Division Algebras and Wireless Communication

arXiv:0906.0997v2 [math.RA] 9 Jun 2009 DIVISION ALGEBRAS AND WIRELESS COMMUNICATION B.A. SETHURAMAN The aim of this note is to bring to the attention of a wide mathematical au- dience the recent application of division algebras to wireless communication. The application occurs in the context of communication involving multiple transmit and receive antennas, a context known in engineering as MIMO, short for multiple in- put, multiple output. While the use of multiple receive antennas goes back to the time of Marconi, the basic theoretical framework for communication using multiple transmit antennas was only published about ten years ago. The progress in the field has been quite rapid, however, and MIMO communication is widely credited with being one of the key emerging areas in telecommunication. Our focus here will be on one aspect of this subject: the formatting of transmit information for optimum reliability. Recall that a division algebra is an (associative) algebra with a multiplicative identity in which every nonzero element is invertible. The center of a division algebra is the set of elements in the algebra that commute with every other element in the algebra; the center is itself just a commutative field, and the division algebra is naturally a vector space over its center. We consider only division algebras that are finite-dimensional as such vector spaces. Commutative fields are trivial examples of these division algebras, but they are by no means the only ones: for instance, class-field theory tells us that over any algebraic number field K, there is a rich supply of noncommutative division algebras whose center is K and are finite-dimensional over K. Interest in MIMO communication began with the papers [20, 9, 22, 10] where it was established that MIMO wireless transmission could be used both to decrease the probability of error as well as to increase the amount of information that can be transmitted. This caught the attention of telecommunication operators, particu- larly since MIMO communication does not require additional resources in the form of either a larger slice of the radio spectrum or else increased transmitted power. The basic setup is as follows: Complex numbers reıφ, encoded as the amplitude (r) and phase (φ) of a radio wave, are sent from t transmit antennas (one number from each antenna), and the encoded signals are then received by r receive antennas. The presence of obstacles in the environment such as buildings causes attenuation of the signals; in addition, the signals are reflected several times and interfere with one another. The combined degradation of the signals is commonly referred to as “fading”, and achieving reliable communication in the presence of fading has The author is supported in part by NSF grant DMS-0700904. The author wishes to thank P. Vijay Kumar for innumerable discussions during the preparation of this article: his counsel was invaluable, his patience monumental. 1 2 B.A. SETHURAMAN been the most challenging aspect of wireless communication. The received and transmitted signals are modeled by the relation Yr×1 = θHr×tXt×1 + Wr×1 where X is a t × 1 vector of transmitted signals, Y is an r × 1 vector of received signals, W is an r × 1 vector of additive noise, H is an r × t matrix that models the fading, and θ is a real number chosen to normalize the transmitted signals so as to fit the available power. Under the most commonly adopted model, the entries of the noise vector W and the channel matrix H are assumed to be Gaussian complex random variables that are independent and identically distributed with zero mean. (A Gaussian complex random variable is one of the form w = x + ıy where x and y are real Gaussian random variables that are independent and have the same mean and variance. The modulus of such a random variable, and in particular the magnitude of of each fading coefficient hij, is then Rayleigh distributed. This model is hence also known as the Rayleigh fading channel model.) It is the presence of fading in the channel that distinguishes this model from more classical channels, where the primary source of disturbance is the additive Gaussian noise W. The transmission typically occurs in blocks of length n: each antenna transmits n times, and the receiver waits to receive all n transmission before processing them. A common engineering model is to assume that r = t = n, so the equation above is accordingly modified to read (1) Yn×n = θHn×nXn×n + Wn×n. Thus, the ith column of Y , X, and W represent (respectively) the received vec- tors, the transmitted information, and the additive noise from the ith transmission. In the model considered here, the fading characteristics of the channel (i.e., the hi,j) are assumed to be known to the receiver, but not to the transmitter (this is known as coherent transmission). A measure of the power available during a single trans- mission from all n antennas, i.e., a single use of the telecommunication channel, is the signal

…(Full text truncated)…

This content is AI-processed based on ArXiv data.