Comments on "Bounds on Probability of Error in Decision Feedback Equalizers"

In the above paper1 a technique was proposed to derive an upper bound on the error probability of a decision feedback equalizer. It involves decomposition of the probability density function of residual intersymbol interference and derivation of Chebyshev-type bounds on the error functionals over the decomposed functions. In this correspondence, we demonstrate that the technique is not applicable in general. The result is not a bound in many cases.     

Cochannel Interference in Partially Coherent Communication

Effects of cochannel interference and synchronization error of the carrier phase on the probability of error in binary communications are considered. Several bounds on the probability of error are proposed. The bounds are easy to compute and do not require complete statistical characterization of the errors. They turn out to be simple linear combinations of error probabilities with no cochannel interferences and no phase errors. Several illustrative examples are given which show that the bounds can be tight.     

Comments on "on Probability of Error in Decision Feedback Equalizer"

The above-mentioned paper [1] derives bounds on the error probability of a decision feedback equalizer (DFE) which are claimed to be significantly tighter than the earlier upper bound of [2]. Unfortunately, [1] makes an unsupportable assumption which reddars the conclusions incorrect. We show that valid bounds can be obtained by the methods of [1], but that for the numerical example of [2] , the resulting upper bound is never tighter than, and sometimes significantly weaker than, that derived in [2]. Similarly, a lower bound derived by the method of [1] is shown for the same numerical example to be very close to the error-free past lower bound previously known.     

线性分组码不可检错误概率新的下限

线性分组码的不可检错误概率是评价自动请求重传(ARQ)系统的重要参数。不可检错误概率的计算涉及码的重量分布,而大部分码的重量分布还不知道,因此,研究不可检错误概率的限显得非常重要。本文得到了GF()上(n,k)线性分组码不可检错误概率的两个新的下限,在一定程度上,改进了Wolf等人的结果。由于新的限与码的重量结构无关,因此可以方便地估计任何线性分组码的不可检错误概率。     

Bounds on the Probability of Error in Digitial Communications

A class of upper bounds and lower bounds on the probability of error in digital communications are derived. Several degrading factors like intersymbol interference, cochannel interference, inaccurate carrier synchronization, and incoherent detection can be considered in obtaining the bounds. Several examples are given and it is shown that many situations not considered so far can be put into the conceptual framework developed here.     

Noisy Feedback in a Binary Sequential Communication Scheme

The effects of a noisy feedback channel on a communication system with the decision and information feedback proposed as by Mancianti et al. are described. Expressions for the expected transmitted energy per symbol, expected decision time, and receiver's error probability are derived in terms of the error probability Pef over the noisy information channel. Because of the complexity of exact calculations, we have analyzed in detail only the case of two repetitions per symbol. Then, in order to give a general picture of the system performance, the asymptotic behavior has been examined.     

结构系统失效概率的近似分析

提出用一阶简单边界的上、下界来估计结构系统失效概率的近似公式。它考虑了结构系统失效模式之间的相关性但仅包含失效模式的一阶概率。算例表明,通常最大误差在5％以内,精度高而计算简便。     

Binary Signal Detection with Decision and Information Feedbacks

This paper analyzes a sequential feedback detection scheme derived by Kramer's nonsequential intermittent one; in addition to the noiseless information feedback channel, it makes use of a noiseless decision feedback channel. Expressions for the normalized average evergy per symbol, average decision time, probability of error, and average power are derived. The comparison with the nonsequential intermittent scheme shows that the proposed one allows an appreciable gain in transmission rate and requires a smaller peak-to-average power ratio.     

Probability of Anomaly Expressions for Random Waveform Registration

Registration by integral-square error correlation of one-dimensional discrete-time waveforms which are treated as random processes with specified autocorrelation functions is considered. An important design parameter for this class of problems is the probability of anomaly (a false dip in the correlation function) because it gives an indication of system immunity to gross registration errors. Explicit expressions for this parameter are not possible, so bounds and appoximations must be derived. Two upper bounds and an approximation for the probability of anomaly are derived here. The use of these expressions is illustrated by an example. The relative utility of these performance indicators is shown for the example by comparison with actual values of the probability of anomaly obtained by computer simulation.     

