Efficiency of optimal conditional importance sampling

In this paper, we analyze the error probability performance of maximal ratio combining (MRC), equal gain combining (EGC), and selection combining (SC) diversity schemes with coherent BPSK signaling on Rayleigh fading channels with Gaussian channel estimation errors. We first show that, with weighting errors, the conditional probability of error is not an explicit function of the signal-to-noise ratio (SNR) at the output of the diversity combiner. We then show that averaging the conditional probability of error with the density function of the SNR at the output of the combiner yields a lower bound on the exact probability of error, independent of the underlying diversity combining scheme. Later, we derive the exact probability of error for MRC, EGC, and SC diversity schemes and show that the exact probability of error with weighting errors is very similar to the case of perfect channel estimation, but with the average SNR per diversity branch, /spl gamma/_, for the case of perfect channel estimation, replaced by the effective SNR, /spl gamma/_/sub /spl rho//, due to weighting errors, which is a function of both /spl gamma/_ and /spl rho/, the magnitude of the normalized cross correlation between the actual and the estimated channel gains. Finally, we show that, as /spl rho/ /spl rarr/ 0, the average probability of error approaches 0.5, irrespective of the order of diversity and the diversity combining rule.

This paper presents a simple closed form expression for the conditional (on the timing error) bit error probability of an otherwise ideal DPSK receiver. Numerical results are given for average bit error probability obtained by averaging the conditional error probability over a Tikhonov PDF which is a typical characterization of the timing error in practical bit synchronizers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Using a Maclaurin series expansion of the conditional (on the carrier phase error) bit error probability of binary and quaternary phase-shift keying (BPSK and QPSK, respectively) and then averaging this result over both the statistics of the carrier phase error and the channel fading, closed-form expressions for the average bit error probability are derived in a form that lend themselves toward obtaining simple formulas for the associated noisy reference loss. Numerical evaluation of this loss based on the use of these formulas is shown to provide excellent accuracy when compared with the exact evaluation, which requires two-fold numerical integration. Although the method is specifically applied to BPSK and QPSK, it is easily extended to other forms of modulation whose conditional bit error probability is known in a closed form.

The union bound is known as the most useful tool for the computation of error probability in digital communication schemes (both coded and non-coded). Given a signal set or codebook C={x/sub 0/,x/sub 1/,...,x/sub M-1/}, the conditional error probability P(e|x/sub 0/) is upper bounded by a sum of pairwise error probabilities (PEP). Here we consider the additive white Gaussian channel, where C is a discrete and finite set of points in the N-dimensional real Euclidean space. In this case, the PEPs are the probabilities of the noise taking a signal on the other side of the hyperplane bounding the decision regions of x/sub 0/ and x/sub i/. An important step toward the reduction of the union bound to its minimal form was made by Verdu, who derived a theorem, commonly referred to as the Verdu-Shields theorem showing how terms can be removed from the upper bound to error probability in the case of binary antipodal transmission over the Gaussian channel with intersymbol interference. In this paper we derive a generalization of the Verdu-Shields theorem which provides a sufficient condition for expurgating a given error sequence x/sub k/ from the union bound.

Command-based importance sampling for statistical model checking

Since NPL only earns income by selling services to customers wanting ‘first-hand’ calibrations, the value (economic benefit) created by the ‘fanout’ is missing from NPL’s revenue. The first part of this document develops an economic model for the value created when using measurement for conformity testing. This model shows how measurement information creates value by reducing mistakes (fewer “false positives” and “false negatives”) in conformity testing. The second part of this document introduces calibration as a perturbation on top of these measurement activities. Moreover, there can be value created from this reduction in uncertainty, which is where the services of NPL and the calibration laboratories play a key role. Previous studies have explored the economic loss from making type-1 errors (“false positives”) and type-2 errors (“false negatives”) in the context of conformance testing. However, such studies do not treat the conditional probabilities of making type-1 and type-2 errors as “choice variables” that are under control of a decision maker. By treating these conditional probabilities as exogenous, they do not internalise the trade-off between type-1 and type-2 errors. In this paper, we address this gap in the literature by endogenizing the conditional probability of a type-1 error (α) and the probability of a type-2 error (β). The decision maker tries to minimise the total cost incurred due to these errors, subject to the trade-off that exists between them. This is an economics study that borrows a little from metrology as opposed to a study by engineers that borrows a little from economics. Firstly, our analysis gives formulae for the relative uncertainty of the measurement process (σ) and a lower bound for the scrap rate (τ) as functions of the conditional probabilities for type-1 and type-2 errors (α and β). Secondly, this analysis yields a formula for the economic benefit created by using ‘first-hand’ calibrations to reduce the uncertainty of the measurement process. Hence, this document provides the mathematics behind a ‘calculator’ giving an estimate of the benefit from ‘first-hand’ calibrations based on assumptions about the parameters. Finally, this model will really come to life once we have estimates of the core parameters from a forthcoming measurement survey.

