Performance Limit of Chaotic Digital Waveform Communication Systems: Approach of Maximizing a Posteriori Probability

In this paper, we investigate a bandpass chaotic waveform communication system in an additive white Gaussian noise channel environment. By making use of the signalspace diagram and maximizing the a posteriori probability, we derive the conditions under which the optimum bit error performance can be achieved. In particular, we make use of a binary communication system to illustrate how the modulator and demodulator should be constructed in order to achieve the best error performance. Finally, the noise performance bound is determined for chaotic digital waveform communications.     

Probability distribution of DPSK in tone interference andapplications to SFH/DPSK

The probability distribution of the received DPSK (differential phase-shift keying) signal under tone jamming is studied. The results should facilitate the analysis of the SFH/DPSK (slow frequency-hopped DPSK) system. The results given are more general than those previously published, in several aspects. First, the differential phase of the transmitted DPSK signal can assume any value. Second, probability distributions are derived instead of a set of probabilities calculated over certain symmetrical regions. This allows analyzing performance for arbitrary selected decision regions as well as determining optimum decision regions for demodulating the DPSK signal. Third, the joint probability distribution of both the magnitude and differential phase of the jammed DPSK signal is given. This can be used in the analysis where both tone jamming and Gaussian noise are considered. These results are used to analyze the error probability performance of a general uncoded SFH/DPSK signal under worst-case tone jamming and Gaussian noise     

The Use of the Fourier-Bessel Series in Calculating Error Probabilities for Digital Communication Systems

Using the theory of spherical symmetric random vectors one can find an expression for the error probability of a wide variety of digital communications systems. These expressions, however, are in the form of Bessel integrals which are usually difficult to solve. In this paper we show how the Fourier-Bessel series can be used to solve the integrals numerically. The calculation error is found to depend on two series parameters which can be manipulated to make the error arbitrarily small. Two examples are used to show the utility of the technique. In the first the probability of error for a CPSK communications system operating in Gaussian noise and cochannel interference is found. In the second the error performance for a multilevel ASK communications system operating in the same corrupting environment is determined. The Fourier-Bessel series technique is a valuable practical tool for solving these and other signal detection problems.     

An optimal signal design for band-limited asynchronous DS-CDMAcommunications

An optimal signal design for band-limited, asynchronous, direct-sequence code-division multiple-access (DS-CDMA) communications with aperiodic random spreading sequences and a conventional matched filter receiver is considered in an additive white Gaussian noise (AWGN) channel. With bandwidth defined in the strict sense, two optimization problems are solved under finite bandwidth and zero interchip interference constraints. First, a chip waveform optimization is performed given the system bandwidth, the data symbol transmission rate, and the processing gain. A technique to characterize a band-limited chip waveform with a finite number of parameters is developed, and it is used to derive optimum chip waveforms which minimize the effect of multiple-access interference (MAI) for any energy and delay profile of users. Next, a joint optimization of the processing gain and the chip waveform is performed, given the system bandwidth and the data symbol transmission rate. A sufficient condition for a system to have lower average probability of bit error for any energy profile is found, and it is used to derive some design strategies. It is shown that the flat spectrum pulse with the processing gain leading to zero excess bandwidth results in the minimum average probability of bit error. Design examples and numerical results are also provided     

Distance distribution of binary codes and the error probability of decoding

We address the problem of bounding below the probability of error under maximum-likelihood decoding of a binary code with a known distance distribution used on a binary-symmetric channel (BSC). An improved upper bound is given for the maximum attainable exponent of this probability (the reliability function of the channel). In particular, we prove that the "random coding exponent" is the true value of the channel reliability for codes rate R in some interval immediately below the critical rate of the channel. An analogous result is obtained for the Gaussian channel.     

Minimum-Error-Based Approximation Model for Symmetric Alpha Stable Distribution

In many communication channels the impulsive noise is usually assumed to be of a symmetric alpha stable (SαS) distribution. Unfortunately, except for the Gaussian, Cauchy, and Lévy laws, the analytical expressions for the probability density functions (PDF) of alpha stable distributions are unknown, resulting in very limited application of this distribution. In a practical system, the bi-parameter Cauchy–Gaussian mixture (BCGM) distribution is used to approximate the PDF of the SαS distribution to tackle this difficulty. In this paper, we derive the optimal mixture ratio of the BCGM model based on the minimum square error criterion and furthermore propose a simplified and robust version of BCGM for the SαS distribution. Numerical simulations show that our proposed model achieves better performance and is more robust than the conventional models, without incurring additional complexity.     

