BEACON: an adaptive set-membership filtering technique with sparseupdates

This paper deals with adaptive solutions to the so-called set-membership filtering (SMF) problem. The SMF methodology involves designing filters by imposing a deterministic constraint on the output error sequence. A set-membership decision feedback equalizer (SM-DFE) for equalization of a communications channel is derived, and connections with the minimum mean square error (MMSE) DFE are established. Further, an adaptive solution to the general SMF problem via a novel optimal bounding ellipsoid (OBE) algorithm called BEACON is presented. This algorithm features sparse updating, wherein it uses about 5-10% of the data to update the parameter estimates without any loss in mean-squared error performance, in comparison with the conventional recursive least-squares (RLS) algorithm. It is shown that the BEACON algorithm can also be derived as a solution to a certain constrained least-squares problem. Simulation results are presented for various adaptive signal processing examples, including estimation of a real communication channel. Further, it is shown that the algorithm can accurately track fast time variations in a nonstationary environment. This improvement is a result of incorporating an explicit test to check if an update is needed at every time instant as well as an optimal data-dependent assignment to the updating weights whenever an update is required     

线性系统集员估计问题研究

研究了线性系统下的Norton和基于OBE两种集员估计算法。Norton算法是通过最小体积或最小迹来优化时间更新阶段和测量更新阶段,但其计算量大、效率低。针对这一不足,OBE算法采用最小半径定界椭球来进行测量阶段的更新,从而简化了算法,减少了计算量。最后通过与传统Kalman滤波算法与Norton集员估计算法相比,验证了基于OBE集员估计算法的有效性。     

Direct estimation of blind zero-forcing equalizers based onsecond-order statistics

Most existing zero-forcing equalization algorithms rely either on higher than second-order statistics or on partial or complete channel identification. We describe methods for computing fractionally spaced zero-forcing blind equalizers with arbitrary delay directly from second-order statistics of the observations without channel identification. We first develop a batch-type algorithm; then, adaptive algorithms are obtained by linear prediction and gradient descent optimization. Our adaptive algorithms do not require channel order estimation, nor rank estimation. Compared with other second-order statistics-based approaches, ours do not require channel identification at all. On the other hand, compared with the CMA-type algorithms, ours use only second-order statistics; thus, no local convergence problem exists, and faster convergence can be achieved. Simulations show that our algorithms outperform most typical existing algorithms     

Eigenanalysis-based blind methods for identification, equalization,and inversion of linear time-invariant channels

Some blind algorithms are presented for estimation of equalizers for linear time invariant (LTI) channels from the eigenvectors of certain rank-one matrices constructed from the second-order statistics of the oversampled received signal. It is shown that the channel can also be identified from the same matrices. It can be shown that in multipath dominated environments, equalizers with symbol spread of only one (referred to as inverters) can be used when sufficient diversity is available. Because of the manner in which structure in the channel distortion is exploited, the proposed identification and equalization algorithms are also applicable to this case. For the same reason, the proposed algorithms do not require estimation of the channel memory (only an upper bound is required). Equalizers of desired delay are estimated directly independent of others     

Frequency-Domain Set-Membership Filtering and Its Applications

Frequency-domain adaptive filtering is appealing in many applications, particularly channel equalization. This paper presents frequency-domain set-membership filtering (F-SMF) and derives adaptive algorithms for F-SMF. The F-SMF is employed to design single-carrier frequency-domain equalizer (SC-FDE). With an unconventional parameter-dependent error-bound specification, an F-SMF algorithm is derived and shown to provide superior performance with sparse updates of parameter estimates. Exploring the feature of sparse updates, we present an innovative parallel adaptive architecture that shares the updating processors and that finds natural appeal in frequency-domain diversity combining and equalization for very dispersive fading channels like those found in broadband wireless communications     

Equalization and semi-blind channel estimation for space-time block coded signals over a frequency-selective fading channel

In this paper, we investigate the equalization and channel identification for space-time block coded signals over a frequency-selective multiple-input multiple-output (MIMO) channel. The equalization has been considered by taking into account the cyclostationarity of space-time block coded signals. The minimum mean square error (MMSE) solutions have been derived for the linear and decision feedback (DF) equalizers. The channel estimation is required for the equalization. With known symbols (as pilot symbols), MIMO channels can be estimated. In addition, due to the redundancy induced by space-time block code, it is possible to identify MIMO channels blindly using the subspace method. We consider both blind and semi-blind channel estimation for MIMO channels. It is shown that the semi-blind channel estimate has fewer estimation errors, and it results in less (bit error rate) performance degradation of the MMSE linear and DF equalizers.     