Importance sampling is a change-of-measure technique for speeding up the simulation of rare events in stochastic systems. In this paper the authors establish a number of properties characterizing optimal importance sampling measures for Markovian systems. The authors show how these properties may be used to compute the optimal measure and give specific results for a tandem queueing system. The authors' approach has no apparent computational advantage over other direct methods, but it suggests a number of heuristic approximations which may lead to computationally attractive methods.

In this paper, the optimal importance sampling, which is a variance reduction technique in Monte Carlo estimation, is theoretically studied in the case of electronic circuits. To the best of our knowledge, this problem has not previously been studied in electronic circuits. In this study, the theoretical basis of the optimal importance sampling is developed in the case where no approximate model of the circuit responses is given. Simulation results show good agreement with the theoretical basis.

In future cellular networks, it will be possible to estimate the channel parameters of non-line-of-sight propagation paths providing unique opportunities for simultaneous localization and mapping (SLAM) with commodity user equipments (UEs). Radio-SLAM generally relies on generating samples of the UE trajectory and constructing a trajectory-conditioned map. To reduce the number of samples and complexity, we propose an iterative method to approximate the optimal sampling density. The numerical results demonstrate that the added computational complexity of the proposed method can be easily justified by the more efficient use of particles. As an outcome, the presented filter nearly achieves the lower bound and still runs in real-time.

Ronald Fisher advocated testing using p-values, Harold Jeffreys proposed use of objective posterior probabilities of hypotheses and Jerzy Neyman recommended testing with fixed error probabilities. Each was quite critical of the other approaches. Most troubling for statistics and science is that the three approaches can lead to quite different practical conclusions. This article focuses on discussion of the conditional frequentist approach to testing, which is argued to provide the basis for a methodological unification of the approaches of Fisher, Jeffreys and Neyman. The idea is to follow Fisher in using p-values to define the "strength of evidence" in data and to follow his approach of conditioning on strength of evidence; then follow Neyman by computing Type I and Type II error probabilities, but do so conditional on the strength of evidence in the data. The resulting conditional frequentist error probabilities equal the objective posterior probabilities of the hypotheses advocated by Jeffreys.

How to use several noisy channels with unknown error probabilities

This paper considers the optimal generator matrices of a binary cyclic code over a binary symmetric channel with crossover probability p rarr 0 when the goal is to minimize the probability of an information bit error. A given code has many encoder realizations and the information bit error probability is a function of this realization. We derive some sufficient conditions for a binary cyclic code to have systematic optimal generator matrices under bounded distance decoding and determine many binary cyclic codes satisfying these properties.

We consider the problem of Linear Programming (LP) decoding of binary linear codes. The LP excess lemma was introduced by the first author, B. Ghazi, and R. Urbanke (IEEE Trans. Inf. Th., 2014) as a technique to trade crossover probability for “LP excess” over the Binary Symmetric Channel. We generalize the LP excess lemma to discrete, binary-input, Memoryless, Symmetric and LLR-Bounded (MSB) channels. As an application, we extend a result by the first author and H. Audah (IEEE Trans. Inf. Th., 2015) on the impact of redundant checks on LP decoding to discrete MSB channels.

In this paper, we propose two novel power allocation strategies for multihop Free-Space Optical (FSO) networks with any number of hops. These strategies correspond to the analytical solutions obtained from minimizing two different upper-bounds on the conditional error probability under the quantum-limit scenario. The performance of the first strategy is evaluated numerically and closed-form expressions of the average probability of error are derived for the second strategy. We also quantify the reduction in the fading variance and scintillation index provided by multihop FSO systems associated with the second power allocation scheme that compromises performance for simplicity. Results show the superiority of the proposed schemes compared to the equal power distribution strategy.

A quantitative theory of choice reaction time (CRT), developed in the context of accurate performance was successfully applied to a speeded experiment. With estimation of a single parameter, mean criterion level, the theory described the reaction time (RT) distributions for correct responses and errors and the details of the speed-accuracy tradeoff. The form of the tradeoff within experiments is not invariant with a shift of criterion when described by conditional error probabilities. However, a new tradeoff function which is invariant under this condition is presented. Descriptions of individual performance and of individual differences are also provided. Error rates are determined primarily by the amount of criterion variability and the ability to inhibit short-latency errors. Slower subjects with relatively high criterion levels tend to make the most errors. Previous presentations of the theory are extended by a more explicit analysis of the implications of the theory for the speed-accuracy tradeoff.