nearest neighbor algorithm for spherical codes from the leech lattice the leech lattice is a regular arrangement of points in 24 dimensional euclidean space that yields an extremely dense packing when equal spheres are centered at these points a subset of the leech lattice can be used as a signal set for the gaussian channel or as representative vectors for a vector quantizer of particular interest are the spherical codes or code books that consist of the points of the leech lattice which lie on a sphere centered at the origin the code points do not have to be stored because they can be obtained from a very small set of basic vectors using permutations of the components in a manner dictated by the words of the extended golay code a nearest neighbor algorithm that works on this is developed to determine the point in the code closet to some arbitrary vector in r sup 24 the performance of this approach when quantizing independent identically distributed gaussian samples is reported

a pyramid vector quantizer the geometric properties of a memoryless laplacian source are presented and used to establish a source coding theorem motivated by this geometric structure a pyramid vector quantizer pvq is developed for arbitrary vector dimension the pvq is based on the cubic lattice points that lie on the surface of an l dimensional pyramid and has simple encoding and decoding algorithms a product code version of the pvq is developed and generalized to apply to a variety of sources analytical expressions are derived for the pvq mean square error mse and simulation results are presented for pvq encoding of several memoryless sources for large rate and dimension pvq encoding of memoryless laplacian gamma and gaussian sources provides rose improvements of 5 64 8 40 and 2 39 db respectively over the corresponding optimum scalar quantizer although suboptimum in a rate distortion sense because the pvq can encode large dimensional vectors it offers significant reduction in rose distortion compared with the optimum lloyd max scalar quantizer and provides an attractive alternative to currently available vector quantizers

hausdorff content and the hardy littlewood maximal operator on metric measure spaces abstract let x d be a complete metric measure space and be a non negative borel regular measure satisfying the doubling condition with some dimensional constant d we prove that the hausdorff content of codimension 0 denoted by h and the hardy littlewood maximal operator m satisfy the strong type inequality x m u p d h c x u p d h 0 u l loc 1 x whenever p max 0 1 d if further satisfies some reverse doubling condition with some other dimensional constant then for the endpoint case p 1 d with 0 d 0 we also obtain the corresponding weak type estimate for h and m the fundamental point in the proofs is to introduce and develop a theory of the dyadic hausdorff content h d which is a choquet capacity comparable to h and has the strong subadditivity property

selecting wavelengths for least squares range estimation we consider the problem of estimating the distance or range between two locations by measuring the phase of multiple sinusoidal signals transmitted between the locations traditional estimators developed for optical interferometry include the beat wavelength and excess fractions methods more recently estimators based on the chinese remainder theorem crt and least squares have appeared recent research suggests the least squares estimator to be the most accurate in many cases the accuracy of all of these range estimators depends upon the wavelengths chosen this leads to the problem of selecting wavelengths that maximize accuracy procedures for selecting wavelengths for the beat wavelength and excess fractions methods have previously been described but procedures for the crt and least squares estimators are yet to be developed in this paper we develop an algorithm to automatically select wavelengths for use with the least square range estimator the algorithm minimizes an optimization criterion connected with the mean square error interesting properties of a particular class of lattices simplify the criterion allowing minimization by depth first search monte carlo simulations indicate that wavelengths that minimize the criterion can result in considerably more accurate range estimates than wavelengths selected by ad hoc means

on the error performance of the a_ n lattices we consider the root lattice a n and derive explicit recursive formulas for the moments of its voronoi cell these formulas enable accurate prediction of the error probability of lattice codes constructed from a n

lattice quantization of phases for equal gain transmission equal gain transmission egt requires only the feedback of phase information and is known to enjoy several implementational advantages compared to maximum ratio transmission mrt this paper investigates the use of lattice quantization for quantizing phases in egt systems in practice the feedback rate is finite and the lattices need to be truncated to have finite code rate the truncation of lattices results in boundary effect hence earlier results developed for untruncated lattices e g fast algorithm and mean square error mse analysis cannot be applied directly exploiting the fact that the quantized signals are phases we show how to quantize using truncated lattices so that there is no boundary effect moreover we propose efficient methods to convert between lattice codewords and binary digits so that the lattice quantization can be used in practical systems furthermore the mean square quantization error is analyzed for several useful truncated lattices we also show how to analyze the signal to noise ratio snr loss when the proposed quantization is applied to an egt system

