design of asymptotically optimal improper constellations with hexagonal packing this paper addresses the problem of designing asymptotically optimal improper constellations with a given circularity coefficient correlation coefficient between the constellation and its complex conjugate the designed constellations are optimal in the sense that at high signal to noise ratio snr and for a large number of symbols yield the lowest probability of error under an average power constraint for additive white gaussian noise channels as the number of symbols grows the optimal constellation is the intersection of the hexagonal lattice with an ellipse whose eccentricity determines the circularity coefficient based on this asymptotic result we propose an algorithm to design finite improper constellations the proposed constellations provide significant snr gains with respect to previous improper designs which were generated through a widely linear transformation of a standard m ary quadrature amplitude modulation constellation as an application example we study the use of these improper constellations by a secondary user in an underlay cognitive radio network

mimo precoding with x and y codes we consider a slow fading multiple input multiple output mimo system with channel state information at both the transmitter and receiver a well known precoding scheme is based upon the singular value decomposition svd of the channel matrix which transforms the mimo channel into parallel subchannels despite having low maximum likelihood decoding mld complexity this svd precoding scheme provides a diversity gain which is limited by the diversity gain of the weakest subchannel we therefore propose x and y codes which improve the diversity gain of the svd precoding scheme but maintain the low mld complexity by jointly coding information across a pair of subchannels in particular subchannels with high diversity gain are paired with those having low diversity gain a pair of subchannels is jointly encoded using a 2 2 real matrix which is fixed a priori and does not change with each channel realization for x codes these rotation matrices are parameterized by a single angle while for y codes these matrices are left triangular matrices moreover we propose x y precoders with the same structure as x y codes but with encoding matrices adapted to each channel realization we observed that x codes precoders are good for well conditioned channels while y codes precoders are good for ill conditioned channels

precoding for outage probability minimization on block fading channels the outage probability limit is a fundamental and achievable lower bound on the word error rate of coded communication systems affected by fading this limit is mainly determined by two parameters the diversity order and the coding gain with linear precoding full diversity on a block fading channel can be achieved without error correcting code however the effect of precoding on the coding gain is not well known mainly due to the complicated expression of the outage probability using a geometric approach this paper establishes simple upper bounds on the outage probability the minimization of which yields to precoding matrices that achieve very good performance for discrete alphabets it is shown that the combination of constellation expansion and precoding is sufficient to closely approach the minimum possible outage achieved by an i i d gaussian input distribution thus essentially maximizing the coding gain

nonequiprobable signaling on the gaussian channel signaling schemes for the gaussian channel based on finite dimensional lattices are considered the signal constellation consists of all lattice points within a region r and the shape of this region determines the average signal power spherical signal constellations minimize average signal power and in the limit as n to infinity the shape gain of the n sphere over the n cube approaches pi e 6 approximately 1 53 db a nonequiprobable signaling scheme is described that approaches this full asymptotic shape gain in any fixed dimension a signal constellation omega is partitioned into t subconstellations omega sub 0 omega sub tau 1 of equal size by scaling a basic region r signal points in the same subconstellation are used equiprobably and a shaping code selects the subconstellation omega sub i with frequency f sub i shaping codes make it possible to achieve any desired fractional bit rate the schemes presented are compared with equiprobable signaling schemes based on voronoi regions of multidimensional lattices for comparable shape gain and constellation expansion ratio the peak to average power ratio of the schemes presented is superior furthermore a simple table lookup is all that is required to address points in the constellations it is also shown that it is possible to integrate coding and nonequiprobable signaling within a common multilevel framework

parabolic evolution equations and nonlinear boundary conditions on etudie le probleme aux valeurs limites et initiales u 7b a t u f t u 7b b t u g t u 0 t t u 0 u 0 qui peut etre considere comme une contrepartie abstraite a des problemes paraboliques semilineaires

backward euler method for abstract time dependent parabolic equations with variable domains we consider semidiscretizations in time based on the backward euler method of an abstract non autonomous parabolic initial value problem where a t d a t subset x to x 0 le t le t is a family of sectorial operators in a banach space x the domains d a t are allowed to depend on t our hypotheses are fulfilled for classical parabolic problems in the l p 1 p infty norms we prove that the semidiscretization is stable in a suitable sense we get optimal estimates for the error even when non homogeneous boundary values are considered in particular the results are applicable to the analysis of the semidiscretizations of time dependent parabolic problems under non homogeneous neumann boundary conditions

avoiding order reduction when integrating reaction diffusion boundary value problems with exponential splitting methods abstract in this paper we suggest a technique to avoid order reduction in time when integrating reaction diffusion boundary value problems under non homogeneous boundary conditions with exponential splitting methods more precisely we consider lie trotter and strang splitting methods and dirichlet neumann and robin boundary conditions beginning from an abstract framework in banach spaces a thorough error analysis after full discretization is performed and some numerical results are shown which corroborate the theoretical results

