A Stochastic Total Least Squares Solution of Adaptive Filtering Problem

An efficient and computationally linear algorithm is derived for total least squares solution of adaptive filtering problem, when both input and output signals are contaminated by noise. The proposed total least mean squares (TLMS) algorithm is designed by recursively computing an optimal solution of adaptive TLS problem by minimizing instantaneous value of weighted cost function. Convergence analysis of the algorithm is given to show the global convergence of the proposed algorithm, provided that the stepsize parameter is appropriately chosen. The TLMS algorithm is computationally simpler than the other TLS algorithms and demonstrates a better performance as compared with the least mean square (LMS) and normalized least mean square (NLMS) algorithms. It provides minimum mean square deviation by exhibiting better convergence in misalignment for unknown system identification under noisy inputs

A Novel Adaptive Algorithm Addresses Potential Problems of Blind Algorithm

A hybrid algorithm called constant modulus least mean square (CMLMS) algorithm is proposed in order to address the potential problems existing with constant modulus algorithm (CMA) about its convergence. It is a two-stage adaptive filtering algorithm and based on least mean square (LMS) algorithm followed by CMA. A hybrid algorithm is theoretically developed and the same is verified through MatLab Software. Theoretical model is verified through simulation and its performance is evaluated in smart antenna in presence of a cochannel interfering signal and additive white Gaussian noise (AWGN) of zero mean. This is also tested in Rayleigh fading channel using digital modulation technique for Bit Error Rate (BER). Finally, a few computer simulations are presented in order to substantiate the theoretical findings with respect to proposed model. Corresponding results obtained with the use of only CMA and LMS algorithms are also presented for further comparison

A Recursive Restricted Total Least-squares Algorithm

We show that the generalized total least squares (GTLS) problem with a singular noise covariance matrix is equivalent to the restricted total least squares (RTLS) problem and propose a recursive method for its numerical solution. The method is based on the generalized inverse iteration. The estimation error covariance matrix and the estimated augmented correction are also characterized and computed recursively. The algorithm is cheap to compute and is suitable for online implementation. Simulation results in least squares (LS), data least squares (DLS), total least squares (TLS), and RTLS noise scenarios show fast convergence of the parameter estimates to their optimal values obtained by corresponding batch algorithms

A Recursive Restricted Total Least-Squares Algorithm

International audienceWe show that the generalized total least squares (GTLS) problem with a singular noise covariance matrix is equivalent to the restricted total least squares (RTLS) problem and propose a recursive method for its numerical solution. The method is based on the generalized inverse iteration. The estimation error covariance matrix and the estimated augmented correction are also characterized and computed recursively. The algorithm is cheap to compute and is suitable for online implementation. Simulation results in least squares (LS), data least squares (DLS), total least squares (TLS), and RTLS noise scenarios show fast convergence of the parameter estimates to their optimal values obtained by corresponding batch algorithms

A fast recursive total least squares algorithm for adaptive FIR filtering

This paper proposes a new fast recursive total least squares (N-RTLS) algorithm to recursively compute the TLS solution for adaptive finite impulse response (FIR) filtering. The N-RTLS algorithm is based on the minimization of the constrained Rayleigh quotient (c-RQ) in which the last entry of the parameter vector is constrained to the negative one. As analysis results on the convergence of the proposed algorithm, we study the properties of the stationary points of the c-RQ. The high computational efficiency of the new algorithm depends on the efficient computation of the fast gain vector (FGV) and the adaptation of the c-RQ. Since the last entry of the parameter vector in the c-RQ has been fixed as the negative one, a minimum point of the c-RQ is searched only along the input data vector, and a more efficient N-RTLS algorithm is obtained by using the FGV. As compared with Davila's RTLS algorithms, the N-RTLS algorithm saves the 6M number of multiplies, divides, and square roots (MADs). The global convergence of the new algorithm is studied by LaSalle's invariance principle. The performances of the relevant algorithms are compared via simulations, and the long-term numerical stability of the N-RTLS algorithm is verified

Adaptive antenna array beamforming using a concatenation of recursive least square and least mean square algorithms

