Importance sampling simulation for evaluating lower-bound symbolerror rate of the Bayesian DFE with multilevel signaling schemes

For the class of equalizers that employs a symbol-decision finite-memory structure with decision feedback, the optimal solution is known to be the Bayesian decision feedback equalizer (DFE). The complexity of the Bayesian DFE, however, increases exponentially with the length of the channel impulse response (CIR) and the size of the symbol constellation. Conventional Monte Carlo simulation for evaluating the symbol error rate (SER) of the Bayesian DFE becomes impossible for high channel signal-to-noise ratio (SNR) conditions. It has been noted that the optimal Bayesian decision boundary separating any two neighboring signal classes is asymptotically piecewise linear and consists of several hyperplanes when the SNR tends to infinity. This asymptotic property can be exploited for efficient simulation of the Bayesian DFE. An importance sampling (IS) simulation technique is presented based on this asymptotic property for evaluating the lower bound SER of the Bayesian DFE with a multilevel pulse amplitude modulation (M-PAM) scheme under the assumption of correct decisions being fed back. A design procedure is developed, which chooses appropriate bias vectors for the simulation density to ensure asymptotic efficiency (AE) of the IS simulation     

Quick simulation: a review of importance sampling techniques incommunications systems

Importance sampling (IS) is a simulation technique which aims to reduce the variance (or other cost function) of a given simulation estimator. In communication systems, this usually, but not always, means attempting to reduce the variance of the bit error rate (BER) estimator. By reducing the variance, IS estimators can achieve a given precision from shorter simulation runs; hence the term “quick simulation.” The idea behind IS is that certain values of the input random variables in a simulation have more impact on the parameter being estimated than others. If these “important” values are emphasized by sampling more frequently, then the estimator variance can be reduced. Hence, the basic methodology in IS is to choose a distribution which encourages the important values. This use of a “biased” distribution will, of course, result in a biased estimator if applied directly in the simulation. However, there is a simple procedure whereby the simulation outputs are weighted to correct for the use of the biased distribution, and this ensures that the new IS estimator is unbiased. Hence, the “art” of designing quick simulations via IS is entirely dependent on the choice of biased distribution. Over the last 50 years, IS techniques have flourished, but it is only in the last decade that coherent design methods have emerged. The outcome of these developments is that at the expense of increasing technical content, modern techniques can offer substantial run-time saving for a very broad range of problems. We present a comprehensive history and survey of IS methods. In addition, we offer a guide to the strengths and weaknesses of the techniques, and hence indicate which techniques are suitable for various types of communications systems. We stress that simple approaches can still yield useful savings, and so the simulation practitioner as well as the technical researcher should consider IS as a possible simulation tool     

Importance sampling for the random phase Gaussian channel

Importance sampling (IS) is developed as a variance reduction technique for Monte Carlo simulation of data communications over random phase additive white Gaussian noise channels. The binary problem (with known performance) is examined initially to determine parameter values and estimate the performance gain of IS. These results can then be applied to intractable m-ary signaling problems through composite IS. An example compares the performance of linear, square-law, and optimum receivers for binary block coded data     

Statistical optimization of dynamic importance sampling parametersfor efficient simulation of communication networks

Importance sampling (IS) is a powerful method for reducing simulation run times when estimating the probabilities of rare events in communication systems using Monte Carlo simulation and is made feasible and effective for the simulation of networks of queues by regenerative techniques. However, using the most favorable IS settings very often makes the length of regeneration cycles infinite or impractically long. To address this problem, a methodology that uses IS dynamically within each regeneration cycle to drive the system back to the regeneration state after an accurate estimate has been obtained is discussed. A statistically based technique for optimizing IS parameter values for simulations of queueing systems, including complex systems with bursty arrival processes, is formulated. A deterministic variant of stochastic simulated annealing (SA), called mean field annealing (MFA), is used to minimize statistical estimates of the IS estimator variance. The technique is demonstrated by evaluating blocking probabilities     

Stochastic gradient optimization of importance sampling for theefficient simulation of digital communication systems

