Chernoff Bound Research Articles

Simple Moment Generating Function Optimisation Technique to Design Optimum Electronic Filter for Underwater Wireless Optical Communication Receiver

This paper introduces a new simple moment-generating function (MGF) design modelling method to conclude an optimum filter to maximize the Q-factor and increase the link communication span. This approach mitigates the pulse temporal dispersion, particularly the underwater wireless optical communication (UWOC) systems. Hence, some form of equalizing filter design is highly desirable. The model solution environment includes a Double Gamma Function (DGF) water channel impulse response, intersymbol interference (ISI), stochastic Poisson process, and additive Gaussian thermal noise (AGTN). The optimal filters exhibit temporal profiles comparable to those derived by published works based on complex Chernoff Bound (CB) and Modified Chernoff Bound (MCB) methods. The results show the impact of the optimum filter at a signal level and optical receiver level utilizing Eye-Diagrams and BER vs. Q-Factor, respectively. The computation involves four different UWOC propagation channel models for Coastal and Harbor waters. One of the main conclusions indicates that the optimum filter manages the temporal dispersion due to the ISI impairment correctly. Also, the proposed optimum filter reduces eye-opening and the corresponding Q-Factor by less than 15% for a five-times increase in pulse width for the same transmitted optical power level.

Improved Chernoff Bound of Gaussian Q-function with ABC algorithm and its QAM applications to DB SC and MRC systems over Beaulieu–Xie channels

In this study, we propose an improved upper bound for the Gaussian Q function by using artificial bee colony algorithm. Then, we investigate the performance of the dual-branch (DB) selection combining (SC) and maximal ratio combining (MRC) systems over the Beaulieu–Xie fading channels. The probability density functions of the instantaneous signal to noise ratio for the considered systems are obtained. Employing the proposed upper bound, we derive closed-form expressions of the error probability for the quadrature amplitude modulation (QAM) techniques such as rectangular QAM (RQAM), cross QAM (XQAM), and hexagonal QAM (HQAM). Furthermore, the asymptotic error probability expressions for the DB SC system are also obtained to simplify the analyses. The effects of some key parameters in the systems are shown in the results. Comparisons of the different modulation types and the different upper bound approaches for the Gaussian Q function are presented. Finally, it has been shown that the upper bound approximation presented in this study can be widely used for many communication applications.

Performance enhancement of hybrid fiber wavelength division multiplexing passive optical network FSO systems using M-ary DPPM techniques under interchannel crosstalk and atmospheric turbulence

The performance of moment generating function approaches, specifically Chernoff bound (CB) and modified Chernoff bound (MCB), is examined and improved in this study. We evaluates and enhances the performance of a wavelength division multiplexing (WDM) technique for free-space optical (FSO) fibre communications based on passive optical network (PON) using the M-ary digital pulse-position modulation (M-ary DPPM) schemes under amplified spontaneous emission (ASE) noise effects, interchannel crosstalk (ICC), and atmospheric turbulence (AT). We use a data rate of 2.5 Gbps for eight channels over a PON/DWDM-FSO optical fibre system in the C-band with 100 GHz channel spacing start from 1550 nm. The results achieve 20 Gbit/s transmissions (2.5 Gbps $$\times $$ 8 channels). This is a technology that can have extended leverage, higher data rates, power-efficient, and is considered an ideal option for the provision of bandwidth for potential access networks. When compared to the CB at a low gain (G = 8), the MCB outperforms the Gaussian approximation at high gain (G = 30). Because of its superior performance at a high gain (G = 30), the MCB offers the tightest limit on the bit error rate (BER). In comparison to an equivalent on–off keying (OOK) non-return-to-zero (NRZ), the M-ary DPPM scheme with a coding level (M) of 2 improves average power about by 2.9 dB at a data rate of 2.5 Gbps on the 1550 nm wavelength and BER of $${10}^{-{9}}$$ . The sensitivity of the M-ary DPPM modulation scheme remains improved over OOK in the presence of ICC. The lower power penalty is predicted to be approximately 0.2–3.0 dB in the DWDM-FSO systems for low coding level M $$=$$ 2. We achieve a lower power penalty at 0.2–3.0 dB for the hybrid fiber DWDM/PON-FSO optical communication system at a BER of $${10}^{-{9}}$$ . At a target BER of $${10}^{-{12}}$$ , the hybrid OOK-NRZ/M-ary DPPM offers about 4–8 dB of optical signal-to-noise-ratio improvements over the M-DPPM of the WDM-PON/FSO link for strong turbulence. The results demonstrated that the M-ary DPPM and the optical relaying amplification technique are powerful treatments for mitigating the impacts of ASE noise, AT, and ICC.

