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Modified Baum-Welch Algorithm for Joint Blind Channel Estimation and Turbo Equalization

Chin-Hung Chen, Boris Karanov, Ivana Nikoloska, Wim van Houtum, Yan Wu, Alex Alvarado

TL;DR

This work tackles blind ISI channel estimation by improving the Baum–Welch EM framework. It introduces a reduced-state BW estimator that links channel parameters to edges in a trellis, cutting the state count in half while preserving performance. Building on this, it couples the estimator with turbo equalization to exploit extrinsic information from the decoder, achieving significantly faster convergence (about 10 EM iterations at $4\ \mathrm{dB}$) and improved estimation accuracy (approximately $\text{MSE}=10^{-4}$ at $6\ \mathrm{dB}$). However, the benefits are condition-dependent: at very low SNRs (e.g., $2\ \mathrm{dB}$) the joint design may be inferior due to unreliable extrinsic information, highlighting the need for robust, unsupervised blind turbo receivers in challenging channels.

Abstract

Blind estimation of intersymbol interference channels based on the Baum-Welch (BW) algorithm, a specific implementation of the expectation-maximization (EM) algorithm for training hidden Markov models, is robust and does not require labeled data. However, it is known for its extensive computation cost, slow convergence, and frequently converges to a local maximum. In this paper, we modified the trellis structure of the BW algorithm by associating the channel parameters with two consecutive states. This modification enables us to reduce the number of required states by half while maintaining the same performance. Moreover, to improve the convergence rate and the estimation performance, we construct a joint turbo-BW-equalization system by exploiting the extrinsic information produced by the turbo decoder to refine the BW-based estimator at each EM iteration. Our experiments demonstrate that the joint system achieves convergence in 10 EM iterations, which is 8 iterations less than a separate system design for a signal-to-noise ratio (SNR) of 4dB. Additionally, the joint system provides improved estimation accuracy with a mean square error (MSE) of $10^{-4}$ for an SNR of 6dB. We also identify scenarios where a joint design is not preferable, especially when the channel is noisy (e.g., SNR=2dB) and the decoder cannot provide reliable extrinsic information for a BW-based estimator.

Modified Baum-Welch Algorithm for Joint Blind Channel Estimation and Turbo Equalization

TL;DR

This work tackles blind ISI channel estimation by improving the Baum–Welch EM framework. It introduces a reduced-state BW estimator that links channel parameters to edges in a trellis, cutting the state count in half while preserving performance. Building on this, it couples the estimator with turbo equalization to exploit extrinsic information from the decoder, achieving significantly faster convergence (about 10 EM iterations at ) and improved estimation accuracy (approximately at ). However, the benefits are condition-dependent: at very low SNRs (e.g., ) the joint design may be inferior due to unreliable extrinsic information, highlighting the need for robust, unsupervised blind turbo receivers in challenging channels.

Abstract

Blind estimation of intersymbol interference channels based on the Baum-Welch (BW) algorithm, a specific implementation of the expectation-maximization (EM) algorithm for training hidden Markov models, is robust and does not require labeled data. However, it is known for its extensive computation cost, slow convergence, and frequently converges to a local maximum. In this paper, we modified the trellis structure of the BW algorithm by associating the channel parameters with two consecutive states. This modification enables us to reduce the number of required states by half while maintaining the same performance. Moreover, to improve the convergence rate and the estimation performance, we construct a joint turbo-BW-equalization system by exploiting the extrinsic information produced by the turbo decoder to refine the BW-based estimator at each EM iteration. Our experiments demonstrate that the joint system achieves convergence in 10 EM iterations, which is 8 iterations less than a separate system design for a signal-to-noise ratio (SNR) of 4dB. Additionally, the joint system provides improved estimation accuracy with a mean square error (MSE) of for an SNR of 6dB. We also identify scenarios where a joint design is not preferable, especially when the channel is noisy (e.g., SNR=2dB) and the decoder cannot provide reliable extrinsic information for a BW-based estimator.

Paper Structure

This paper contains 11 sections, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. HMM, BCJR, and the Baum-Welch algorithm
  4. Hidden Markov model
  5. BCJR inference algorithm
  6. Baum-Welch learning algorithm
  7. Receiver Design
  8. Modified Baum-Welch Channel Estimator
  9. Turbo Equalization
  10. Numerical Simulations
  11. Conclusions

Figures (3)

  • Figure 1: System block of a convolutionally coded bit-interleaved BPSK transmitter over a linear ISI channel with AWGN.
  • Figure 2: Receiver block diagram of a joint turbo-BW-Equalization system design.
  • Figure 3: Trellis representations of BW-based channel estimators. $\phi_l = \{\hat{\mu}_l,\hat{\sigma}_l^2\}$ denotes the channel parameters set, which is associated with individual states in Fig. \ref{['fig:trellis1']} and state pairs (edges) in Fig. \ref{['fig:trellis2']}.