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Deep Learning Based Sphere Decoding

Mostafa Mohammadkarimi, Mehrtash Mehrabi, Masoud Ardakani, Yindi Jing

TL;DR

The performance achieved by the proposed DL-based sphere decoding algorithm is very close to the optimal maximum likelihood decoding (MLD) over a wide range of signal-to-noise ratios (SNRs), while the computational complexity, compared to existing sphere decoding variants, is significantly reduced.

Abstract

In this paper, a deep learning (DL)-based sphere decoding algorithm is proposed, where the radius of the decoding hypersphere is learned by a deep neural network (DNN). The performance achieved by the proposed algorithm is very close to the optimal maximum likelihood decoding (MLD) over a wide range of signal-to-noise ratios (SNRs), while the computational complexity, compared to existing sphere decoding variants, is significantly reduced. This improvement is attributed to DNN's ability of intelligently learning the radius of the hypersphere used in decoding. The expected complexity of the proposed DL-based algorithm is analytically derived and compared with existing ones. It is shown that the number of lattice points inside the decoding hypersphere drastically reduces in the DL-based algorithm in both the average and worst-case senses. The effectiveness of the proposed algorithm is shown through simulation for high-dimensional multiple-input multiple-output (MIMO) systems, using high-order modulations.

Deep Learning Based Sphere Decoding

TL;DR

The performance achieved by the proposed DL-based sphere decoding algorithm is very close to the optimal maximum likelihood decoding (MLD) over a wide range of signal-to-noise ratios (SNRs), while the computational complexity, compared to existing sphere decoding variants, is significantly reduced.

Abstract

In this paper, a deep learning (DL)-based sphere decoding algorithm is proposed, where the radius of the decoding hypersphere is learned by a deep neural network (DNN). The performance achieved by the proposed algorithm is very close to the optimal maximum likelihood decoding (MLD) over a wide range of signal-to-noise ratios (SNRs), while the computational complexity, compared to existing sphere decoding variants, is significantly reduced. This improvement is attributed to DNN's ability of intelligently learning the radius of the hypersphere used in decoding. The expected complexity of the proposed DL-based algorithm is analytically derived and compared with existing ones. It is shown that the number of lattice points inside the decoding hypersphere drastically reduces in the DL-based algorithm in both the average and worst-case senses. The effectiveness of the proposed algorithm is shown through simulation for high-dimensional multiple-input multiple-output (MIMO) systems, using high-order modulations.

Paper Structure

This paper contains 14 sections, 2 theorems, 36 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations
  3. dnn for Deep Learning
  4. System Model
  5. dl-based Sphere Decoding
  6. Training Phase
  7. Decoding Procedure
  8. Intuition behind dl-based Sphere Decoding
  9. Expected Complexity of the dl-based Sphere Decoding
  10. Asymptotic Complexity Analysis
  11. Simulation Results
  12. Simulation Setup
  13. Simulation Results
  14. Conclusion

Key Result

Theorem 4.1

(Universal Approximation Theorem): Let $\varphi(\cdot)$$\mathbb{R} \rightarrow \mathbb{R}$ be a nonconstant, bounded and continuous function. Then, given any $\epsilon > 0$ and any function $f: \mathbb{I}_m \rightarrow \mathbb{R}$, where $\mathbb{I}_m$ is a compact subset of $\mathbb{R}^m$, there ex where $F( {\bf{x}} )$ is an approximate realization of the function $f$. This result holds even if

Figures (8)

  • Figure 1: A typical dnn with three hidden layers.
  • Figure 2: The empirical pdf of the radiuses learnt by the designed nn at $\gamma=24$ dB for $64$-QAM.
  • Figure 3: Performance comparison of the proposed dl-based sphere decoding algorithm and its SE variate ($q=3$), the spi vikalo2005sphere, and SE-spi in zhao2005sphere.
  • Figure 4: Average decoding time versus BER. The corresponding SNR of markers (left to right) for 64-QAM and 16-QAM are $\{28,26,24,22,20\}$ dB and $\{22,20,18,16,14\}$ dB, respectively.
  • Figure 5: Maximum decoding time versus BER. The corresponding SNR of markers (left to right) for 64-QAM and 16-QAM are $\{28,26,24,22,20\}$ dB and $\{22,20,18,16,14\}$ dB, respectively.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Theorem 4.1
  • Lemma 1
  • proof