Phase-Locked-Loop Detection of Binary PSK Signals Utilizing Decision Feedback

A modified quadrature channel phase-locked-loop demodulator for binary phase-shift-keyed signals is presented. The modification, called decision feedback, is a practical implementation of decisiondirected channel measurement. Noise analyses are carried out for both additive and phase noise. An optimum loop filter is derived and the analyses include determination of probability of error for the system. The results of an analog simulation of the decision feedback technique are also included.     

基于密度权重的可靠性灵敏度分析方法简

 为了提高可靠性灵敏度求解数字模拟法的效率,提出了一种变量空间确定性低偏差均匀抽样与样本点处联合概率密度函数构造权重相结合的方法,来估计可靠性灵敏度。该方法通过均匀样本点处联合概率密度函数的权重保证了可靠性灵敏度的估计值收敛于真值,而由低偏差抽样代替原问题中的联合概率密度抽样则可以保证更低的误差阶以及在小失效概率条件下抽得的样本有更高的可能性落入失效域,从而保证了所提方法具有更高的收敛速度。另外,所提方法可以采用与独立变量相同的步骤来估计相关变量情况下的可靠性灵敏度,计算简便,适用范围广。算例充分证明了所提方法的优越性。     

基于密度权重的可靠性灵敏度分析方法

为了提高可靠性灵敏度求解数字模拟法的效率,提出了一种变量空间确定性低偏差均匀抽样与样本点处联合概率密度函数构造权重相结合的方法,来估计可靠性灵敏度。该方法通过均匀样本点处联合概率密度函数的权重保证了可靠性灵敏度的估计值收敛于真值,而由低偏差抽样代替原问题中的联合概率密度抽样则可以保证更低的误差阶以及在小失效概率条件下抽得的样本有更高的可能性落入失效域,从而保证了所提方法具有更高的收敛速度。另外,所提方法可以采用与独立变量相同的步骤来估计相关变量情况下的可靠性灵敏度,计算简便,适用范围广。算例充分证明了所提方法的优越性。     

A Note on Bounds for Target Tracking with pd>1

Recently several new results for Cramer-Rao lower bounds (CRLBs) in dynamical systems have been developed. Several different approaches and approximations have been presented. For the general case of target tracking with a detection probability smaller than one and possibly in the presence of false measurements, two main approaches have been presented. The first approach is the information reduction factor (IRF) approach. The second approach is the enumeration (ENUM) approach, also referred to as the conditioning approach. It has been found that the ENUM approach leads to a strictly larger covariance matrix than the IRF approach, however, still providing a lower bound on the attainable error covariance. Thus, the ENUM approach provides a strictly tighter bound on the attainable performance. It has been conjectured that these bounds converge to one another in the limit or equivalently after an initial transition stage. We demonstrate, using some recent results from the modified Riccati equation (MRE) and by means of counter examples, that this conjecture does not hold true in general. We also demonstrate that the conjecture does hold true in the special case of deterministic target motion, or equivalently in the absence of process noise. Furthermore, we show that the detection probability has an influence on the limiting behaviors of the bounds. Moreover, we show that the MRE approximation provides a very good and computationally efficient approximation of the ENUM bound. The various results are illustrated by means of representative examples.     

