# Unsupervised BER estimation based on Stochastic Expectation Maximization algorithm using Kernel method

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## Monte-Carlo simulation

The Monte-Carlo method is the most widely used technique for estimating the BER of a communication system [JBS00, Jer84]. This technique is implemented by passing N data symbols through a model of the studied digital system and by counting the number of errors that occur at receiver. The simulation will include pseudo random data and noise sources, along with the models of the devices that process the signal present in the studied system. A number of symbols are processed by the simulation, and the experimental BER is then estimated.
Let us consider a communication system transmitting BPSK symbols over an AWGN channel. Let (bi)1≤i≤N ∈ { 1, +1} be a set of N independent transferred data. For AWGN channel, the standard baseband system model can be expressed as : s = g b + n, (2.1).
where s and b are the received and transmitted signals respectively, g is the channel gain, n is the additive noise. Let (Xi)1≤i≤N be the corresponding soft output before the decision at the receiver. Thus, Xi = si, i = 1, . . . , N. The hard decision is given by : ˆ (2.2) bi = sign(Xi).

### Importance Sampling method

As previously discussed, small BER requires a large number of data symbols. This is often considered as a fatal weakness of the classical Monte-Carlo method, especially for Spread Spectrum (SS) communication systems [QGP99] (e.g., CDMA system) that every transmitted bit needs to be modulated by the spread spectrum codes with a large number of bits.
A widely used method that can reduce BER simulation complexity for SS commu-nication systems is a modified Monte-Carlo method, called Importance Sampling (IS) method [Wik13a,And99]. In [CHD09], a BER estimation method based on Importance Sampling applied to Trapping Sets has been proposed. For Importance Sampling method, the statistics of the noise sources in the system are biased in some manner so that bit errors occur with greater probability, thereby reducing the required execution time. As an example, for a BER equal to 10 5, we may artificially “degrade” the channel performance to increase the BER to 10 2.

#### Quasi-analytical estimation

The above methods consist in analyzing the entire received waveform (data + noise) at the output of receiver. Now we consider solving the BER estimation problem in two steps :
– One deals with the transmitted signal ;
– The other deals with the noise contribution to the waveform.
Particularly, we assume that :
– The noise is referred to as an Equivalent Noise Source (ENS) ;
– The probability density function of the ENS is known and specifiable.
Therefore, we can assume that the system performance can be closely evaluated by an ENS having a suitable distribution. This method is called the Quasi-Analytical (QA) estimation [Jer84]. By taking into account the noiseless waveform, we can compute the BER with the ENS statistics. More specifically, we let the simulation itself compute the eﬀect of signal fluctuations in the absence of noise, and then superimpose the noise on the noiseless waveform.
The assumption of the noise statistics leads to a great reduction in computation eﬀort. The usefulness of the QA estimation will depend on how closely the assumption matches reality [SPKK99]. However, except for the linear system, the ENS statistics may be very diﬃcult to predict before the fact.

