Sphere decoding complexity exponent for decoding full rate codes over the quasi-static MIMO channel

TL;DR

This paper analyzes the high-SNR asymptotic complexity of sphere decoding for full rate MIMO codes, introducing a complexity exponent that characterizes the computational effort needed for near-optimal decoding.

Contribution

It introduces a complexity exponent for sphere decoding in quasi-static MIMO channels, providing bounds and exact expressions for various codes, including the recently developed threaded CDA codes.

Findings

The SD complexity exponent describes the minimal complexity for near-ML performance.

Explicit formulas are derived for large families of codes with diverse performance traits.

For threaded CDA codes, the complexity exponent varies non-monotonically with multiplexing gain.

Abstract

In the setting of quasi-static multiple-input multiple-output (MIMO) channels, we consider the high signal-to-noise ratio (SNR) asymptotic complexity required by the sphere decoding (SD) algorithm for decoding a large class of full rate linear space-time codes. With SD complexity having random fluctuations induced by the random channel, noise and codeword realizations, the introduced SD complexity exponent manages to concisely describe the computational reserves required by the SD algorithm to achieve arbitrarily close to optimal decoding performance. Bounds and exact expressions for the SD complexity exponent are obtained for the decoding of large families of codes with arbitrary performance characteristics. For the particular example of decoding the recently introduced threaded cyclic division algebra (CDA) based codes -- the only currently known explicit designs that are uniformly optimal with respect to the diversity multiplexing tradeoff (DMT) -- the SD complexity exponent is shown to take a particularly concise form as a non-monotonic function of the multiplexing gain. To date, the SD complexity exponent also describes the minimum known complexity of any decoder that can provably achieve a gap to maximum likelihood (ML) performance which vanishes in the high SNR limit.