Optimal training design for MIMO OFDM systems in mobile wireless channels

This paper describes a least squares (LS) channel estimation scheme for multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems based on pilot tones. We first compute the mean square error (MSE) of the LS channel estimate. We then derive optimal pilot sequences and optimal placement of the pilot tones with respect to this MSE. It is shown that the optimal pilot sequences are equipowered, equispaced, and phase shift orthogonal. To reduce the training overhead, an LS channel estimation scheme over multiple OFDM symbols is also discussed. Moreover, to enhance channel estimation, a recursive LS (RLS) algorithm is proposed, for which we derive the optimal forgetting or tracking factor. This factor is found to be a function of both the noise variance and the channel Doppler spread. Through simulations, it is shown that the optimal pilot sequences derived in this paper outperform both the orthogonal and random pilot sequences. It is also shown that a considerable gain in signal-to-noise ratio (SNR) can be obtained by using the RLS algorithm, especially in slowly time-varying channels.     

Numeric Variable Forgetting Factor RLS Algorithm for Second-Order Volterra Filtering

Nonlinear adaptive filtering techniques for system identification (based on the Volterra model) are widely used for the identification of nonlinearities in many applications. In this correspondence, the improved tracking capability of a numeric variable forgetting factor recursive least squares (NVFF-RLS) algorithm is presented for first-order and second-order time-varying Volterra systems under a nonstationary environment. The nonlinear system tracking problem is converted into a state estimation problem of the time-variant system. The time-varying Volterra kernels are governed by the first-order Gauss–Markov stochastic difference equation, upon which the state-space representation of this system is built. In comparison to the conventional fixed forgetting factor recursive least squares algorithm, the NVFF-RLS algorithm provides better channel estimation as well as channel tracking performance in terms of the minimum mean square error (MMSE) for first-order and second-order Volterra systems. The NVFF-RLS algorithm is adapted to the time-varying signals by using the updating prediction error criterion, which accounts for the nonstationarity of the signal. The demonstrated simulation results manifest that the proposed method has good adaptability in the time-varying environment, and it also reduces the computational complexity.     

Blind adaptive MIMO receivers for space-time block-coded DS-CDMA systems in multipath channels using the constant modulus criterion

We propose blind adaptive multi-input multi-output (MIMO) linear receivers for DS-CDMA systems using multiple transmit antennas and space-time block codes (STBC) in multipath channels. A space-time code-constrained constant modulus (CCM) design criterion based on constrained optimization techniques is considered and recursive least squares (RLS) adaptive algorithms are developed for estimating the parameters of the linear receivers. A blind space-time channel estimation method for MIMO DS-CDMA systems with STBC based on a subspace approach is also proposed along with an efficient RLS algorithm. Simulations for a downlink scenario assess the proposed algorithms in several situations against existing methods.     

基于压缩感知的分布式协同估计算法

为了降低分布式协同估计算法的计算量并改善其收敛性能,提出了基于压缩感知(CS)和递归最小二乘(RLS)的分布式协同估计算法.该算法在传统RLS分布式协同估计算法的基础上引入压缩感知技术,首先在压缩域中进行递归最小二乘运算,然后利用压缩感知重构算法得到未知参数向量的估计值.提出的算法能够在增量式策略和两种模式的扩散式策略下实现对未知向量的有效估计.理论分析和仿真结果表明,该算法一方面降低了RLS分布式协同估计算法的计算量,另一方面保持较快的收敛速度与良好的均方误差性能.     

Variable forgetting factor linear least squares algorithm forfrequency selective fading channel estimation

In this article, the variable forgetting factor linear least squares algorithm is presented to improve the tracking capability of channel estimation. A linear channel model with respect to time change describes a time-varying channel more accurately than a conventional stationary channel model. To reduce the estimation error due to model mismatch, we incorporate the modified variable forgetting factor into the proposed algorithm. Compared to the existing algorithms-exponentially windowed recursive least squares algorithm with the optimal forgetting factor and linear least squares algorithm-the proposed method makes a remarkable improvement in a fast fading environment. The effects of channel parameters such as signal-to-noise ratio and fading rate are investigated by computer simulations     

Tracking Performance of Least Squares MIMO Channel Estimation Algorithm

In this paper, the tracking performance analysis of the least squares (LS) multiple-input multiple-output (MIMO) channel estimation and tracking algorithm is presented. MIMO channel estimation is a novel application of the LS algorithm that presents near-optimum performance by Karami and Shiva in 2003 and 2006. In this paper, the mean square error (MSE) of tracking of the LS MIMO channel estimator algorithm is derived as a closed-form function of the Doppler shift, forgetting factor, channel rank, and the length of training sequences. In the analysis, all training symbols are considered as randomly generated equal-power vectors on the unit circle, or in other words, phase-shift keying (PSK) signaling. By evaluating this function, some insights into the tracking behavior of the LS MIMO channel estimator are achieved. Then, the calculated tracking error is compared with the tracking error derived from Monte Carlo simulation for quaternary-PSK-based training signals to verify the validation of the presented analysis. Finally, the optimum forgetting factor is derived to minimize the error function, and it is shown that the optimum forgetting factor is highly dependent on the training length, Doppler shift, and E_b/N_o. Also, it is concluded that in low E_b/N_o values, the number of transmitter antennas has negligible effect on the optimal value of the forgetting factor.     

