the strong simplex conjecture is false the design of m average energy constrained signals in additive white gaussian noise is addressed the long standing strong simplex conjecture which postulates that the regular simplex signal set maximizes the probability of correct detection under an average energy constraint is disproven a signal set is presented that performs better than the regular simplex signal set at low signal to noise ratios for all m spl ges 7 this leads to the result that for all m spl ges 7 there is no signal set of m signals which is optimal at all signal to noise ratios furthermore the optimal signal set at low signal to noise ratios is not an equal energy set for any m spl ges 7 the regular simplex is shown to be the unique signal set which maximizes the minimum distance between signals it follows that a signal set which maximizes the minimum distance is not necessarily optimum however the regular simplex is shown to be globally optimum in the sense of uniquely maximizing the union bound on error probability at all signal to noise ratios

fundamental limits of parallel optical wireless channels capacity results and outage formulation multi channel mc optical wireless communication owc systems employing wave division multiplexing for outdoors free space optical communications or multi user time division multiple access for indoors visible light communications e g can be modeled as parallel channels multi input multi output owc systems can also be transformed possibly with some performance loss to parallel channels using pre post coding studying the performance of such mc owc systems requires characterizing the capacity of the underlying parallel channels in this paper upper and lower bounds on the capacity of constant parallel owc channels with a total average intensity constraint are derived then this paper focuses on finding intensity allocations that maximize the lower bounds given channel state information at the transmitter csit due to its nonconvexity the karush kuhn tucker conditions are used to describe a list of candidate allocations instead searching exhaustively for the best solution low complexity near optimal algorithms are proposed the resulting optimized lower bound nearly coincides with capacity at high signal to noise ratio snr under a quasi static channel model and in the absence of csit outage probability upper and lower bounds are derived those bounds also meet at high snr thus characterizing the outage capacity in this regime finally the results are extended to a system with both average and peak intensity constraints

on the capacity of free space optical intensity channels upper and lower bounds are derived on the capacity of the free space optical intensity channel this channel has a nonnegative input representing the transmitted optical intensity which is corrupted by additive white gaussian noise to preserve the battery and for safety reasons the input is constrained in both its average and its peak power for a fixed ratio of the allowed average power to the allowed peak power the difference between the upper and the lower bound tends to zero as the average power tends to infinity and their ratio tends to one as the average power tends to zero when only an average power constraint is imposed on the input the difference between the bounds tends to zero as the allowed average power tends to infinity and their ratio tends to a constant as the allowed average power tends to zero

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

calculation of mutual information for partially coherent gaussian channels with applications to fiber optics the mutual information between a complex valued channel input and its complex valued output is decomposed into four parts based on polar coordinates an amplitude term a phase term and two mixed terms numerical results for the additive white gaussian noise awgn channel with various inputs show that at high signal to noise ratio snr the amplitude and phase terms dominate the mixed terms for the awgn channel with a gaussian input analytical expressions are derived for high snr the decomposition method is applied to partially coherent channels and a property of such channels called spectral loss is developed spectral loss occurs in nonlinear fiber optic channels and it may be one effect that needs to be taken into account to explain the behavior of the capacity of nonlinear fiber optic channels

on ternary phase shift keyed signaling corresp in agreement with a recent paper by pierce it is shown that ternary phase shift keyed psk signals attain the lowest bit error rate at high signal to noise ratio snr if more than 4 3 binary digits can be mapped to a symbol this point is further demonstrated by using a 3b 2t code which maps 1 5 bits to a symbol the spectrum and bit error rate are evaluated for the 3b 2t coded ternary psk signal and compared with those of quaternary phase shift keyed qpsk signal

shaping and nonequiprobable signaling for intensity modulated signals theory of shaping and nonequiprobable signaling which has been developed for conventional electrical signals must be modified to treat intensity modulated im signals we show that for im signals the optimum shape of the constellation bounding region in n dimensional n d space is an n d simplex as n spl rarr spl infin the maximum achievable shape gain is 1 33 db in terms of transmitted power and the resulting marginal signaling distribution on the one dimensional 1 d constituent constellation is exponential we also investigate the tradeoffs between shaping and its negative consequences and find that a 1 db shape gain can be achieved while incurring reasonable increases in peak to average power ratio and constellation expansion ratio