Importance sampling (IS) techniques offer the potential for large speed-up factors for bit error rate (BER) estimation using Monte Carlo (MC) simulation. To obtain these speed-up factors, the IS parameters specifying the simulation probability density function (PDF) must be carefully chosen. With the increased complexity in communication systems, analytical optimization of the IS parameters can be virtually impossible. We present a new IS optimization algorithm based on stochastic gradient techniques. The formulation of the stochastic gradient descent (SGD) algorithm is more general and system-independent than other existing IS methodologies, and its applicability is not restricted to a specific PDF or biasing scheme. The effectiveness of the SGD algorithm is demonstrated by two examples of communication systems where the IS techniques have not been applied before. The first example is a communication system with diversity combining, slow nonselective Rayleigh fading channel, and noncoherent envelope detection. The second example is a binary baseband communication system with a static linear channel and a recursive least square (RLS) linear equalizer in the presence of additive white Gaussian noise (AWGN)     

Estimation variance bounds of importance sampling simulations indigital communication systems

In practical applications of importance sampling (IS) simulation, two basic problems are encountered, that of determining the estimation variance and that of evaluating the proper IS parameters needed in the simulations. The authors derive new upper and lower bounds on the estimation variance which are applicable to IS techniques. The upper bound is simple to evaluate and may be minimized by the proper selection of the IS parameter. Thus, lower and upper bounds on the improvement ratio of various IS techniques relative to the direct Monte Carlo simulation are also available. These bounds are shown to be useful and computationally simple to obtain. Based on the proposed technique, one can readily find practical suboptimum IS parameters. Numerical results indicate that these bounding techniques are useful for IS simulations of linear and nonlinear communication systems with intersymbol interference in which bit error rate and IS estimation variances cannot be obtained readily using prior techniques     

A stochastic importance sampling methodology for the efficientsimulation of adaptive systems in frequency nonselective Rayleigh fadingchannels

We present an IS stochastic technique for the efficient simulation of adaptive systems which employ diversity in the presence of frequency nonselective slow Rayleigh fading and additive, white, Gaussian noise. The computational efficiency is achieved using techniques based on importance sampling (IS). We utilize a stochastic gradient descent (SGD) algorithm to determine the near-optimal IS parameters that characterize the dominant fading process. After accounting for the overhead of the optimization algorithm, average speed-up factors of up to six orders of magnitude [over conventional Monte Carlo (MC)] were attained for error probabilities as low as 10^-11 for a fourth-order diversity model     

Performance analysis of a land mobile satellite system using importance sampling

Land mobile communication using satellites has become widespread. However, estimation of its average bit-error rate is still challenging due to the complexity of various system components, such as the nonlinearity of traveling wave tube amplifiers (TWTAs), intersymbol interference, spread spectrum under fading, or jamming conditions. The analytical approach using closed form solutions is very difficult if not impossible. Monte Carlo (MC) simulation can offer an alternative method of performance estimation; however, the use of MC simulation is often limited by an excessive computational burden. Importance sampling (IS) is an efficient simulation method that can greatly reduce this computational overhead although it requires additional modeling and a careful biasing scheme. We propose an integrated IS model which combines error event simulation, a conditional IS method, and an asymptotically efficient method for the effective estimation of the error rate. Simulation results conditioned on different channel parameters such as the backoff of a TWTA, jamming signal strength, and elevation angle of the low-Earth orbit satellite are provided. Comparisons between the proposed method and the MC method show that a great reduction in simulation time, as well as increased accuracy, can be achieved with the IS method.     

Importance sampling for error event analysis of HMM frequency linetrackers

This paper considers the problem of designing efficient and systematic importance sampling (IS) schemes for the performance study of hidden Markov model (HMM) based trackers. Importance sampling (IS) is a powerful Monte Carlo (MC) variance reduction technique, which can require orders of magnitude fewer simulation trials than ordinary MC to obtain the same specified precision. We present an IS technique applicable to error event analysis of HMM based trackers. Specifically, we use conditional IS to extend our work in another of our paper to estimate average error event probabilities. In addition, we derive upper bounds on these error probabilities, which are then used to verify the simulations. The power and accuracy of the proposed method is illustrated by application to an HMM frequency tracker     

