Improved stochastic gradient adaptive filter with gradient adaptivestep size

The performance of a stochastic gradient adaptive filter can be significantly improved by introducing a forgetting factor. The complexity of the original algorithm can also be reduced by using only the signs of error signals and input signals in the gradient adaptive step size computation     

A multistep size (MSS) frequency domain adaptive filter

A multistep size frequency-domain adaptive filter capable of tracking both stationary and nonstationary signals is proposed. This algorithm makes use of the simple structure of the least mean square error algorithm to update the filter coefficients and process the signal. It then proceeds further by incorporating a set of three step sizes and a knowledge-based strategy to select the best set of filter coefficients and the optimum step size iteratively. The main advantage of the MSS algorithm over the conventional LMS algorithm is that better performance is achieved without the knowledge of the input signal characteristics, such as signal power, degree of nonstationarity, signal-to-noise ratio, and stability bounds. Experimentally, the MSS algorithm is tested under various signal environments. The transient characteristics of the step sizes are found to be in agreement with previous theoretical studies of nonstationary characteristics of the LMS algorithm. In order to reduce the complexity, the conventional frequency-domain block LMS structure is also modified so that the MSS algorithm can be embedded more efficiently by exploiting the block structure     

Acquisition performance of an adaptive receiver for DS-CDMA

This paper presents a single-user code timing estimation algorithm for direct-sequence code-division multiple access that is based on processing the weight vector of an adaptive filter. The filter weight vector can be shown to adapt in the mean to a scaled time-shifted version of the spreading code of the desired user. Therefore, our algorithm requires very little side information in order to form its estimate. The acquisition performance of the algorithm is investigated when the filter is adapted using the least mean square (LMS) or the recursive least square (RLS) algorithm. The proposed algorithm is shown through experimental results to be resistant to the near-far problem when the RLS adaptation algorithm is used, but not when the LMS algorithm is used. However, the performance of this code-acquisition technique is still substantially better than the traditional correlator-based approach, even when the computationally simple LMS algorithm is used. As an extension to the basic timing estimator algorithm, we consider the effect of frequency synchronization error on the performance of the timing estimate. As expected, frequency-offset error degrades the performance of the timing estimate. However, a modified version of the adaptive filter is presented to combat this effect     

Analysis of the dynamics of a memoryless nonlinear gradient IIR adaptive notch filter

A nonlinear dynamical model of a memoryless nonlinear gradient IIR adaptive notch filter for estimating the frequency of a noisy sinusoid is derived. The model is verified through simulations, where simulated responses of the estimated frequency are compared with the responses obtained from the model with good agreement. Convergence properties of the filter are studied using the model, and maximum step sizes and initial frequency ranges for convergence are determined. The performance of the adaptive filter in tracking a time-varying signal frequency is also examined.     

A variable step size LMS algorithm

A least-mean-square (LMS) adaptive filter with a variable step size is introduced. The step size increases or decreases as the mean-square error increases or decreases, allowing the adaptive filter to track changes in the system as well as produce a small steady state error. The convergence and steady-state behavior of the algorithm are analyzed. The results reduce to well-known results when specialized to the constant-step-size case. Simulation results are presented to support the analysis and to compare the performance of the algorithm with the usual LMS algorithm and another variable-step-size algorithm. They show that its performance compares favorably with these existing algorithms     

Low-complexity adaptive step size constrained constant modulus SG algorithms for adaptive beamforming

In this paper, we propose two low-complexity adaptive step size mechanisms to enhance the performance of stochastic gradient (SG) algorithms for adaptive beamforming. The beamformer is designed according to the constrained constant modulus (CCM) criterion and the proposed mechanisms are employed in the SG algorithm for implementation. A complexity comparison is provided to show their advantages over existing methods, and a sufficient condition for the convergence of the mean weight vector is established. Theoretical expressions of the excess mean-squared error (EMSE), in both the steady-state and tracking cases, are derived based on the energy conservation approach. The effects of multiple access interference (MAI) and additive noise are considered. Simulation experiments are presented for both the stationary and non-stationary scenarios, illustrating that the proposed algorithms achieve superior performance compared with existing methods, and verifying the accuracy of the analyses.     

A complex adaptive notch filter using modified gradient algorithm

A modified gradient algorithm is developed for improving the convergence speed of a first-order complex adaptive IIR notch filter, which is used for estimating an unknown frequency of a complex sinusoidal signal embedded in white Gaussian noise. The new cost function using new error criterion is presented and analyzed theoretically. The proposed technique can significantly improve the convergence speed as compared with a complex notch filter using plain gradient algorithm. The computer simulations are conducted to demonstrate the validity of the proposed complex adaptive notch filter.     

Analysis of the frequency domain adaptive filter

The purpose of this note is to demonstrate significant analytical simplifications for studying the behavior of adaptive filtering in the frequency domain as opposed to studying the behavior of adaptive filtering in the time domain. A closed form expression, for the single complex weight in the frequency domain adaptive filter, is presented which allows significant statistical analysis to be performed. The mean-square error of the filter is evaluated as a function of the algorithm step size and the signal and noise powers.     

