Blind Equalization via Particle Filtering Yuki Yoshida, Kazunori Hayashi, Hideaki Sakai Department of System Science, Graduate School of Informatics, Kyoto University
Historical Remarks A sequential Monte Carlo methodology Monte-Carlo Filter (Kitagawa, 1993) Bootstrap Filter (Gordon, 1993) Poor man s Monte Carlo (Hammersley, 1954) Condensation (Isard and Blake, 1998) Significant development of computers makes it possible to realize Bayesian approach in Simple way
Historical Remarks (Cont d) The basic ideas are the Bayesian formulation and sequential importance sampling method (SIS) The capability of coping with difficult nonlinear and/or non-gaussian problems Linear Gaussian State-space Model Non-linear Non-Gaussian State-space Model Generalized State-space Model Chain Structure Graphical Model
Applications in Wireless Communications Communication Applications Single-User System Detection in Flat Fading OFDM Equalization Multiple Access System Synchronization Time-invariant Channels CDMA Space-Time Coding Time-variant Channels ( Particle Filtering, Signal Processing Magazine, 2003)
Blind Equalization via PF Examples of Blind Equalization method via PF for Time-Invariant channels - J. Míguez and P. M. Djuruć, Blind Equalization by sequential importance sampling, 2002 for Time-variant channels - A. Doucet, S. J. Godsill, and C. Andrieu, On sequential Monte Carlo sampling methods for Baysian filtering, 2000 for OFDM systems - Z. Yang and X. Wang, A sequential Mote Carlo blind receiver for OFDM in frequency-selective fading channels, 2002
Signal Model Channel FIR Filter AWGN : Binary signal ht: Channel coefficients L: Length of the channel
Signal Model (Cont d) The state-space model: State equation: Observation equation: ex)
Recursive Computation of the Posterior Probability The MAP estimation of the transmitted sequence at time T : (Maximum A Posteriori) Due to computational complexity, It is desirable to solve the posterior distribution recursively Obtaining from as is observed
Recursive Computation of the Posterior Probability (Cont d) Let us consider the following decomposition... Posterior at time t Likelihood Posterior at time t-1 This provides the sequential computation of the posterior, while the likelihood can be analytically derived
Recursive Computation of the Posterior Probability (Cont d) The Likelihood Function: Recalling... Observation equation:
Recursive Computation of the Posterior Probability (Cont d) The Likelihood Function: If we assume Gaussian prior for h: The posterior channel densities are also Gaussian
Recursive Computation of the Posterior Probability (Cont d) The Likelihood Function: Gaussian Channel Update: The MAP estimation of the channel at time t!
Recursive Computation of the Posterior Probability (Cont d) The Likelihood Function: Gaussian Rao-Blackwellization Channel Update: The MAP estimation of the channel at time t!
On Implementation Our goal is... Now, we can compute this part recursively! How to implement the computation? Application of Sequential Importance Sampling algorithm (SIS) Let us begin with an Importance Sampling method (IS)
Description of Conditional pdf True distribution Linear spline Monte Carlo (Particles) Gaussian sum Particles + Weights Monte-Carlo approximation with samples drawn from importance function and their importance weights
IS Algorithm To obtain Monte-Carlo approximation of the real pdf with M samples drawn from importance function and their importance weights Draw M samples: Importance function Importance weights:
SIS Algorithm The modified IS method where it becomes possible to build the MC estimate sequentially as new observations arrive. Employing the importance function that can be factorized as Likelihood
SIS Algorithm (Cont d) Likelihood At time T...
SIS Algorithm (Cont d) Optimal importance function: Likelihood Recalling, Correspondingly,
Simulation Results Mod./ Demod. QPSK (differential coding) Observation: T=200 Channel: 3-path rayleigh fading channels # of Particle M=200 ( Particle Filtering, Signal Processing Magazine, 2003)
ｖｓ Average BER # of Particles # of Particles BER
On Further Applications in Wireless Communications DS P Non-Gaussian Noise Time-variant Channels Suitable for wireless communications? An alo g Fr on t-e nd
Other Bayesian Approaches Is the Particle Filtering the best solution? From the view point of Learning Theory... Functional approximation MCMC (Markov chain Monte Carlo) Gaussian Approximation (Laplace s Approximation) Gibbs Sampling, M-H Sampling Variational Bayesian Learning (EM Algorithm) Particle Filter
Other topics Suitability for Iterative channel decoder PF bit LLR computer π-1 Symbol Prob. computer π Channel Decoder (Resampling)... Thank You