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Minimum mean square distance estimation of a subspace  [PDF]
Olivier Besson,Nicolas Dobigeon,Jean-Yves Tourneret
Statistics , 2011, DOI: 10.1109/TSP.2011.2166548
Abstract: We consider the problem of subspace estimation in a Bayesian setting. Since we are operating in the Grassmann manifold, the usual approach which consists of minimizing the mean square error (MSE) between the true subspace $U$ and its estimate $\hat{U}$ may not be adequate as the MSE is not the natural metric in the Grassmann manifold. As an alternative, we propose to carry out subspace estimation by minimizing the mean square distance (MSD) between $U$ and its estimate, where the considered distance is a natural metric in the Grassmann manifold, viz. the distance between the projection matrices. We show that the resulting estimator is no longer the posterior mean of $U$ but entails computing the principal eigenvectors of the posterior mean of $U U^{T}$. Derivation of the MMSD estimator is carried out in a few illustrative examples including a linear Gaussian model for the data and a Bingham or von Mises Fisher prior distribution for $U$. In all scenarios, posterior distributions are derived and the MMSD estimator is obtained either analytically or implemented via a Markov chain Monte Carlo simulation method. The method is shown to provide accurate estimates even when the number of samples is lower than the dimension of $U$. An application to hyperspectral imagery is finally investigated.
On the Minimum Distance of Generalized Spatially Coupled LDPC Codes  [PDF]
David G. M. Mitchell,Michael Lentmaier,Daniel J. Costello Jr
Mathematics , 2013,
Abstract: Families of generalized spatially-coupled low-density parity-check (GSC-LDPC) code ensembles can be formed by terminating protograph-based generalized LDPC convolutional (GLDPCC) codes. It has previously been shown that ensembles of GSC-LDPC codes constructed from a protograph have better iterative decoding thresholds than their block code counterparts, and that, for large termination lengths, their thresholds coincide with the maximum a-posteriori (MAP) decoding threshold of the underlying generalized LDPC block code ensemble. Here we show that, in addition to their excellent iterative decoding thresholds, ensembles of GSC-LDPC codes are asymptotically good and have large minimum distance growth rates.
On Generalized Minimum Distance Decoding Thresholds for the AWGN Channel  [PDF]
Christian Senger,Vladimir Sidorenko,Victor Zyablov
Mathematics , 2009,
Abstract: We consider the Additive White Gaussian Noise channel with Binary Phase Shift Keying modulation. Our aim is to enable an algebraic hard decision Bounded Minimum Distance decoder for a binary block code to exploit soft information obtained from the demodulator. This idea goes back to Forney and is based on treating received symbols with low reliability as erasures. This erasing at the decoder is done using a threshold, each received symbol with reliability falling below the threshold is erased. Depending on the target overall complexity of the decoder this pseudo-soft decision decoding can be extended from one threshold T to z>1 thresholds T_1<...1 thresholds case, starting with a geometric interpretation of the optimal threshold location problem and using an approach from Zyablov.
The Euclidean Algorithm for Generalized Minimum Distance Decoding of Reed-Solomon Codes  [PDF]
Sabine Kampf,Martin Bossert
Mathematics , 2010, DOI: 10.1109/CIG.2010.5592677
Abstract: This paper presents a method to merge Generalized Minimum Distance decoding of Reed-Solomon codes with the extended Euclidean algorithm. By merge, we mean that the steps taken to perform the Generalized Minimum Distance decoding are similar to those performed by the extended Euclidean algorithm. The resulting algorithm has a complexity of O(n^2).
Minimum Distance Distribution of Irregular Generalized LDPC Code Ensembles  [PDF]
Ian P. Mulholland,Mark F. Flanagan,Enrico Paolini
Mathematics , 2013,
Abstract: In this paper, the minimum distance distribution of irregular generalized LDPC (GLDPC) code ensembles is investigated. Two classes of GLDPC code ensembles are analyzed; in one case, the Tanner graph is regular from the variable node perspective, and in the other case the Tanner graph is completely unstructured and irregular. In particular, for the former ensemble class we determine exactly which ensembles have minimum distance growing linearly with the block length with probability approaching unity with increasing block length. This work extends previous results concerning LDPC and regular GLDPC codes to the case where a hybrid mixture of check node types is used.
