Implementation of the Trigonometric LMS Algorithm using Original Cordic Rotation

The LMS algorithm is one of the most successful adaptive filtering algorithms. It uses the instantaneous value of the square of the error signal as an estimate of the mean-square error (MSE). The LMS algorithm changes (adapts) the filter tap weights so that the error signal is minimized in the mean square sense. In Trigonometric LMS (TLMS) and Hyperbolic LMS (HLMS), two new versions of LMS algorithms, same formulations are performed as in the LMS algorithm with the exception that filter tap weights are now expressed using trigonometric and hyperbolic formulations, in cases for TLMS and HLMS respectively. Hence appears the CORDIC algorithm as it can efficiently perform trigonometric, hyperbolic, linear and logarithmic functions. While hardware-efficient algorithms often exist, the dominance of the software systems has kept those algorithms out of the spotlight. Among these hardware- efficient algorithms, CORDIC is an iterative solution for trigonometric and other transcendental functions. Former researches worked on CORDIC algorithm to observe the convergence behavior of Trigonometric LMS (TLMS) algorithm and obtained a satisfactory result in the context of convergence performance of TLMS algorithm. But revious researches directly used the CORDIC block output in their simulation ignoring the internal step-by-step rotations of the CORDIC processor. This gives rise to a need for verification of the convergence performance of the TLMS algorithm to investigate if it actually performs satisfactorily if implemented with step-by-step CORDIC rotation. This research work has done this job. It focuses on the internal operations of the CORDIC hardware, implements the Trigonometric LMS (TLMS) and Hyperbolic LMS (HLMS) algorithms using actual CORDIC rotations. The obtained simulation results are highly satisfactory and also it shows that convergence behavior of HLMS is much better than TLMS.Comment: 12 pages, 5 figures, 1 table. Published in IJCNC; http://airccse.org/journal/cnc/0710ijcnc08.pdf, http://airccse.org/journal/ijc2010.htm

Modified filtered-x hierarchical lms algorithm with sequential partial updates for active noise control

In the field of active noise control (ANC), a popular method is the modified filtered-x LMS algorithm. However, it has two drawbacks: Its computational complexity higher than that of the conventional FxLMS, and its convergence rate that could still be improved. Therefore, we propose an adaptive strategy which aims at speeding up the convergence rate of an ANC system dealing with periodic disturbances. This algorithm consists in combining the organization of the filter weights in a hierarchy of subfilters of shorter length and their sequential partial updates (PU). Our contribution is threefold: (1) we provide the theoretical basis of the existence of a frequency-depend-ent parameter, called gain in step-size. (2) The theoretical upper bound of the step-size is compared with the limit obtained from simulations. (3) Additional experiments show that this strategy results in a fast algorithm with a computational complexity close to that of the conventional FxLMS

Inference on Treatment Effects After Selection Amongst High-Dimensional Controls

We propose robust methods for inference on the effect of a treatment variable on a scalar outcome in the presence of very many controls. Our setting is a partially linear model with possibly non-Gaussian and heteroscedastic disturbances. Our analysis allows the number of controls to be much larger than the sample size. To make informative inference feasible, we require the model to be approximately sparse; that is, we require that the effect of confounding factors can be controlled for up to a small approximation error by conditioning on a relatively small number of controls whose identities are unknown. The latter condition makes it possible to estimate the treatment effect by selecting approximately the right set of controls. We develop a novel estimation and uniformly valid inference method for the treatment effect in this setting, called the "post-double-selection" method. Our results apply to Lasso-type methods used for covariate selection as well as to any other model selection method that is able to find a sparse model with good approximation properties. The main attractive feature of our method is that it allows for imperfect selection of the controls and provides confidence intervals that are valid uniformly across a large class of models. In contrast, standard post-model selection estimators fail to provide uniform inference even in simple cases with a small, fixed number of controls. Thus our method resolves the problem of uniform inference after model selection for a large, interesting class of models. We illustrate the use of the developed methods with numerical simulations and an application to the effect of abortion on crime rates

Performance of RLMS algorithm in adaptive array beam forming

This paper examines the performance of an adaptive linear array employing the new RLMS algorithm, which consists of a recursive least square (RLS) section followed by a least mean square (LMS) section. The performance measures used are output and input signal-to-interference plus noise ratios (SINR), side lobe level (SLL), and SINRo as a function of the direction of arrival of the interfering signal. Computer simulation results show that the performance of RLMS is superior to either the RLS or LMS based on these measures, particularly when operating with low input SINR