Automated screening system for retinal health using bi-dimensional empirical mode decomposition and integrated index

Posterior Segment Eye Diseases (PSED) namely Diabetic Retinopathy (DR), glaucoma and Age-related Macular Degeneration (AMD) are the prime causes of vision loss globally. Vision loss can be prevented, if these diseases are detected at an early stage. Structural abnormalities such as changes in cup-to-disc ratio, Hard Exudates (HE), drusen, Microaneurysms (MA), Cotton Wool Spots (CWS), Haemorrhages (HA), Geographic Atrophy (GA) and Choroidal Neovascularization (CNV) in PSED can be identified by manual examination of fundus images by clinicians. However, manual screening is labour-intensive, tiresome and time consuming. Hence, there is a need to automate the eye screening. In this work Bi-dimensional Empirical Mode Decomposition (BEMD) technique is used to decompose fundus images into 2D Intrinsic Mode Functions (IMFs) to capture variations in the pixels due to morphological changes. Further, various entropy namely Renyi, Fuzzy, Shannon, Vajda, Kapur and Yager and energy features are extracted from IMFs. These extracted features are ranked using Chernoff Bound and Bhattacharyya Distance (CBBD), Kullback–Leibler Divergence (KLD), Fuzzy-minimum Redundancy Maximum Relevance (FmRMR), Wilcoxon, Receiver Operating Characteristics Curve (ROC) and t-test methods. Further, these ranked features are fed to Support Vector Machine (SVM) classifier to classify normal and abnormal (DR, AMD and glaucoma) classes. The performance of the proposed eye screening system is evaluated using 800 (Normal=400 and Abnormal=400) digital fundus images and 10-fold cross validation method. Our proposed system automatically identifies normal and abnormal classes with an average accuracy of 88.63%, sensitivity of 86.25% and specificity of 91% using 17 optimal features ranked using CBBD and SVM-Radial Basis Function (RBF) classifier. Moreover, a novel Retinal Risk Index (RRI) is developed using two significant features to distinguish two classes using single number. Such a system helps to reduce eye screening time in polyclinics or community-based mass screening. They will refer the patients to main hospitals only if the diagnosis belong to the abnormal class. Hence, the main hospitals will not be unnecessarily crowded and doctors can devote their time for other urgent cases.

Model complexities of shallow networks representing highly varying functions

Model complexities of shallow (i.e., one-hidden-layer) networks representing highly varying multivariable {−1,1}-valued functions are studied in terms of variational norms tailored to dictionaries of network units. It is shown that bounds on these norms define classes of functions computable by networks with constrained numbers of hidden units and sizes of output weights. Estimates of probabilistic distributions of values of variational norms with respect to typical computational units, such as perceptrons and Gaussian kernel units, are derived via geometric characterization of variational norms combined with the probabilistic Chernoff Bound. It is shown that almost any randomly chosen {−1,1}-valued function on a sufficiently large d-dimensional domain has variation with respect to perceptrons depending on d exponentially.

Improved bit error rate evaluation for optically pre-amplified free-space optical communication systems in turbulent atmosphere

The use of an optical pre-amplifier at the receiver of free-space optical (FSO) communication systems necessitates the consideration of both turbulence and amplified spontaneous emission (ASE) noise in performance modelling. Until now, this problem has been typically approached by the use of a Gaussian approximation (GA) for the conditional probability of error with averaging then performed using the received irradiance probability density function (PDF) (governed by scintillation). However, the GA is deficient as it only uses the first two moments. Moment generating function (MGF) techniques, notably the Chernoff bound (CB) and modified Chernoff bound (MCB), are used to evaluate the bit error rate performance of an optically pre-amplified FSO system in the presence of both atmospheric turbulence and ASE noise. The MGF-based methods incorporate a fuller statistical description of the signal and noise processes encountered in the optically pre-amplified case when compared to the GA. The lognormal, gamma–gamma, K and negative exponential distributions have been used to characterise the weak, moderate, strong and saturated turbulence regimes. The MCB gives the tightest bound upon the BER compared to the CB, particularly at lower gains, and, as it also can be exceeded by the GA at higher gains, it is a logical method to use in general.

In-band crosstalk penalties in optical networks with narrow optical and electric filtering

Optical cross-connects (OXC) introduce crosstalk and intersymbolic interference due to filtering in the optical signal. We show experimentally and by simulation that higher optical signal-to-noise ratio (OSNR) penalties are obtained for the combined effect of both impairments compared to the sum of the penalties taken independently. Furthermore, the Modified Chernoff Bound is applied for performance estimation, with good accuracy and fast computation time. By using the correct OSNR penalty, the maximum number of cascaded OXC is reduced by 17% relatively to when the penalties are considered separately, for a 2 dB maximum penalty.