Bit Error Probability and Bit Outage Rate in Chaos Communication

This paper investigates the notion of the probability of bit error (PBE) and its distribution in chaos-based communication systems; these are seen as being the fundamental quantities to both the well-known bit error rate (BER) and the new concept in chaos communications of bit outage rate (BOR). The form of the distribution illustrates the degree to which bit error rate is a stable representation of performance. Bit outage rate is another measure of performance which gives practically helpful information about bit error. For a simple coherent chaos-shift-keying system the distribution of bit error probability is derived exactly, and theoretically exact formulas for the bit outage rate and bit error rate are presented. Two specific cases are developed to obtain useful qualitative and quantitative information. The cases concern independent Gaussian spreading, as a lower benchmark and logistic map spreading, as typical of effective chaotic spreading. Comparisons are obtained between these spreading distributions and between different extents of their spreading, calibrated against per bit signal to noise ratio. A general conclusion is that bit outage and bit error rates are complementary measures of performance.     

User selection scheme with limited feedback processing and outdated CSI in multiuser relay networks

A new user selection strategy is investigated and analyzed in a multiuser relaying environment in the presence of co‐channel interference. The proposed selection scheme aims at avoiding unnecessary feedback load processing, in cases where a target threshold, in the received instantaneous signal‐to‐noise ratio, is exceeded. Assuming that perfect channel state information is available, closed‐form lower bound expressions are derived for the cumulative distribution function of the output signal to interference plus noise ratio. Moreover, the impact of outdated channel state information on the system's performance is also investigated. In addition, under the assumption of high signal‐to‐noise ratio conditions, simplified approximated expressions are also provided for the cumulative distribution functions of the output signal to interference plus noise ratio, which are employed to study the outage probability and bit error probability performance of the system. It is shown that with the proposed approach, a significant reduction in feedback load processing is achieved, with only a slight loss in performance. Copyright © 2016 John Wiley & Sons, Ltd.     

实时高精度雷达噪声信号产生方法与FPGA实现

针对各种典型分布的噪声信号在雷达系统半实物仿真和噪声雷达波形设计中的实际需求，基于Box-Muller算法提出了一种能够实时产生多种分布特性噪声的信号生成方法。首先使用线性反馈移位寄存器产生均匀分布随机序列，然后利用能够实现高精度、低延迟的坐标旋转数字算法(Coordinate Rotation Digital Computer,CORDIC)实现Box-Muller变换中复杂函数的快速计算，将均匀随机序列转换成高斯分布随机序列，最后利用高斯随机序列经过相关数学运算得到其他复杂分布的随机序列，在此基础上产生具有各种分布特性的噪声信号。基于Xilinx XC7VX415T 现场可编程门阵列(Field Programmable Gate Array,FPGA)芯片的实验结果表明，所提方法在保证小数位数据精度为20 b时，可实时产生速率为2.5 Gb/s的高精度随机数据序列，进而生成服从多种函数分布下的噪声信号。所提方法整体资源占用率较少，具有较强的实用性。     

Two‐pilot channel estimation scheme for amplify‐and‐forward relay networks

In amplify‐and‐forward relay networks, the equivalent channel to the destination node is not independent of equivalent noise and the equivalent noise does not follow a Gaussian distribution. Therefore, it is difficult to directly estimate the equivalent channel based on traditional optimal rules. In this paper, we propose a two‐pilot estimation (TPE) scheme that decomposes a non‐Gaussian noise channel estimation problem into two channel estimation problems in Gaussian noise. In TPE scheme, the relay‐destination channel is first estimated by one pilot and the other pilot is used to estimate the equivalent channel with the aid of the estimated relay‐destination channel. Simulation results show that the TPE scheme can achieve less estimation error and larger system throughput than other existing channel estimators in slow fading case. Copyright © 2012 John Wiley & Sons, Ltd.     