indexing the output points of an lbvq used for image transform coding a new method for indexing the points of a lattice based vector quantizer lbvq used to quantize the dct coefficients of images is presented with this method a large number of lattice points can be selected as codewords as a result the quality of the compressed data can be very high the problem is that the large number of points results in a high bit rate to reduce the bit rate a shorter representation is assigned to the more frequently used lattice points these points are grouped and a prefix code is used to index these lattice points our method outperforms jpeg particularly in the case of images with high frequency components

a fast encoding method for lattice codes and quantizers in an earlier paper the authors described a very fast method which for the root lattices a n d n e n their duals and certain other lattices finds the closest lattice point to an arbitrary point of the underlying space if the lattices are used as codes for a gaussian channel the algorithm provides a fast decoding procedure or if they are used as vector quantizers the algorithm performs the analog to digital conversion efficiently the present paper offers a solution to the inverse problem for the same lattices the encoding problem for channel codes or the digital to analog part of quantizing namely given an integer k to find the kth code vector and to the closely related problem of finding the index k of a given code vector

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

shaping multidimensional signal spaces i optimum shaping shell mapping the structure of the regions which provide the optimum tradeoff between gamma sub s shaping gain and cer constellation expansion ratio and between gamma sub s and par peak to average power ratio in a finite dimensional space is introduced analytical expressions are derived for the corresponding tradeoff curves in general the initial parts of the curves have a steep slope this means that an appreciable portion of the maximum shaping gain corresponding to a spherical region can be achieved with a small value of cer sub s par the technique of shell mapping is introduced this is a change of variable which maps the optimum shaping region to a hypercube truncated within a simplex this mapping is a useful tool in computing the performance and also in facilitating the addressing of the optimum shaping region using the shell mapping a practical addressing scheme is presented that achieves a point on the optimum tradeoff curves for dimensionalities around 12 the point achieved is located near the knee of the corresponding tradeoff curve for larger dimensionalities a general shaping region with two degrees of freedom is used this region provides more flexibility in selecting the tradeoff point

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

pyramidal lattice vector quantization for multiscale image coding introduces a new image coding scheme using lattice vector quantization the proposed method involves two steps biorthogonal wavelet transform of the image and lattice vector quantization of wavelet coefficients in order to obtain a compromise between minimum distortion and bit rate we must truncate and scale the lattice suitably to meet this goal we need to know how many lattice points lie within the truncated area we investigate the case of laplacian sources where surfaces of equal probability are spheres for the l sup 1 metric pyramids for arbitrary lattices we give explicit generating functions for the codebook sizes for the most useful lattices like z sup n d sub n e sub s spl and sub 16

shaping of multidimensional signal constellations using a lookup table this paper describes a lookup table for the addressing of an optimally shaped constellation the method is based on partitioning the subconstellations into shaping macro shells of integer bit rate and increasing average energy the macro shells do not need to have an equal number of points a lookup table is used to select a subset of the partitions in the cartesian product space by devising appropriate partitioning merging rules we obtain suboptimum schemes of very low addressing complexity and small performance degradation the performance is computed using the weight distribution of an optimally shaped constellation

shaping using variable size regions constellation shaping is extended to provide shaping gains without resorting to high dimensional constellations this is accomplished by dividing the constellation into unequally sized constellations and selecting these constellations on an equiprobable basis a design example is provided demonstrating the simplicity and power of the approach

