scholarly journals A New Efficient Expression for the Conditional Expectation of the Blind Adaptive Deconvolution Problem Valid for the Entire Range ofSignal-to-Noise Ratio

Entropy ◽  
2019 ◽  
Vol 21 (1) ◽  
pp. 72 ◽  
Author(s):  
Monika Pinchas
Keyword(s):  
Probability Density Function ◽  
Closed Form ◽  
Maximum Entropy ◽  
Conditional Expectation ◽  
Entropy Density ◽  
Approximation Technique ◽  
Density Approximation ◽  
Signal To Noise ◽  
Computational Burden ◽  
Blind Adaptive

In the literature, we can find several blind adaptive deconvolution algorithms based on closed-form approximated expressions for the conditional expectation (the expectation of the source input given the equalized or deconvolutional output), involving the maximum entropy density approximation technique. The main drawback of these algorithms is the heavy computational burden involved in calculating the expression for the conditional expectation. In addition, none of these techniques are applicable for signal-to-noise ratios lower than 7 dB. In this paper, I propose a new closed-form approximated expression for the conditional expectation based on a previously obtained expression where the equalized output probability density function is calculated via the approximated input probability density function which itself is approximated with the maximum entropy density approximation technique. This newly proposed expression has a reduced computational burden compared with the previously obtained expressions for the conditional expectation based on the maximum entropy approximation technique. The simulation results indicate that the newly proposed algorithm with the newly proposed Lagrange multipliers is suitable for signal-to-noise ratio values down to 0 dB and has an improved equalization performance from the residual inter-symbol-interference point of view compared to the previously obtained algorithms based on the conditional expectation obtained via the maximum entropy technique.

scholarly journals Edgeworth Expansion Based Model for the Convolutional Noise pdf

2014 ◽  
Vol 2014 ◽  
pp. 1-19
Author(s):  
Yonatan Rivlin ◽  
Monika Pinchas
Keyword(s):  
Probability Density Function ◽  
Lagrange Multipliers ◽  
Conditional Expectation ◽  
Edgeworth Expansion ◽  
Entropy Density ◽  
Approximation Technique ◽  
Density Approximation ◽  
Mean Square ◽  
Simulation Results ◽  
Probability Density Function Pdf

Recently, the Edgeworth expansion up to order 4 was used to represent the convolutional noise probability density function (pdf) in the conditional expectation calculations where the source pdf was modeled with the maximum entropy density approximation technique. However, the applied Lagrange multipliers were not the appropriate ones for the chosen model for the convolutional noise pdf. In this paper we use the Edgeworth expansion up to order 4 and up to order 6 to model the convolutional noise pdf. We derive the appropriate Lagrange multipliers, thus obtaining new closed-form approximated expressions for the conditional expectation and mean square error (MSE) as a byproduct. Simulation results indicate hardly any equalization improvement with Edgeworth expansion up to order 4 when using optimal Lagrange multipliers over a nonoptimal set. In addition, there is no justification for using the Edgeworth expansion up to order 6 over the Edgeworth expansion up to order 4 for the 16QAM and easy channel case. However, Edgeworth expansion up to order 6 leads to improved equalization performance compared to the Edgeworth expansion up to order 4 for the 16QAM and hard channel case as well as for the case where the 64QAM is sent via an easy channel.

scholarly journals 16QAM Blind Equalization via Maximum Entropy Density Approximation Technique and Nonlinear Lagrange Multipliers

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
R. Mauda ◽  
M. Pinchas
Keyword(s):  
Maximum Entropy ◽  
Lagrange Multipliers ◽  
Signal To Noise Ratio ◽  
Minimum Mean Square Error ◽  
Blind Equalization ◽  
Entropy Density ◽  
High Signal ◽  
Approximation Technique ◽  
Density Approximation ◽  
Simulation Results

Recently a new blind equalization method was proposed for the 16QAM constellation input inspired by the maximum entropy density approximation technique with improved equalization performance compared to the maximum entropy approach, Godard’s algorithm, and others. In addition, an approximated expression for the minimum mean square error (MSE) was obtained. The idea was to find those Lagrange multipliers that bring the approximated MSE to minimum. Since the derivation of the obtained MSE with respect to the Lagrange multipliers leads to a nonlinear equation for the Lagrange multipliers, the part in the MSE expression that caused the nonlinearity in the equation for the Lagrange multipliers was ignored. Thus, the obtained Lagrange multipliers were not those Lagrange multipliers that bring the approximated MSE to minimum. In this paper, we derive a new set of Lagrange multipliers based on the nonlinear expression for the Lagrange multipliers obtained from minimizing the approximated MSE with respect to the Lagrange multipliers. Simulation results indicate that for the high signal to noise ratio (SNR) case, a faster convergence rate is obtained for a channel causing a high initial intersymbol interference (ISI) while the same equalization performance is obtained for an easy channel (initial ISI low).

scholarly journals Improved Approach for the Maximum Entropy Deconvolution Problem

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 547
Author(s):  
Shay Shlisel ◽  
Monika Pinchas
Keyword(s):  
Maximum Entropy ◽  
Density Function ◽  
Edgeworth Expansion ◽  
Entropy Density ◽  
Point Of View ◽  
Convergence Time ◽  
Approximation Technique ◽  
Gaussian Density ◽  
Deconvolution Problem ◽  
Equalization Method