Intersymbol interference error bounds with application to ideal bandlimited signaling

An upper bound is derived for the probability of error of a digital communication system subject to intersymbol interference and Gaussian noise. The bound is applicable to multilevel as well as binary signals and to all types of intersymbol interference. The bound agrees with the exponential portion of a normal distribution in which the larger intersymbol interference components subtract from the signal amplitude, and the smaller components add to the noise power. The results are applied to the case of random binary signaling with sinx/xpulses. It is shown that such signals are not so sensitive to timing error as is commonly believed, nor does the total signal amplitude become very large with significant probability. However, the error probability does grow very rapidly as the system bandwidth is reduced below the Nyquist band.     

Maximum likelihood localization of sources in noise modeled as astable process

This paper introduces a new class of robust beamformers which perform optimally over a wide range of non-Gaussian additive noise environments. The maximum likelihood approach is used to estimate the bearing of multiple sources from a set of snapshots when the additive interference is impulsive in nature. The analysis is based on the assumption that the additive noise can be modeled as a complex symmetric α-stable (SαS) process. Transform-based approximations of the likelihood estimation are used for the general SαS class of distributions while the exact probability density function is used for the Cauchy case. It is shown that the Cauchy beamformer greatly outperforms the Gaussian beamformer in a wide variety of non-Gaussian noise environments, and performs comparably to the Gaussian beamformer when the additive noise is Gaussian. The Cramer-Rao bound for the estimation error variance is derived for the Cauchy case, and the robustness of the SαS beamformers in a wide range of impulsive interference environments is demonstrated via simulation experiments     

Worst case additive noise for binary-input channels andzero-threshold detection under constraints of power and divergence

Additive-noise channels with binary inputs and zero-threshold detection are considered. We study worst case noise under the criterion of maximum error probability with constraints on both power and divergence with respect to a given symmetric nominal noise distribution. Particular attention is focused on the cases of a) Gaussian nominal distributions and b) asymptotic increase in worst case error probability when the divergence tolerance tends to zero     

Distribution of the phase angle between two vectors perturbed byGaussian noise II

The Case I, Case II, and Case III distributions for the phase angle between two vectors perturbed by Gaussian noise are restated in terms of integrals with integrands containing only exponentials. Added to these are a Case IV distribution in which one of the vectors is noise free; and Case V, Case VI, and Case VII distributions for the instantaneous radian frequency that results when the time between the two vectors goes to zero. In addition, existing and new forms for the probability density functions corresponding to each of the distributions are given. The results are applied to digital FM with a limiter/discriminator receiver to easily obtain the bit error probability without coding and the corresponding probability density function needed in a Chernoff bound approach to study the bit error probability with coding     

Symbol error probability evaluation for PSK modulation schemes in generalized non‐Gaussian noise channels

The average symbol error probability (ASEP) performance over the faded radio frequency (RF) link when the noise is additive in nature and has generalized Laplacian distribution is evaluated in this paper. The additive white generalized Laplacian noise (AWGLN) distribution can model different impulsive and non‐Gaussian noise environments often encountered in practice and provides a robust alternative to Gaussian distribution. A new expression for evaluating the exact symbol error probability over a multilevel M‐ary PSK‐modulated AWGLN channel is derived. Based on the obtained expression, the ASEP for the single‐hop RF link that models the shadowing and fading conditions over the RF channel by a generalized –K (GK) distribution is derived. Further, the error performance of a decode‐and‐forward relayed and GK‐distributed two‐hop RF link is discussed, and the results are validated through numerical plots.     

Synthesis and analysis of algorithms for digital signal recognition in conditions of deforming distortions and additive noise

The problem of digital signal recognition has been considered in conditions of deforming distortions of the waveform of these signals and additive Gaussian noise. A mathematical model for introducing deformations of the known or random waveform signals is proposed for synthesizing recognition algorithms. The model is based on introducing the nonlinear deformation operator as an operator of permutations with repetitions of elements of the initial discrete signal with addition of additive noise component caused by quantization errors of continuous deformation function. Two recognition algorithms were synthesized and investigated. The first is an optimal one based on the exact calculation of likelihood functions, and the second is a quasi-optimal algorithm based on using the Gaussian approximation of likelihood functions. These algorithms were simulated for different variants of the specified values of deforming distortions in the form of determinate functions and in the form of random function realizations. The experimental error probability was compared with its theoretical estimate at different values of signal-to-noise ratio.     