design of spherical lattice space time codes in this paper we propose a systematic procedure for designing spherical lattice space time codes by employing stochastic optimization techniques we design lattice codes which are well matched to the fading statistics as well as to the decoder used at the receiver the decoders we consider here include the optimal albeit of highest decoding complexity maximum likelihood ml decoder the suboptimal lattice decoders as well as the suboptimal lattice reduction aided lra decoders having the lowest decoding complexity for each decoder our design methodology can be tailored to obtain low error rate lattice codes for arbitrary fading statistics and signal to noise ratios snrs of interest further we obtain fundamental lower bounds on the error probabilities yielded by lattice and lra decoders and characterize their asymptotic behavior

tables of sphere packings and spherical codes the theta function of a sphere packing gives the number of centers at each distance from the origin the theta functions of a number of important packings a n d n e n the leech lattice and others and tables of the first fifty or so of their coefficients are given in this paper

image coding with uniform and piecewise uniform vector quantizers new lattice vector quantizer design procedures for nonuniform sources that yield excellent performance while retaining the structure required for fast quantization are described analytical methods for truncating and scaling lattices to be used in vector quantization are given and an analytical technique for piecewise linear multidimensional companding is presented the uniform and piecewise uniform lattice vector quantizers are then used to quantize the discrete cosine transform coefficients of images and their objective and subjective performance and complexity are contrasted with other lattice vector quantizers and with lbg training mode designs

coset codes for partial response channels or coset codes with spectral nulls known coset codes are adapted for use on partial response channels or to generate signals with spectral nulls by using coset precoding and running digital sum feedback any desired tradeoff can be achieved between the power and spectra of the relevant sequences up to the optimum tradeoff possible a fundamental theorem specifying this optimum tradeoff is given a maximum likelihood sequence estimation mlse decoder for the original code may be used for the adapted code and such a decoder then attains the minimum squared distance of the original code these methods sometimes generate codes with greater minimum squared distance than that of the original code this distance can be attained by augmented decoders although such decoders inherently require long decoding delays and may be subjected to quasi catastrophic error propagation the authors conclude that at least for sequences supporting large numbers of bits per symbol coset codes can be adapted to achieve effectively the same performance and complexity on partial response channels or for sequences with spectral nulls as they do in the ordinary memoryless case

signal codes convolutional lattice codes the coded modulation scheme proposed in this paper has a simple construction an integer sequence representing the information is convolved with a fixed continuous valued finite impulse response fir filter to generate the codeword a lattice point due to power constraints the code construction includes a shaping mechanism inspired by precoding techniques such as the tomlinson harashima filter we naturally term these codes convolutional lattice codes or alternatively signal codes due to the signal processing interpretation of the code construction surprisingly properly chosen short fir filters can generate good codes with large minimal distance decoding can be done efficiently by sequential decoding or for better performance by bidirectional sequential decoding error analysis and simulation results indicate that for the additive white gaussian noise awgn channel convolutional lattice codes with computationally reasonable decoders can achieve low error rate close to the channel capacity

lattice and trellis quantization with lattice and trellis bounded codebooks high rate theory for memoryless sources high rate lattice and trellis quantizers for nonuniform sources are introduced and analyzed the performance of these quantizers is determined by two separable quantities the granular gain and the boundary gain which are determined by the shapes of the granular cells and of the support region respectively the granular gain and boundary gain are the duals of shaping and coding gain in data transmission applications using this duality it is shown for gaussian sources that the ultimate achievable boundary gain with high rate lattice bounded lattice codebooks is the same as the ultimate gain that can be obtained from variable rate entropy coding it is observed that if lattice codebooks can achieve the ultimate granular gain of 0 255 b per dimension then lattice bounded lattice codebooks can approach the rate distortion limit the performance of lattice quantizers is compared to that of optimum vector quantizers

geometric source coding and vector quantization a geometric formulation is presented for source coding and vector quantizer design motivated by the asymptotic equipartition principle the authors consider two broad classes of source codes and vector quantizers elliptical codes and quantizers based on the gaussian density function and pyramid codes and quantizers based on the laplacian density function elliptical and weighted pyramid vector quantizers are developed by selecting codewords as points in a lattice that lie on or near a specified ellipse or pyramid the combination of geometric structure and lattice basis allows simple encoding and decoding algorithms