Remerciements
Contents
Abstract
Résumé
Acronyms
1 Introduction
1.1 Overview
1.2 Requirement of real-time on-line Bit Error Rate estimation
1.3 Thesis organization
2 State of the art for Bit Error Rate estimation
2.1 Overview of conventional Bit Error Rate estimation techniques
2.1.1 Monte-Carlo simulation
2.1.2 Importance Sampling method
2.1.3 Tail Extrapolation method
2.1.4 Quasi-analytical estimation
2.1.5 BER estimation based on Log-Likelihood Ratio
2.1.6 Conclusion of BER estimation methods
2.2 Probability Density Function estimation
2.2.1 Introduction to PDF estimation
2.2.2 Parametric density estimation : Maximum Likelihood Estimation
2.2.3 Non-parametric density estimation
2.2.3.1 Empirical density estimation
2.2.3.2 Histogram
2.2.3.3 General formulation of non-parametric density estimation
2.2.3.4 Introduction to Kernel Density Estimation
2.2.3.4.1 Naïve estimator : Parzen window
2.2.3.4.2 Smooth Kernels
2.2.4 Semi-parametric density estimation
2.2.4.1 Introduction to Gaussian Mixture Model
2.2.4.2 Difficulties of Mixture Models
2.3 BER calculation with PDF estimate
2.3.1 Theoretical BER : BER estimation based on parametric PDF estimation
2.3.2 Practical situation : necessity of non-parametric or semiparametric PDF estimation
2.4 Conclusion
3 Bit Error Rate estimation based on Kernel method
3.1 Properties of Kernel-based PDF estimator
3.1.1 Bias and variance of Kernel estimator
3.1.2 MSE and IMSE of Kernel estimator
3.1.3 Kernel selection
3.1.4 Bandwidth (smoothing parameter) selection
3.1.4.1 Subjective selection
3.1.4.2 Selection with reference to some given distribution : optimal smoothing parameter
3.2 BER estimation based on Kernel method
3.2.1 PDF estimation based on Kernel method
3.2.2 Smoothing parameters optimization in practical situation
3.2.2.1 Curve fitting method
3.2.2.2 Newton’s method
3.2.3 BER calculation with Kernel-based PDF estimates
3.2.4 MSE of Kernel-based soft BER estimator
3.3 Simulation results of BER estimation based on Kernel method
3.3.1 Sequence of BPSK symbol over AWGN and Rayleigh channels .
3.3.2 CDMA system
3.3.3 Turbo coding system
3.3.4 LDPC coding system
3.4 Conclusion
4 Bit Error Rate estimation based on Gaussian Mixture Model
4.1 Missing data of component assignment
4.1.1 K-means clustering
4.1.1.1 Principle of K-means clustering
4.1.1.2 K-means clustering algorithm : KMA
4.1.2 Probabilistic clustering as a mixture of models
4.2 BER estimation based on Gaussian Mixture Model
4.2.1 Introduction to Expectation-Maximization algorithm
4.2.1.1 Jensen’s inequality
4.2.1.2 Principle of Expectation-Maximization algorithm .
4.2.2 Expectation-Maximization algorithm for Gaussian Mixture Model
4.2.2.1 Estimation step
4.2.2.2 Maximization step
4.2.2.2.1 Calculation of μk
4.2.2.2.2 Calculation of σ2 k
4.2.2.2.3 Calculation of αk
4.2.3 Example of GMM-based PDF estimation using Expectation Maximization algorithm
4.2.4 BER calculation with GMM-based PDF estimates
4.2.5 Optimal choice of the number of Gaussian components
4.2.6 Conclusion of Gaussian Mixture Model-based BER estimationusing Expectation-Maximization algorithm
4.3 Simulation results of BER estimation based on Gaussian Mixture Model
4.3.1 Sequence of BPSK symbol over AWGN and Rayleigh channels .
4.3.2 CDMA system with decorrelator-based receiver
4.3.3 Turbo coding system
4.3.4 LDPC coding system
4.4 Conclusion
5 Unsupervised Bit Error Rate Estimation
5.1 Unsupervised BER estimation based on Stochastic Expectation Maximization algorithm using Kernel method
5.1.1 Initialization
5.1.2 Estimation step
5.1.3 Maximization step
5.1.4 Stochastic step
5.1.5 Conclusion for SEM-based unsupervised BER estimation using Kernel method
5.2 Unsupervised BER estimation based on Stochastic Expectation Maximization algorithm using Gaussian Mixture Model
5.3 Simulation results
5.3.1 Sequence of BPSK symbol over AWGN channel
5.3.2 CDMA system with standard receiver
5.3.3 Turbo coding and LDPC coding systems
5.4 Conclusion
Conclusion and perspectives
Appendix
Appendix A
Résumé de la thèse
Publications
Bibliography

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