Plain Gradient Algorithm Research Articles

Performance Comparison of a Second-order adaptive IIR Notch Filter based on Plain Gradient Algorithm

This paper presents performance analysis of a second-order adaptive IIR notch filter with constrained poles and zeros in direct form II structure that is adapted by algorithms of indirect and direct frequency estimations based on plain gradient. In this work, the steady-state bias properties and stability bounds of both algorithms are derived in closed form. The simulated processes are performed in order to verify the analytical results. Furthermore, these results of the proposed filter are also compared with those results of the direct form I adaptive IIR notch filter under the same parameters. It has been revealed that the direct form II structure has given the steady-state performance over the direct form I structure.

Loop analysis of adaptive notch filters

Adaptive notch filters have been the focus of intense research for more than three decades. Low computational requirements and good performance make them attractive for tracking frequency modulated signals. Despite the extensive literature on adaptive notch filters, algorithm extensions and new models for describing the dynamics of the notch frequency continue to be proposed. In this study, the equivalence between adaptive notch filters using a plain gradient (PG) algorithm and frequency lock loops (FLLs) with exponential filtering is established. FLL theory is then used to analyse the noise performance and signal tracking capabilities of adaptive notch filters. A linear model describing the dynamics of the filter adaptation process is derived and the concepts of loop and Doppler bandwidths are introduced. Criteria based on the loop and Doppler bandwidths are suggested to set the PG adaptation step. Finally, algorithm extensions based on the FLL theory are proposed. Theoretical results are supported by Monte Carlo simulations which show the validity of the analysis performed.

Two‐stage optimisation algorithm for adaptive IIR notch filter

The performances of the linear prediction (LP) algorithm and the indirect plain gradient (IPG) algorithm for the adaptive infinite-impulse-response (IIR) notch filter are theoretically studied, and the two phenomena in these two algorithms are revealed. First, the LP algorithm has a faster convergence speed than the IPG algorithm when the initial adjusted value is far away from the optimal solution. Secondly, the IPG algorithm has a much smaller estimation bias and mean squared error than the LP algorithm in a steady state. On the basis of these facts, a two-stage optimisation algorithm is proposed to make full use of the advantages of both the LP algorithm and the IPG algorithm. Simulation results show that the proposed algorithm can improve the convergence rate while keeping the same steady state as the IPG algorithm.

Steady-state analysis of a complex adaptive notch filter using modified gradient algorithm

Based on power spectral density (PSD) analytical technique, mean square error (MSE) (or variance) of the frequency estimate of a first-order complex adaptive IIR notch filter (ANF) using modified complex plain gradient (MCPG) algorithm is investigated in this paper. The steady-state expression for MSE is derived in closed form. A quantitative analysis for the estimation MSE has been carried out. It has been revealed that the MSE of frequency estimate is independent of an input frequency of a complex sinusoid. In addition, computer simulations are treated to corroborate the theoretical analysis and the relationships between MSE and system parameters are shown.

Mean square error analysis of unbiased modified plain gradient algorithm for second-order adaptive IIR notch filter

In this paper, theoretical analysis for deriving the estimation of mean square error (MSE) at steady-state of an adaptive IIR notch filter using unbiased modified plain gradient (UMPG) algorithm is presented in closed form. Moreover, the stability bound of the algorithm is also derived. To confirm the analytical results, the computer simulations are conducted to corroborate the effectiveness of the UMPG algorithm. Furthermore, the performances of the algorithm are also compared with the modified plain gradient (MPG) algorithm, unbiased plain gradient (UPG) algorithm and simplified lattice algorithm (SLA).

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.

Unbiased plain gradient algorithm for a second-order adaptive IIR notch filter with constrained poles and zeros

This article proposes an unbiased plain gradient algorithm for a second-order adaptive IIR notch filter with constrained poles and zeros. The proposed algorithm employs removing a dominant parameter that produces inherent bias. By using this technique, the performances are improved with slight expense in computational complexity. In this paper, theoretical analysis for deriving the estimations of bias and mean square error (MSE) at steady state are presented in closed form. Moreover, the stability bound of the algorithm is also derived. To confirm the analytical results, the computer simulations are provided to corroborate the effectiveness of the proposed algorithm. Furthermore, the performances of the algorithm are also compared with the plain gradient (PG) and modified plain gradient (MPG) algorithms.

