[Paper Review] On the Average Complexity of Sphere Decoding in Lattice Space-Time Coded MIMO Channel

Walid Abediseid|arXiv (Cornell University)|Mar 8, 2011

Advanced Wireless Communication Techniques18 references3 citations

TL;DR

This paper analyzes the average computational complexity of sphere decoding in lattice space-time (LAST) coded MIMO systems, deriving an upper bound on the tail distribution of complexity that is dominated by outage probability at high SNR. It identifies a cut-off multiplexing gain below which average complexity remains bounded, revealing a fundamental tradeoff between performance, complexity, and system parameters like SNR, dimensions, and codeword length.

ABSTRACT

The exact average complexity analysis of the basic sphere decoder for general space-time codes applied to multiple-input multiple-output (MIMO) wireless channel is known to be difficult. In this work, we shed the light on the computational complexity of sphere decoding for the quasi-static, LAttice Space-Time (LAST) coded MIMO channel. Specifically, we drive an upper bound of the tail distribution of the decoder's computational complexity. We show that, when the computational complexity exceeds a certain limit, this upper bound becomes dominated by the outage probability achieved by LAST coding and sphere decoding schemes. We then calculate the minimum average computational complexity that is required by the decoder to achieve near optimal performance in terms of the system parameters. Our results indicate that there exists a cut-off rate (multiplexing gain) for which the average complexity remains bounded.

Motivation & Objective

• To analyze the average computational complexity of sphere decoding in quasi-static LAST-coded MIMO channels.
• To derive an upper bound on the tail distribution of the decoder’s complexity, especially when computations exceed a threshold.
• To determine the minimum average complexity required to achieve near-optimal performance in terms of system parameters (SNR, M, N, T).
• To investigate the existence of a cut-off multiplexing gain for which average complexity remains bounded.
• To compare sphere decoding with lattice sequential decoding, quantifying the performance-complexity tradeoff.

Proposed method

• Derives an upper bound on the tail distribution of sphere decoder complexity using lattice theory and the geometry of numbers.
• Models the MIMO channel using a real Gaussian vector channel model with input-output relation y = Mx + e.
• Applies the MMSE-DFE preprocessing to improve decoding efficiency and reduce complexity.
• Uses the Fincke-Pohst sphere decoding algorithm without radius reduction or restarting for analysis.
• Relies on the asymptotic behavior of the complexity distribution at high SNR, linking it to the outage probability of LAST coding schemes.
• Compares sphere decoding with lattice sequential decoding via the ratio of average complexities, γ = L_sphere / L_sequential.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the sphere decoder’s computational complexity tail distribution in LAST-coded MIMO systems?
• RQ2 How does the decoder’s complexity scale with system parameters such as SNR, number of antennas (M, N), and codeword length T?
• RQ3 Is there a multiplexing gain threshold (cut-off rate) below which the average complexity remains bounded?
• RQ4 What is the performance-complexity tradeoff between sphere decoding and lattice sequential decoding?
• RQ5 How does MMSE-DFE preprocessing affect the average complexity of sphere decoding in LAST-coded systems?

Key findings

• The tail distribution of sphere decoder complexity is upper bounded by the asymptotic outage probability of the LAST-coded MIMO system at high SNR.
• A cut-off multiplexing gain exists below which the average complexity of the sphere decoder remains bounded, regardless of SNR.
• For a 3×3 LAST-coded MIMO system with T=5 and R=4 bpcu, the sphere decoder’s average complexity is approximately 31 times higher than that of the lattice sequential decoder at 30 dB SNR.
• As SNR increases beyond 30 dB, the complexity ratio γ approaches 1, indicating diminishing gains in complexity reduction for sequential decoding.
• Lattice-reduction-aided MMSE-DFE decoding incurs a performance gap of about 2.7 dB compared to the sphere decoder at high SNR, especially for larger T.
• The average number of computations required to terminate the search when the channel is not in outage is derived in terms of ρ, M, N, and T, showing rapid decay at high SNR.