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

constellation design for a multicarrier optical wireless communication channel a block wise constellation design is presented for optical communication systems with multi subcarrier modulation msm intensity modulation im and direct detection dd the dc bias traditionally used only for compensating the negative peaks of the transmitter side signals is treated as an information carrying basis in our proposed scheme called msm jdcm designs are done for both flat fading and frequency selective fading scenarios and following a principle of high dimensional sphere packing to simplify the problem we apply the following methods first we use bounds on the waveform s maximum and minimum second we use the maximum and minimum constraints on a set of sufficient samples of waveforms third we relax non convex distance constraints into convex ones by iterative linearizations with the msm jdcm we minimize electrical power optical power and peak power with a common target bit error rate ber analysis shows that the msm jdcm offers significant power gains over msm normal and msm spss the short term peak to average power ratio papr and long term papr constraints are combined with the msm jdcm to mitigate the nonlinear distortion caused by high power amplifier and laser diode which is another novelty of our scheme to attain lower ber a binary switching algorithm bsa is applied to find the improved constellation labeling

optical intensity modulated direct detection channels signal space and lattice codes traditional approaches to constructing constellations for electrical channels cannot be applied directly to the optical intensity channel this work presents a structured signal space model for optical intensity channels where the nonnegativity and average amplitude constraints are represented geometrically lattice codes satisfying channel constraints are defined and coding and shaping gain relative to a baseline are computed an effective signal space dimension is defined to represent the precise impact of coding and shaping on bandwidth average optical power minimizing shaping regions are derived in some special cases example lattice codes are constructed and their performance on an idealized point to point wireless optical link is computed bandwidth efficient schemes are shown to have promise for high data rate applications but require greater average optical power

optimizing constellations for single subcarrier intensity modulated optical systems we optimize modulation formats for the additive white gaussian noise channel with nonnegative input also known as the intensity modulated direct detection channel with and without confining them to a lattice structure our optimization criteria are the average electrical average optical and peak power the nonnegative constraint on the input to the channel is translated into a conical constraint in signal space and modulation formats are designed by sphere packing inside this cone some dense packings are found which yield more power efficient modulation formats than previously known for example at a spectral efficiency of 1 5 bit s hz the modulation format optimized for average electrical power has a 2 55 db average electrical power gain over the best known format to achieve a symbol error rate of 10 6 the corresponding gains for formats optimized for average and peak optical power are 1 35 and 1 72 db respectively using modulation formats optimized for peak power in average power limited systems results in a smaller power penalty than when using formats optimized for average power in peak power limited systems we also evaluate the modulation formats in terms of their mutual information to predict their performance in the presence of capacity achieving error correcting codes and finally show numerically and analytically that the optimal modulation formats for reliable transmission in the wideband regime have only one nonzero point

bandlimited optical intensity modulation under average and peak power constraints bandlimited optical intensity channels arising in applications such as visible light communications require that all signals satisfy a bandwidth constraint as well as average peak and non negativity amplitude constraints in this paper a 2 d signal space for bandlimited optical intensity channels is presented a novel feature of this model is that the non negativity and peak constraints are relaxed and the signal space parameterizes the probability of the negative or peak amplitude excursions in the output although the intensity channel only supports non negative amplitudes the impact of clipping on system performance is shown to be negligible if the likelihood of the negative excursion is small a tractable approximation using finite series is used to accurately compute the probability of clipping under average and peak optical power constraints the optical power and spectral efficiencies using hexagonal lattice constellations are computed schemes designed in this paper have higher average and peak optical power gains than m ary pulse amplitude modulation pam using previously established techniques for spectral efficiencies greater than 2 5 and 3 5 bits s hz respectively for high spectral efficiency e g greater than 6 bits s hz the proposed scheme attains a more than 2 db average and peak optical power gain over 16 pam using the previous approaches