computation of the exact bit error rate of coherent m ary psk with gray code bit mapping the problem of calculating the average bit error probability bep of coherent m ary phase shift keying psk over a gaussian channel has been studied previously in the literature a solution to the problem for systems using a binary reflected gray code brgc to map bits to symbols was first presented by p j lee see ibid vol com 34 p 488 91 1986 we show that the results obtained by lee are incorrect for m spl ges 16 we show that the reason for this is an invalid assumption that the bit error rate ber is independent of the transmitted symbols an assumption which has also propagated to textbooks we give a new expression for the ber of m psk systems using the brgc and compare this with lee s results

low density graph codes for coded cooperation on slow fading relay channels we study low density parity check ldpc codes with iterative decoding on block fading bf relay channels we consider two users that employ coded cooperation a variant of decode and forward with a smaller outage probability than the latter an outage probability analysis for discrete constellations shows that full diversity can be achieved only when the coding rate does not exceed a maximum value that depends on the level of cooperation we derive a new code structure by extending the previously published full diversity root ldpc code designed for the bf point to point channel to exhibit a rate compatibility property which is necessary for coded cooperation we estimate the asymptotic performance through a new density evolution analysis and the word error rate performance is determined for finite length codes we show that our code construction exhibits near outage limit performance for all block lengths and for a range of coding rates up to 0 5 which is the highest possible coding rate for two cooperating users

coded modulation in the block fading channel coding theorems and code construction we consider coded modulation schemes for the block fading channel in the setting where a codeword spans a finite number n of fading degrees of freedom we show that coded modulations of rate r bit per complex dimension over a finite signal set spl chi spl sube spl copf of size 2 sup m achieve the optimal rate diversity tradeoff given by the singleton bound spl delta n m r 1 spl lfloor n 1 r m spl rfloor for r spl isin 0 m spl rfloor furthermore we show also that the popular bit interleaved coded modulation achieves the same optimal rate diversity tradeoff we present a novel coded modulation construction based on blockwise concatenation that systematically yields singleton bound achieving turbo like codes defined over an arbitrary signal set spl chi spl sub spl copf the proposed blockwise concatenation significantly outperforms conventional serial and parallel turbo codes in the block fading channel we analyze the ensemble average performance under maximum likelihood ml decoding of the proposed codes by means of upper bounds and tight approximations we show that differently from the additive white gaussian noise awgn and fully interleaved fading cases belief propagation iterative decoding performs very close to ml on the block fading channel for any signal to noise ratio snr and even for relatively short block lengths we also show that at constant decoding complexity per information bit the proposed codes perform close to the information outage probability for any block length while standard block codes e g obtained by trellis termination of convolutional codes have a gap from outage that increases with the block length this is a different and more subtle manifestation of the so called interleaving gain of turbo codes

multiple antenna fading channels with arbitrary inputs characterization and optimization of the information rate we investigate the high snr capacity of multiple antenna fading coherent channels driven by inputs that are constrained to obey a certain equiprobable discrete distribution rather than the optimal capacity achieving gaussian one in particular we capitalize on the machinery of asymptotic analysis most notably the asymptotic expansion of integrals in order to construct high snr expansions to the average minimum mean squared error mmse and via the relation between mmse and mutual information high snr expansions to the average mutual information thereby providing immediately a high snr expansion to the so called constrained capacity we use the expansions to characterize the constrained capacity of various multiple antenna fading coherent channels including rayleigh fading models ricean fading models and antenna correlated models the analysis unveils in detail the impact of the number of transmit and receive antennas transmit and receive antenna correlation line of sight components and the geometry of the signaling scheme on the reliable information transmission rate we also use the expansions to optimize the constrained capacity of multiple antenna fading coherent channels as an example we focus on the derivation of design criteria for space time signaling schemes

the recursive pyramid algorithm for the discrete wavelet transform the recursive pyramid algorithm rpa is a reformulation of the classical pyramid algorithm pa for computing the discrete wavelet transform dwt the rpa computes the n point dwt in real time running dwt using just l log n spl minus 1 words of storage as compared with o n words required by the pa l is the length of the wavelet filter the rpa is combined with the short length fir filter algorithms to reduce the number of multiplications and additions

second order asymptotics of mutual information a formula for the second order expansion of the input output mutual information of multidimensional channels as the signal to noise ratio snr goes to zero is obtained while the additive noise is assumed to be gaussian we deal with very general classes of input and channel distributions as special cases these channel models include fading channels channels with random parameters and channels with almost gaussian noise when the channel is unknown at the receiver the second term in the asymptotic expansion depends not only on the covariance matrix of the input signal but also on the fourth mixed moments of its components the study of the second order asymptotics of mutual information finds application in the analysis of the bandwidth power tradeoff achieved by various signaling strategies in the wideband regime