vector quantization for entropy coding of image subbands vector quantization for entropy coding of image subbands is investigated rate distortion curves are computed with mean square error as a distortion criterion the authors show that full search entropy constrained vector quantization of image subbands results in the best performance but is computationally expensive lattice quantizers yield a coding efficiency almost indistinguishable from optimum full search entropy constrained vector quantization orthogonal lattice quantizers were found to perform almost as well as lattice quantizers derived from dense sphere packings an optimum bit allocation rule based on a lagrange multiplier formulation is applied to subband coding coding results are shown for a still image

the covering radius of the leech lattice we investigate the points in 24 dimensional space that are at maximal distance from the leech lattice i e the deep holes in that lattice the distance of such a point from the leech lattice is 1 sqrt 2 times the minimal distance between the lattice points furthermore there are 23 inequivalent types of deep hole one for each of the 23 even unimodular 24 dimensional lattices found by niemeier

the root lattices as low bit rate vector quantizers vector quantizers based on the a and d lattices are explored using the mean square error mse fidelity criterion for independent identically distributed sources with normal laplacian and gamma probability density functions code books are studied for uniform coding rates of up to 2 b sample and two and four dimensions the expected mse performance of the quantizers is better than that of optimal nonuniform lloyd max quantizers while the design algorithm presented offers only modest improvements in mse performance for a normally distributed source improvement gains of up to several decibels can be achieved when coding laplacian and gamma sources

lda lattices without dithering achieve capacity on the gaussian channel this paper deals with low density construction a lda lattices which are obtained via construction a from non binary low density parity check codes more precisely a proof is provided that voronoi constellations of lda lattices achieve capacity of the awgn channel under lattice encoding and decoding for every signal to noise ratio greater than 1 this is obtained after showing the same result for more general construction a lattice constellations the theoretical analysis is carried out in a way that allows to describe how the prime number underlying construction a behaves as a function of the lattice dimension moreover no dithering is required in the transmission scheme simplifying some previous solutions of the problem remarkably capacity is achievable with lda lattice codes whose parity check matrices have constant row and column hamming weights some expansion properties of random bipartite graphs constitute an extremely important tool for dealing with sparse matrices and allow to find a lower bound for the minimum euclidean distance of lda lattices in our ensemble

universal bound on the performance of lattice codes we present a lower bound on the probability of symbol error for maximum likelihood decoding of lattices and lattice codes on a gaussian channel the bound is tight for error probabilities and signal to noise ratios of practical interest as opposed to most existing bounds that become tight asymptotically for high signal to noise ratios the bound is also universal it provides a limit on the highest possible coding gain that may be achieved at specific symbol error probabilities using any lattice or lattice code in n dimensions in particular it is shown that the effective coding gains of the densest known lattices are much lower than their nominal coding gains the asymptotic as n spl rarr spl infin behavior of the new bound is shown to coincide with the shannon 1948 limit for gaussian channels

non equilibrium behaviour of a many particle process density profile and local equilibria one considers a simple exclusion particle jump process on where the underlying one particle motion is a degenerate random walk that moves only to the right one starts with the configuration in which the left halfline is completely occupied and the right one free it is shown that the number of particles at time t between site u t and v t divided by t converges a s to int limits u nu f w dw where f might be called the density profile it is explicitely determined and shown to be an affine function secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u converges weakly to the bernoulli measure with density f u as the time tends to infinity

enumeration encoding and decoding algorithms for pyramid cubic lattice and trellis codes a pyramid source code is a code that assigns equal length binary strings to all reproduction codevectors of equal weighted spl epsiv sub 1 norm a pyramid source encoding is partitioned into two concatenated mappings the first from source word to reproduction codeword within a codebook the second from the reproduction codevector to a binary string the first mapping allows distortion and is accomplished using lattice quantization or trellis coded quantization the second mapping is noiseless and is denoted as enumeration efficient pyramid enumeration encoding and decoding algorithms are presented for use with fixed rate or variable rate pyramid lattice and trellis codes

trellis coded modulation with multidimensional constellations trellis coded modulation schemes using four eight or 16 dimensional constellations have a number of potential advantages over the usual two dimensional schemes a smaller constituent two dimensional constellation easier tolerance to phase ambiguities and a better trade off between complexity and coding gain a number of such schemes are presented and evaluated starting with a variety of multidimensional lattices we show how to select multidimensional constellations how to partition them into subsets how to construct trellis codes using those subsets and how to map hits to constellation points simplifications of the viterbi decoding algorithm are presented we conclude that there are multidimensional trellis coded modulation schemes that perform better for the same complexity than do two dimensional schemes