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

average power reduction techniques for multiple subcarrier intensity modulated optical signals we describe two classes of simple effective techniques for reducing the average optical power requirement in intensity modulated optical systems using multiple bpsk or qpsk subcarriers the first class of techniques involves block coding between the information bits to be transmitted and the symbol amplitudes modulated onto the subcarriers in order to increase the minimum value of the multiple subcarrier electrical waveform the second class of techniques involves replacing the fixed dc bias by a bias signal that varies on a symbol by symbol basis these two classes of techniques can be applied separately or in tandem the reduction in power requirement increases with the number of subcarriers and with eight subcarriers can be as high as about 3 6 db and 3 2 db with bpsk and qpsk respectively the techniques described are applicable as long as all subcarriers originate from a single transmitter and are symbol synchronized

vertex operator algebras and associative algebras letvbe a vertex operator algebra we construct a sequence of associative algebrasan v n 0 1 2 such thatan v is a quotient ofan 1 v and a pair of functors between the category ofan v modules which are notan 1 v modules and the category of admissiblev modules these functors exhibit a bijection between the simple modules in each category we also show thatvis rational if and only if allan v are finite dimensional semisimple algebras

error rate analysis for peaky signaling over fading channels in this paper the performance of signaling strategies with high peak to average power ratio is analyzed in both coherent and noncoherent fading channels two modulation schemes namely on off phase shift keying oopsk and on off frequency shift keying oofsk are considered the optimal detector structures are identified and analytical expressions for the error probabilities are obtained for arbitrary constellation sizes numerical techniques are employed to compute the error rates it is concluded that increasing the peakedness of the signals results in reduced error rates for a given power level and hence equivalently improves the energy efficiency for fixed error probabilities

on quantum galois theory the goals of the present paper are to initiate a program to systematically study and rigorously establish what a physicist might refer to as the operator content of orbifold models to explain what this might mean and to clarify the title of the paper we will assume that the reader is familiar with the algebraic formulation of 2 dimensional cft in the guise of vertex operator algebras voa see b flm and dm for more information on this point in the paper dvvv several ideas are proposed concerning the structure of a holomorphic orbifold in other words if v is a holomorphic voa and if g is a finite group of automorphisms of v then the sub voa v g of g invariants is itself a voa and the subject of dvvv is very much concerned with speculation on the nature of the v modules it turns out to be more useful at least for purpose of inductive proofs to take v to be a simple voa we will then see that v g is also simple whenever g is a finite group of automorphisms of v one consequence of our main results is the following

on the tensor structure of modules for compact orbifold vertex operator algebras suppose v g is the fixed point vertex operator subalgebra of a compact group g acting on a simple abelian intertwining algebra v we show that if all irreducible v g modules contained in v live in some braided tensor category of v g modules then they generate a tensor subcategory equivalent to the category mathrm rep g of finite dimensional representations of g with associativity and braiding isomorphisms modified by the abelian 3 cocycle defining the abelian intertwining algebra structure on v additionally we show that if the fusion rules for the irreducible v g modules contained in v agree with the dimensions of spaces of intertwiners among g modules then the irreducibles contained in v already generate a braided tensor category of v g modules these results do not require rigidity on any tensor category of v g modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known for example they apply when v g is c 2 cofinite but not necessarily rational when v g is both c 2 cofinite and rational and v is a vertex operator algebra we use the equivalence between mathrm rep g and the corresponding subcategory of v g modules to show that v is also rational as another application we show that a certain category of modules for the virasoro algebra at central charge 1 admits a braided tensor category structure equivalent to mathrm rep su 2 up to modification by an abelian 3 cocycle

lattice codes for amplified direct detection optical systems theories of shaping for lattice codes have been developed for systems optical or non optical using coherent detection with additive white gaussian noise awgn and for direct detection optical systems with awgn this paper considers shaping for amplified direct detection optical systems in which signal spontaneous beat noise a form of signal dependent noise is dominant an a dimensional a d signal is formed by modulating the intensities squares of field magnitudes of a sequence of n time disjoint pulses in field magnitude coordinates signal energy is represented by a l2 norm and the optimal constellation bounding region is the nonnegative orthant bounded by an n sphere under a continuous approximation as nrarrinfin the ultimate shape gain is 1 53 db and the induced signaling distribution on the constituent 1 d constellation becomes half gaussian in practice the ultimate shape gain can be approached when the 1 d constellation follows a truncated half gaussian distribution we investigate the tradeoffs between shape gain and increases in constellation expansion ratio or peak to average power ratio we compare our shaping results with those for coherent detection systems and direct detection optical systems with awgn