower triangular matrix and then applies the stair shape sparsity to the upper and lower triangular matrices this enables us to avoid the operation of zero fill ins for the inverse of an asymmetric matrix the introduced sparsity provides an efficient procedure for the solution of a sparse system of linear equations

achievable rates for multiple descriptions consider a sequence of independent identically distributed i i d random variables x l x 2 cdots x n and a distortion measure d x i x i on the estimates x i of x i two descriptions i x in 1 2 cdots 2 nr 1 and j x in 1 2 cdots 2 nr 2 are given of the sequence x x 1 x 2 cdots x n from these two descriptions three estimates i x x2 j x and hat x o i x j x are formed with resulting expected distortions e frac 1 n sum n k 1 d x k hat x mk d m m 0 1 2 we find that the distortion constraints d 0 d 1 d 2 are achievable if there exists a probability mass distribution p x p hat x 1 hat x 2 hat x 0 x with ed x hat x m leq d m such that r 1 i x hat x 1 r 2 i x hat x 2 where i cdot denotes shannon mutual information these rates are shown to be optimal for deterministic distortion measures

optical orthogonal codes their bounds and new optimal constructions a v k spl lambda sub a spl lambda sub c optical orthogonal code ooc c is a family of 0 1 sequences of length v and weight k satisfying the following two correlation properties 1 spl sigma sub 0 spl les t spl les v 1 x sub t x sub t i spl les spl lambda sub a for any x x sub 0 x sub 1 spl middot spl middot spl middot x sub v 1 spl isin c and any integer i not equivalent 0 mod v and 2 spl sigma sub 0 spl les t spl les v 1 x sub t y sub t i spl les spl lambda sub b for any x x sub 0 x sub 1 spl middot spl middot spl middot x sub v 1 spl isin c y y sub 0 y sub 1 spl middot spl middot spl middot y sub v 1 spl isin c with x spl ne y and any integer i where the subscripts are taken modulo v the study of optical orthogonal codes is motivated by an application in optical code division multiple access communication systems in this paper upper bounds on the size of an optical orthogonal code are discussed several new constructions for optimal optical orthogonal codes with weight k spl ges 4 and correlation constraints spl lambda sub a spl lambda sub c 1 are described by means of optimal cyclic packings many new infinite series of such optimal optical orthogonal codes are thus produced

perfect reconstruction filter banks with rational sampling factors an open problem namely how to construct perfect reconstruction filter banks with rational sampling factors is solved such filter banks have n branches each one having a sampling factor of p i qi and their sum equals one in this way the well known theory of filter banks with uniform band splitting is extended to allow for nonuniform divisions of the spectrum this can be very useful in the analysis of speech and music the theory relies on two transforms the first transform leads to uniform filter banks having polyphase components as individual filters the other results in a uniform filter bank containing shifted versions of same filters this in turn introduces dependencies in design and is left for future work as an illustration several design examples for the 2 3 1 3 case are given filter banks are then classified according to the possible ways in which they can be built it is shown that some cases cannot be solved even with ideal filters with real coefficients

large families of asymptotically optimal two dimensional optical orthogonal codes nine new two dimensional optical orthogonal codes 2 d oocs are presented here all sharing the common feature of a code size that is much larger in relation to the number of time slots than those of constructions appearing previously in the literature each of these constructions is either optimal or asymptotically optimal with respect to either the original johnson bound or else a nonbinary version of the johnson bound introduced in this paper the first five codes are constructed using polynomials over finite fields the first construction is optimal while the remaining four are asymptotically optimal the next two codes are constructed using rational functions in place of polynomials and these are asymptotically optimal the last two codes also asymptotically optimal are constructed by composing two of the above codes with a constant weight binary code also presented is a three dimensional optical orthogonal code 3 d ooc that exploits the polarization dimension finally phase encoded optical cdma is considered and construction of two efficient codes are provided