Adaptive MLSE equalizers with parametric tracking for multipathfast-fading channels

We use the parametric channel identification algorithm proposed by Chen and Paulraj (see Proc. IEEE Vehicular Technology Conf., p.710-14, 1997) and by Chen, Kim and Liang (see IEEE Trans. Veh. Technol., p.1923-35, 1999) to adaptively track the fast-fading channels for the multichannel maximum likelihood sequence estimation (MLSE) equalizer using multiple antennas. Several commonly-used channel tracking schemes, decision-directed recursive least square (DD/RLS), per-survivor processing recursive least square (PSP/RLS) and other reduced-complexity MLSE algorithms are considered. An analytic lower bound for the multichannel MLSE equalizer with no channel mismatch in the time-varying specular multipath Rayleigh-fading channels is derived. Simulation results that illustrate the performance of the proposed algorithms working with various channel tracking schemes are presented, and then these results are compared with the analytic bit error rate (BER) lower bound and with the conventional MLSE equalizers directly tracking the finite impulse response (FIR) channel tap coefficients. We found that the proposed algorithm always performs better than the conventional adaptive MLSE algorithm, no matter what channel tracking scheme is used. However, which is the best tracking scheme to use depends on the scenario of the system     

A new system identification method for fast echo cancelerinitialization

Proposes a new recursive version of an earlier technique for fast initialization of data-driven echo cancelers (DDECs). The speed of convergence and the covariance of the estimate of the proposed technique are comparable to the recursive least squares (RLS) algorithm, however, the computational complexity is no greater than the least mean square (LMS) algorithm. Analysis of computational complexity and the estimation error is also provided. Simulation results based on both floating-point and fixed-point arithmetic illustrate a remarkable improvement in terms of speed of convergence and steady-state error over the computationally comparable LMS algorithm     

基于可变遗忘因子RLS在OFDM系统的载波频偏估计

在时变信道下正交频分复用（OFDM）系统中,通过导频辅助,提出基于可变遗忘因子RLS（VFF-RLS）的载波频偏（CFO）估计改进算法。针对传统RLS（CFF-RLS）算法中遗忘因子无法同时满足CFO估计收敛速度和收敛精度的缺陷,本文设计了线性变化遗忘因子(LFF)和非线性变化遗忘因子(NLFF) 两种可变遗忘因子方案来提升CFO估计性能。仿真结果显示：在低信噪比的情形下,基于VFF-RLS算法的CFO估计性能明显优于基于CFF-RLS算法的CFO估计性能。     

Adaptive Filtering Techniques using Cyclic Prefix in OFDM Systems for Multipath Fading Channel Prediction

This paper presents adaptive channel prediction techniques for wireless orthogonal frequency division multiplexing (OFDM) systems using cyclic prefix (CP). The CP not only combats intersymbol interference, but also precludes requirement of additional training symbols. The proposed adaptive algorithms exploit the channel state information contained in CP of received OFDM symbol, under the time-invariant and time-variant wireless multipath Rayleigh fading channels. For channel prediction, the convergence and tracking characteristics of conventional recursive least squares (RLS) algorithm, numeric variable forgetting factor RLS (NVFF-RLS) algorithm, Kalman filtering (KF) algorithm and reduced Kalman least mean squares (RK-LMS) algorithm are compared. The simulation results are presented to demonstrate that KF algorithm is the best available technique as compared to RK-LMS, RLS and NVFF-RLS algorithms by providing low mean square channel prediction error. But RK-LMS and NVFF-RLS algorithms exhibit lower computational complexity than KF algorithm. Under typical conditions, the tracking performance of RK-LMS is comparable to RLS algorithm. However, RK-LMS algorithm fails to perform well in convergence mode. For time-variant multipath fading channel prediction, the presented NVFF-RLS algorithm supersedes RLS algorithm in the channel tracking mode under moderately high fade rate conditions. However, under appropriate parameter setting in \(2\times 1\) space–time block-coded OFDM system, NVFF-RLS algorithm bestows enhanced channel tracking performance than RLS algorithm under static as well as dynamic environment, which leads to significant reduction in symbol error rate.     

