accurate evaluation of bit error rates of optical communication systems using the gram charlier series the probability densities and cumulative distribution functions of decision statistics of optical communications systems are expanded as a gram charlier g c series leading to arbitrarily accurate systematic evaluation of bit error rates bers and optimal decision thresholds of optical communication systems the method displays negligible computational complexity and is applicable whenever the moment or cumulant generating functions of the decision statistics are analytically available we applied the technique to a birth and death markovian model of a direct detection receiver with optical preamplifier in a two level amplitude shift keying system the modal expansion series rapidly converged whereas the alternative saddlepoint approximation method predicted a ber which deviated by 7 from the g c result

relationship between the saddlepoint approximation and the modified chernoff bound optical communication a comparison is made of two methods for the computation of the probability that the sum of a gaussian and a nongaussian random variable exceeds a certain threshold value the two methods the saddlepoint approximation sap and the modified chernoff bound are closely related to one another for the computation of error probabilities in digital optical communications where both methods can be applied the modified chernoff bound is found to lie close above the saddlepoint approximation for all cases of practical interest the ratio of signal dependent to additive noise has been established as the parameter dominating the bound tightness since the sap has been reported to be extremely precise this result indicates that the more versatile upper bound can be used with confidence both for performance evaluation and for optimization of optical communication systems

computing bit error probabilities for avalanche photodiode receivers by large deviations theory a bit error probability analysis of direct detection optical receivers is presented employing avalanche photodiodes an asymptotic analysis for large signal intensities is presented this analysis provides some useful insight into the balance between the poisson statistics the avalanche gain statistics and the gaussian thermal noise the conjugate distribution is developed it is obtained by applying the large deviation exponential twisting formula it is demonstrated that this conjugate distribution can be used to obtain numerically efficient monte carlo estimates of the bit error probability via the importance sampling method

a class of optimum importance sampling strategies abstract the problem of estimating expectations of functionals of random vectors via simulation is investigated as an alternative to simple averaging an efficient form of monte carlo simulations known as importance sampling can be employed to produce arbitrarily accurate estimates of the expectation of interest the fundamental problem in importance sampling is to determine the appropriate statistics under which to conduct the simulation these statistics are typically determined from a constraint class by minimizing the variance of the importance sampling estimator over the class of admissible statistics in previous work the authors showed that minimizing the importance sampling variance is equivalent to finding the distribution which is closest to the unconstrained optimal distribution with respect to a specific information theoretic f divergence or ali silvey distance in this work the authors determine the optimal biasing density from a constraint class whose controlling parameter is fundamental in the performance analysis of importance sampling in addition it is shown that the constrained optimal distribution from this class minimizes every statistical distance measure to the global optimal distribution and as a consequence is optimal with respect to a large family of nonlinear cost functions salient features of this distribution are 1 unlike the unconstrained optimal solution this distribution can be made independent of the parameter to be estimated and thus can admit to implementation and 2 this distribution renders performance gains which can be made arbitrarily close to the optimal gains applications to estimating probabilities of rare events e g error rates in communication systems will be presented further analysis will show that in this case the savings over monte carlo simulations become unbounded as the probability of the rare event diminishes

additive local spectrum compressors let l x be the algebra of all bounded linear operators on an infinite dimensional complex banach space x we characterize additive continuous maps from l x onto itself which compress the local spectrum and the convexified local spectrum at a nonzero fixed vector additive continuous maps from l x onto itself that preserve the local spectral radius at a nonzero fixed vector are also characterized

nonparametric importance sampling abstract importance sampling is a widely used variance reduction simulation technique for the evaluation of high dimensional integrals a key step in the implementation of importance sampling is to choose a proper distribution function from which pseudorandom numbers are generated parametric sampling distributions if available at all are often inadequate for high dimensional integrals over irregular regions one possible remedy is to use a nonparametric method to estimate the unknown optimal sampling function we show that the nonparametric approach yields integral estimates that converge faster than estimates obtained from parametric approaches we also demonstrate that an adaptive method which has been used successfully in parametric settings does not yield better results than simple one step methods in the nonparametric setting

