Trieu-Kien Truong

The use of finite fields to compute convolutions

A transform is defined in the Galois field of q^2 elements GF(q^2) , a finite field analogous to the field of complex numbers, when q is a prime such that (--1) is not a quadratic residue. It is shown that the action of this transform over GF(q^2) is equivalent to the discrete Fourier transform of a sequence of complex integers of finite dynamic range. If q is a Mersenne prime, one can utilize the fast Fourier transform (FFT) algorithm to yield a fast convolution without the usual roundoff problem of complex numbers.

A comparison of VLSI architecture of finite field multipliers using dual, normal, or standard bases

Three different finite-field multipliers are presented: (1) a dual-basis multiplier due to E.R. Berlekamp (1982); the Massey-Omura normal basis multiplier; and (3) the Scott-Tavares-Peppard standard basis multiplier. These algorithms are chosen because each has its own distinct features that apply most suitably in particular areas. They are implemented on silicon chips with NMOS technology so that the multiplier most desirable for VLSI implementation can readily be ascertained. >

Audio classification and categorization based on wavelets and support vector Machine

In this paper, an improved audio classification and categorization technique is presented. This technique makes use of wavelets and support vector machines (SVMs) to accurately classify and categorize audio data. When a query audio is given, wavelets are first applied to extract acoustical features such as subband power and pitch information. Then, the proposed method uses a bottom-up SVM over these acoustical features and additional parameters, such as frequency cepstral coefficients, to accomplish audio classification and categorization. A public audio database (Muscle Fish), which consists of 410 sounds in 16 classes, is used to evaluate the performances of the proposed method against other similar schemes. Experimental results show that the classification errors are reduced from 16 (8.1%) to six (3.0%), and the categorization accuracy of a given audio sound can achieve 100% in the Top 2 matches.

Spectral representation of fractional Brownian motion in n dimensions and its properties

Fractional Brownian motion (fBm) provides a useful model for processes with strong long-term dependence, such as 1/f/sup /spl beta// spectral behavior. However, fBms are nonstationary processes so that the interpretation of such a spectrum is still a matter of speculation. To facilitate the study of this problem, another model is provided for the construction of fBm from a white-noise-like process by means of a stochastic or Ito integral in frequency of a stationary uncorrelated random process. Also a generalized power spectrum of the nonstationary fBm process is defined. This new approach to fBm can be used to compute all of the correlations, power spectra, and other properties of fBm. In this paper, a number of these fBm properties are developed from this model such as the T/sup H/ law of scaling, the power law of fractional order, the correlation of two arbitrary fBms, and the evaluation of the fractal dimension under various transformations. This new treatment of fBm using a spectral representation is extended also, for the first time, to two or more topological dimensions in order to analyze the features of isotropic n-dimensional fBm. >

A VLSI design for a trace-back Viterbi decoder

A systolic Viterbi decoder for convolutional codes is developed which uses the trace-back method to reduce the amount of data needed to be stored in registers. It is shown that this new algorithm requires a smaller chip size and achieves a faster decoding time than other existing methods. >

Decoding the (47,24,11) quadratic residue code

The techniques needed to decode the (47,24,11) quadratic residue (QR) code differ from the schemes developed for cyclic codes. By finding certain nonlinear relations between the known and unknown syndromes for this special code, two methods are developed to decode up to the true minimum distance of the (47,24,11) QR code. These algorithms can be utilized to decode effectively the 1/2 -rate (48,24,12) QR code for correcting five errors and detecting six errors.

The algebraic decoding of the (41, 21, 9) quadratic residue code

A new algebraic approach for decoding the quadratic residue (QR) codes, in particular the (41, 21, 9) QR code, is presented. The key ideas behind this decoding technique are a systematic application of the Sylvester resultant method to the Newton identities associated with the syndromes to find the error-locator polynomial, and next a method for determining error locations by solving certain quadratic, cubic, and quartic equations over GF(2/sup m/) in a new way which uses Zechs logarithms for the arithmetic. The logarithms developed for Zechs logarithms save a substantial amount of computer memory by storing only a table of Zechs logarithms. These algorithms are suitable for implementation in a programmable microprocessor or special-purpose VLSI chip. It is expected that the algebraic methods developed can apply generally to other codes such as the BCH and Reed-Solomon codes. >

Algebraic decoding of (71, 36, 11), (79, 40, 15), and (97, 49, 15) quadratic residue codes

Recently, a new algebraic decoding algorithm for quadratic residue (QR) codes was proposed by Truong et al. Using that decoding scheme, we now develop three decoders for the QR codes with parameters (71, 36, 11), (79, 40, 15), and (97, 49, 15), which have not been decoded before. To confirm our results, an exhaustive computer simulation has been executed successfully.

Algebraic decoding of the (32, 16, 8) quadratic residue code

An algebraic decoding algorithm for the 1/2-rate (32, 16, 8) quadratic residue (QR) code is found. The key idea of this algorithm is to find the error locator polynomial by a systematic use of the Newton identities associated with the code syndromes. The techniques developed extend the algebraic decoding algorithm found recently for the (32, 16, 8) QR code. It is expected that the algebraic approach developed here and by M. Elia (1987) applies also to longer QR codes and other BCH-type codes that are not fully decoded by the standard BCH decoding algorithm. >

Fourier analysis and signal processing by use of the Mobius inversion formula

A novel Fourier technique for digital signal processing is developed. This approach to Fourier analysis is based on the number-theoretic method of the Mobius inversion of series. The Fourier transform method developed is shown also to yield the convolution of two signals. A computer simulation shows that this method for finding Fourier coefficients is quite suitable for digital signal processing. It competes with the classical FFT (fast Fourier transform) approach in terms of accuracy, complexity, and speed. >