Adaptive IIR Filters for Single Interference Suppression in a BPSK DS CDMA System In Rayleigh Fading Channel

In this paper, effect of a single narrow-band interference (NBI) on bit error rate (BER) performance for a binary phase shift keying (BPSK) synchronous direct-sequence code-division multiple access (DS CDMA) communication system operating in a frequency nonselective Rayleigh fading channel is analyzed. Second-order adaptive infinite impulseresponse (IIR) notch filters with plain gradient algorithm (GA) for suppression of NBI in the DS CDMA system are proposed. A general closed-form BER expression for the DS CDMA system with NBI suppression second order adaptive IIR notch filters is derived based on the standard Gaussian approximation (SGA) method. BER expressions are then derived for the allpass filter-based adaptive IIR notch filter and adaptive IIR notch filter with constrained poles and zeros, the two structures that are commonly found in literature. It is observed that both adaptive IIR notch filter structures exhibit comparable BER performance. Extensive computer simulation results are presented to verify the accuracy and limitations of the analysis.

Statistical performance of the memoryless nonlinear gradient algorithm for the constrained adaptive IIR notch filter

Gradient-type algorithms for the adaptive infinite-impulse response (IIR) notch filters are very attractive in terms of both performance and computational requirements for various real-life applications. This paper presents, in detail, a statistical analysis of the memoryless nonlinear gradient (MNG) algorithm applied to the well-known second-order adaptive IIR notch filter with constrained poles and zeros. This analysis is based on a proper use of Taylor series expansion and nonlinearization of output signals of the notch and gradient filters. Two difference equations are derived first for the convergence in the mean and mean square senses, respectively. Two closed-form expressions, one for the steady-state estimation bias and the other for the mean-square error, are then derived based on the difference equations, with the former valid for both fast and slow adaptations and the latter valid for slow adaptation only. A closed-form coarse stability bound for the step size parameter of the algorithm is also derived. Extensive simulations are performed to reveal the validity and limitations of the analytical findings. Comparisons between the MNG and the conventional plain gradient algorithm are also made.

Plain gradient based direct frequency estimation using second-order constrained adaptive IIR notch filter

An investigation of the problem of direct frequency estimation with a constrained second-order adaptive IIR notch filter using the plain gradient algorithm is reported. The closed form expressions for the bias and mean square error of the proposed algorithm are derived. Simulations show that the proposed algorithm yields a faster convergence speed than the conventional indirect frequency estimation algorithms.

Performance analysis of the sign algorithm for a constrained adaptive IIR notch filter

Adaptive infinite impulse response (IIR) notch filters are very attractive in terms of their reasonable performances and low computational requirements. Generally, it is very difficult to assess their performances analytically due to their IIR nature. This paper analyzes in detail the steady-state performance of the sign algorithm (SA) for a well-known adaptive IIR notch filter with constrained poles and zeros. Slow adaptation and Gaussianity of the notch filter output are assumed for the sake of analysis. Two difference equations are first established for the convergences in the mean and mean square in the vicinity of the steady state of the algorithm. Steady-state estimation error or bias and mean square error (MSE) of the SA are then derived in closed forms. A coarse stability bound is also derived for the algorithm. Theory-based comparison between the algorithm and the plain gradient (PG) algorithm is done in some detail. Extensive simulations are conducted to demonstrate the validity of the analytical results for both slow and relatively fast adaptations.

Tracking properties of a gradient-based second-order adaptive IIR notch 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.

Steady-state analysis of a plain gradient algorithm for a second-order adaptive IIR notch filter with constrained poles and zeros

Gradient-type algorithms for adaptive infinite-impulse response (IIR) notch filters are very attractive in terms of performance and computation cost. However, it is generally quite difficult to assess their performances analytically. There are several trials to analyze adaptive algorithms, such as the sign and the plain gradient algorithms for some types of adaptive IIR notch filters, but the analysis techniques used cannot be directly applied to different types of adaptive IIR notch filters. This brief presents closed form expressions for steady-state estimation bias and mean square error (MSE) of a well known plain gradient (LMS-like) second-order adaptive IIR notch filter with constrained poles and zeros. First, theoretical expressions for output signals of the notch filter and its corresponding gradient filter at their steady states are developed based on the Taylor series expansions of transfer functions of these two filters in the vicinity of the sinusoidal signal frequency difference equations for convergences in the mean and mean square are then established by using these output signals, from which the steady-state bias and MSE of the algorithm are derived. Stability bound of the algorithm is also investigated based on the difference equations. Extensive simulations are provided to support the analytical findings.

A new memoryless nonlinear gradient algorithm for a second-order adaptive IIR notch filter and its performance analysis

Adaptive infinite impulse-response (IIR) notch filters have been widely studied for many years. However, further efforts are still to be made to pursue new algorithms which work better than the conventional plain gradient algorithm but have little increase in complexity. In this paper, we employ the gradient linearization, Taylor series expansion, and calculus of variations to derive a novel memoryless nonlinear gradient algorithm for a second-order adaptive IIR notch filter, which improves the estimation performance considerably. Approximate closed-form expressions for the stability bounds on the step size parameter and the steady-state coefficient variance are also derived. Extensive simulations indicate the significant improvement that may be achieved using the new algorithm, and verify the validity of the analytical results.