Adaptive Minimum Symbol Error Rate Beamforming Assisted Detection for Quadrature Amplitude Modulation

We consider beamforming assisted detection for multiple antenna aided multiuser systems that employ the bandwidth efficient quadrature amplitude modulation scheme. A minimum symbol error rate (MSER) design is proposed for the beamforming assisted receiver, and it is shown that this MSER design provides significant performance enhancement, in terms of achievable symbol error rate, over the standard minimum mean square error (MMSE) design. A sample-by-sample adaptive algorithm, referred to as the least symbol error rate, is derived for adaptive implementation of the MSER beamforming solution. The proposed adaptive MSER scheme is evaluated in simulation using Rayleigh fading channels, in comparison with the adaptive MMSE benchmarker.     

Adaptive minimum symbol-error-rate decision feedback equalization for multilevel pulse-amplitude modulation

The design of decision feedback equalizers (DFEs) is typically based on the minimum mean square error (MMSE) principle as this leads to effective adaptive implementation in the form of the least mean square algorithm. It is well-known, however, that in certain situations, the MMSE solution can be distinctly inferior to the optimal minimum symbol error rate (MSER) solution. We consider the MSER design for multilevel pulse-amplitude modulation. Block-data adaptive implementation of the theoretical MSER DFE solution is developed based on the Parzen window estimate of a probability density function. Furthermore, a sample-by-sample adaptive MSER algorithm, called the least symbol error rate (LSER), is derived for adaptive equalization applications. The proposed LSER algorithm has a complexity that increases linearly with the equalizer length. Computer simulation is employed to evaluate the proposed alternative MSER design for equalization application with multilevel signaling schemes.     

Performance of Adaptive Modulation Scheme for Adaptive Minimum Symbol Error Rate Beamforming Receiver

This paper considers adaptive beamforming receiver that support multiple users, each having one transmit antenna. In certain circumstances, symbol error rate (SER) performance of the beamforming receiver degrades severely. In order to minimize the SER, minimum symbol error rate (MSER) beamforming receiver is utilized. Then, we propose an adaptive modulation scheme for the receiver to maintain the average SER below the target SER while maximizing the average throughput. The scheme uses the information on the direction of arrival and the average signal-to-noise ratio to decide the appropriate modulation mode. For comparison, the proposed scheme is also applied to minimum mean square error (MMSE) beamforming receiver system. Simulations were carried out in the presence of single and two interferers. Simulation results show that the performance of the proposed algorithm employing MSER beamforming is superior to its MMSE counterpart, with the largest advantage of 0.21 in the outage probability.     

单载波频域均衡系统中一种简化的IBDFE算法

在单载波频域均衡系统中,线性均衡算法虽然简单易行,但是其抑制噪声干扰和符号间干扰的能力有限,因此需要引入非线性的反馈和迭代机制以进一步提升系统性能.迭代块判决反馈均衡（Iterative Block Decision Feedback Equalization,IBDFE）就是一种行之有效的非线性算法,但其缺点是计算复杂度高.鉴于此,在IBDFE结构的基础上,利用最小均方误差准则,推导出了一种新的简化算法.之后,对简化后算法和现有低复杂度算法的均方误差（Mean Square Error,MSE）性能进行了理论分析和比较,并在两种无线多径衰落信道下对三种不同的算法进行了仿真.结果表明,在所给信道条件下,这种算法在迭代两次后已基本收敛.同时,仿真也验证了MSE分析的结论.最后,对算法复杂度的分析和比较表明,提出的简化算法相比传统IBDFE算法,其误比特率性能有所下降,但能有效地减小计算量.     