Adaptive importance sampling [digital communication]

Parametric adaptive importance sampling (IS) algorithms that adapt the IS density to the system of interest during the course of the simulation are discussed. This approach removes the burden of selecting the IS density from the system designer. The performance of two such algorithms is investigated for both linear and nonlinear systems operating in Gaussian noise. In addition, the algorithms are shown to converge to the optimum improved importance sampling density for the special case of a linear system with Gaussian noise     

用重要性抽样方法估计数字光纤通信链路的误码率

 本文使用重要性抽样(IS)方法,对数字光纤通信链路的误码率参量进行了估计.并且,在设计IS偏置方案时采用了LDT技术,即把最优的LDT的α-扭转(指数扭转)密度选取为IS偏置密度.计算机模拟实验表明,这种基于LDT技术的IS方法具有很高的精确度和计算效率以及充分的可靠性,特别适合于极低误码率的估计.此方法为数字光纤通信链路的计算机辅助建模、分析和设计(CAMAD)提供了一种有用的研究工具.     

An algorithmic approach to the optimization of importance samplingparameters in digital communication system simulation

Importance sampling is recognized as a potentially powerful method for reducing simulation runtimes when estimating the bit error rate (BER) of communications systems using Monte Carlo simulation. Analytically, minimizing the variance of the importance sampling (IS) estimator with respect to the biasing parameters has typically yielded solutions for systems for which the BER could be found analytically. A technique for finding an asymptotically optimal set of biasing parameter values, in the sense that as the resolution of the search and the number of runs used both approach infinity, the algorithm converges to the true optimum, is proposed. The algorithm determines the amount of biasing that minimizes a statistical measure of the variance of the BER estimate and exploits a theoretically justifiable relationship, for small sample sizes, between the BER estimate and the amount of biasing. The translation biasing scheme is considered, although the algorithm is applicable to other parametric IS techniques. Only mild assumptions are required of the noise distribution and system. Experimentally, improvement factors ranging from two to eight orders of magnitude are obtained for a number of distributions for both linear and nonlinear systems with memory     

动态贝叶斯网络在战场态势估计中的应用

贝叶斯网络用于态势估计时,系统参数不能及时调整,无法对未来时刻进行预测。为解决这一问题,对动态贝叶斯网络在战场态势评估中的应用进行了研究,引入时间因素,建立了网络模型,分析了概率参数与推理的过程,并利用卡尔曼滤波模型法对推理进行仿真实验。实验结果表明了动态模型推理的有效性。动态贝叶斯网络可以有效利用侦察数据中的时间信息,实时动态地处理影响分析和决策的种种因素,对指挥员的作战决策具有极大的参考价值。     

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.     

Simulation properties of the Bayesian and maximum likelihood estimators of availability

Little has been written about availability from the Bayesian point of view (3). In this study a closed form Bayesian availability estimator is investigated. The degree of bias and small variation of the estimator are obtained through computer simulation. Sampling distributions of the maximum likelihood and Bayesian estimations of availability are obtained by Efron's bootstrap methods. The Bayesian estimator is good for small sample size problems when the failure time and repairing time are exponentially distributed.     

Efficient estimation of cell blocking probability for ATM systems

The assessment of the cell loss performance of networks using asynchronous transfer mode (ATM) via Monte Carlo simulation incurs an enormous computational burden due to the need to estimate an event that has a very small probability of occurrence. Although importance sampling (IS) techniques have been proven useful in simulations of rate events related to bit error rate in digital communications and false alarm rate in radar systems, its application to ATM queuing problems with correlated input traffic has yet to be demonstrated. It is established that significant computational savings can be obtained using IS for correlated traffic by using regenerative properties of the underlying system and biasing the conditional arrival process. The results show that IS can reduce the computational burden by more than three orders of magnitude. Extensions of the methodology to more complex arrival processes are discussed. The foundation for applying IS to ATM systems given can be used to study congestion control as well as networks of ATM queues in the future     

Glucose effectiveness and insulin sensitivity from the minimalmodels: consequences of undermodeling assessed by Monte Carlo simulation