In recent years, adaptive or smart antennas have become a key component for various wireless applications, such as radar, sonar and cellular mobile communications including worldwide interoperability for microwave access (WiMAX). They lead to an increase in the detection range of radar and sonar systems, and the capacity of mobile radio communication systems. These antennas are used as spatial filters for receiving the desired signals coming from a specific direction or directions, while minimizing the reception of unwanted signals emanating from other directions.Because of its simplicity and robustness, the LMS algorithm has become one of the most popular adaptive signal processing techniques adopted in many applications, including antenna array beamforming. Over the last three decades, several improvements have been proposed to speed up the convergence of the LMS algorithm. These include the normalized-LMS (NLMS), variable-length LMS algorithm, transform domain algorithms, and more recently the constrained-stability LMS (CSLMS) algorithm and modified robust variable step size LMS (MRVSS) algorithm. Yet another approach for attempting to speed up the convergence of the LMS algorithm without having to sacrifice too much of its error floor performance, is through the use of a variable step size LMS (VSSLMS) algorithm. All the published VSSLMS algorithms make use of an initial large adaptation step size to speed up the convergence. Upon approaching the steady state, smaller step sizes are then introduced to decrease the level of adjustment, hence maintaining a lower error floor. This convergence improvement of the LMS algorithm increases its complexity from 2N in the case of LMS algorithm to 9N in the case of the MRVSS algorithm, where N is the number of array elements.An alternative to the LMS algorithm is the RLS algorithm. Although higher complexity is required for the RLS algorithm compared to the LMS algorithm, it can achieve faster convergence, thus, better performance compared to the LMS algorithm. There are also improvements that have been made to the RLS algorithm families to enhance tracking ability as well as stability. Examples are, the adaptive forgetting factor RLS algorithm (AFF-RLS), variable forgetting factor RLS (VFFRLS) and the extended recursive least squares (EX-KRLS) algorithm. The multiplication complexity of VFFRLS, AFF-RLS and EX-KRLS algorithms are 2.5N2 + 3N + 20 , 9N2 + 7N , and 15N3 + 7N2 + 2N + 4 respectively, while the RLS algorithm requires 2.5N2 + 3N .All the above well known algorithms require an accurate reference signal for their proper operation. In some cases, several additional operating parameters should be specified. For example, MRVSS needs twelve predefined parameters. As a result, its performance highly depends on the input signal.In this study, two adaptive beamforming algorithms have been proposed. They are called recursive least square - least mean square (RLMS) algorithm, and least mean square - least mean square (LLMS) algorithm. These algorithms have been proposed for meeting future beamforming requirements, such as very high convergence rate, robust to noise and flexible modes of operation. The RLMS algorithm makes use of two individual algorithm stages, based on the RLS and LMS algorithms, connected in tandem via an array image vector. On the other hand, the LLMS algorithm is a simpler version of the RLMS algorithm. It makes use of two LMS algorithm stages instead of the RLS – LMS combination as used in the RLMS algorithm.Unlike other adaptive beamforming algorithms, for both of these algorithms, the error signal of the second algorithm stage is fed back and combined with the error signal of the first algorithm stage to form an overall error signal for use update the tap weights of the first algorithm stage.Upon convergence, usually after few iterations, the proposed algorithms can be switched to the self-referencing mode. In this mode, the entire algorithm outputs are swapped, replacing their reference signals. In moving target applications, the array image vector, F, should also be updated to the new position. This scenario is also studied for both proposed algorithms. A simple and effective method for calculate the required array image vector is also proposed. Moreover, since the RLMS and the LLMS algorithms employ the array image vector in their operation, they can be used to generate fixed beams by pre-setting the values of the array image vector to the specified direction.The convergence of RLMS and LLMS algorithms is analyzed for two different operation modes; namely with external reference or self-referencing. Array image vector calculations, ranges of step sizes values for stable operation, fixed beam generation, and fixed-point arithmetic have also been studied in this thesis. All of these analyses have been confirmed by computer simulations for different signal conditions. Computer simulation results show that both proposed algorithms are superior in convergence performances to the algorithms, such as the CSLMS, MRVSS, LMS, VFFRLS and RLS algorithms, and are quite insensitive to variations in input SNR and the actual step size values used. Furthermore, RLMS and LLMS algorithms remain stable even when their reference signals are corrupted by additive white Gaussian noise (AWGN). In addition, they are robust when operating in the presence of Rayleigh fading. Finally, the fidelity of the signal at the output of the proposed algorithms beamformers is demonstrated by means of the resultant values of error vector magnitude (EVM), and scatter plots. It is also shown that, the implementation of an eight element uniform linear array using the proposed algorithms with a wordlength of nine bits is sufficient to achieve performance close to that provided by full precision