The multi-user detection under fading channel

This paper proposes a method of blind multi-user detection algorithm based on signal sub-space estimation under the fading channels in the present of impulse noise. This algorithm adapts recursive least square (RLS) filter that can estimate the coefficients using only the signature waveform. In addition, to strengthen the ability of resisting the impulse noise, a new suppressive factor is induced, which can suppress the amplitude of the impulse, and improve the ability of convergence speed. Simulation results show that new RLS algorithm is more robust against consecutive impulse noise and have better convergence ability than conventional RLS. In addition, Compared to the least mean square (LMS) detector, the new robust RLS sub-space based method has better multi-address-inference (MAI) suppressing performance, especially, when channel degrades.     

Optimization of Recursive Least Square-Based Adaptive Linear Equalizer for ZigBee Transceiver

An efficient technique to compensate for the channel detrimental effects in ZigBee systems is introduced in this paper. The proposed methodology relies on adding a recursive least square (RLS) based adaptive linear equalizer (ALE) to the physical layer of the receiver side. The performance of the RLS based ALE is investigated inside the ZigBee system under different multipath fading situations: Rician and Rayleigh. Moreover, the paper proposes a methodology for deciding the RLS based ALE’s design parameters. The design procedure depends on solving multiple objectives optimizing function based on genetic algorithms (GAs). The ALE’s parameters are chosen, such that the system experiences minimum bit error rate (BER) with fast convergence response. For design verification purposes, the ZigBee transceiver is modeled in MATLAB Simulink and tested under different fading and signal to noise ratios. In addition, the performance of the RLS adaptation algorithm is compared with the least mean square (LMS) one. The results show that the RLS based ALE provides better ZigBee performance with less BER and fast adaptation response.     

Improved Soft Iterative Channel Estimation for Turbo Equalization of Time Varying Frequency Selective Channels

In this paper, we present computationally efficient iterative channel estimation algorithms for Turbo equalizer-based communication receiver. Least Mean Square (LMS) and Recursive least Square (RLS) algorithms have been widely used for updating of various filters used in communication systems. However, LMS algorithm, though very simple, suffers from a relatively slow and data dependent convergence behaviour; while RLS algorithm, with its fast convergence rate, finds little application in practical systems due to its computational complexity. Variants of LMS algorithm, Variable Step Size Normalized LMS (VSSNLMS) and Multiple Variable Step Size Normalized LMS algorithms, are employed through simulation for updating of channel estimates for turbo equalization in this paper. Results based on the combination of turbo equalizer with convolutional code as well as with turbo codes alongside with iterative channel estimation algorithms are presented. The simulation results for different normalized fade rates show how the proposed channel estimation based-algorithms outperformed the LMS algorithm and performed closely to the well known Recursive least square (RLS)-based channel estimation algorithm.     

基于平行因子分解的大规模MIMO盲信道估计

针对单小区大规模多输入多输出（Multiple-Input Multiple-Output,MIMO）系统上行链路,提出了一种基于平行因子（Parallel Factor,PARAFAC）模型的信道估计方法。在基站端,将接收信号构造成PARAFAC模型,利用大规模MIMO系统中信道的渐近正交的性质,提出了一种基于约束二线性迭代最小二乘算法(Constrained Blinear Alternating Least Squares,CBALS),从而实现了盲信道估计。理论分析及仿真结果表明,所提方法与传统最小二乘方法相比,不仅提高了频带利用率而且具有更高的估计精度;与已有的二线性交替最小二乘方法(BALS)相比,所提算法有更快的收敛速度。     

Adaptive MAP receiver via the EM algorithm and message passings for MIMO-OFDM mobile communications

This paper proposes two new types of maximum a posteriori probability (MAP) receivers for multiple-input-multiple-output and orthogonal frequency-division multiplexing mobile communications with a channel coding such as the low-density parity-check code. One proposed receiver employs the expectation-maximization algorithm so as to improve performance of approximated MAP detection. Differently from a conventional receiver employing the minimum mean-square estimation (MMSE) algorithm, it applies the recursive least squares (RLS) algorithm to the channel estimation in order to track a fast fading channel. For the purpose of further improvement, the other proposed receiver applies a new adaptive algorithm that can be derived from the message passing on factor graphs. The algorithm exploits all detected signals but one of targeted time, and can gain a considerable advantage over the MMSE and RLS. Computer simulations show that the first proposed receiver is superior in channel-tracking ability to the conventional receiver employing the MMSE. Furthermore, it is demonstrated that the second proposed receiver remarkably outperforms both the conventional and the first proposed ones.     

Recursive Least Squares Parameter Estimation for a Class of Output Nonlinear Systems Based on the Model Decomposition

In this paper, we study the parameter estimation problem of a class of output nonlinear systems and propose a recursive least squares (RLS) algorithm for estimating the parameters of the nonlinear systems based on the model decomposition. The proposed algorithm has lower computational cost than the existing over-parameterization model-based RLS algorithm. The simulation results indicate that the proposed algorithm can effectively estimate the parameters of the nonlinear systems.     