水下复合信道对GMSK无线光通信系统性能的影响

The absorption and scattering of light in seawater channel cause signal attenuation, and the turbulence of seawater causes signal amplitude fluctuation, both of which will reduce the bit error rate (BER) performance of underwater wireless optical communication (UWOC) system. The effects of the two channel characteristics on the signal performance were considered comprehensively, and a method was proposed to equate the transmission distance and turbulence probability density function to the system signal-to-noise ratio (SNR) and turbulence noise, and then the signal attenuation and turbulence noise were combined into the signal waveform to establish the underwater composite channel signal transmission model. According to the experimental system parameters, the signal transmission waveforms of Gaussian minimum frequency shift keying (GMSK) modulation under composite channel were simulated, and the one-bit difference demodulation algorithm was used to compare the demodulated waveforms with the original waveform, and the influence relationships of composite channel on the system BER performance was analyzed. The simulation experiment results show that, compared with on-off keying modulation (OOK), pulse position modulation (PPM), GMSK system can obtain the SNR gain of 3.3 dB, 4.8 dB respectively only in the attenuation channel with seawater attenuation coefficient of 0.151 m?1. Under the composite channel, GMSK modulation performance is superior to OOK modulation and PPM modulation. When the water attenuation coefficient is 0.151 m?1, and turbulence intensity variance is smaller than 0.16, GMSK modulation system has no error rate limit, the system BER is decided by signal attenuation and turbulence noise and Gaussian noise together, GMSK modulation achieves SNR gain of 4.35 dB compared with PPM modulation. Furthermore, turbulence intensity variance is greater than 0.16, system BER arrives limit, which value is determined by the turbulence intensity, and the limit value of BER increases nonlinearly with the increase of turbulence intensity.     

Analysis of SER of MIMO‐MRC systems with imperfect channel estimation in the presence of non‐Rayleigh CCIs

In this paper, we study the average symbol error rate (SER) for a multiple input multiple output (MIMO) maximal ratio combining (MRC) system with Rayleigh fading desired signal in the presence of non‐Rayleigh fading co‐channel interferers (CCIs) and additive white Gaussian noise (AWGN). To simulate the actual environments of wireless transmission, we assume that the transceiver only obtains imperfect channel estimation (ICE). For the cases of Nakagami and Rician fading CCIs, analytic expressions for the SER have been derived approximately by introducing the modified signal‐to‐interference and noise power ratio (SINR) that can be obtained by averaging the CCI term in the original SINR over the distribution of ICE of intended user. These formulas can provide important reference of design of MIMO diversity systems. Numerical simulations verify the effectiveness of these formulas. Copyright © 2010 John Wiley & Sons, Ltd.     

Performance of optimum wavelet waveform for DS‐CDMA chip waveform over QS‐AWGN channel

In this paper, we search for a better chip waveform based on orthogonal wavelets for direct sequence‐code division multiple access (DS‐CDMA) signals to improve the probability of error (P_e) performance with minimal signal bandwidth variations. First, we derive the P_e expression over a quasi‐synchronous additive white Gaussian noise channel for DS‐CDMA signals, which use various pulse shaping waveforms including orthogonal wavelets as chip waveforms. It is observed that this expression depends on the chip waveform. Then, we design an optimum wavelet by using the relationship between wavelets and filter coefficients to reduce the probability of error. The DS‐CDMA system using the optimum wavelet waveform results in a lower probability of error than those using the conventional chip waveforms such as raised cosine, half‐sine and rectangular waveforms. Especially, the P_e of the optimum wavelet‐based scheme becomes significantly better than those of the conventional chip waveforms‐based schemes under the heavy loading that is the case for commercial wireless systems. When the systems work with full load (i.e. the number of users equals the processing gain), the optimum wavelet‐based system results in 0.5, 2.1 and 4 dB better SNR values than those of the raised cosine, half‐sine and rectangular‐based systems, respectively, for a P_e value of 10^?3. Copyright © 2005 John Wiley & Sons, Ltd.     

Optimization of Trellis Codes with Multilevel Amplitude Modulation with Respect to an Error Probability Criterion

An easily computed upper bound on the error probability of a data communications system with combined trellis coding and multilevel/phase modulation is derived, assuming an additive white Gaussian noise channel and maximum-likelihood decoding. This bound is used to search for codes obtained by set-partitioning that minimize the bound for a fixed number of trellis states. Only amplitude modulated signals typically used in voiceband modem applications are considered. The signal levels that minimize the error probability bound subject to an average power constraint are presented for some specific codes.