a new class of two channel biorthogonal filter banks and wavelet bases proposes a novel framework for a new class of two channel biorthogonal filter banks the framework covers two useful subclasses i causal stable iir filter banks ii linear phase fir filter banks there exists a very efficient structurally perfect reconstruction implementation for such a class filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity the properties of such a class are discussed in detail the design of the analysis synthesis systems reduces to the design of a single transfer function very simple design methods are given both for fir and iir cases zeros of arbitrary multiplicity at aliasing frequency can be easily imposed for the purpose of generating wavelets with regularity property in the iir case two new classes of iir maximally flat filters different from butterworth filters are introduced the filter coefficients are given in closed form the wavelet bases corresponding to the biorthogonal systems are generated the authors also provide a novel mapping of the proposed 1 d framework into 2 d the mapping preserves the following i perfect reconstruction ii stability in the iir case iii linear phase in the fir case iv zeros at aliasing frequency v frequency characteristic of the filters

an incomplete inverse as a preconditioner for the conjugate gradient method abstract this paper introduces a new preconditioner with a super convergence for the conjugate gradient method for solving a system of linear equations in a banded structure the idea is based upon an innovative method of the author to decompose a symmetric matrix into its inverse and the preconditioner is the result of an incomplete decomposition i e an approximation to the inverse by the presented preconditioner the rate of convergence can be significantly improved examples to demonstrate the improvement of rate of convergence and the speedup implemented on an alliant fx8 computer are included

a sparse approximate inverse preconditioner for parallel preconditioning of general sparse matrices a factored sparse approximate inverse is computed and used as a preconditioner for solving general sparse matrices the algorithm is derived from a matrix decomposition algorithm for inverting dense nonsymmetric matrices several strategies and special data structures are proposed to implement the algorithm efficiently sparsity patterns of the factored inverse are exploited to reduce computational cost the preconditioner possesses greater inherent parallelism than traditional preconditioners based on incomplete lu factorizations numerical experiments are used to show the effectiveness and efficiency of the proposed sparse approximate inverse preconditioner

combinatorial constructions for optimal two dimensional optical orthogonal codes optical orthogonal codes oocs have been designed for ocdma a one dimensional 1 d optical orthogonal code 1 d ooc is a set of one dimensional binary sequences having good auto and cross correlations one limitation of 1 d ooc is that the length of the sequence increases rapidly when the number of users or the weight of the code is increased which means large bandwidth expansion is required if a big number of codewords is needed to lessen this problem two dimensional 2 d coding also called multiwavelength oocs was invested a two dimensional 2 d optical orthogonal code 2 d ooc is a set of utimesv matrices with 0 1 elements having good auto and cross correlations recently many researchers are working on constructions and designs of 2 d oocs in this paper we shall reveal the combinatorial properties of 2 d oocs and give an equivalent combinatorial description of a 2 d ooc based on this we are able to use combinatorial methods to obtain many optimal 2 d oocs

design of multiple description scalar quantizers the design of scalar quantizers for communication systems that use diversity to overcome channel impairments is considered the design problem is posed as an optimization problem and necessary conditions for optimality are derived a design algorithm a generalization of s p lloyd s 1962 algorithm for quantizer design is developed unlike a single channel scalar quantizer the performance of a multiple description scalar quantizer is dependent on the index assignment the problem of index assignment is addressed good index assignments performance results and sample quantizer designs are presented for a memoryless gaussian source comparisons are made with rate distortion bounds for the multiple description problem

new construction of multiwavelength optical orthogonal codes we investigate multiwavelength optical orthogonal codes mwoocs for optical code division multiple access particularly we present a new construction method for mn spl lambda 2 spl lambda mwoocs with the number of available wavelengths m codeword length n and constant hamming weight spl lambda 2 that have autocorrelation and cross correlation values not exceeding spl lambda in the proposed scheme there is no constraint on the relationship between the number of available wavelengths and the codeword length and it is also possible to use an arbitrary spl lambda we show that the constructed code is optimal especially for spl lambda 1 finally we analyze the bit error rate of the new code and compare it with that of other optical codes