constellation shaping for pragmatic turbo coded modulation with high spectral efficiency we propose a new turbo encoding scheme for high spectral efficiency with performance close to the gaussian channel capacity the scheme combines nonuniform signaling on a gaussian channel with pragmatic turbo coded modulation tcm for simple and flexible implementation a variable rate turbo code is followed by a huffmann code mapping onto nonequiprobable points in a quadrature amplitude modulation constellation the rate of the turbo code is matched to the huffmann code by variable puncturing such that both the input bit rate and the output symbol rate are constant it is shown that the new scheme provides shaping gains of 0 6 and 0 9 db at rates 2 and 3 b dimension respectively compared with the equiprobable pragmatic tcm and reach about 1 db from the continuous input gaussian channel capacity

constellation labeling for linear encoders this paper investigates optimal constellation labeling in the context of the edge profile a constellation s edge profile lists the minimum distance edge for each binary symbol error the paper introduces the symmetric ultracomposite su labeling structure and shows that this structure provides undominated edge profiles for 2 sup n psk 2 sup n pam and 2 sup 2n point square qam the su structure is a generalization of the commonly used reflected binary gray code with the proper choice of basis vectors su labeling can support either set partition or gray code labeling of 2 sup n psk 2 sup n pam and 2 sup 2n point square qam notably there are gray code and set partition labelings that do not have the su structure these labelings yield inferior edge profiles the su structure does not apply to cross constellations however for any standard cross constellation with 32 or more points a quasi su labeling structure can approximate the su structure with the correct choice of basis quasi su labelings produce quasi gray labelings however the quasi su structure cannot support set partition labeling in fact the quasi su structure provides a better edge profile than standard set partition labeling thus for cross constellations there is a choice between edge profile optimality and the group structure provided by set partitioning here the correct choice depends on whether the encoder trellis has parallel branches

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

the hadamard transform a tool for index assignment we show that the channel distortion for maximum entropy encoders due to noise on a binary symmetric channel is minimized if the vector quantizer can be expressed as a linear transform of a hypercube the index assignment problem is regarded as a problem of linearizing the vector quantizer we define classes of index assignments with related properties within which the best index assignment is found by sorting not searching two powerful algorithms for assigning indices to the codevectors of nonredundant coding systems are presented one algorithm finds the optimal solution in terms of linearity whereas the other finds a very good but suboptimal solution in a very short time

broadcast coded modulation multilevel and bit interleaved construction the capacity of the awgn broadcast channel is achieved by superposition coding but the superposition of individual coded modulations expands the modulation alphabet and distorts its configuration hierarchical modulations which appear in the literature mostly in the context of unequal error protection can approach only a few isolated points on the boundary of the broadcast capacity region this paper studies multilevel coding mlc for constellation constrained broadcast coded modulation the conditions under which multilevel codes can achieve the constellation constrained capacity of the awgn broadcast channel are derived we propose a pragmatic multilevel design technique with near constellation constrained capacity performance where the coupling of the superposition inner and outer codes is localized to each bit level it is shown that this can be further relaxed to a code coupling on only one bit level with a little or no penalty under natural labeling the rate allocation problem between the bit levels of the two users is studied and a pragmatic method is proposed again with near capacity performance in further pursuit of lower complexity a hybrid mlc bit interleaved coded modulation is proposed whose performance is shown to be very close to the boundary of the constellation constrained capacity region simulation results show that good point to point ldpc codes produce excellent performance in the proposed coded modulation framework

bit interleaved coded modulation revisited a mismatched decoding perspective we revisit the information theoretic analysis of bit interleaved coded modulation bicm by modeling the bicm decoder as a mismatched decoder the mismatched decoding model is well defined for finite yet arbitrary block lengths and naturally captures the channel memory among the bits belonging to the same symbol we give two independent proofs of the achievability of the bicm capacity calculated by caire et al where bicm was modeled as a set of independent parallel binary input channels whose output is the bitwise log likelihood ratio our first achievability proof uses typical sequences and shows that due to the random coding construction the interleaver is not required the second proof is based on the random coding error exponents with mismatched decoding where the largest achievable rate is the generalized mutual information moreover the generalized mutual information of the mismatched decoder coincides with the infinite interleaver bicm capacity we show that the error exponent and hence the cutoff rate of the bicm mismatched decoder is upper bounded by that of coded modulation and may thus be lower than in the infinite interleaved model for binary reflected gray mapping in gaussian channels the loss in error exponent is small we also consider the mutual information appearing in the analysis of iterative decoding of bicm with extrinsic information transfer exit charts if the symbol metric has knowledge of the transmitted symbol exit mutual information admits a representation as a pseudo generalized mutual information which is in general not achievable a different symbol decoding metric for which the extrinsic side information refers to the hypothesized symbol induces a generalized mutual information lower than the coded modulation capacity in this case perfect extrinsic side information turns the mismatched decoder error exponent into that of coded modulation

