Hari Krishna

A coding theory approach to error control in redundant residue number systems. I. Theory and single error correction

A coding theory approach to error control in redundant residue number systems (RRNSs) is presented. The concepts of Hamming weight, minimum distance, weight distribution, and error detection and correction capabilities in redundant residue number systems are introduced. The necessary and sufficient conditions for the desired error control capability are derived from the minimum distance point of view. Closed-form expressions are derived for approximate weight distributions. Computationally efficient procedures are described for correcting single errors. A coding theory framework is developed for redundant residue number systems, and an efficient numerical procedure is derived for a single error correction. >

A coding theory approach to error control in redundant residue number systems. II. Multiple error detection and correction

For pt.I see ibid., vol.39, no.1, p.8-17 (1992). The coding theory approach to error control in redundant residue number systems (RRNSs) is extended by deriving computationally efficient algorithms for correcting multiple errors, single-burst-error, and detecting multiple errors. These algorithms reduce the computational complexity of the previously known algorithms by at least an order of magnitude. >

A New Error Control Scheme for Hybrid ARQ Systems

This paper introduces the concept of a generalized hybrid ARQ (GH-ARQ) scheme for adaptive error control in digital communication systems. This technique utilizes the redundant information available upon successive retransmissions in an efficient manner so as to provide high throughput during poor channel conditions. A new class of linear codes is proposed for the GH-ARQ system application. The main feature of this class of codes is that the encoder/decoder configuration does not change as the length of the code is varied. As a result, the receiver uses the same decoder for decoding the received information after every retransmission while the error correcting capability of the code increases, thereby leading to an improved performance and minimum complexity for the overall system implementation.

On theory and fast algorithms for error correction in residue number system product codes

The authors develop a coding theory approach to error control in residue number system product codes. Based on this coding theory framework, computationally efficient algorithms are derived for correcting single errors, double errors, and multiple errors, and simultaneously detecting multiple errors and additive overflow. These algorithms have lower computational complexity than previously known algorithms by at least an order of magnitude. In addition, it is noted that all the literature published thus far deals almost exclusively with single error correction. >

The Levinson recurrence and fast algorithms for solving Toeplitz systems of linear equations

This work brings together classical polynomial theory as it relates to the Levinson recurrence for a Hermitian Toeplitz operator and matrix theory as it relates to the class of Hermitian centro-Hermitian matrices. A new computationally efficient alternative is presented to the Levinson recurrence on either the Hermitian or skew-Hermitian polynomial spaces. This approach also leads to an entirely new algorithm for solving systems of linear equations when the coefficient matrix is Hermitian Toeplitz or real symmetric Toeplitz. Analysis of the computational complexity of the algorithms presented is also performed, and it is shown that these algorithms lead to significant improvements in the computational complexity as compared to the previously best-known recursive algorithms. They also provide further insight into the mathematical properties of the structurally rich Toeplitz matrices.

Digital Signal Processing Algorithms: Number Theory, Convolutions, Fast Fourier Transforms, and Applications