on the choice of alternative measures in importance sampling with markov chains in the simulation of markov chains importance sampling involves replacing the original transition matrix say p with a suitably chosen transition matrix q that tends to visit the states of interest more frequently the likelihood ratio of p relative to q is an important random variable in the importance sampling method it always has expectation one and for any interesting pair of transition matrices p and q there is a sample path length that causes the likelihood ratio to be close to zero with probability close to one this may cause the variance of the importance sampling estimator to be larger than the variance of the traditional estimator we develop sufficient conditions for ensuring the tightness of the distribution of the logarithm of the likelihood ratio for all sample path lengths and we show that when these conditions are satisfied the likelihood ratio is approximately lognormally distributed with expected value one these conditions can be used to eliminate some choices of the alternative transition matrix q that are likely to result in a variance increase we also show that if the likelihood ratio is to remain well behaved for all sample path lengths the alternative transition matrix q has to approach the original transition matrix p as the sample path length increases the practical significance of this result is that importance sampling can be difficult to apply successfully in simulations that involve long sample paths

jordan triple maps abstract let a be a standard operator algebra on a banach space of dimension 1 and b be an algebra over the field of rational numbers let k be a non zero rational number in this note we prove that if a bijection a b satisfies kaba k a b a for all a b a or k abc cba k a b c c b a for all a b c a then is additive

additive spectrum compressors abstract additive bijections a b which compress the spectrum between two unital standard operator algebras are characterized applications to local approximate anti multiplications are also given

a not so simple local multiplier algebra we construct an af algebra a such that its local multiplier algebra mloc a does not agree with mloc mloc a thus answering a question raised by g k pedersen in 1978

on large deviations theory and asymptotically efficient monte carlo estimation it is shown how large deviations theory can be applied to construct an asymptotically efficient simulation distribution for monte carlo simulation using the importance sampling technique a sufficient and necessary condition is given for the asymptotic efficiency of the candidate simulation distributions this is done in the multidimensional setting that is required by many practical simulation problems the result obtained is applied primarily in two areas first the generalization of previous work dealing with functionals of markov chains is discussed a second area of application is the simulation of nonlinear systems with gaussian inputs as an example of a system with gaussian inputs the simulation of a digital communication channel with nonlinear isi intersymbol interference characteristic of satellite data links is considered in the case of linear isi the asymptotically efficient exponentially twisted distribution turns out to agree with a method previously proposed by d lu and k yao 1988 the large deviations point of view provides some useful insight on how to extent the lu and yao method to nonlinear channels

the local multiplier algebra of a c algebra ii abstract let a be a separable c algebra and let m loc a be the local multiplier algebra of a it is shown that every minimal closed prime ideal of m loc a is primitive if prim a has a dense g consisting of closed points for instance if prim a is a t 1 space and a is unital then m loc a is its own local multiplier algebra and has only inner derivations the same is true for m loc m loc a if a is non unital if a is postliminal then m loc m loc a is the regular completion of a which is an aw algebra

second order local multiplier algebras of continuous trace c algebras abstract we determine the injective envelope and local multiplier algebra of a continuous trace c algebra a that arises from a continuous hilbert bundle over an arbitrary locally compact hausdorff space in addition we show that the second order local multiplier algebra m loc 2 a of any such algebra a is injective

additive maps on standard operator algebras preserving parts of the spectrum let a and b be standard operator algebras on infinite dimensional complex banach spaces x and y respectively and let be a unital additive surjection from a onto b we introduce thirteen parts of the spectrum for elements in a and b and prove that if preserves any one of these parts of the spectrum then it is either an isomorphism or an anti isomorphism

the local multiplier algebra of a c algebra with finite dimensional irreducible representations abstract we describe up to isomorphism the local multiplier algebra m loc a of a c algebra a which admits only finite dimensional irreducible representations m loc a in this case is a finite or countable direct product of c algebras of the form c x n m n where each space x n is stonean in particular m loc a is an a w algebra of type i so it admits only inner derivations and it coincides with the injective envelope i a of a

