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Towards Explainable Machine Learning: The Effectiveness of Reservoir Computing in Wireless Receive Processing

Shashank Jere, Karim Said, Lizhong Zheng, Lingjia Liu

TL;DR

This work applies the echo state network (ESN) as a channel equalizer and provides a first principles-based signal processing understanding of its operation and paves the way for optimized initialization of the ESN model weights, which are traditionally untrained and randomly initialized.

Abstract

Deep learning has seen a rapid adoption in a variety of wireless communications applications, including at the physical layer. While it has delivered impressive performance in tasks such as channel equalization and receive processing/symbol detection, it leaves much to be desired when it comes to explaining this superior performance. In this work, we investigate the specific task of channel equalization by applying a popular learning-based technique known as Reservoir Computing (RC), which has shown superior performance compared to conventional methods and other learning-based approaches. Specifically, we apply the echo state network (ESN) as a channel equalizer and provide a first principles-based signal processing understanding of its operation. With this groundwork, we incorporate the available domain knowledge in the form of the statistics of the wireless channel directly into the weights of the ESN model. This paves the way for optimized initialization of the ESN model weights, which are traditionally untrained and randomly initialized. Finally, we show the improvement in receive processing/symbol detection performance with this optimized initialization through simulations. This is a first step towards explainable machine learning (XML) and assigning practical model interpretability that can be utilized together with the available domain knowledge to improve performance and enhance detection reliability.

Towards Explainable Machine Learning: The Effectiveness of Reservoir Computing in Wireless Receive Processing

TL;DR

This work applies the echo state network (ESN) as a channel equalizer and provides a first principles-based signal processing understanding of its operation and paves the way for optimized initialization of the ESN model weights, which are traditionally untrained and randomly initialized.

Abstract

Deep learning has seen a rapid adoption in a variety of wireless communications applications, including at the physical layer. While it has delivered impressive performance in tasks such as channel equalization and receive processing/symbol detection, it leaves much to be desired when it comes to explaining this superior performance. In this work, we investigate the specific task of channel equalization by applying a popular learning-based technique known as Reservoir Computing (RC), which has shown superior performance compared to conventional methods and other learning-based approaches. Specifically, we apply the echo state network (ESN) as a channel equalizer and provide a first principles-based signal processing understanding of its operation. With this groundwork, we incorporate the available domain knowledge in the form of the statistics of the wireless channel directly into the weights of the ESN model. This paves the way for optimized initialization of the ESN model weights, which are traditionally untrained and randomly initialized. Finally, we show the improvement in receive processing/symbol detection performance with this optimized initialization through simulations. This is a first step towards explainable machine learning (XML) and assigning practical model interpretability that can be utilized together with the available domain knowledge to improve performance and enhance detection reliability.
Paper Structure (13 sections, 1 theorem, 11 equations, 6 figures)

This paper contains 13 sections, 1 theorem, 11 equations, 6 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Wireless Channel
  4. The Conventional Echo State Network (ESN)
  5. Model Interpretability in RC
  6. Signal Processing Preliminaries for the ESN
  7. Optimum Orthogonal Basis Set
  8. From Optimum Basis to Optimum ESN Design
  9. Experimental Validation
  10. Improvement in Symbol Error Rate (SER) Performance
  11. Conclusion and Future Work
  12. Construction of Complex-Valued Optimum Basis
  13. Proof of Lemma \ref{['lemma:covariance_matrix_rank']}

Key Result

Lemma 1

For an $L$-tap channel impulse response $\mathbf{h}$ following an i.i.d. Gaussian distribution, the covariance matrix of $\Tilde{\mathbf{v}}$ has an $\epsilon$-rank given by $\mathop{\mathrm{rank}}\nolimits_{\epsilon} \left( \mathbb{E}[(\Tilde{\mathbf{v}} - \mathbb{E}[\Tilde{\mathbf{v}}]) (\Tilde{\m

Figures (6)

  • Figure 1: Modeling a neuron in the reservoir as a single-pole IIR filter.
  • Figure 2: Flowchart outlining the derivation of "optimum" ESN model weights (conventionally untrained) from available channel statistics.
  • Figure 3: Symbol detection performance comparison for exponentially decaying PDP channel with 16-QAM modulation.
  • Figure 4: Symbol detection performance comparison for exponentially decaying PDP channel with 64-QAM modulation.
  • Figure 5: Symbol detection performance comparison for 3GPP CDL-D channel with QPSK modulation.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Lemma 1