The probability density function (pdf) valid for the Gaussian case is often applied for describing the convolutional noise pdf in the blind adaptive deconvolution problem, although it is known that it can be applied only at the latter stages of the deconvolution process, where the convolutional noise pdf tends to be approximately Gaussian. Recently, the deconvolutional noise pdf was approximated with the Edgeworth Expansion and with the Maximum Entropy density function for the 16 Quadrature Amplitude Modulation (QAM) input but no equalization performance improvement was seen for the hard channel case with the equalization algorithm based on the Maximum Entropy density function approach for the convolutional noise pdf compared with the original Maximum Entropy algorithm, while for the Edgeworth Expansion approximation technique, additional predefined parameters were needed in the algorithm. In this paper, the Generalized Gaussian density (GGD) function and the Edgeworth Expansion are applied for approximating the convolutional noise pdf for the 16 QAM input case, with no need for additional predefined parameters in the obtained equalization method. Simulation results indicate that improved equalization performance is obtained from the convergence time point of view of approximately 15,000 symbols for the hard channel case with our new proposed equalization method based on the new model for the convolutional noise pdf compared to the original Maximum Entropy algorithm. By convergence time, we mean the number of symbols required to reach a residual inter-symbol-interference (ISI) for which reliable decisions can be made on the equalized output sequence.

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

2021 ◽  
Vol 48 (3) ◽  
pp. 91-96
Author(s):  
Shigeo Shioda
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Infinite Number ◽  
Stable Distribution ◽  
Random Variable ◽  
Cauchy Distribution ◽  
Closed Form Expression ◽  
Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

scholarly journals Consistency and Generalization Bounds for Maximum Entropy Density Estimation

Entropy ◽  
2013 ◽  
Vol 15 (12) ◽  
pp. 5439-5463 ◽  
Author(s):  
Shaojun Wang ◽  
Russell Greiner ◽  
Shaomin Wang
Keyword(s):  
Maximum Entropy ◽  
Density Estimation ◽  
Entropy Density ◽  
Generalization Bounds

scholarly journals A Novel Structure for a Multi-Bernoulli Filter without a Cardinality Bias

Electronics ◽  
2019 ◽  
Vol 8 (12) ◽  
pp. 1484
Author(s):  
Weijian Si ◽  
Hongfan Zhu ◽  
Zhiyu Qu
Keyword(s):  
Detection Probability ◽  
Posterior Probability ◽  
Density Approximation ◽  
Signal To Noise ◽  
Generating Functional ◽  
Significant Bias ◽  
Novel Structure ◽  
High Detection ◽  
Multi Target Tracking ◽  
Bernoulli Filter

The original multi-target multi-Bernoulli (MeMBer) filter for multi-target tracking (MTT) is shown analytically to have a significant bias in its cardinality estimation. A novel cardinality balance multi-Bernoulli (CBMeMBer) filter reduces the cardinality bias by calculating the exact cardinality of the posterior probability generating functional (PGFl) without the second assumption of the original MeMBer filter. However, the CBMeMBer filter can only have a good performance under a high detection probability, and retains the first assumption of the MeMBer filter, which requires measurements that are well separated in the surveillance region. An improved MeMBer filter proposed by Baser et al. alleviates the cardinality bias by modifying the legacy tracks. Although the cardinality is balanced, the improved algorithm employs a low clutter density approximation. In this paper, we propose a novel structure for a multi-Bernoulli filter without a cardinality bias, termed as a novel multi-Bernoulli (N-MB) filter. We remove the approximations employed in the original MeMBer filter, and consequently, the N-MB filter performs well in a high clutter intensity and low signal-to-noise environment. Numerical simulations highlight the improved tracking performance of the proposed filter.

Maximum-likelihood maximum-entropy constrained probability density function estimation for prediction of rare events

AIChE Journal ◽  
2014 ◽  
Vol 60 (3) ◽  
pp. 1013-1026 ◽  
Author(s):  
Taha Mohseni Ahooyi ◽  
Masoud Soroush ◽  
Jeffrey E. Arbogast ◽  
Warren D. Seider ◽  
Ulku G. Oktem
Keyword(s):  
Maximum Likelihood ◽  
Probability Density Function ◽  
Probability Density ◽  
Maximum Entropy ◽  
Density Function ◽  
Rare Events ◽  
Function Estimation ◽  
Density Function Estimation

scholarly journals On the Density and Moments of the Time of Ruin with Exponential Claims

2003 ◽  
Vol 33 (1) ◽  
pp. 11-21 ◽  
Author(s):  
Steve Drekic ◽  
Gordon E. Willmot
Keyword(s):  
Probability Density Function ◽  
Laplace Transform ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Classical Model ◽  
Time Of Ruin ◽  
The Time Of Ruin

The probability density function of the time of ruin in the classical model with exponential claim sizes is obtained directly by inversion of the associated Laplace transform. This result is then used to obtain explicit closed-form expressions for the moments. The form of the density is examined for various parameter choices.

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