The unlabeled (cold) minimal model (MM) and the labeled (hot) minimal model (HMM) are a powerful tool to investigate in vivo metabolism from a standard intravenous glucose tolerance test (IVGTT) or hot IVGTT (HIVGTT). They allow to estimate metabolic indexes of the glucose-insulin system, namely glucose effectiveness (GE) and insulin sensitivity (IS) (of uptake and production those of MM, and of uptake only these of HMM). Here, the consequences of the single-compartment glucose kinetics approximation used in the MM's are investigated via Monte Carlo simulation, using a physiologic reference model (RM) of the system, RM allows to generate noisy synthetic plasma concentrations of glucose, tracer glucose, and insulin during IVGTT and HIVGTT, which are then analyzed with MM and HMM. The MM and HMM CE and IS are then compared with the RM ones. Results of 400 runs show that: (1) correlation of MM GE with the RM index is weak; (2) MM IS is well correlated with the RM index, but severely underestimates it; (3) HMM clearance rate is correlated with RM clearance; and (4) HMM IS is well correlated and only slightly overestimates the RM index. These results demonstrate that GE of MM is most affected by the single-compartment approximation and the indexes of HMM are more robust than those of MM     

Developments in the Theory and Application of Importance Sampling

The assessment of bit error rate (BER) performance of a digital communication system via computer simulation has traditionally been done using the Monte Carlo method. For very low BER, this method requires excessive computer time. This time can be substantially reduced by using extrapolation based on importance sampling (IS). In applying IS to a complex system, many considerations must be addressed, chief among which is the reliability (variance) of the estimator as a function of the system particulars. We discuss a number of these considerations and, specifically, derive a number of expressions for the variance. We find that the variance improvement may be severely limited by the dimensionality (or memory) of the system. We describe a means for circumventing this limitation through the definition of a statistically equivalent impulse response. For a linear system, this amounts to the ordinary impulse response. The simulation can be structured to estimate the equivalent impulse response using statistical regression. This new approach has been implemented and found to yield significant runtime improvement over conventional importance sampling for linear systems of large dimensionality. We believe this technique will work also for mildly nonlinear systems, as might be encountered in typical satellite Communications.     

一种重要抽样仿真方法的研究及应用

介绍了蒙特-卡洛仿真原理,分析了改进型的蒙特-卡洛仿真即重要抽样(IS)的原理及其在通信中的应用,特别是在高速光纤通信的偏振模色散(PMD)仿真中的应用价值.实验证明,采用重要抽样的仿真方法可以得到大差分群时延(DGD)值的小概率事件,并且系统配置比用蒙特-卡洛的仿真方法简单得多.     

A Fresh Look at the Bayesian Bounds of the Weiss-Weinstein Family

Minimal bounds on the mean square error (MSE) are generally used in order to predict the best achievable performance of an estimator for a given observation model. In this paper, we are interested in the Bayesian bound of the Weiss–Weinstein family. Among this family, we have Bayesian CramÉr-Rao bound, the Bobrovsky–MayerWolf–ZakaÏ bound, the Bayesian Bhattacharyya bound, the Bobrovsky–ZakaÏ bound, the Reuven–Messer bound, and the Weiss–Weinstein bound. We present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator (MMSEE) and on a constrained optimization problem. With this approach, we obtain a useful theoretical framework to derive new Bayesian bounds. For that purpose, we propose two bounds. First, we propose a generalization of the Bayesian Bhattacharyya bound extending the works of Bobrovsky, Mayer–Wolf, and ZakaÏ. Second, we propose a bound based on the Bayesian Bhattacharyya bound and on the Reuven–Messer bound, representing a generalization of these bounds. The proposed bound is the Bayesian extension of the deterministic Abel bound and is found to be tighter than the Bayesian Bhattacharyya bound, the Reuven–Messer bound, the Bobrovsky–ZakaÏ bound, and the Bayesian CramÉr–Rao bound. We propose some closed-form expressions of these bounds for a general Gaussian observation model with parameterized mean. In order to illustrate our results, we present simulation results in the context of a spectral analysis problem.