efficient simulation of value at risk with heavy tailed risk factors simulation of small probabilities has important applications in many disciplines the probabilities considered in value at risk var are moderately small however the variance reduction techniques developed in the literature for var computation are based on large deviations methods which are good for very small probabilities modeling heavy tailed risk factors using multivariate t distributions we develop a new method for var computation we show that the proposed method minimizes the variance of the importance sampling estimator exactly whereas previous methods produce approximations to the exact solution thus the proposed method consistently outperforms existing methods derived from large deviations theory under various settings the results are confirmed by a simulation study

a randomized bias technique for the importance sampling simulation of bayesian equalizers an importance sampling is simulation technique is presented for bayesian equalizers based on the large deviations theory approach developed by sadowsky and bucklew see ibid vol 36 no 5 p 579 1990 the resulting simulation density consists of a sum of exponentially twisted distributions for the additive gaussian channel this simulation density is equivalent to the conventional mean shift noise biasing is method but with the bias vector chosen from a fixed set in a random manner in order to properly select the bias vectors the asymptotic decision boundary of the bayesian equalizer is first determined it is shown that the boundary is formed by multiple hyperplanes and that the appropriate bias vectors are orthogonal to the hyperplanes the simulation technique is then extended to the recursive symbol by symbol detector of abend and fritchman a f algorithm proposed in 1970 and simulation results are presented for both recursive and non recursive equalizers

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

modified chernoff bounds for pam systems with noise and interference corresp by using principles of analytic continuation upper and lower bounds on the error probability of a canonical binary system corrupted by additive interference and independent zero mean gaussian noise are derived the bounds which are simple functions require only the evaluation or bounding of the moment generating function of the interference for large signal to noise ratios snr and even for moderately large interference the bounds are shown to be tight and useful

efficient importance sampling for events of moderate deviations with applications summary we propose a method for finding the alternative distribution in importance sampling the alternative distribution is optimal in the sense that the asymptotic variance is mini mised for estimating tail probabilities of asymptotically normal statistics our contribution to importance sampling is three fold to begin with we obtain an explicit expression for the mean of the optimal alternative distribution and the expression motivates a recursive approximation algorithm secondly a new multi dimensional exponential tilting formula is presented lastly a conservative estimator of the variance is given to facilitate a quick comparison among different stratified sampling schemes in conjunction with importance sampling several numerical examples illustrating the efficacy of the proposed method are also included these results indicate that the proposed method is considerably more efficient than the method based on large deviations theory and the efficiency gain is more significant in higher dimensions

maps completely preserving spectral functions abstract let x y be infinite dimensional complex banach spaces and a b be standard operator algebras on x and y respectively in this paper we show that surjective maps completely preserving certain spectral function from a to b are isomorphisms where stands for any one of 13 spectral functions l r l r p c p c p c ap s and ap s

adaptive threshold estimation via extreme value theory determining a detection threshold to automatically maintain a low false alarm rate is a challenging problem in a number of different applications the underlying parametric assumptions of most automatic target detection algorithms are invalid therefore thresholds derived using these incorrect distribution assumptions do not produce desirable results when applied to real sensor data monte carlo methods for threshold determination work well but tend to perform poorly when targets are present in order to mitigate these effects we propose an algorithm using extreme value theory through the use of the generalized pareto distribution gpd and a kolmogorov smirnov statistical test unlike previous work based on gpd estimates this algorithm incorporates a way to adaptively maintain low false alarm rates in the presence of targets both synthetic and real world detection results demonstrate the usefulness of this algorithm