A Fast Generalized Minimum Distance Decoder for Reed-Solomon Codes Based on the Extended Euclidean Algorithm  [PDF]
Sabine Kampf,Martin Bossert
Mathematics , 2010, DOI: 10.1109/ISIT.2010.5513705
Abstract: This paper presents a method to determine a set of basis polynomials from the extended Euclidean algorithm that allows Generalized Minimum Distance decoding of Reed-Solomon codes with a complexity of O(nd).
Improved Minimum Entropy Filtering for Continuous Nonlinear Non-Gaussian Systems Using a Generalized Density Evolution Equation  [PDF]
Mifeng Ren,Jianhua Zhang,Fang Fang,Guolian Hou,Jinliang Xu
Entropy , 2013, DOI: 10.3390/e15072510
Abstract: This paper investigates the filtering problem for multivariate continuous nonlinear non-Gaussian systems based on an improved minimum error entropy (MEE) criterion. The system is described by a set of nonlinear continuous equations with non-Gaussian system noises and measurement noises. The recently developed generalized density evolution equation is utilized to formulate the joint probability density function (PDF) of the estimation errors. Combining the entropy of the estimation error with the mean squared error, a novel performance index is constructed to ensure the estimation error not only has small uncertainty but also approaches to zero. According to the conjugate gradient method, the optimal filter gain matrix is then obtained by minimizing the improved minimum error entropy criterion. In addition, the condition is proposed to guarantee that the estimation error dynamics is exponentially bounded in the mean square sense. Finally, the comparative simulation results are presented to show that the proposed MEE filter is superior to nonlinear unscented Kalman filter (UKF).
A Generalized Floor Bound for the Minimum Distance of Geometric Goppa Codes and its Application to Two-Point Codes  [PDF]
Benjamin Lundell,Jason McCullough
Mathematics , 2004,
Abstract: We prove a new bound for the minimum distance of geometric Goppa codes that generalizes two previous improved bounds. We include examples of the bound to one and two point codes over both the Suzuki and Hermitian curves.
Generalized Wald-type Tests based on Minimum Density Power Divergence Estimators  [PDF]
Ayanendranath Basu,Abhijit Mandal,Nirian Martin,Leandro Pardo
Statistics , 2014, DOI: 10.1080/02331888.2015.1016435
Abstract: In testing of hypothesis the robustness of the tests is an important concern. Generally, the maximum likelihood based tests are most efficient under standard regularity conditions, but they are highly non-robust even under small deviations from the assumed conditions. In this paper we have proposed generalized Wald-type tests based on minimum density power divergence estimators for parametric hypotheses. This method avoids the use of nonparametric density estimation and the bandwidth selection. The trade-off between efficiency and robustness is controlled by a tuning parameter $\beta$. The asymptotic distributions of the test statistics are chi-square with appropriate degrees of freedom. The performance of the proposed tests are explored through simulations and real data analysis.
Zero-Forcing and Minimum Mean-Square Error Multiuser Detection in Generalized Multicarrier DS-CDMA Systems for Cognitive Radio  [cached]
Lie-Liang Yang,Li-Chun Wang
EURASIP Journal on Wireless Communications and Networking , 2008, DOI: 10.1155/2008/541410
Abstract: In wireless communications, multicarrier direct-sequence code-division multiple access (MC DS-CDMA) constitutes one of the highly flexible multiple access schemes. MC DS-CDMA employs a high number of degrees-of-freedom, which are beneficial to design and reconfiguration for communications in dynamic communications environments, such as in the cognitive radios. In this contribution, we consider the multiuser detection (MUD) in MC DS-CDMA, which motivates lowcomplexity, high flexibility, and robustness so that the MUD schemes are suitable for deployment in dynamic communications environments. Specifically, a range of low-complexity MUDs are derived based on the zero-forcing (ZF), minimum mean-square error (MMSE), and interference cancellation (IC) principles. The bit-error rate (BER) performance of the MC DS-CDMA aided by the proposed MUDs is investigated by simulation approaches. Our study shows that, in addition to the advantages provided by a general ZF, MMSE, or IC-assisted MUD, the proposed MUD schemes can be implemented using modular structures, where most modules are independent of each other. Due to the independent modular structure, in the proposed MUDs one module may be reconfigured without yielding impact on the others. Therefore, the MC DS-CDMA, in conjunction with the proposed MUDs, constitutes one of the promising multiple access schemes for communications in the dynamic communications environments such as in the cognitive radios.
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