asymptotic comparison of ml and map detectors for multidimensional constellations a classical problem in digital communications is to evaluate the symbol error probability sep and bit error probability bep of a multidimensional constellation over an additive white gaussian noise channel in this paper we revisit this problem for nonequally likely symbols and study the behavior of the optimal maximum a posteriori map detector at asymptotically high signal to noise ratios exact closed form asymptotic expressions for sep and bep for arbitrary constellations and input distributions are presented the well known union bound is proven to be asymptotically tight under general conditions the performance of the practically relevant maximum likelihood ml detector is also analyzed although the decision regions with map detection converge to the ml regions at high signal to noise ratios the ratio between the map and ml detectors in terms of both sep and bep approaches a constant which depends on the constellation and a priori probabilities necessary and sufficient conditions for asymptotic equivalence between the map and ml detectors are also presented

comments on phase shifting for nonseparable 2 d haar wavelets in their recent paper see ibid vol 17 no 7 p 1061 8 2008 alnasser and foroosh derive a wavelet domain in band method for phase shifting of 2 d ldquononseparablerdquo haar transform coefficients their approach is parametrical to the a priori known image translation in this correspondence we show that the utilized transform is in fact the separable haar discrete wavelet transform dwt as such wavelet domain phase shifting can be performed using previously proposed phase shifting approaches that utilize the overcomplete dwt odwt if the given image translation is mapped to the phase component and in band position within the odwt

new algebraic constructions of rotated z sup n lattice constellations for the rayleigh fading channel in this correspondence we present various families of full diversity rotated z sup n lattice constellations based on algebraic number theory constructions we are able to give closed form expressions of their minimum product distance using the corresponding algebraic properties

constellation mappings for two dimensional signaling of nonuniform sources the design of two dimensional constellation mappings for the transmission of binary nonuniform memoryless sources over additive white gaussian noise channels using standard m ary psk and qam modulation schemes is investigated the main application of this problem is the incorporation of an adaptive mapping assignment in modem devices that employ fixed psk qam modulation schemes for the transmission of heterogenous data such as multimedia information containing various levels of nonuniformity in general the optimal mapping depends on both the probability distribution of the input signals and the signal to noise ratio snr in the channel in addition to the geometry of the signal constellation we show that constellation mappings which follow the objective of minimizing the average symbol energy and given this maximizing the decoding probability of the most likely signals can yield symbol error rate and bit error rate performance that is substantially better than gray encoding maps gains as high as 3 5 db in snr e sub b n sub 0 are obtained for highly nonuniform sources finally we note that the mappings techniques result in nonzero mean constellations and briefly consider their performances when they are converted to zero mean constellations by shifting in this case we observe that the shifted zero mean gray map outperforms our shifted maps for small to medium sized constellations m spl les 32 but not for larger sizes

semigroups and nonlinear evolution equations abstract in the main part of this paper we survey some recent results on a semigroup approach to parabolic equations under nonlinear boundary conditions in particular we give a detailed motivation and a heuristic derivation of a generalized variation of constant formula which is the basis for the whole method in the appendix we prove some new results about semigroups in interpolation and extrapolation spaces

bit interleaved coded modulation with mismatched decoding metrics bit interleaved coded modulation bicm has become the de facto coding standard for communication systems recently bicm has been cast as a mismatched decoding scheme due to the assumption of independent bit metrics in addition to this inherent mismatch practical demodulators may produce mismatched decoding metrics because of implementation constraints such as clipping and metric approximation to reduce computational complexity in this paper we investigate bicm with such metrics in line with recent works on this topic we adopt the generalized mutual information gmi as the pertinent performance measure first we show that level dependent scaling of logarithmic bit metrics can improve the bicm gmi second we propose a uniform metric scaling which can lead to an improved performance of mismatched sum product symbol by symbol decoding even if the gmi is not changed third we investigate general metric mismatch correction methods and analyze their effects in terms of the gmi by means of three application examples we illustrate that metric mismatch correction including metric scaling can significantly increase bicm rates