performance of multiuser detection for optical cdma i error probabilities characterization of optimum multiuser performance in intensity modulation direct detection optical cdma channels is addressed analysis is based on some simple bounds on optimum performance provided by receivers with side information or by a linear decorrelating receiver numerical comparisons of the performance bounds are obtained via application of saddlepoint approximations to the multiuser photocount statistics general analytical relations comparing these approximations to similar asymptotic approximations based on the chernoff bound are also derived the numerical results illustrate the fundamental importance of two of the performance bounds in addition the numerical error probability results verify the relevance of the asymptotically based performance analysis for optical cdma in part ii

stochastic systems with small noise analysis and simulation a phase locked loop example systems with wide bandwidth noise inputs are a common occurrence in stochastic control and communication theory and elsewhere e g tracking or synchronization systems such as phase locked loops pll one is often interested in calculating such quantities as the probability of escape from a desired error set in some time interval or the mean time for such escape diffusion approximations the system obtained in the limit bandwidth as the to infty are often used for this since they are easier to analyze when the noise effects in the physical system are small one is tempted to do an asymptotic analysis noise intensity to 0 on the diffusion approximation and use this for the desired estimates on the original system such a procedure does not work in general the double limit bandwidth to infty intensity to 0 is not always justified under quite broad conditions on the noise processes it is justified for the systems studied here we study a particular form of the pll owing to

conditional importance sampling estimators we give a unified presentation of the conditional importance sampling estimators we show that they are always better than their nonconditional counterparts we then present the large deviation theory associated with these estimators in particular we give conditional simulation distributions that are optimal in the sense that they are efficient interestingly enough these distributions will not in general be the usual exponential shifts we give examples showing how to use the theory developed

spectrum preserving linear maps abstract let x and y be banach spaces we show that a spectrum preserving surjective linear map from b x to b y is either of the form t ata 1 for an isomorphism a of x onto y or the form t bt b 1 for an isomorphism b of x onto y

importance sampling for tanner trees this correspondence presents an importance sampling is simulation scheme for the soft iterative decoding on loop free multiple layer trees it is shown that this scheme is asymptotically efficient in that for an arbitrary tree and a given estimation precision the required number of samples is inversely proportional to the noise standard deviation this work has its application in the simulation of low density parity check ldpc codes

direct sequence spread spectrum multiple access communications with random signature sequences a large deviations analysis a direct sequence spread spectrum multiple access bit error probability analysis is developed using large deviations theory let m denote the number of interfering spread spectrum signals and let n denote the signature sequence length then the large deviations limit is as n to infinity with m fixed a tight asymptotic expression for the bit error probability is proven and in addition recent large deviations results with the importance sampling monte carlo estimation technique are applied to obtain accurate and computationally efficient estimates of the bit error probability for finite values of m and n the large deviations point of view is compared also to the conventional asymptotics of central limit theory and the associated gaussian approximation the gaussian approximation is accurate and the ratio m n is moderately large and all signals have roughly equal power in the near far situation however the gaussian approximation is quite poor in contrast large deviations techniques are more accurate in the near far situation and it is here that these methods provide some important practical insight

bit error rates for ultrafast apd based optical receivers exact and large deviation based asymptotic approaches exact analysis as well as asymptotic analysis based on large deviation theory ldt are developed to compute the bit error rate ber for ultrafast avalanche photodiode apd based optical receivers assuming on off keying and direct detection the effects of intersymbol interference isi resulting from the apd s stochastic avalanche buildup time as well as the apd s dead space are both included in the analysis isi becomes a limiting factor as the transmission rate approaches the detector s bandwidth in which case the bit duration becomes comparable to apd s avalanche buildup time further the effect of dead space becomes significant in high speed apds that employ thin avalanche multiplication regions while the exact ber analysis at the generality considered here has not been reported heretofore the asymptotic analysis is a major generalization of that developed by letaief and sadowsky ieee trans inform theory vol 38 1992 in which the ldt was used to estimate the ber assuming apds with an instantaneous response negligible avalanche buildup time and no dead space these results are compared with those obtained using the common gaussian approximation approach showing the inadequacy of the guassian approximation when isi noise has strong presence