Minimum Bayes Risk Adaptive Linear Equalizers

This paper introduces Bayes risk (expected loss) as a criterion for linear equalization. Since the probability of error is equal to the Bayes risk (BR) for a particular binary loss function, this work is a natural generalization of previous works on minimum probability of error (PE) equalizers. Adaptive equalization algorithms are developed that minimize the BR. Like the minimum PE equalizers, the BR algorithms have low computational complexity which is comparable to that of the LMS algorithm. The advantage of the BR criterion is that the loss function can be specified in a manner that accelerates adaptive equalizer convergence relative to the minimum PE adaptive algorithm as illustrated in simulation examples. Besides introducing a new criterion, this paper provides another independent contribution to the field of PE minimizing equalization. While most prior works focus on $M$-ary QAM type modulations with rectangular decision regions, this paper uses upper bounds on the probabilities of certain events to yield tractable mathematics that apply to two-dimensional constellations with arbitrarily shaped decision regions. The resulting adaptive algorithm use the full information available in the phase of the error signal, whereas previous algorithms use a quantized version of this error phase.      

Stochastic Unbiased Minimum Mean Error Rate Algorithm for Decision Feedback Equalizers

Capitalizing on a well-known minimum mean-square error (MMSE) property for decision feedback equalization (DFE) along with the use of stochastic gradient approach, we formulate an adaptive minimum error rate (MER) algorithm for DFE over M-ary PAM channels to be named as stochastic unbiased minimum mean-error rate (SUMMER). Comparisons are made between our algorithm and existing MER algorithms in the literature. Also, by invoking the central limit theorem, we present an analytical proof that an unbiased MMSE equalizer will approach an MER equalizer when the equalizer length approaches infinity; thereby, we obtain a lower bound expression for MER.     

On Blind Equalization of Biorthogonal Signaling

Motivated by increasing interest in energy efficient modulations, we provide the first look at adaptive equalization of biorthogonal signaling. While this modulation has historically been considered only for use in narrowband systems without intersymbol interference (ISI), recent attention has been given to its use in ISI channels. Due to the fact that biorthogonal modulation (BOM) results in a source that is not i.i.d., however, classical blind adaptive equalization techniques cannot be directly applied to equalization of BOM signals. We first examine minimum mean-squared error (MMSE) and least mean squares (LMS)-based equalizers, and identify some peculiarities that arise in equalization of BOM signals when compared to more traditional modulations like binary phase shift keying (BPSK). Next, we present two novel blind algorithms for the adaptive equalization of BOM signals: LTBOMB and TROMBONE. We discuss the convergence properties of these algorithms, and demonstrate their performance with numerical simulations     

The Kernel Least-Mean-Square Algorithm

The combination of the famed kernel trick and the least-mean-square (LMS) algorithm provides an interesting sample-by-sample update for an adaptive filter in reproducing kernel Hilbert spaces (RKHS), which is named in this paper the KLMS. Unlike the accepted view in kernel methods, this paper shows that in the finite training data case, the KLMS algorithm is well posed in RKHS without the addition of an extra regularization term to penalize solution norms as was suggested by Kivinen [Kivinen, Smola and Williamson, ldquoOnline Learning With Kernels,rdquo IEEE Transactions on Signal Processing, vol. 52, no. 8, pp. 2165-2176, Aug. 2004] and Smale [Smale and Yao, ldquoOnline Learning Algorithms,rdquo Foundations in Computational Mathematics, vol. 6, no. 2, pp. 145-176, 2006]. This result is the main contribution of the paper and enhances the present understanding of the LMS algorithm with a machine learning perspective. The effect of the KLMS step size is also studied from the viewpoint of regularization. Two experiments are presented to support our conclusion that with finite data the KLMS algorithm can be readily used in high dimensional spaces and particularly in RKHS to derive nonlinear, stable algorithms with comparable performance to batch, regularized solutions.     

