67,570 research outputs found
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
Low-complexity RLS algorithms using dichotomous coordinate descent iterations
In this paper, we derive low-complexity recursive least squares (RLS) adaptive filtering algorithms. We express the RLS problem in terms of auxiliary normal equations with respect to increments of the filter weights and apply this approach to the exponentially weighted and sliding window cases to derive new RLS techniques. For solving the auxiliary equations, line search methods are used. We first consider conjugate gradient iterations with a complexity of O(N-2) operations per sample; N being the number of the filter weights. To reduce the complexity and make the algorithms more suitable for finite precision implementation, we propose a new dichotomous coordinate descent (DCD) algorithm and apply it to the auxiliary equations. This results in a transversal RLS adaptive filter with complexity as low as 3N multiplications per sample, which is only slightly higher than the complexity of the least mean squares (LMS) algorithm (2N multiplications). Simulations are used to compare the performance of the proposed algorithms against the classical RLS and known advanced adaptive algorithms. Fixed-point FPGA implementation of the proposed DCD-based RLS algorithm is also discussed and results of such implementation are presented
Variational Integrators for the Gravitational N-Body Problem
This paper describes a fourth-order integration algorithm for the
gravitational N-body problem based on discrete Lagrangian mechanics. When used
with shared timesteps, the algorithm is momentum conserving and symplectic. We
generalize the algorithm to handle individual time steps; this introduces
fifth-order errors in angular momentum conservation and symplecticity. We show
that using adaptive block power of two timesteps does not increase the error in
symplecticity. In contrast to other high-order, symplectic, individual
timestep, momentum-preserving algorithms, the algorithm takes only forward
timesteps. We compare a code integrating an N-body system using the algorithm
with a direct-summation force calculation to standard stellar cluster
simulation codes. We find that our algorithm has about 1.5 orders of magnitude
better symplecticity and momentum conservation errors than standard algorithms
for equivalent numbers of force evaluations and equivalent energy conservation
errors.Comment: 31 pages, 8 figures. v2: Revised individual-timestepping description,
expanded comparison with other methods, corrected error in predictor
equation. ApJ, in pres
Multigrid elliptic equation solver with adaptive mesh refinement
In this paper we describe in detail the computational algorithm used by our
parallel multigrid elliptic equation solver with adaptive mesh refinement. Our
code uses truncation error estimates to adaptively refine the grid as part of
the solution process. The presentation includes a discussion of the orders of
accuracy that we use for prolongation and restriction operators to ensure
second order accurate results and to minimize computational work. Code tests
are presented that confirm the overall second order accuracy and demonstrate
the savings in computational resources provided by adaptive mesh refinement.Comment: 12 pages, 9 figures, Modified in response to reviewer suggestions,
added figure, added references. Accepted for publication in J. Comp. Phy
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