Optimal Two-Dimensional Lattices for Precoding of Linear Channels

• Published in: IEEE transactions on wireless communications Vol. 12; no. 5; pp. 2104 - 2113
• Main Authors: Kapetanovic, D., Cheng, H. V., Wai Ho Mow, Rusek, F.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.05.2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Channels; Codes; Coding, codes; Communication systems; Constellation diagram; Electrical Engineering, Electronic Engineering, Information Engineering; Elektroteknik och elektronik; Engineering and Technology; Exact sciences and technology; Hafnium; Hexagonal lattice; Information, signal and communications theory; Lattices; linear channel; Mathematical analysis; Mathematical models; Modulation, demodulation; Mutual information; Optimization; precoding; Radiocommunications; Signal and communications theory; Signal to noise ratio; Symbols; Systems, networks and services of telecommunications; TECHNOLOGY; Teknik; TEKNIKVETENSKAP; Telecommunications; Telecommunications and information theory; Transmission and modulation (techniques and equipments); Transmitters. Receivers; Two-dimensional lattices; Vectors; Vectors (mathematics); Singularity; AWGN; Quadrature amplitude modulation; Coding circuit; Transmitter; Two dimensional model; Linear channel; Mean value; precoding; Communication theory; Complex variable method; M ary modulation; Two-dimensional lattices; Coding; Telecommunication system; Minimal distance; Pretreatment
• ISSN: 1536-1276; 1558-2248; 1558-2248
• DOI: 10.1109/TWC.2013.050313.120452

Consider the communication system model y = HFx + n, where H and F are the channel and precoder matrices, x is a vector of data symbols drawn from some lattice-type constellation, such as M-QAM, n is an additive white Gaussian noise vector and y is the received vector. It is assumed that both the transmitter and the receiver have perfect knowledge of the channel matrix H and that the transmitted signal Fx is subject to an average energy constraint. The columns of the matrix HF can be viewed as the basis vectors that span a lattice, and we are interested in the precoder F that maximizes the minimum distance of this lattice. This particular problem remains open within the theory of lattices and the communication theory. This paper provides the complete solution for any nonsingular M × 2 channel matrix H. For real-valued matrices and vectors, the solution is that HF spans the hexagonal lattice. For complex-valued matrices and vectors, the solution is that HF, when viewed in four-dimensional real-valued space, spans the Schlafli lattice D 4 .