Low-Complexity Block Turbo Equalization for OFDM Systems in Time-Varying Channels

We propose low-complexity block turbo equalizers for orthogonal frequency-division multiplexing (OFDM) systems in time-varying channels. The presented work is based on a soft minimum mean-squared error (MMSE) block linear equalizer (BLE) that exploits the banded structure of the frequency-domain channel matrix, as well as a receiver window that enforces this banded structure. This equalization approach allows us to implement the proposed designs with a complexity that is only linear in the number of subcarriers. Three block turbo equalizers are discussed: two are based on a biased MMSE criterion, while the third is based on the unbiased MMSE criterion. Simulation results show that the proposed iterative MMSE BLE achieves a better bit error rate (BER) performance than a previously proposed iterative MMSE serial linear equalizer (SLE). The proposed equalization algorithms are also tested in the presence of channel estimation errors.      

Entropy minimization for supervised digital communications channelequalization

This paper investigates the application of error-entropy minimization algorithms to digital communications channel equalization. The pdf of the error between the training sequence and the output of the equalizer is estimated using the Parzen windowing method with a Gaussian kernel, and then, the Renyi's quadratic entropy is minimized using a gradient descent algorithm. By estimating Renyi's entropy over a short sliding window, an online training algorithm is also introduced. Moreover, for a linear equalizer, an orthogonality condition for the minimum entropy solution that leads to an alternative fixed-point iterative minimization method is derived. The performance of linear and nonlinear equalizers trained with entropy and mean square error (MSE) is compared. As expected, the results of training a linear equalizer are very similar for both criteria since, even if the input noise is non-Gaussian, the output filtered noise tends to be Gaussian. On the other hand, for nonlinear channels and using a multilayer perceptron (MLP) as the equalizer, differences between both criteria appear. Specifically, it is shown that the additional information used by the entropy criterion yields a faster convergence in comparison with the MSE     

Space-time turbo equalization in frequency-selective MIMO channels

A computationally efficient space-time turbo equalization algorithm is derived for frequency-selective multiple-input-multiple-output (MIMO) channels. The algorithm is an extension of the iterative equalization algorithm by Reynolds and Wang (see Signal Processing, vol.81, no.5, p.989-995, 2001) for frequency-selective fading channels and of iterative multiuser detection for code-division multiple-access (CDMA) systems by Wang and Poor (see IEEE Trans. Commun., vol.47, p.1046-1061, 1999). The proposed algorithm is implemented as a MIMO detector consisting of a soft-input-soft-output (SISO) linear MMSE detector followed by SISO channel decoders for the multiple users. The detector first forms a soft replica of each composite interfering signal using the log likelihood ratio (LLR), fed back from the SISO channel decoders, of the transmitted coded symbols and subtracts it from the received signal vector. Linear adaptive filtering then takes place to suppress the interference residuals: filter taps are adjusted based on the minimum mean square error (MMSE) criterion. The LLR is then calculated for adaptive filter output. This process is repeated in an iterative fashion to enhance signal-detection performance. This paper also discusses the performance sensitivity of the proposed algorithm to channel-estimation error. A channel-estimation scheme is introduced that works with the iterative MIMO equalization process to reduce estimation errors.     

Fractionally spaced equalization using CMA: robustness to channelnoise and lack of disparity

In the noise-free case, the fractionally spaced equalization using constant modulus (FSE-CM) criterion has been studied previously. Its minima were shown to achieve perfect equalization when zero-forcing (ZF) conditions are satisfied and to be able to still achieve fair equalization when there is lack of disparity. However, to our best knowledge, the effect of additive channel noise on the FSE-CM cost-function minima has not been studied. In this paper, we show that the noisy FSE-CM cost function is subject to a smoothing effect with respect to the noise-free cost function, the result of which is a tradeoff between achieving zero forcing and noise enhancement. Furthermore, we give an analytical closed-form expression for the loss of performance due to the noise in terms of input-output mean square error (MSE). Under the ZF conditions, the FSE-CM MSE is shown to be mostly due to output noise enhancement and not to residual intersymbol interference (ISI). When there is lack of disparity, an irreducible amount of ISI appears independently of the algorithm. It is the lower equalizability bound for given channel conditions and equalizer length-the so-called minimum MSE (MMSE). The MMSE lower bound is the sum of the MMSE and of additional MSE mostly due to noise enhancement. Finally, we compare the FSE-CM MSE to this lower bound     