complete to overcomplete discrete wavelet transforms theory and applications a new transform is proposed that derives the overcomplete discrete wavelet transform odwt subbands from the critically sampled dwt subbands complete representation this complete to overcomplete dwt codwt has certain advantages in comparison to the conventional approach that performs the inverse dwt to reconstruct the input signal followed by the a spl grave trous or the lowband shift algorithm specifically the computation of the input signal is not required as a result the minimum number of downsampling operations is performed and the use of upsampling is avoided the proposed codwt computes the odwt subbands by using a set of prediction filter matrices and filtering and downsampling operators applied to the dwt this formulation demonstrates a clear separation between the single rate and multirate components of the transform this can be especially significant when the codwt is used in resource constrained environments such as resolution scalable image and video codecs to illustrate the applicability of the proposed transform in these emerging applications a new scheme for the transform calculation is proposed and existing coding techniques that benefit from its usage are surveyed the analysis of the proposed codwt in terms of arithmetic complexity and delay reveals significant gains as compared with the conventional approach

signal shaping for bicm at low snr the generalized mutual information gmi of bit interleaved coded modulation bicm systems sometimes called the bicm capacity is investigated at low signal to noise ratio snr the combinations of input alphabet input distribution and binary labeling that achieve the shannon limit 1 59 db are completely characterized the main conclusion is that a bicm system with probabilistic shaping achieves the shannon limit at low snr if and only if it can be represented as a zero mean linear projection of a hypercube hence probabilistic shaping offers no extra degrees of freedom to optimize the low snr bicm gmi in addition to what is provided by geometrical shaping the analytical conclusions are confirmed by numerical results which also show that for a fixed input alphabet probabilistic shaping can improve the bicm gmi in the low and medium snr range

the near shift invariance of the dual tree complex wavelet transform revisited abstract the dual tree complex wavelet transform dt c wt is an enhancement of the conventional discrete wavelet transform dwt due to a higher degree of shift invariance and a greater directional selectivity finding its applications in signal and image processing this paper presents a quantitative proof of the superiority of the dt c wt over the dwt in case of modulated wavelets

a new description of trellis codes a trellis code is a sliding window method of encoding a binary data stream as a sequence of real or complex numbers that are input to a noisy transmission channel ungerboeck has constructed simple trellis codes that provide the same noise immunity as is given by increasing the power of uncoded transmission by factors ranging from two to four his method is to specify an underlying convolutional code and a rule mapping by set partitioning that maps the output of this code onto a fixed signal constallation a new description of a trellis code is given that combines these two steps into one the new description is analytic rather than graphical many practical codes can be described very simply and strict bounds on performance can be obtained a method for differential encoding trellis codes is presented that was suggested by the authors representation

bit interleaved coded modulation in the wideband regime the wideband regime of bit interleaved coded modulation bicm in gaussian channels is studied the taylor expansion of the coded modulation capacity for generic signal constellations at low signal to noise ratio snr is derived and used to determine the corresponding expansion for the bicm capacity simple formulas for the minimum energy per bit and the wideband slope are given bicm is found to be suboptimal in the sense that its minimum energy per bit can be larger than the corresponding value for coded modulation schemes the minimum energy per bit using standard gray mapping on m pam or m 2 qam is given by a simple formula and shown to approach 0 34 db as m increases using the low snr expansion a general tradeoff between power and bandwidth in the wideband regime is used to show how a power loss can be traded off against a bandwidth gain

three dimensional subband coding with motion compensation three dimensional 3 d frequency coding is an alternative approach to hybrid coding concepts used in today s standards the first part of this paper presents a study on concepts for temporal axis frequency decomposition along the motion trajectory in video sequences it is shown that if a two band split is used it is possible to overcome the problem of spatial inhomogeneity in the motion vector field mvf which occurs at the positions of uncovered and covered areas in these cases original pixel values from one frame are placed into the lowpass band signal while displaced frame difference values are embedded into the highpass band this technique is applicable with arbitrary mvf s examples with block matching and interpolative motion compensation are given derivations are first performed for the example of two tap quadrature mirror filters qmf s and then generalized to any linear phase qmf s with two band analysis and synthesis stages arranged as cascade structures higher resolution frequency decompositions are realizable in the second part of the paper encoding of the temporal axis subband signals is discussed a parallel filterbank scheme was used for spatial subband decomposition and adaptive lattice vector quantization was employed to approach the entropy rate of the 3 d subband samples coding results suggest that high motion video sequences can be encoded at significantly lower rates than those achievable with conventional hybrid coders main advantages are the high energy compaction capability and the nonrecursive decoder structure in the conclusion the scheme is interpreted more generally viewed as a motion compensated short time spectral analysis of video sequences which can adapt to the quickness of changes although a 3 d multiresolution representation of the picture information is produced a true multiresolution representation of motion information based on spatio temporal decimation and interpolation of the mvf is regarded as the still missing part

avoiding the order reduction of runge kutta methods for linear initial boundary value problems a new strategy to avoid the order reduction of runge kutta methods when integrating linear autonomous nonhomogeneous initial boundary value problems is presented the solution is decomposed into two parts one of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction a numerical illustration is given this idea applies to practical problems where spatial discretization is also required leading to the full order both in space and time