Bit-serial VLSI implementation of delayed LMS adaptive FIR filters

The delayed least-mean-square (DLMS) algorithm is useful for adaptive finite impulse response (FIR) filtering applications where high throughput rates are required. In the paper, a bit-serial bit-level systolic array based on new schemes for multiplication and inner-product computation is presented to implement DLMS adaptive N-tap FIR filters. The architecture is highly regular, modular, and thus well-suited to VLSI implementation. It has an efficiency of 100% and a throughput rate of one filter output per 2B cycles, where B is the word length of input data. In addition, the proposed array uses a small delay of [(4B+N/2+4)/2B] in the filter coefficient adaptation, where [x] is the smallest integer greater than or equal to x. This ensures that the DLMS algorithm can have good performance under proper selection of the step size. Based on a conservative design technique of static complementary metal oxide semiconductor (CMOS) logic, it is shown that the proposed system can be realized in a single chip for most practical applications     

Exact expectation analysis of the LMS adaptive filter

In almost all analyses of the least mean square (LMS) adaptive filter, it is assumed that the filter coefficients are statistically independent of the input data currently in filter memory, an assumption that is incorrect for shift-input data. We present a method for deriving a set of linear update equations that can be used to predict the exact statistical behavior of a finite-impulse-response (FIR) LMS adaptive filter operating upon finite-time correlated input data. Using our method, we can derive exact bounds upon the LMS step size to guarantee mean and mean-square convergence. Our equation-deriving procedure is recursive and algorithmic, and we describe a program written in the MAPLE symbolic-manipulation software package that automates the derivation for arbitrarily-long adaptive filters operating on input data with stationary statistics. Using our analysis, we present a search algorithm that determines the exact step size mean-square stability bound for a given filter length and input correlation statistics. Extensive computer simulations indicate that the exact analysis is more accurate than previous analyses in predicting adaptation behavior. Our results also indicate that the exact step size bound is much more stringent than the bound predicted by the independence assumption analysis for correlated input data     

基于凸组合的同步长最大均方权值偏差自适应滤波算法

针对NLMS和PNLMS滤波器对时变信道跟踪能力差的缺点,提出了一种同步长凸组合最大均方权值偏差(MSD,mean square deviation)算法。该算法将同步长的NLMS和PNLMS 2种不同类型的自适应滤波器进行凸组合,以最大均方权值偏差为准则,使新的滤波器能够在外界信道特性(稀疏、非稀疏和模糊态)时变的情况下,保持良好的随动性能,并在收敛的各个阶段均保持快速且稳定的均方特性。理论推导和仿真实验表明:该算法与NLMS、PNLMS和IPNLMS算法相比,在稀疏和非稀疏状态时能够保持四者中最快的收敛速度,并且在模糊状态时算法性能优于其余三者。另外,该算法仍保持较好的稳态均方性能。     

Steady-state behavior of the improved normalized subband adaptive filter algorithm and its improvement in under-modeling

In practice, adaptive filter could work in an under-modeling scenario, meaning that its length is less than that of the unknown system. In this realistic situation, therefore, the existing analysis for the improved normalized subband adaptive filter (INSAF) algorithm is not applicable. To this end, this paper analyzes the mean square steady-state performance of the INSAF for under-modeling. In addition, we propose a variable step size INSAF algorithm suitable for under-modeling scenario, to obtain fast convergence rate and low steady-state error. Simulation results have supported our theoretical analysis and proposed algorithm.     

Analysis of stochastic gradient identification ofWiener-Hammerstein systems for nonlinearities with Hermite polynomialexpansions

This paper investigates the statistical behavior of a sequential adaptive gradient search algorithm for identifying an unknown Wiener-Hammerstein (1958) system (WHS) with Gaussian inputs. The WHS nonlinearity is assumed to be expandable in a series of orthogonal Hermite polynomials. The sequential procedure uses (1) a gradient search for the unknown coefficients of the Hermite polynomials, (2) an LMS adaptive filter to partially identify the input and output linear filters of the WHS, and (3) the higher order terms in the Hermite expansion to identify each of the linear filters. The third step requires the iterative solution of a set of coupled nonlinear equations in the linear filter coefficients. An alternative scheme is presented if the two filters are known a priori to be exponentially shaped. The mean behavior of the various gradient recursions are analyzed using small step-size approximations (slow learning) and yield very good agreement with Monte Carlo simulations. Several examples demonstrate that the scheme provides good estimates of the WHS parameters for the cases studied     

Orthogonal weight perturbation algorithms in partially adaptive arrays

Single port adaptive algorithms have been proposed to circumvent the need to implement a separate receiver for each adaptive element in an adpative antenna. In this paper, transient and steady-state performance of a bounded single port perturbation algorithm is obtained and compared with that of the least mean square (LMS) algorithm. Results are developed both with and without a hardlimiter on one input to the correlator. Techniques for adaptively changing both the algorithm step size and perturbation amplitude are given. Simulation results are presented and compared with theoretical results.     