Imperfect Carrier Recovery Effect on Filtered PSK Signals

Coherent demodulation of a PSK signal requires the generation of a local carrier phase reference. Methods are given to determine the detection loss caused by noisy phase recovery and its use in the coherent detection of filtered BPSK and QPSK signals. It is assumed that the phase noise can have a static part and a random component with a Tikhonov-type distribution. The static part is mostly due to offset frequency tracking of the PLL used to recover the carrier, while the random component is due to thermal noise present in the carrier recovery loop and is also due to the random nature of the phase modulation. It is shown that the probability of error of BPSK and QPSK can be expressed as a finite sum of a set of strictly alternating converging series when the number of ISI terms is finite. Upper and lower bounds on the probability of error have been derived when this number becomes infinite and we show how this error rate can be computed with any desired accuracy. Numerical results are presented for various values of static error and phase noise variance when the transmit and receive filters are 4-pole Butterworth filters. For filtered PSK signals and for a bit error rate of 10-6, our results show that the additional degradation in presentday receiver systems due to imperfect carrier recovery can be less than 0.1 dB for BPSK and less than 1 dB for QPSK.     

Energy Minimization of M-Ary Orthogonal Signals in Communication Systems with Feedback

A communication system with a feedback channel which makes use of orthogonal M-ary signals is considered. The problem of its optimization, by setting a maximum time Tm to decide, is analyzed particularly so that the average energy per bit required to obtain a pre-fixed probability of error results as a minimum. The results, shown on graphs, allow one to evaluate the maximum decision time Tm and, afterwards, the maximum transmission rate.     

Digital Communication Systems in Impulsive Atmospheric Radio Noise

Analyses are presented of the performance of binary and M-ary coherent and noncoherent communication systems operating in the impulsive atmospheric radio noise environment. The receiver is usually a maximum likelihood detector for white Gaussian interference and therefore has the form of a parallel bank of matched filters followed by decision circuitry. By employing a Poisson or generalized Shot noise model for the impulsive noise with a suitable probability density function (pdf), closed-form expressions and bounds of error probabilities for M-ary noncoherent and coherent amplitude-shift keying (ASK), phase-shift keying (PSK), and frequency-shift keying (FSK) systems are obtained and the results discussed.     

Model Error and the Direction-Finder Problem

The error introduced into many direction-finder (DF) algorithms by the use of projections is discussed. Upper bounds for this error are calculated and formulas for calculating these bounds are extended to the case of nonconformal transformations.     

Dimension reduction for array processing with robust interference cancellation

We present a robust solution for data reduction in array processing. The purpose is to reduce the computation and improve the performance of applied signal processing algorithms by mapping the data into a lower dimension beamspace (BS) through a transformation. Nulls steering to interference are incorporated into a transformation using the subspace projection technique, and the BS spatial spectrum estimation accuracy is evaluated and maximized with a measure. The derived transformation tries to preserve the full-dimension Cramer-Rao bounds (CRBs) for the parameters of interest while rejecting undesired signals effectively. When compared with an optimal method and an adaptive approach, simulation results show that significant improvements are obtained in terms of BS direction-of-arrival (DOA) estimation root-mean-squared error (RMSE), bias, and resolution probability.     

Angle measurement in the presence of mainbeam interference

The problem of reducing interference impinging on an antenna array when the sources lie in the main beam is addressed. Adaptive antenna arrays are incorporated to form adapted sum and difference beams in which the interference signals are suppressed. Monopulse error curves are then obtained, providing the necessary distortion correction curves across the entire mainbeam tracking angle region. New Cramer-Rao (C-R) bounds on the angle estimation error are derived with generalized assumptions on the signal amplitude and phase. The bounds previously derived by others are valid under different conditions. With these generalized assumptions on the signal characteristics, a Monte Carlo simulation is performed, based on the estimation procedure presented, to determine the angle estimation error. These errors are compared with the C-R bounds. Good performance is shown for sufficient S/N₀ and angular separation between the target and the interference sources     

Decision-Directed Detector for Overlapping PCM/NRZ Signals

A decision-directed (DD) technique for the detection of overlapping PCM/NRZ signals in the presence of white Gaussian noise is investigated. The performance of the DD detector is represented by probability of error PE versus input signal-to-noise ratio (SNR). To examine how much improvement in performance can beachieved with this technique, PE's with and without DD feedback are evaluated in parallel. Further, analytical results are compared with those found by Monte Carlo simulations. The results are shown in good agreement.