Effect of Least Square Channel Estimation Errors on Achievable Rate in MIMO Fading Channels

The achievable rate of a multiple input multiple output (MIMO) fading channel is derived in the form of generalized mutual information (GMI) when least square (LS) channel estimation is employed to gain channel state information at receivers. The derived average GMI of a channel with LS channel estimation is compared with the known average GMI (equivalently, the lower bound on the ergodic capacity) of a channel with linear minimum mean square error channel estimation. An effective approximation method is also proposed to ease the numerical calculation of the average GMI of a MIMO channel with LS channel estimation.     

Robust and Improved Channel Estimation Algorithm for MIMO-OFDM Systems

Multiple-input multiple-output (MIMO) system using orthogonal frequency division multiplexing (OFDM) technique has become a promising method for reliable high data-rate wireless transmission system in which the channel is dispersive in both time and frequency domains. Due to multiple cochannel interferences in a MIMO system, the accuracy of channel estimation is a vital factor for proper receiver design in order to realize the full potential performance of the MIMO-OFDM system. A robust and improved channel estimation algorithm is proposed in this paper for MIMO-OFDM systems based on the least squares (LS) algorithm. The proposed algorithm, called improved LS (ILS), employs the noise correlation in order to reduce the variance of the LS estimation error by estimating and suppressing the noise in signal subspace. The performance of the ILS channel estimation algorithm is robust to the number of antennas in transmit and receive sides. The new algorithm attains a significant improvement in performance in comparison with that of the regular LS estimator. Also, with respect to mean square error criterion and without using channel statistics, the ILS algorithm achieves a performance very close to that of the minimum mean square error (MMSE) estimator in terms of the parameters used in practical MIMO-OFDM systems. A modification of the ILS algorithm, called modified ILS (MILS), is proposed based on using the second order statistical parameters of channel. Analytically, it is shown that the MILS estimator achieves the exact performance of the MMSE estimator. Due to no specific data sequences being required to perform the estimation, in addition to the training mode, the proposed channel estimation algorithms can also be extended and used in the tracking mode with decision-aided method.     

Performance of adaptive MC-CDMA detectors in rapidly fading Rayleigh channels

Multicarrier code-division multiple access (MC-CDMA) combines multicarrier transmission with direct sequence spread spectrum. Different approaches have been adopted which do not assume a perfectly known channel. We examine the forward-link performance of decision-directed adaptive detection schemes, with and without explicit channel estimation, for MC-CDMA systems operating in fast fading channels. We analyze theoretically the impact of channel estimation errors by first considering a simpler system employing a threshold orthogonality restoring combining (TORC) detector with a Kalman channel estimator. We show that the performance deteriorates significantly as the channel fading rate increases and that the fading rate affects the selection of system parameters. We examine the performance of more realistic schemes based on the minimum mean square error (MMSE) criterion using least mean square (LMS) and recursive least square (RLS) adaptation. We present a discussion which compares the decision-directed and pilot-aided approaches and explores the tradeoffs between channel estimation overhead and performance. We find that there is a fading rate range where each method provides a good tradeoff between performance and overhead. We conclude that the MMSE per carrier decision-directed detector with RLS estimation combines good performance in low to moderate fading rates, robustness in parameter variations, and relatively low complexity and overhead. For higher fading rates, however, only pilot-symbol-aided detectors are appropriate.     

RLS Algorithm with Variable Fogetting Factor for Decision Feedback Equalizer over Time-Variant Fading Channels

In a high-rate indoor wireless personal communication system, the delay spread due to multipath propagation results in intersymbol interference (ISI) which can significantly increase the transmission bit error rate (BER). Decision feedback equalizer (DFE) is an efficient approach to combating the ISI. Recursive least squares (RLS) algorithm with a constant forgetting factor is often used to update the tap-coefficient vector of the DFE for ISI-free transmission. However, using a constant forgetting factor may not yield the optimal performance in a nonstationary environment. In this paper, an adaptive algorithm is developed to obtain a time-varying forgetting factor. The forgetting factor is used with the RLS algorithm in a DFE for calculating the tap-coefficient vector in order to minimize the squared equalization error due to input noise and due to channel dynamics. The algorithm is derived based on the argument that, for optimal filtering, the equalization errors should be uncorrelated. The adaptive forgetting factor can be obtained based on on-line equalization error measurements. Computer simulation results demonstrate that better transmission performance can be achieved by using the RLS algorithm with the adaptive forgetting factor than that with a constant forgetting factor previously proposed for optimal steady-state performance or a variable forgetting factor for a near deterministic system.