基于最大相关熵准则的恒模盲均衡算法

针对恒模算法(constant modulus algorithm, CMA)在脉冲噪声环境下性能退化的问题,本文基于最大相关熵准则(maximum correntropy criterion, MCC)对恒模算法中基于最小均方误差(mean square error, MSE)准则的代价函数进行修正,推导出适用于脉冲噪声环境的基于MCC准则的恒模盲均衡算法(MCC_CMA)。该算法利用通信信号的恒模特性,首先得到发送信号与均衡器输出信号模值的误差信号,再通过使模值误差信号的相关熵最大来获得其迭代误差调节项,避免了传统高阶统计量算法在脉冲噪声环境下性能退化的问题。对高斯噪声以及α-稳定分布和混合高斯分布两种脉冲噪声环境下的信道均衡问题的仿真实验表明,相对于经典的自适应恒模盲均衡算法,MCC_CMA算法不依赖噪声的先验知识就能获得较快的收敛速度、较低的剩余码间干扰和误码率,并且在不同脉冲强度的脉冲噪声环境下都能够得到较好的均衡结果,表明MCC_CMA算法具有很好的鲁棒性。      

Adaptive minimum bit-error-rate filtering

Adaptive filtering has traditionally been developed based on the minimum mean square error (MMSE) principle and has found ever-increasing applications in communications. The paper develops adaptive filtering based on an alternative minimum bit error rate (MBER) criterion for communication applications. It is shown that the MBER filtering exploits the non-Gaussian distribution of filter output effectively and, consequently, can provide significant performance gain in terms of smaller bit error rate (BER) over the MMSE approach. Adopting the classical Parzen window or kernel density estimation for a probability density function (pdf), a block-data gradient adaptive MBER algorithm is derived. A stochastic gradient adaptive MBER algorithm is further developed for sample-by-sample adaptive implementation of the MBER filtering. Extension of the MBER approach to adaptive nonlinear filtering is also discussed.     

Fast adaptive equalization/diversity combining for time-varyingdispersive channels

We examine adaptive equalization and diversity combining methods for fast Rayleigh-fading frequency selective channels. We assume a block adaptive receiver in which the receiver coefficients are obtained from feedforward channel estimation. For the feedforward channel estimation, we propose a novel reduced dimension channel estimation procedure, where the number of unknown parameters are reduced using a priori information of the transmit shaping filter's impulse response. Fewer unknown parameters require a shorter training sequence. We obtain least-squares, maximum-likelihood, and maximum a posteriori (MAP) estimators for the reduced dimension channel estimation problem. For symbol detection, we propose the use of a matched filtered diversity combining decision feedback equalizer (DFE) instead of a straightforward diversity combining DFE. The matched filter form has lower computational complexity and provides a well-conditioned matrix inversion. To cope with fast time-varying channels, we introduce a new DFE coefficient computation algorithm which is obtained by incorporating the channel variation during the decision delay into the minimum mean square error (MMSE) criterion. We refer to this as the non-Toeplitz DFE (NT-DFE). We also show the feasibility of a suboptimal receiver which has a lower complexity than a recursive least squares adaptation, with performance close to the optimal NT-DFE     

OFDM调制系统中一种鲁棒的MMSE信道均衡算法

在OFDM调制系统中,由于多载波的存在,载波间ICI对信道均衡结果影响很大,同时,采用MMSE信道均衡算法对不同的信道模型也存在均衡误差不稳定的情况。本文提出一种鲁棒的MMSE信道均衡算法,该算法的核心是针对不同信道对信道的PDP进行归一化处理,从而得到在不同类型信道下稳定且鲁棒的信道均衡性能。     

