scholarly journals Improved Computing-Efficiency Least-Squares Algorithm with Application to All-Pass Filter Design

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Lo-Chyuan Su ◽  
Yue-Dar Jou ◽  
Fu-Kun Chen
Keyword(s):  
Least Squares ◽  
Filter Design ◽  
Linear Equations ◽  
Hankel Matrix ◽  
Cholesky Decomposition ◽  
Matrix Inversion ◽  
System Of Linear Equations ◽  
Computationally Efficient ◽  
Pass Filter ◽  
Simulation Results

All-pass filter design can be generally achieved by solving a system of linear equations. The associated matrices involved in the set of linear equations can be further formulated as a Toeplitz-plus-Hankel form such that a matrix inversion is avoided. Consequently, the optimal filter coefficients can be solved by using computationally efficient Levinson algorithms or Cholesky decomposition technique. In this paper, based on trigonometric identities and sampling the frequency band of interest uniformly, the authors proposed closed-form expressions to compute the elements of the Toeplitz-plus-Hankel matrix required in the least-squares design of IIR all-pass filters. Simulation results confirm that the proposed method achieves good performance as well as effectiveness.

scholarly journals A Representation of Nonhomogeneous Quadratic Forms with Application to the Least Squares Solution

2011 ◽  
Vol 2011 ◽  
pp. 1-5 ◽  
Author(s):  
Czesław Stępniak
Keyword(s):  
Least Squares ◽  
Quadratic Forms ◽  
Differential Calculus ◽  
Linear Models ◽  
Linear Equations ◽  
Algebraic Approach ◽  
System Of Linear Equations ◽  
Least Squares Solution ◽  
Inconsistent System ◽  
Usual Solution

The least squares problem appears, among others, in linear models, and it refers to inconsistent system of linear equations. A crucial question is how to reduce the least squares solution in such a system to the usual solution in a consistent one. Traditionally, this is reached by differential calculus. We present a purely algebraic approach to this problem based on some identities for nonhomogeneous quadratic forms.

scholarly journals A novel interpretation of least squares solution

1992 ◽  
Vol 15 (1) ◽  
pp. 41-46
Author(s):  
Jack-Kang Chan
Keyword(s):  
Least Squares ◽  
Source Localization ◽  
Convex Combination ◽  
Linear Equations ◽  
Singular Values ◽  
Statistical Interpretation ◽  
System Of Linear Equations ◽  
Overdetermined System ◽  
The Matrix ◽  
Cramer’S Rule

We show that the well-known least squares (LS) solution of an overdetermined system of linear equations is a convex combination of all the non-trivial solutions weighed by the squares of the corresponding denominator determinants of the Cramer's rule. This Least Squares Decomposition (LSD) gives an alternate statistical interpretation of least squares, as well as another geometric meaning. Furthermore, when the singular values of the matrix of the overdetermined system are not small, the LSD may be able to provide flexible solutions. As an illustration, we apply the LSD to interpret the LS-solution in the problem of source localization.

scholarly journals LINEAR ALGEBRA AND OPTIMIZATION BASED CONTROLLER DESIGN FOR TRAJECTORY TRACKING OF TYPICAL CHEMICAL PROCESS

2014 ◽  
Vol 44 (4) ◽  
pp. 313-318
Author(s):  
M. E. SERRANO ◽  
G. J. E. SCAGLIA ◽  
P. ABALLAY ◽  
O. A. ORTIZ ◽  
V. MUT
Keyword(s):  
Chemical Process ◽  
Controller Design ◽  
Linear Equations ◽  
Tracking Error ◽  
Optimal Operation ◽  
System Of Linear Equations ◽  
Plant Model ◽  
Control Actions ◽  
Simulation Results ◽  
Tracking Trajectory

This paper presents a new controller design to tracking trajectory of a typical chemical process. The plant model is represented by numerical methods and, from this approach; the control actions for an optimal operation of the system are obtained. Its main advantage is that the condition for the tracking error tends to zero and the calculation of control actions, are obtained solving a system of linear equations. The proofs of convergence to zero of the tracking error are presented. Simulation results show the good performance of the proposed control system.

An Efficient and Simple Algorithm for Matrix Inversion

2012 ◽  
pp. 21-28
Author(s):  
Ahmad Farooq ◽  
Khan Hamid
Keyword(s):  
Execution Time ◽  
Numerical Experiments ◽  
Linear Equations ◽  
Simple Algorithm ◽  
Matrix Inversion ◽  
Test Problems ◽  
Computer Implementation ◽  
System Of Linear Equations ◽  
Memory Utilization

In this paper, a new algorithm is proposed for finding inverse and determinant of a given matrix in one instance. The algorithm is straightforward in understanding and manual calculations. Computer implementation of the algorithm is extremely simple and is quite efficient in time and memory utilization. The algorithm is supported by an example. The number of multiplication/division performed by the algorithm is exactly; however, its efficiency lies in the simplicity of coding and minimal utilization of memory. Simple applicability and reduced execution time of the method is validated form the numerical experiments performed on test problems. The algorithm is applicable in the cases of pseudo inverses for non-square matrices and solution of system of linear equations with minor modification.

An Efficient and Simple Algorithm for Matrix Inversion

2010 ◽  
Vol 1 (1) ◽  
pp. 20-27 ◽  
Author(s):  
Ahmad Farooq ◽  
Khan Hamid
Keyword(s):  
Execution Time ◽  
Numerical Experiments ◽  
Linear Equations ◽  
Simple Algorithm ◽  
Matrix Inversion ◽  
Test Problems ◽  
Computer Implementation ◽  
System Of Linear Equations ◽  
Memory Utilization

In this paper, a new algorithm is proposed for finding inverse and determinant of a given matrix in one instance. The algorithm is straightforward in understanding and manual calculations. Computer implementation of the algorithm is extremely simple and is quite efficient in time and memory utilization. The algorithm is supported by an example. The number of multiplication/division performed by the algorithm is exactly; however, its efficiency lies in the simplicity of coding and minimal utilization of memory. Simple applicability and reduced execution time of the method is validated form the numerical experiments performed on test problems. The algorithm is applicable in the cases of pseudo inverses for non-square matrices and solution of system of linear equations with minor modification.

Active Low Pass Filter Design Based on Chaos Theory

2013 ◽  
Vol 380-384 ◽  
pp. 3822-3825
Author(s):  
Jing Chang ◽  
Yan Fang Feng ◽  
Wei Feng Li
Keyword(s):  
Chaos Theory ◽  
Filter Design ◽  
Simulation Software ◽  
Effective Solution ◽  
Matlab Simulation ◽  
Output Frequency ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass ◽  
Simulation Results

The circuit in this paper is an effective solution to interference and noise of input signal during data acquisition and the circuit designed is based on an active low pass circuit with output frequency from 0~100Hz. The low pass circuit consisted mainly of an OPAMP TL084. Chaos theory is applied in this active nonlinear filter.The designed circuit is analysed by Matlab simulation software and the simulation results show the method can effectively improve effects of the filtering circuit.

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