multidimensional coded modulation in block fading channels we study coded modulation over multidimensional signal sets in nakagami m block fading channels we consider the optimal diversity reliability exponent of the error probability when the multidimensional constellation is obtained as the rotation of complex plane signal constellations we show that multidimensional rotations of full dimension achieve the optimal diversity reliability exponent also achieved by gaussian constellations rotations of full dimension induce a large decoding complexity and in some cases it might be beneficial to use multiple rotations of smaller dimension we also study the diversity reliability exponent in this case which yields the optimal rate diversity complexity tradeoff in block fading channels with discrete inputs

mutual information and minimum mean square error in gaussian channels this paper deals with arbitrarily distributed finite power input signals observed through an additive gaussian noise channel it shows a new formula that connects the input output mutual information and the minimum mean square error mmse achievable by optimal estimation of the input given the output that is the derivative of the mutual information nats with respect to the signal to noise ratio snr is equal to half the mmse regardless of the input statistics this relationship holds for both scalar and vector signals as well as for discrete time and continuous time noncausal mmse estimation this fundamental information theoretic result has an unexpected consequence in continuous time nonlinear estimation for any input signal with finite power the causal filtering mmse achieved at snr is equal to the average value of the noncausal smoothing mmse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr

precoding by pairing subchannels to increase mimo capacity with discrete input alphabets we consider gaussian multiple input multiple output mimo channels with discrete input alphabets we propose a non diagonal precoder based on the x codes in to increase the mutual information the mimo channel is transformed into a set of parallel subchannels using singular value decomposition svd and x codes are then used to pair the subchannels x codes are fully characterized by the pairings and a 2 2 real rotation matrix for each pair parameterized with a single angle this precoding structure enables us to express the total mutual information as a sum of the mutual information of all the pairs the problem of finding the optimal precoder with the above structure which maximizes the total mutual information is solved by i optimizing the rotation angle and the power allocation within each pair and ii finding the optimal pairing and power allocation among the pairs it is shown that the mutual information achieved with the proposed pairing scheme is very close to that achieved with the optimal pre coder by cruz et al and is significantly better than mercury waterfllling strategy by lozano et al our approach greatly simplifies both the precoder optimization and the detection complexity making it suitable for practical applications

bridging mixture model estimation and information bounds using i mmse we derive bounds on mutual information for arbitrary estimation problems in additive noise modeled using gaussian mixtures previous work exploiting the i minimum mean squared error mmse formula to formulate a bridge between bounds on the mmse for gaussian mixture model estimation problems and bounds on the mutual information are generalized to allow arbitrary noise modeling a novel upper bound on estimation information is also developed for the general estimation case in addition limits are analyzed to develop bounds on arbitrary entropy asymptotic behavior of all bounds and bound errors with some results bridged back to the mmse domain

optimal alphabets and binary labelings for bicm at low snr optimal binary labelings input distributions and input alphabets are analyzed for the so called bit interleaved coded modulation bicm capacity paying special attention to the low signal to noise ratio snr regime for 8 ary pulse amplitude modulation pam and for 0 75 bit symbol the folded binary code results in a higher capacity than the binary reflected gray code brgc and the natural binary code nbc the 1 db gap between the additive white gaussian noise awgn capacity and the bicm capacity with the brgc can be almost completely removed if the input symbol distribution is properly selected first order asymptotics of the bicm capacity for arbitrary input alphabets and distributions dimensions mean variance and binary labeling are developed these asymptotics are used to define first order optimal foo constellations for bicm i e constellations that make bicm achieve the shannon limit 1 59 db it is shown that the eb n0 required for reliable transmission at asymptotically low rates in bicm can be as high as infinity that for uniform input distributions and 8 pam there are only 72 classes of binary labelings with a different first order asymptotic behavior and that this number is reduced to only 26 for 8 ary phase shift keying psk a general answer to the question of foo constellations for bicm is also given using the hadamard transform it is found that for uniform input distributions a constellation for bicm is foo if and only if it is a linear projection of a hypercube a constellation based on pam or quadrature amplitude modulation input alphabets is foo if and only if they are labeled by the nbc if the constellation is based on psk input alphabets instead it can never be foo if the input alphabet has more than four points regardless of the labeling

space time coding techniques with bit interleaved coded modulations for mimo block fading channels the space time bit interleaved coded modulation st bicm is an efficient technique to obtain high diversity and coding gain on a block fading multiple input multiple output mimo channel its maximum likelihood ml performance is computed under ideal interleaving conditions which enables a global optimization taking into account channel coding thanks to a diversity upper bound derived from the singleton bound an appropriate choice of the time dimension of the space time coding is possible which maximizes diversity while minimizing complexity based on the analysis an optimized interleaver and a set of linear precoders called dispersive nucleo algebraic dna precoders are proposed the proposed precoders have good performance with respect to the state of the art and exist for any number of transmit antennas and any time dimension with turbo codes they exhibit a frame error rate which does not increase with frame length