Soft-Input Soft-Output Equalizers for Turbo Receivers: A Statistical Physics Perspective

Many algorithms in signal processing and digital communications must deal with the problem of computing the probabilities of the hidden state variables given the observations, i.e., the inference problem, as well as with the problem of estimating the model parameters. Such an inference and estimation problem is encountered, for e.g., in adaptive turbo equalization/demodulation where soft information about the transmitted data symbols has to be inferred in the presence of the channel uncertainty, given the received signal samples and a priori information provided by the decoder. An exact inference algorithm computes the a posteriori probability (APP) values for all transmitted symbols, but the computation of APPs is known to be an NP-hard problem, thus, rendering this approach computationally prohibitive in most cases. In this paper, we show how many of the well-known low-complexity soft-input soft-output (SISO) equalizers, including the channel-matched filter-based linear SISO equalizers and minimum mean square error (MMSE) SISO equalizers, as well as the expectation-maximization (EM) algorithm-based SISO demodulators in the presence of the Rayleigh fading channel, can be formulated as solutions to a variational optimization problem. The variational optimization is a well-established methodology for low-complexity inference and estimation, originating from statistical physics. Importantly, the imposed variational optimization framework provides an interesting link between the APP demodulators and the linear SISO equalizers. Moreover, it provides a new set of insights into the structure and performance of these widely celebrated linear SISO equalizers while suggesting their fine tuning as well.     

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.      

Block channel equalization in the frequency domain

In this paper, channel equalization algorithms processing two samples of the received signal per channel symbol and operating in the frequency domain are described in a unifying framework. First, minimum mean-square error linear and decision-feedback equalizers are derived, and a synthesis technique based on the well-known Levinson-Durbin algorithm is proposed for the latter. Then, iterative linear and decision-feedback equalization algorithms for turbo processing are devised. Performance results for both uncoded and coded phase-shift keying transmissions show the efficacy of the proposed equalization techniques and their superiority over other existing frequency-domain equalization strategies.     

Channel Equalization Using Adaptive Lattice Algorithms

In this paper, a study of adaptive lattice algorithms as applied to channel equalization is presented. The orthogonalization properties of the lattice algorithms make them appear promising for equalizing channels which exhibit heavy amplitude distortion. Furthermore, unlike the majority of other orthogonalization algorithms, the number of operations per update for the adaptive lattice equalizers is linear with respect to the number of equalizer taps.     

Performance bounds for multistep prediction-based blindequalization

Blind equalization attempts to remove the interference caused by a communication channel without using any known training sequences. Blind equalizers may be implemented with linear prediction-error filters (PEFs). For many practical channel types, a suitable delay at the output of the equalizer allows for achieving a small estimation error. The delay cannot be controlled with one-step predictors. Consequently, multistep PEF-based algorithms have been suggested as a solution to the problem. The derivation of the existing algorithms is based on the assumption of a noiseless channel, which results in zero-forcing equalization. We consider the effects of additive noise at the output of the multistep PEF. Analytical error bounds for two PEF-based blind equalizers in the presence of noise are derived. The obtained results are verified with simulations. The effect of energy concentration in the channel impulse response on the error bound is also addressed     

最低误码率非线性均衡器的快速自适应学习算法

针对最低误码率非线性均衡器的参数在线自适应学习问题,本文提出基于拟牛顿方法的快速自适应学习算法。采用Parzen窗函数方法估计误码率,通过设定切换条件,使参数学习在滑窗随机梯度法与滑窗拟牛顿法之间切换。这既增加了新算法的数值稳定性,又可提高收敛速度。通过对拟牛顿方法进行修改,还使新算法既可以在线自适应学习,也可用于高维参数的快速学习。仿真采用最低误码率非线性均衡器对通信系统进行干扰抑制和信道均衡,结果表明了新算法的高效性。     

Blind MMSE Equalization of FIR/IIR Channels Using Oversampling and Multichannel Linear Prediction

A linear‐prediction‐based blind equalization algorithm for single‐input single‐output (SISO) finite impulse response/infinite impulse response (FIR/IIR) channels is proposed. The new algorithm is based on second‐order statistics, and it does not require channel order estimation. By oversampling the channel output, the SISO channel model is converted to a special single‐input multiple‐output (SIMO) model. Two forward linear predictors with consecutive prediction delays are applied to the subchannel outputs of the SIMO model. It is demonstrated that the partial parameters of the SIMO model can be estimated from the difference between the prediction errors when the length of the predictors is sufficiently large. The sufficient filter length for achieving the optimal prediction is also derived. Based on the estimated parameters, both batch and adaptive minimum‐mean‐square‐error equalizers are developed. The performance of the proposed equalizers is evaluated by computer simulations and compared with existing algorithms.     