Stochastic analysis of the LMS algorithm with a saturationnonlinearity following the adaptive filter output

This paper presents a statistical analysis of the least mean square (LMS) algorithm with a zero-memory scaled error function nonlinearity following the adaptive filter output. This structure models saturation effects in active noise and active vibration control systems when the acoustic transducers are driven by large amplitude signals. The problem is first defined as a nonlinear signal estimation problem and the mean-square error (MSE) performance surface is studied. Analytical expressions are obtained for the optimum weight vector and the minimum achievable MSE as functions of the saturation. These results are useful for adaptive algorithm design and evaluation. The LMS algorithm behavior with saturation is analyzed for Gaussian inputs and slow adaptation. Deterministic nonlinear recursions are obtained for the time-varying mean weight and MSE behavior. Simplified results are derived for white inputs and small step sizes. Monte Carlo simulations display excellent agreement with the theoretical predictions, even for relatively large step sizes. The new analytical results accurately predict the effect of saturation on the LMS adaptive filter behavior     

自适应MPAM解调方法研究

提出了一种新型的基于自适应滤波器权系数解调多进制幅度调制(MPAM)信号的方法.文中以常用的最小均方误差自适应算法(LMS)为例,讨论了新型的MPAM自适应解调的过程及其性能.该解调算法不需要自适应滤波器完成收敛,从而降低了采样频率和处理速度.给出的理论性能与仿真结果表明,MPAM自适应解调的误码率仿真结果与理论值吻合非常好;而且该方法具有抗干扰性能强、输出响应快、便于数字信号处理(DSP)技术实现等特点,在相同的采样频率下其误码率优于相关解调的误码率.     

A parallel binary structured LMS algorithm for transversal adaptive filters

The adaptation process in digital filters requires extensive calculation. This computation makes adaptation a slow and time consuming process. Simple, but versatile, parallel algorithms for adaptive filters, suitable for VLSI implementation, are in demand. In this paper a regular and modular parallel algorithm for an adaptive filter is presented. This parallel structure is based on the Gradient Vector Estimation Algorithm, which minimizes the Mean Square Error. In the parallel method, the tap weights of the adaptive filter are updated everys steps, whereas in the recursive algorithms, the tap weights are updated at each step. Fors step update, bit strings of lengths are used to derive the terms with which the tap weights of the adaptive filter are calculated. The algorithm presented computes the tap weights at timen+s as a function of the tap weights at timen, the inputs from timen+1 ton+s−1, and the desired output from timen+1 ton+s−1. The algorithm also can be mapped to a VLSI architecture that is both regular and modular and allows either expansion of the order of the filter or the degree of parallelism obtainable. A comparison between the performance of the sequential LMS algorithm, Fast Exact LMS algorithm, and the parallel binary structured LMS algorithm is presented.     

Block-based fuzzy step size nlms algorithms for subband adaptive channel equalisation

Judicious selection of the step size parameter is crucial for adaptive algorithms to strike a good balance between convergence speed and misadjustment. The fuzzy step size (FSS) technique has been shown to improve the performance of the classical fixed step size and variable step size (VSS) normalised least mean square (NLMS) algorithms. The performance of the FSS technique in the context of subband adaptive equalisation is analysed and two novel block-based fuzzy step size (BFSS) strategies for the NLMS algorithm, namely fixed block fuzzy step size (FBFSS) and adaptive block fuzzy step size (ABFSS) are proposed. By exploiting the nature of gradient noise inherent in stochastic gradient algorithms, these strategies are shown to substantially reduce the computational complexity of the conventional FSS technique without sacrificing the convergence speed and steady-state performance. Instead of updating the step size at every iteration, the proposed techniques adjust the step size based on the instantaneous squared error once over a block length. Design methodology and guidelines that lead to good performance for the algorithms are given.     

Optimal variable step size for the LMS/Newton algorithm withapplication to subband adaptive filtering

An efficient approach for the computation of the optimum convergence factor for the LMS (least mean square)/Newton algorithm applied to a transversal FIR structure is proposed. The approach leads to a variable step size algorithm that results in a dramatic reduction in convergence time. The algorithm is evaluated in system identification applications where two alternative implementations of the adaptive filter are considered: the conventional transversal FIR realization and adaptive filtering in subbands     

Tracking properties of a gradient-based second-order adaptive IIRnotch filter with constrained poles and zeros

Gradient-type adaptive IIR notch filters have many attractive merits for various real-life applications since they require a small number of computations and yet demonstrate practical performance. However, it is generally quite difficult to assess their performance analytically. Their tracking properties, in particular, have not yet been investigated. In this paper, the tracking performance of a plain gradient (PG) algorithm is analyzed in detail for a second-order adaptive IIR notch filter with constrained poles and zeros, which takes a linear chirp signal as its input. First, two sets of difference equations for the frequency tracking error and mean square error (MSE) are established in the sense of convergence in the mean and convergence in the mean square, respectively. Closed-form expressions for the asymptotic tracking error and MSE are then derived from these difference equations. An optimum step-size parameter for the algorithm is also evaluated based on the minimization of the asymptotic tracking error or the tracking MSE. It is discovered that the asymptotic tracking error may be driven to zero for a positive chirp rate by selecting a proper step size, which is an interesting property for a real-valued adaptive filtering algorithm. Extensive simulations are performed to support the analytical findings