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

an efficient block based addressing scheme for the nearly optimum shaping of multidimensional signal spaces we introduce an efficient addressing scheme for the nearly optimum shaping of a multidimensional signal constellation the 2 d two dimensional subspaces are partitioned into k energy shells of equal cardinality the average energy of a 2 d shell can be closely approximated by a linear function of its index in an n 2n d space we obtain k sup n shaping clusters of equal cardinality shaping is achieved by selecting t spl les k sup n of the n d clusters with the least sum of the 2 d indices this results in a set of t integer n tuples with the components in the range 0 k 1 and the sum of the components being at most a given number l the problem of addressing is to find a one to one mapping between the set of such n tuples and the set of integers 0 t 1 such that the mapping and its inverse can be easily implemented in the proposed scheme the n d clusters are grouped into blocks of identical binary weight vectors this results in a simple rule for the addressing of points within the blocks the addressing of the blocks is based on some recursive relationship which allows us to decompose the problem into simpler parts the overall scheme requires a modest amount of memory and has a small computational complexity

piecewise uniform vector quantizers the companding model for quantizer design and analysis has been widely applied in the scalar quantization case however if the signal to be quantized is a vector then the optimum companding system can be designed for only a limited number of distributions on the other hand multidimensional piecewise linear companders can be designed for any signal density generating quantizers that are uniform on each region of the compander these systems while not optimal can have asymptotic performance arbitrarily close to the optimum their analysis and implementation can be simpler than those of optimal systems piecewise linear companders for asymptotic multidimensional quantization are analyzed and a method for their design is suggested

uniform and piecewise uniform lattice vector quantization for memoryless gaussian and laplacian sources lattice vector quantizer design procedures for nonuniform sources are presented the procedures yield lattice vector quantizers with excellent performance and retaining the structure required for fast quantization analytical methods for truncating and scaling lattices to be used in vector quantizations are given and their utility is demonstrated for independent and identically distributed i i d gaussian and laplacian sources an analytical technique for piecewise linear multidimensional compandor designs is evaluated for i i d gaussian and laplacian sources by comparing its performance to that of the other vector quantizers

a novel fingerprint image compression technique using wavelets packets and pyramid lattice vector quantization a novel compression algorithm for fingerprint images is introduced using wavelet packets and lattice vector quantization a new vector quantization scheme based on an accurate model for the distribution of the wavelet coefficients is presented the model is based on the generalized gaussian distribution we also discuss a new method for determining the largest radius of the lattice used and its scaling factor for both uniform and piecewise uniform pyramidal lattices the proposed algorithms aim at achieving the best rate distortion function by adapting to the characteristics of the subimages in the proposed optimization algorithm no assumptions about the lattice parameters are made and no training and multi quantizing are required we also show that the wedge region problem encountered with sharply distributed random sources is resolved in the proposed algorithm the proposed algorithms adapt to variability in input images and to specified bit rates compared to other available image compression algorithms the proposed algorithms result in higher quality reconstructed images for identical bit rates

fast quantizing and decoding and algorithms for lattice quantizers and codes for each of the lattices a n n geq 1 d n n geq 2 e 6 e 7 e 8 and their duals a very fast algorithm is given for finding the closest lattice point to an arbitrary point if these lattices are used for vector quantizing of uniformly distributed data the algorithm finds the minimum distortion lattice point if the lattices are used as codes for a gaussian channel the algorithm performs maximum likelihood decoding