MIMO DFE equalization for multitone DS/SS systems over multipathchannels

The combination of multitone modulation with direct sequence spectrum spreading (DS/SS) has been introduced in the past. The performance of a correlation receiver has been evaluated for a multipath channel and in the presence of an additional multiple access interference. We analyze the problem of decision feedback equalization (DFE) for such a system. In order to understand the potential of the system with equalization, we first study the steady-state behavior of the equalizer for a minimum mean square error (MMSE) criterion. The investigation is carried out for a receiver made of a bank of filters matched to both the symbol shape and the channel, and for a two path channel. Assuming transmission of binary phase shift keying (BPSK) symbols, an exact expression of the bit error probability is obtained in the form of an integral. Then adaptive least mean square (LMS) and recursive least square (RLS) structures are derived. The performance of the adaptive RLS algorithm is demonstrated by means of computer simulations     

Spatial-temporal equalization for IS-136 TDMA systems with rapiddispersive fading and cochannel interference

In this paper, we investigate spatial-temporal equalization for IS-136 time-division multiple-access (TDMA) cellular/PCS systems to suppress intersymbol interference and cochannel interference and improve communication quality. This research emphasizes channels with large Doppler frequency (up to 184 Hz), delay dispersion under one symbol duration, and strong cochannel interference. We first present the structure of the optimum spatial-temporal decision-feedback equalizer (DFE) and linear equalizer and derive closed-form expressions for the equalizer parameters and mean-square error (MSE) for the case of known channel parameters. Since the channel can change within an IS-136 time slot, the spatial-temporal equalizer requires parameter tracking techniques. Therefore, we present three parameter tracking algorithms: the diagonal loading minimum MSE algorithm, which uses diagonal loading to improve tracking ability, the two-stage tracking algorithm, which uses diagonal loading in combination with a reduced complexity architecture, and the simplified two-stage tracking algorithm, which further reduces complexity to one M×M and one 3×3 matrix inversion for weight calculation with M antennas. For a four-antenna system, the simplified two-stage tracking algorithm can attain a 10^-2 bit error rate (BER) when the channel delay spread is half of the symbol duration and the signal-to-interference ratio (SIR) of the system is as low as 5 dB, making it a computationally feasible technique to enhance system performance for IS-136 TDMA systems     

Multiuser Detection in SDMA–OFDM Wireless Communication System Using Complex Multilayer Perceptron Neural Network

The space division multiple access–orthogonal frequency division multiplexing (SDMA–OFDM) wireless system has become very popular owing high spectral efficiency and high load capability. The optimal maximum likelihood multiuser detection (MUD) technique suffers from high computational complexity. On the other hand the linear minimum mean square error (MMSE) MUD techniques yields poor performance and also fails to detect users in overload scenario, where the number of users are more than that of number of receiving antennas. By contrast, the differential evolution algorithm (DEA) aided minimum symbol error rate (MSER) MUD can sustain in overload scenario as it can directly minimizes probability of error rather than mean square error. However, all these classical techniques are still complex as these do channel estimation and multiuser detection sequentially. In this paper, complex multi layer perceptron (CMLP) neural network model is suggested for MUD in SDMA–OFDM system as it do both channel approximation and MUD simultaneously. Simulation results prove that the CMLP aided MUD performs better than the MMSE and MSER techniques in terms of enhanced bit error rate performance with low computational complexity.     

Minimum mean-squared error decision-feedback equalization fordigital subscriber line transmission with possibly correlated line codes

The author presents a theory on MMSE (minimum mean-squared error) decision-feedback equalization which augments previously published results by allowing both a correlated symbol sequence and a fractionally spaced DFE (decision-feedback equalizer) forward filter. This theory facilitates calculating the potential DSL (digital subscriber line) transmission performance in cases of correlated line codes, especially for situations where one or both of the DFE filters are infinite in length. The situation of an infinite-length DFE is of interest because it provides information on the limit of MMSE equalization and can thus serve as a benchmark against which the performance of a finite-length DFE may be compared. The author also presents a few numerical examples of the performance of MMSE decision-feedback equalization in DSL transmission at ISDN (integrated services digital network) basic access rates with several well-known line codes