ber analysis of bpsk signals in ricean faded cochannel interference the bit error rate of a binary phase shift keying bpsk signal in ricean faded cochannel interference is studied a precise bit error rate expression is derived for a bandlimited bpsk signal corrupted by an arbitrary number of asynchronous ricean faded interfering signals an error rate estimation based on a saddlepoint approximation is also provided it is shown that this approximation is highly accurate for the special case when there is one synchronous ricean faded interfering signal in an interference limited environment an asymptotic error rate analysis based on the saddle point approximation reveals that fading in the cochannel interfering signal can have different impacts on the desired user s error rate performance our study shows that when the signal to interference ratio sir is sufficiently large the error floor of the desired user signal decreases with an increase of the rice factor in the interfering user s fading channel however the opposite phenomenon can happen when the sir is small

large deviations theory and efficient simulation of excessive backlogs in a gi gi m queue the problem of using importance sampling to estimate the average time to buffer overflow in a stable gi gi m queue is considered using the notion of busy cycles estimation of the expected time to buffer overflow is reduced to the problem of estimating p sub n p buffer overflow during a cycle where n is the buffer size the probability p sub n is a large deviations probability p sub n vanishes exponentially fast as n to infinity a rigorous analysis of the method is presented it is demonstrated that the exponentially twisted distribution of s parekh and j walrand 1989 has the following strong asymptotic optimality property within the nonparametric class of all gi gi importance sampling simulation distributions as n to infinity the computational cost of the optimal twisted distribution of large deviations theory grows less than exponentially fast and conversely all other gi gi simulation distributions incur a computational cost that grows with strictly positive exponential rate

constrained solutions in importance via robust statistics the problem of estimating estimating expectations of functions of random vectors via simulation is investigated monte carlo simulations also known as simple averaging have been used as a direct means of estimation a technique known as importance sampling can be used to modify the simulation via weighted averaging in the hope that the estimate will converge more rapidly to the expected value than standard monte carlo simulations a constrained optimal solution to the problem of minimizing the variance of the importance sampling estimator is derived this is accomplished by finding the distribution which is closest to the unconstrained optimal solution in the ali silvey sense s ali et al 1966 the solution from the constraint class is shown to be the least favorable density function in terms of bayes risk against the optimal density function examples of constraint classes which include epsilon mixture show that the constrained optimal solution can be made arbitrarily close to the optimal solution applications to estimating probability of error in communication systems are presented

outage exponent a unified performance metric for parallel fading channels the parallel fading channel which consists of finite number of subchannels is very important because it can be used to formulate many practical communication systems the outage probability on the other hand is widely used to analyze the relationship among the communication efficiency reliability signal to noise ratio snr and channel fading to the best of our knowledge the previous works only studied the asymptotic outage performance of the parallel fading channels which are only valid for a large number of subchannels or high snrs in this paper a unified performance metric which we shall refer to as the outage exponent will be proposed our approach is mainly based on the large deviations theory and meijer s g function it is shown that the proposed outage exponent is not only an accurate estimation of the outage probability for any number of subchannels any snr and any target transmission rate but also provides an easy way to compute the outage capacity finite snr diversity multiplexing tradeoff and snr gain the asymptotic performance metrics such as the delay limited capacity ergodic capacity and diversity multiplexing tradeoff can be directly obtained by letting the number of subchannels or snr tend to infinity similar to gallager s error exponent a reliable function for parallel fading channels which illustrates a fundamental relationship between the transmission reliability and efficiency can also be defined from the outage exponent therefore the proposed outage exponent provides a complete and comprehensive performance measure for parallel fading channels

additive mappings preserving operators of rank one abstract let b x be the algebra of all bounded linear operators on a nontrivial real or complex banach space and let f x be the subalgebra of all finite rank operators a characterization of additive mappings on f x which preserve operators of rank one or projections of rank one is given in the real case such mappings are automatically linear