Introduction Outline The Organization PART I: Computational Number Theory Computational Number Theory Groups, Rings, and Fields Elements of Number Theory Integer Rings and Fields Chinese Remainder Theorem for Integers Number Theory for Finite Integer Rings Polynomial Algebra Algebra of Polynomials over a Field Roots of a Polynomial Polynomial Fields and Rings The Chinese Remainder Theorem for Polynomials CRT-P in Matrix Form Lagrange Interpolation Polynomial Algebra over GF(p) Order of an Element Theoretical Aspects of Discrete Fourier Transform and Convolution The Discrete Fourier Transform Basic Formulation of Convolution Bounds on the Multiplicative Complexity Basic Formulation of Convolution Algorithms Matrix Exchange Property Cyclotomic Polynomial Factorization and Associated Fields Cyclotomic Polynomial Factorization over Complex and Real Numbers Cyclotomic Polynomial Factorization over Rational Numbers Cyclotomic Fields and Cyclotomic Polynomial Factorizations Extension Fields of Cyclotomic Fields and Cyclotomic Polynomial Factorization A Preview of Applications to Digital Signal Processing Cyclotomic Polynomial Factorization in Finite Fields Cyclotomic Polynomial Factorization Factorization of (un - 1) over GF (p) Primitive Polynomials over GF (p) Complex Finite Fields and Cyclotomic Polynomial Factorization Finite Integer Rings: Polynomial Algebra and Cyclotomic Factorization Polynomial Algebra over a Ring Lagrange Interpolation Number Theoretic Transforms Monic Polynomial Factorization Extension of CRT-P over Finite Integer Rings Polynomial Algebra and CRT-PR: The Complex Case Number Theoretic Transforms: The Complex Case Pseudo Number Theoretic Transforms Polynomial Algebra and Direct Sum Properties in Integer Polynomial Rings PART II: Convolution Algorithms Thoughts on Part II Fast Algorithms for Acyclic Convolution CRT-P Based Fast Algorithms for One-Dimensional Acyclic Convolution Casting the Algorithm in Bilinear Formulation Multidimensional Approaches to One-Dimensional Acyclic Convolution Multidimensional Acyclic Convolution Algorithms Nesting and Split Nesting Algorithms for Multidimensional Convolution Acyclic Convolution Algorithms over Finite Fields and Rings Fast One-Dimensional Cyclic Convolution Algorithms Bilinear Forms and Cyclic Convolution Cyclotomic Polynomials and Related Algorithms over Re and C Cyclotomic Polynomials and Related Algorithms over Z Other Considerations Complex Cyclotomic Polynomials and Related Algorithms over CZ The Agarwal-Cooley Algorithm Cyclic Convolution Algorithms over Finite Fields and Rings Two- and Higher Dimensional Cyclic Convolution Algorithms Polynomial Formulation and an Algorithm Improvements and Related Algorithms Discrete Fourier Transform Based Algorithms Algorithms Based on Extension Fields Algorithms for Multidimensional Cyclic Convolution Algorithms for Two-Dimensional Cyclic Convolution in Finite Integer Rings Validity of Fast Algorithms over Different Number Systems Introduction Mathematical Preliminaries Chinese Remainder Theorem over Finite Integer Rings Interrelationships among Algorithms over Different Number Systems Analysis of Two-Dimensional Cyclic Convolution Algorithms Fault Tolerance for Integer Sequences A Framework for Fault Tolerance Mathematical Structure of C over Z(M) Coding Techniques over Z(q) Examples and SFC-DFD Codes PART III: Fast Fourier Transform (FFT) Algorithms Thoughts on Part III Fast Fourier Transform: One-Dimensional Data Sequences The DFT: Definitions and Properties Raders FFT Algorithm, n=p, p an Odd Prime Raders FFT Algorithm, n=pc, p an Odd Prime Cooley-Tukey FFT Algorithm, n=a . b FFT Algorithms for n a Power of 2 The Prime Factor FFT n=a . b, (a,b) =1 The Winograd FFT Algorithm Fast Fourier Transform: Multidimensional Data Sequences The Multidimensional DFT: Definition and Properties FFT for n=p, p an Odd Prime Multidimensional FFT Algorithms for n a Power of 2 Matrix Formulation of Multidimensional DFT and Related Algorithms Polynomial Version of Raders Algorithm Polynomial Transform Based FFT Algorithms PART IV: Recent Results on Algorithms in Finite Integer Rings Thoughts on Part IV Paper One: A Number Theoretic Approach to Fast Algorithms for Two-Dimensional Digital Signal Processing in Finite Integer Rings Paper Two: On Fast Algorithms for One-Dimensional Digital Signal Processing in Finite Integer and Complex Integer Rings Paper Three: Cyclotomic Polynomial Factorization in Finite Integer Rings with Applications to Digital Signal Processing Paper Four: Error Control Techniques for Data Sequences Defined in Finite Integer Rings A. Small Length Acyclic Convolution Algorithms B. Classification of Cyclotomic Polynomials Index

Rings, fields, the Chinese remainder theorem and an extension-Part I: theory

The much celebrated Chinese Remainder Theorem has been widely employed in designing fast computationally efficient algorithms in the field of digital signal processing. It has two versions. One is over a ring of integers and the second is over a ring of polynomials with Coefficients defined over a field. In this research work, we extend the Chinese Remainder Theorem to the case of a ring of polynomials with coefficients defined over a finite ring of integers. The entire work is closely related to the already established results on finite fields. This extension is expected to serve as a keystone in the future design of number-theoretic algorithms for performing some of the most computationally intensive tasks. This approach is superior to the number-theoretic-transforms in the sense that the limitations on both the word length and the sequence length are completely removed. In fact, the number-theoretic-transforms may be considered as a very special case of our general approach. Furthermore, the computations required in this work. Which inherits all the merits of the Chinese Remainder Theorem, can be performed in parallel. >

Split Levinson algorithm is weakly stable

The authors explore the numerical stability properties of the split Levinson algorithm for computing the predictor polynomial associated with a positive-definite real symmetric Toeplitz matrix. Various bounds on the residual vector are derived for the fixed-point and floating-point implementation of the algorithm. These bounds are similar in form to the bounds derived by G. Cybenko (1980) for the Levinson algorithm and are obtained by converting a three-term recurrence for the error vector to an equivalent two-term recurrence. The split Levinson algorithm is shown to be weakly stable.<<ETX>>

New split Levinson, Schur, and lattice algorithms for digital signal processing

The mathematical structure associated with the split algorithms for computing the reflection coefficients for a given real symmetric positive-definite Toeplitz matrix is analyzed. A novel form of three-term recurrence relation is derived and computationally efficient alternatives to the Levinson-Durbin, Schur, and lattice algorithms are obtained. The computational complexity of the proposed algorithms is the same as those of the split algorithms described in recent literature. These algorithms provide further insight into the mathematical properties of the structurally rich Toeplitz matrices.<<ETX>>

Computational aspects of the bilinear transformation based algorithm for S-plane to Z-plane mapping

The author analyzes the computational complexity of an algorithm by F.D. Groutage et al. (ibid., vol.AC-32, no.7, p.635-7, July 1987) for performing the transformation of a continuous transfer function to a discrete equivalent by a bilinear transformation. Groutage et al. defend their method by noting that their technique is not limited to the bilinear transformation. Rather, it can be extended to any higher-order integration rule (Simpson, Runge-Kutta, etc.), or to any higher-order expansion of the ln function. In general, using the method, s can be any appropriate mapping function s=f (z). >