Precise Performance Characterization of Precoded Integer Forcing Applied to Two Parallel Channels

Abstract

Precoded integer-forcing equalization is a low-complexity transmission and reception scheme, primarily designed for open-loop communication. The data is encoded into independent streams, all using the same linear code, after which linear precoding is applied, while integer-forcing equalization is applied at the receiver side. Previous works have established that this architecture achieves channel capacity up to a finite gap for general multiple-input multiple-output Gaussian channels, as well as obtained tighter bounds for the special case of diagonal (parallel) channels. The present work provides a precise performance characterization when integer-forcing equalization is applied to two parallel channels and where precoding is done using the full-diversity rotation matrix cyclo <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> . It is shown that this scheme achieves capacity up to a gap bounded by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\log _{2}({\scriptstyle ^{\scriptstyle 9}}\hspace {-0.224em}/\hspace {-0.112em}{\scriptstyle 5})$ </tex-math></inline-formula> bits per complex channel use when the standard integer-forcing receiver is used, and the gap is reduced to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\log _{2}({\scriptstyle ^{\scriptstyle 5}}\hspace {-0.224em}/\hspace {-0.112em}{\scriptstyle 4})$ </tex-math></inline-formula> bits per complex channel use when its successive decoding variant is applied. In addition, a full characterization of the basis transformation matrices used by the integer-forcing receiver is derived. These turn out to consist only of Fibonacci numbers and are explicitly determined as a function of the condition number. The results obtained are also highly relevant to lattice-reduction detection schemes.