Performance analysis of optically preamplified DC‐coupled burst mode receivers

AbstractBit error rate and threshold acquisition penalty evaluation is performed for an optically preamplified DC‐coupled burst mode receiver using a moment generating function (MGF) description of the signal plus noise. The threshold itself is a random variable and is also described using an appropriate MGF. Chernoff bound (CB), modified Chernoff bound (MCB) and the saddle‐point approximation (SPA) techniques make use of the MGF to provide the performance analyses. This represents the first time that these widely used approaches to receiver performance evaluation have been applied to an optically preamplified burst mode receiver and it is shown that they give threshold acquisition penalty results in good agreement with a prior existing approach, whilst having the facility to incorporate arbitrary receiver filtering, receiver thermal noise and non‐ideal extinction ratio. A traditional Gaussian approximation (GA) is also calculated and comparison shows that it is clearly less accurate (it exceeds the upper bounds provided by CB and MCB) in the realistic cases examined. It is deduced, in common with the equivalent continuous mode analysis, that the MCB is the most sensible approach. Copyright © 2009 John Wiley & Sons, Ltd.

Unitary Space–Time Constellation Design Based on the Chernoff Bound of the Pairwise Error Probability

Unitary space-time constellation design is considered for noncoherent multiple-antenna communications, where neither the transmitter nor the receiver knows the fading coefficients of the channel. By employing the Clarke's subdifferential theorem of the sum of the kappa largest singular values of a unitary matrix, we present a numerical optimization procedure for finding unitary space-time signal constellations of any dimension. The Chernoff bound of the pairwise error probability is used directly as a design criterion. The constellations are found by performing gradient descent search on a family ldquosurrogaterdquo functions that converge to the maximum pairwise error probability. The complexity of the search procedure increases with the dimension and the size of the constellation, but it can be considered to be acceptable for an off-line design procedure. Since the designed constellations are unstructured, and, thus, require an exhaustive search over all codewords in decoding, their main practical value is to serve as constellation design performance benchmarks. We compare the performance of the new constellations to that of some other well-known constellations. Computer simulation results illustrate typically about 0.4-3.5 dB performance gains.

A Data-Bearing Approach for Pilot-Embedding Frameworks in Space-Time Coded MIMO Systems

Space-time (ST) coded MIMO systems employing coherent detectors crucially require channel state information. This paper presents a novel pilot-embedding framework for channel estimation and data detection by exploiting the null-space property and the orthogonality property of the data-bearer and pilot matrices. The ST data matrix is firstly projected onto the data bearer matrix, which is a null-space of the pilot matrix, and the resulting matrix and the pilot matrix are combined for transmitting. The data and pilot extractions are achieved independently through linear transformations by exploiting the null-space property. The unconstrained maximum-likelihood (ML) and linear minimum mean-squared error (lmmse) estimators are explored for channel estimation. Then the ML approach for data detection is developed by exploiting the orthogonality property. The mean-squared error (mse) of channel estimation, Cramer-Rao lower bound (CRLB), and the Chernoff's bound of a pair-wise error probability for ST codes are analyzed for examining the performance of the proposed scheme. The optimum power allocation scheme for data and pilot parts is also considered. Three data-bearer and pilot structures, including time-multiplexing (TM)-based, ST-block-code (STBC)-based, and code-multiplexing (CM)-based, are proposed. Simulation results show that the CM-based structure provides superior performance for nonquasi-static flat Rayleigh fading channels, while these three structures yield similar performances for quasi-static flat Rayleigh fading channels

Evaluation of Polarisation Mode Dispersion in Longhaul Digital Optical Fiber Transmission System

The paper investigates the birefringence of a single mode fiber. The suitable value of birefringence for fiber application has been identified. Evolution of polarisation mode dispersion (PMD) and its analysis has been presented. An upper bound for the probability of error due to PMD in terms of Chernoff Bound (CB) and Modified Chernoff Bound (MCB) has been evaluated. The further analysis elaborates upon the dependence of mean power penalty on the birefringence of the fiber. The mean penalty dependence on the birefringence of a highly birefringent dispersion shifted fiber at 1.55 μm is calculated. This dependence obeys square law distribution and highlights the need of birefringence of the order of 10-2 in order to have lower penalty due to PMD.