signal constellations for bit interleaved coded modulation bit interleaved coded modulation bicm is a bandwidth efficient coding technique consisting of serial concatenation of binary error correcting coding bit by bit interleaving and high order modulation bicm is capable of achieving excellent error performance provided that powerful codes such as for example turbo codes or low density parity check ldpc codes are employed we address the problem of finding the signal sets that are the most suitable ones for designing power efficient bicm schemes over an additive white gaussian noise awgn channel to this end we exploit the expression of the bicm capacity limit and evaluate it for several 8 and 16 ary constellations the bit error rate ber performance of some bicm schemes made up of turbo codes and various signal sets is also investigated by computer simulations so as to illustrate the theoretical results we show that for spectral efficiencies of practical interest the most attractive signal sets are those for which gray mapping is possible provided that their symbol error rate performance is sufficiently close to the optimum this explains why some constellations having a simple structure such as 8 psk and 16 qam perform very well when combined with a powerful code at the same time the constellations displaying optimal error performance without coding are generally not of interest for bicm

gray coding for multilevel constellations in gaussian noise the problem of finding the optimal labeling bit to symbol mapping of multilevel coherent phase shift keying psk pulse amplitude modulation pam and quadrature amplitude modulation qam constellations with respect to minimizing the bit error probability bep over a gaussian channel is addressed we show that using the binary reflected gray code brgc to label the signal constellation results in the lowest possible bep for high enough signal energy to noise ratios and analyze what is high enough in this sense it turns out that the brgc is optimal for psk and pam systems whenever the target bep is at most a few percent which covers most systems of practical interest new and simple closed form expressions are presented for the bep of psk pam and qam using the brgc

mmse dimension if n is standard gaussian the minimum mean square error mmse of estimating a random variable x based on snr x n vanishes at least as fast as 1 snr as snr we define the mmse dimension of x as the limit as snr of the product of snr and the mmse mmse dimension is also shown to be the asymptotic ratio of nonlinear mmse to linear mmse for discrete absolutely continuous or mixed distribution we show that mmse dimension equals renyi s information dimension however for a class of self similar singular x e g cantor dis tribution we show that the product of snr and mmse oscillates around information dimension periodically in snr db we also show that these results extend considerably beyond gaussian noise under various technical conditions

on channel capacity per unit cost memoryless communication channels with arbitrary alphabets where each input symbol is assigned a cost are considered the maximum number of bits that can be transmitted reliably through the channel per unit cost is studied it is shown that if the input alphabet contains a zero cost symbol then the capacity per unit cost admits a simple expression as the maximum normalized divergence between two conditional output distributions the direct part of this coding theorem admits a constructive proof via stein s lemma on the asymptotic error probability of binary hypothesis tests single user multiple access and interference channels are studied

mimo gaussian channels with arbitrary inputs optimal precoding and power allocation in this paper we investigate the linear precoding and power allocation policies that maximize the mutual information for general multiple input multiple output mimo gaussian channels with arbitrary input distributions by capitalizing on the relationship between mutual information and minimum mean square error mmse the optimal linear precoder satisfies a fixed point equation as a function of the channel and the input constellation for non gaussian inputs a nondiagonal precoding matrix in general increases the information transmission rate even for parallel noninteracting channels whenever precoding is precluded the optimal power allocation policy also satisfies a fixed point equation we put forth a generalization of the mercury waterfilling algorithm previously proposed for parallel noninterfering channels in which the mercury level accounts not only for the non gaussian input distributions but also for the interference among inputs

evaluating the performance of convolutional codes over block fading channels this correspondence considers union upper bound techniques for error control codes with limited interleaving over block fading rician channels a modified bounding technique is presented that relies on limiting the conditional union bound before averaging over the fading process this technique although analytically not very attractive provides tight and hence useful numerical results

high snr asymptotics of mutual information for discrete constellations with applications to bicm asymptotic expressions of the mutual information between any discrete input and the corresponding output of the scalar additive white gaussian noise channel are presented in the limit as the signal to noise ratio snr tends to infinity asymptotic expressions of the symbol error probability sep and the minimum mean square error mmse achieved by estimating the channel input given the channel output are also developed it is shown that for any input distribution the conditional entropy of the channel input given the output mmse and sep have an asymptotic behavior proportional to the gaussian q function the argument of the q function depends only on the minimum euclidean distance med of the constellation and the snr and the proportionality constants are functions of the med and the probabilities of the pairs of constellation points at med the developed expressions are then generalized to study the high snr behavior of the generalized mutual information gmi for bit interleaved coded modulation bicm by means of these asymptotic expressions the long standing conjecture that gray codes are the binary labelings that maximize the bicm gmi at high snr is proven it is further shown that for any equally spaced constellation whose size is a power of two there always exists an anti gray code giving the lowest bicm gmi at high snr