saddle point bit error rate computations for optical communication systems incorporating equalizers bit error rate computations for optical communication systems incorporating equalizers and under both noise and intersymbol interference isi are discussed an accurate method based on saddle point approximation is used previous work using saddle point approximation considered only basic integration and dump detection the use of equalizers other than integration and dump complicates the computation because of the complexity of the moment generating function involved when isi is strong use of an equalizer such as raised cosine filtering is important for isi reduction to help optical receiver design this paper explains the use of saddle point approximation when an equalizer is used in addition to integration and dump cosine and raised cosine equalizers are considered for isi reduction to study the interdependence between the received waveform and equalizer used two different input pulses are also considered to verify the computation accuracy exact contour integration based on the numerical quadrature method is performed results show that saddle point approximation can provide a fast and accurate computation from the bit error rate ber performance computed it is found that the traditional use of matched filtering has a poor performance when either isi is strong or the apd gain is large and the use of a proper equalizer can significantly improve the performance in other words the best choice of an equalizer design depends on the input waveform amount of isi and the apd gain

spectrum preserving additive maps abstract let x and y be complex banach spaces we show that a spectrum preserving surjective additive map from b x to b y is either of the form f ata 1 for a linear isomorphism a of x onto y or of the form t bt b 1 for a linear isomorphism b of x onto y

uniformly efficient importance sampling for the tail distribution of sums of random variables successful efficient rare event simulation typically involves using importance sampling tailored to a specific rare event however in applications one may be interested in simultaneous estimation of many probabilities or even an entire distribution in this paper we address this issue in a simple but fundamental setting specifically we consider the problem of efficient estimation of the probabilities p sn na for large n for all a lying in an interval a where sn denotes the sum of n independent identically distributed light tailed random variables importance sampling based on exponential twisting is known to produce asymptotically efficient estimates when a reduces to a single point we show however that this procedure fails to be asymptotically efficient throughout a when a contains more than one point we analyze the best performance that can be achieved using a discrete mixture of exponentially twisted distributions and then present a method using a continuous mixture we show that a continuous mixture of exponentially twisted probabilities and a discrete mixture with a sufficiently large number of components produce asymptotically efficient estimates for all a a simultaneously

further results on differential detection of gmsk signals we evaluate the performance in terms of bit error rate of 1 and 2 bit differential detectors for gaussian minimum shift keying gmsk signals by using a very fast numerical technique which makes use of the saddle point approximation moreover the performance of each detector is optimized by using a closed form expression of the optimum threshold in the slicer

direct evaluation of dynamical large deviation rate functions using a variational ansatz we describe a simple form of importance sampling designed to bound and compute large deviation rate functions for time extensive dynamical observables in continuous time markov chains we start with a model defined by a set of rates and a time extensive dynamical observable we construct a reference model a variational ansatz for the behavior of the original model conditioned on atypical values of the observable direct simulation of the reference model provides an upper bound on the large deviation rate function associated with the original model an estimate of the tightness of the bound and if the ansatz is chosen well the exact rate function the exact rare behavior of the original model does not need to be known in advance we use this method to calculate rate functions for currents and counting observables in a set of network and lattice models taken from the literature straightforward ansatze yield bounds that are tighter than bounds obtained from level 2 5 of large deviations via approximations that involve uniform scalings of rates we show how to correct these bounds in order to recover the rate functions exactly our approach is complementary to more specialized methods and offers a physically transparent framework for approximating and calculating the likelihood of dynamical large deviations

analysis of avalanche diode receivers by saddlepoint integration the error probability in a binary fiber optical communication system whose receiver utilizes an avalanche diode detector is calculated by applying the method of numerical saddlepoint integration to laplace inversion integrals for the probabilities of errors of the first and second kinds it is shown how to choose the contours of integration for efficient numerical quadrature the effects of shot noise intersymbol interference and additive gaussian noise are taken into account and the decision level on the receiver output is set to minimize the average probability of error examples show how the method can be used in receiver design to select system parameters such as the output pulsewidth and the sampling time in order to minimize the probability of error