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.      

On-line blind multichannel equalization based on mutually referenced filters

This paper presents a novel approach to the blind linear equalization of possibly nonminimum phase and time-varying communication channels. In the context of channel diversity, we introduce the concept of mutually referenced equalizers (MREs) in which several filters are considered, the outputs of which act as training signals for each other. A corresponding (constrained) multidimensional mean-square error (MSE) cost function is derived, the minimization of which is shown to be a necessary and sufficient condition for equalization. The links with a standard linear prediction problem are demonstrated. The proposed technique exhibits properties of important practical concern: 1) the proposed algorithm is globally convergent. 2) Simple closed-form solutions exist for the MREs, but the MREs also lend themselves readily to adaptive implementation. In particular, the recursive least-squares (RLS) algorithm can be used to offer optimal convergence rate. 3) The MRE method provides a solution for all equalization delays, which results in robustness properties with respect to SNR and ill-defined channel lengths.     

一种新的神经网络均衡器:结构、算法与性能

本文根据克服数字通信中码间干扰（ISI）的最佳均衡解一般表达式，提出了一种新的自适应神经网络均衡器结构，然后导出了基于该结构的一种自适应算法和相应的学习规则，最后对提出的自适应神经网络均衡器性能进行了计算机模拟，模拟结果与分析表明：本文提出的神经网络均衡器用于实现最佳信道均衡非常有效，比传统线性均衡器和Gibson等人[1]提出的多层感知均衡器（MLPE）性能更优越，更具实用性．     

Global convergence of fractionally spaced Godard (CMA) adaptiveequalizers

The Godard (1980) or constant modulus algorithm (CMA) equalizer is perhaps the best known and the most popular scheme for blind adaptive channel equalization. Most published works on blind equalization convergence analysis are confined to T-spaced equalizers with real-valued inputs. The common belief is that analysis of fractionally spaced equalizers (FSEss) with complex inputs is a straightforward extension with similar results. This belief is, in fact, untrue. We present a convergence analysis of Godard/CMA FSEs that proves the important advantages provided by the FSE structure. We show that an FSE allows the exploitation of the channel diversity that supports two important conclusions of great practical significance: (1) a finite-length channel satisfying a length-and-zero condition allows Godard/CMA FSE to be globally convergent, and (2) the linear FSE filter length need not be longer than the channel delay spread. Computer simulation demonstrates the performance improvement provided by the adaptive Godard FSE     

Determination of Tap Positions for Sparse Equalizers

Sparse equalizers, in which only a small subset of the filter taps is selected to be nonzero, were recently proposed as a low-complexity solution for receivers operating in wireless frequency-selective channels with sparse power profiles. The performance of the sparse equalizer heavily depends on its tap-positioning algorithm. This paper presents efficient low-complexity algorithms for determination of sparse equalizer tap positions based on a forward sequential search. We develop low-complexity metrics for the evaluation of the candidate tap positions in the search space as well as methods to effectively reduce the search space size. The proposed algorithms are shown to be superior over previously proposed algorithms in a wide range of channel conditions. Actually, the proposed algorithms yield, in most of the tested cases, performance identical to the optimal, prohibitively complex, tap-positioning algorithm. The main emphasis is on linear equalization suitable for wideband code-division multiple-access systems but the algorithm can be extended to a variety of equalization schemes and channels.     

Adaptive set-membership identification in O(m)time for linear-in-parameters models

Some fundamental contributions to the theory and applicability of optimal bounding ellipsoid (OBE) algorithms for signal processing are described. All reported OBE algorithms are placed in a general framework that demonstrates the relationship between the set-membership principles and least square error identification. Within this framework, flexible measures for adding explicit adaptation capability are formulated and demonstrated through simulation. Computational complexity analysis of OBE algorithms reveals that they are of O(m²) complexity per data sample with m the number of parameters identified. Two very different approaches are described for rendering a specific OBE algorithm, the set-membership weighted recursive least squares algorithm, of O(m) complexity. The first approach involves an algorithmic solution in which a suboptimal test for innovation is employed. The performance is demonstrated through simulation. The second method is an architectural approach in which complexity is reduced through parallel competition