It is shown that a small change in the argument of Harper and Murty implies that there are at most two real quadratic fields with class-number one and without euclidean algorithm.
It is shown that a small change in the argument of Harper and Murty implies that there are at most two real quadratic fields with class-number one and without euclidean algorithm.
Extracting main melody from polyphonic music is one of the most appealing and challenging tasks in music information retrieval (MIR). In this paper, a new melody extraction method based on a modified euclidean algorit...
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Extracting main melody from polyphonic music is one of the most appealing and challenging tasks in music information retrieval (MIR). In this paper, a new melody extraction method based on a modified euclidean algorithm (MEA) is proposed. Firstly, the instantaneous frequency is adopted to gain better frequency discrimination, and the frame-wise pitch candidates are estimated based on the modified euclidean algorithm. Next, the candidate trajectories are formed using these potential candidates, and padded by the candidate one octave above or below if there is a gap at some isolated frames. Finally, the melodic contours are extracted using the melody smoothness and salience principle. The proposed modified euclidean algorithm can deal with diverse coprime harmonic combinations, and work well at low computational cost and memory requirement. The experimental results show that the proposed method can extract main melody extraction effectively with few pitch candidates. (C) 2016 Elsevier Ltd. All rights reserved.
It is well known that the euclidean algorithm or its equivalent, continued fractions, can be used to find the error locator polynomial and the error evaluator polynomial in Berlekamp's key equation that is needed ...
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It is well known that the euclidean algorithm or its equivalent, continued fractions, can be used to find the error locator polynomial and the error evaluator polynomial in Berlekamp's key equation that is needed to decode a Reed-Solomon (RS) code. In the paper, a simplified procedure is developed and proved to correct erasures as well as errors by replacing the initial condition of the euclidean algorithm by the erasure locator polynomial and the Forney syndrome polynomial. By this means, the errata locator polynomial and the errata evaluator polynomial can be obtained simultaneously and simply, by the euclidean algorithm only. With this improved technique, the complexity of time-domain Reed-Solomon decoders for correcting both errors and erasures is reduced substantially from previous approaches. As a consequence, decoders for correcting both errors and erasures of RS codes can be made more modular, regular, simple, and naturally suitable for both VLSI and software implementation. An example illustrating this modified decoding procedure is given for a (15, 9) RS code.
In this paper, we will define a euclidean -like norm and a division algorithm for a non-Noetherian Bezout domain, k[y] + x.k(y)[x], where k is a field. And we will show that the euclidean algorithm for that domain alw...
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In this paper, we will define a euclidean -like norm and a division algorithm for a non-Noetherian Bezout domain, k[y] + x.k(y)[x], where k is a field. And we will show that the euclidean algorithm for that domain always terminates. As its application, we will give an algorithm to find the normal form of any matrix in GL(2)(k[x, y]) over k, with respect to the amalgamated free product structure. (C) 2016 Elsevier Inc. All rights reserved.
An approach to high resolution spectral analysis using rational signal models is presented. This technique is based on a fast euclidean algorithm for polynomials that has a computational complexity on the order of n l...
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An approach to high resolution spectral analysis using rational signal models is presented. This technique is based on a fast euclidean algorithm for polynomials that has a computational complexity on the order of n log/sup 2/ n. Some background material on Pade approximation and the euclidean algorithm is presented. Some examples are presented, and the results are evaluated using the exact Cramer-Rao lower bound. This method is shown to perform well in the presence of white and colored noise.
We analyse the behaviour of the euclidean algorithm applied to pairs (g,f) of univariate nonconstant polynomials over a finite field $\mathbb{F}_{q}$ of q elements when the highest degree polynomial g is fixed. Consid...
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We analyse the behaviour of the euclidean algorithm applied to pairs (g,f) of univariate nonconstant polynomials over a finite field $\mathbb{F}_{q}$ of q elements when the highest degree polynomial g is fixed. Considering all the elements f of fixed degree, we establish asymptotically optimal bounds in terms of q for the number of elements f that are relatively prime with g and for the average degree of $\gcd(g,f)$ . We also exhibit asymptotically optimal bounds for the average-case complexity of the euclidean algorithm applied to pairs (g,f) as above.
A modified euclidean decoding algorithm to solve the Berlekamp's key equation of Reed-Solomon code for correcting errors, is presented in this paper. It is derived to solve the error locator and evaluator polynomi...
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A modified euclidean decoding algorithm to solve the Berlekamp's key equation of Reed-Solomon code for correcting errors, is presented in this paper. It is derived to solve the error locator and evaluator polynomials simultaneously without performing the operations of polynomial division and field element inversion. In this algorithm, the number of iterations used to solve the equation is fixed, and also the weights used to reduce the degree of the error evaluator polynomial at each iteration can be extracted from the coefficient of fixed degree. Therefore, this proposed algorithm saves many controlling circuits, and provides module architecture with regularity. As a result it is simple and easy to implement, and in addition it can be easily configured for various applications. (C) 2003 Elsevier Inc. All rights reserved.
A novel method based on euclidean algorithm is proposed to solve the problem of blind recognition of binary Bose-Chaudhuri-Hocquenghem (BCH) codes in non-cooperative applications. By carrying out iterative euclidean d...
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A novel method based on euclidean algorithm is proposed to solve the problem of blind recognition of binary Bose-Chaudhuri-Hocquenghem (BCH) codes in non-cooperative applications. By carrying out iterative euclidean divisions on the demodulator output bit-stream, the proposed method can determine the codeword length and generator polynomial of unknown BCH code. The computational complexity is derived asO(n(3)). Simulation results show the efficiency of the proposed method.
Given a polynomial f (x) = a(0)x(n) + a(1)x(n-1) + ... + a(n) with positive coefficients a(k), and a positive integer M <= n, we define an infinite generalized Hurwitz matrix H-M(f) : = (a(Mj-i))(i, j). We prove th...
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Given a polynomial f (x) = a(0)x(n) + a(1)x(n-1) + ... + a(n) with positive coefficients a(k), and a positive integer M <= n, we define an infinite generalized Hurwitz matrix H-M(f) : = (a(Mj-i))(i, j). We prove that the polynomial f (z) does not vanish in the sector {z is an element of C : vertical bar arg(z)vertical bar < pi/M} whenever the matrix H-M is totally non-negative. This result generalizes the classical Hurwitz' Theorem on stable polynomials (M = 2), the Aissen-Edrei-Schoenberg-Whitney theorem on polynomials with negative real roots (M = 1), and the Cowling-Thron theorem (M = n). In this connection, we also develop a generalization of the classical euclidean algorithm, of independent interest per se.
We study the ergodic properties of the additive euclidean algorithm f defined in R(2)(+). A natural extension of f is obtained using the action of SL(2, Z) on a subset of SL(2, R). We prove that, while f is an ergodic...
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We study the ergodic properties of the additive euclidean algorithm f defined in R(2)(+). A natural extension of f is obtained using the action of SL(2, Z) on a subset of SL(2, R). We prove that, while f is an ergodic transformation with an infinite invariant measure equivalent to the Lebesgue measure, the invariant measure is not unique up to scalar multiples, and in fact there is a continuous family of such measures.
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