Tight bounds for the tail of the packet waiting time distribution in buffered networks

In this paper we consider the end to end delay through a series of queues in which the service time is fixed and equal to the transmission time of constant length data packets at constant bitrate, for example real time delay sensitive traffic in IP or ATM networks. In these cases, algorithms are required either for on-line admission control or for planning and provisioning of service partitions within such networks. Here we develop concise, accurate and easy to use solutions for the end to end delay through this series of queues, and provide numerical results quantifying the high degree of accuracy obtained. Additionally we compare against a Chernoff Bound approach, showing that our solutions are more accurate. Copyright © 2001 John Wiley & Sons, Ltd.

An optically preamplified intersatellite PPM receiver employing maximum likelihood detection

An original analysis, facilitated by the use of Gaussian approximation (GA) and Chernoff bound (CB) techniques, is presented for an optically preamplified intersatellite pulse position modulation (PPM) receiver employing maximum likelihood detection (MLD). The theoretical results, from calculations performed at a wavelength of 1.54 /spl mu/m and bit rate of 25 Mb/s demonstrate that the receiver is approximately 1.5 dB more sensitive than an earlier optically preamplified PPM receiver and also more sensitive than the fundamental limit of optically preamplified on-off keyed (OOK) NRZ signalling. Thus, it is demonstrated that the proposed receiver configuration is a strong contender for implementation in future laser intersatellite communication systems.

A Moment Generating Function Formulation for an Optically Preamplified PPM Intersatellite System

A scheme for the utilisation of optically preamplified pulse position modulation (PPM) receivers in intersatellite laser communications is presented. To facilitate higher bit-rate operation an analysis is presented for a PPM system employing a receiver which integrates over a time slot and compares the result with a fixed threshold. When operating at the SITM-1 bit-rate of 155 Mbit/s and using conservative parameters our model demonstrates a possible receiver sensitivity of 50.7 photons/bit (at a BER of 10 -6 ) in the presence of background noise, indicating the potential of this receiver configuration for use in satellite communications. The analysis is facilitated by the use of a moment generating function approach which enables the use of tight upper bounds on the bit error probability to be derived using Chernoff bound (CB) techniques. We compare results obtained from this technique with those obtained by applying a Gaussian approximation (GA) and find that the GA is consistently less conservative for the cases considered (by 0.5 dB) at a BER of 10 -6 . This nevertheless suggests a reasonable level of accuracy and such a GA could be a useful tool for obtaining a first order approximation.

The Moment Bound Is Tighter Than Chernoff's Bound for Positive Tail Probabilities

The Moment Bound Is Tighter Than Chernoff's Bound for Positive Tail Probabilities

The Moment Bound is Tighter than Chernoff's Bound for Positive Tail Probabilities

Abstract Chernoff's bound on P[X ⩾ t] is used almost universally when a tight bound on tail probabilities is required. In this article we show that for all positive t and for all distributions, the moment bound is tighter than Chernoff's bound. By way of example, we demonstrate that the improvement is often substantial.

Linear Direct-Detection Fiber-Optic Receiver Optimization in the Presence of Intersymbol Interference

A technique is developed for optimizing communication receivers by minimizing a Chernoff bound (CB) on the Bayes cost function. Being couched in terms of moment generating functions, the procedure is well suited to optical systems. Optimal CB receivers are derived and shown to coincide exactly with known minimum cost Bayes receivers for certain special cases of both electrical and optical communications for which closed form analytic solutions are possible. This provides endorsement for the method, which is then applied to more complex practical cases of linear optical fiber receivers in which both colored Gaussian noise and intersymbol interference play a part. The CB receivers are shown to offer some improvement compared with a zero forcing raised cosine equalizing receiver such as is commonly adopted, and all the evidence suggests that CB receivers achieve performance very close to the best possible with a linear receiver.

Gilat — Raubenheimer method for k - space integration

Model complexities of shallow (i.e., one-hidden-layer) networks representing highly varying multivariable {−1,1}-valued functions are studied in terms of variational norms tailored to dictionaries of network units. It is shown that bounds on these norms define classes of functions computable by networks with constrained numbers of hidden units and sizes of output weights. Estimates of probabilistic distributions of values of variational norms with respect to typical computational units, such as perceptrons and Gaussian kernel units, are derived via geometric characterization of variational norms combined with the probabilistic Chernoff Bound. It is shown that almost any randomly chosen {−1,1}-valued function on a sufficiently large d-dimensional domain has variation with respect to perceptrons depending on d exponentially.

Filters for the Detection of Binary Signaling: Optimization Using the Chernoff Bound

The probability of error in the detection of binary signaling by a linear receiver is upper bounded by the Chernoff bound, and a general relationship is obtained whose solution gives a receiver filter and threshold which minimize this bound. As an example, the detection of Poisson signaling in the presence of additive white-Gaussian noise is considered and the resulting filter lies between the optimum filter for Poisson signaling and a matched filter.