Covariant Lyapunov vectors of a quasi-geostrophic baroclinic model: analysis of instabilities and feedbacks

The classical approach for studying atmospheric variability is based on defining a background state and studying the linear stability of the small fluctuations around such a state. Weakly non-linear theories can be constructed using higher order expansions terms. While these methods have undoubtedly great value for elucidating the relevant physical processes, they are unable to follow the dynamics of a turbulent atmosphere. We provide a first example of extension of the classical stability analysis to a non-linearly evolving quasi-geostrophic flow. The so-called covariant Lyapunov vectors (CLVs) provide a covariant basis describing the directions of exponential expansion and decay of perturbations to the non-linear trajectory of the flow. We use such a formalism to re-examine the basic barotropic and baroclinic processes of the atmosphere with a quasi-geostrophic beta-plane two-layer model in a periodic channel driven by a forced meridional temperature gradient ΔT. We explore three settings of ΔT, representative of relatively weak turbulence, well-developed turbulence, and intermediate conditions. We construct the Lorenz energy cycle for each CLV describing the energy exchanges with the background state. A positive baroclinic conversion rate is a necessary but not sufficient condition of instability. Barotropic instability is present only for few very unstable CLVs for large values of ΔT. Slowly growing and decaying hydrodynamic Lyapunov modes closely mirror the properties of the background flow. Following classical necessary conditions for barotropic/baroclinic instability, we find a clear relationship between the properties of the eddy fluxes of a CLV and its instability. CLVs with positive baroclinic conversion seem to form a set of modes for constructing a reduced model of the atmosphere dynamics

Adaptive array beam forming using a combined RLS-LMS algorithm

A new adaptive algorithm, called RLMS, which combines the use of recursive least square (RLS) and least mean square (LMS), is proposed for array beam forming. The convergence of the RLMS algorithm is analyzed, in terms of mean square error, in the presence of additive white Gaussian noise. Computer simulation results show that the convergence performance of RLMS is superior to either RLS or LMS operating on its own. Furthermore, the convergence of RLMS is quite insensitive to changes in either signal-to-noise ratio, or the initial value of the input correlation matrix for the RLS section, or the step size adopted for the LMS section

The universal Teukolsky equations and black hole perturbations in higher-derivative gravity

We reduce the study of perturbations of rotating black holes in higher-derivative extensions of general relativity to a system of decoupled radial equations that stem from a set of universal Teukolsky equations. We detail a complete computational strategy to obtain these decoupled equations in general higher-derivative theories. We apply this to six-derivative gravity to compute the shifts in the quasinormal mode frequencies with respect to those of Kerr black holes in general relativity. At linear order in the angular momentum we reproduce earlier results obtained with a metric perturbation approach. In contrast with this earlier work, however, the method given here applies also to post-merger black holes with significant spin, which are of particular observational interest.Comment: 50 pages, 5 figures. v2: we fixed an error in our code and this led to improved results for the QNMs reported in section 6. The rest of the sections remain unchanged up to small adjustements. Conclusions unchanged. Version sent to the journal. We provide an ancillary Mathematica notebook with the modified Teukolsky radial equations for the (l,m)=(2,3) and (3,3) modes in six-derivative gravit

A New LLMS Algorithm for Antenna Array Beamforming

A new adaptive algorithm, called LLMS, which employs an array image factor, AI, sandwiched in between two Least Mean Square (LMS) sections, is proposed for different applications of array beamforming. The convergence of LLMS algorithm is analyzed, in terms of mean square error, in the presence of Additive White Gaussian Noise (AWGN) for two different modes of operation; namely with either an external reference or self-referencing. Unlike earlier LMS based schemes, which make use of step size adaptation to enhance their performance, the proposed algorithm derives its overall error signal by feeding back the error signal from the second LMS stage to combine with that of the first LMS section.This results in LLMS being less sensitive to variations in input signal-to-noise ratio as well as the step sizes used. Computer simulation results show that the proposed LLMS algorithm is superior in convergence performance over the conventional LMS algorithm as well some of the more recent LMS based algorithms, such as constrained-stability LMS (CSLMS), and Modified Robust Variable Step Size LMS (MRVSS) algorithms. Also, the operation of LLMS remains stable even when its reference signal is corrupted by AWGN. It is also shown that LLMS performs well in the presence of Rayleigh fading