a shift invariant discrete wavelet transform this article presents a unifying approach to the derivation and implementation of a shift invariant wavelet transform of one and two dimensional 1 d and 2 d discrete signals starting with mallat s 1989 multiresolution wavelet representation mrwar it presents an analytical process through which a shift invariant orthogonal discrete wavelet transform called the multiscale wavelet representation mswar is obtained the coefficients in the mswar are shown to be inclusive of those in the mrwar with the implication that the derived representation is invertible the computational complexity of the mswar is quantified in terms of the required convolutions and its implementation is shown to be equivalent to the filter upsampling technique

optimum power allocation for parallel gaussian channels with arbitrary input distributions the mutual information of independent parallel gaussian noise channels is maximized under an average power constraint by independent gaussian inputs whose power is allocated according to the waterfilling policy in practice discrete signaling constellations with limited peak to average ratios m psk m qam etc are used in lieu of the ideal gaussian signals this paper gives the power allocation policy that maximizes the mutual information over parallel channels with arbitrary input distributions such policy admits a graphical interpretation referred to as mercury waterfilling which generalizes the waterfilling solution and allows retaining some of its intuition the relationship between mutual information of gaussian channels and nonlinear minimum mean square error mmse proves key to solving the power allocation problem

avoiding the order reduction when solving second order in time pdes with fractional step runge kutta nystrom methods we study some of the main features of fractional step runge kutta nystrom methods when they are used to integrate initial boundary value problems of second order in time in combination with a suitable spatial discretization we focus our attention on the order reduction phenomenon which appears if classical boundary conditions are taken at the internal stages this drawback is specially hard when time dependent boundary conditions are considered in this paper we present an efficient technique very simple and computationally cheap which allows us to avoid the order reduction such technique consists in modifying the boundary conditions for the internal stages of the method

spectral fractional step runge kutta discretizations for initial boundary value problems with time dependent boundary conditions in this paper we develop a technique for avoiding the order reduction caused by nonconstant boundary conditions in the methods called splitting alternating direction or more generally fractional step methods such methods can be viewed as the combination of a semidiscrete in time procedure with a special type of additive runge kutta method which is called the fractional step runge kutta method and a standard space discretization which can be of type finite differences finite elements or spectral methods among others spectral methods have been chosen here to complete the analysis of convergence of a totally discrete scheme of this type of improved fractionary steps the numerical experiences performed also show the increase of accuracy that this technique provides

comparison of the achievable rates in ofdm and single carrier modulation with i i d inputs we compare the maximum achievable rates in single carrier sc and orthogonal frequency division multiplexing ofdm modulation schemes under the practical assumptions of independent identically distributed finite alphabet inputs and linear intersymbol interference with additive gaussian noise we show that the shamai laroia approximation serves as a bridge between the two rates while it is well known that this approximation is often a lower bound on the sc achievable rate it is revealed to also essentially upper bound the ofdm achievable rate we apply information estimation relations in order to rigorously establish this result for both general input distributions and to sharpen it for commonly used pulse amplitude modulation pam and quadratic amplitude modulation constellations to this end novel bounds on minimum mean square error estimation of pam inputs to a scalar gaussian channel are derived which may be of general interest our results show that under reasonable assumptions optimal sc schemes may offer spectral efficiency significantly superior to that of ofdm motivating further research of such systems

motion estimation using low band shift method for wavelet based moving picture coding the discrete wavelet transform dwt has several advantages of multiresolution analysis and subband decomposition which has been successfully used in image processing however the shift variant property is intrinsic due to the decimation process of the wavelet transform and it makes the wavelet domain motion estimation and compensation inefficient to overcome the shift variant property a low band shift method is proposed and a motion estimation and compensation method in the wavelet domain is presented the proposed method has a superior performance to the conventional motion estimation methods in terms of the mean absolute difference mad as well as the subjective quality the proposed method can be a model method for the motion estimation in wavelet domain just like the full search block matching in the spatial domain

on the optimality of the binary reflected gray code this paper concerns the problem of selecting a binary labeling for the signal constellation in m psk m pam and m qam communication systems gray labelings are discussed and the original work by frank gray is analyzed as is noted the number of distinct gray labelings that result in different bit error probability grows rapidly with increasing constellation size by introducing a recursive gray labeling construction method called expansion the paper answers the natural question of what labeling among all possible constellation labelings will give the lowest possible average probability of bit errors for the considered constellations under certain assumptions on the channel the answer is that the labeling proposed by gray the binary reflected gray code is the optimal labeling for all three constellations which has surprisingly never been proved before