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Division Algebras and Wireless Communication B Division Algebras and Wireless Communication B. A. Sethuraman he aim of this note is to bring to the at- algebras that are finite-dimensional as such vector tention of a wide mathematical audience spaces. Commutative fields are trivial examples the recent application of division alge- of these division algebras, but they are by no bras to wireless communication. The means the only ones: for instance, class-field application occurs in the context of theory tells us that over any algebraic number Tcommunication involving multiple transmit and field K, there is a rich supply of noncommutative receive antennas, a context known in engineer- division algebras whose center is K and are finite- ing as MIMO—short for multiple input, multiple dimensional over K. output. While the use of multiple receive anten- Interest in MIMO communication began with the nas goes back to the time of Marconi, the basic papers [21, 10, 24, 11], in which it was established theoretical framework for communication using that MIMO wireless transmission could be used multiple transmit antennas was only published both to decrease the probability of error and to about ten years ago. The progress in the field has increase the amount of information that can be been quite rapid, however, and MIMO communica- transmitted. This result caught the attention of tion is widely credited with being one of the key telecommunication operators, particularly since emerging areas in telecommunication. Our focus MIMO communication does not require additional here will be on one aspect of this subject: the resources in the form of either a larger slice of the formatting of transmit information for optimum radio spectrum or increased transmitted power. reliability. The basic setup is as follows: Complex numbers ıφ Recall that a division algebra is an (associative) Re , encoded as the amplitude (R) and phase (φ) algebra with a multiplicative identity in which of a radio wave, are sent from t transmit antennas every nonzero element is invertible. The center of (one number from each antenna), and the encoded a division algebra is the set of elements in the signals are then received by r receive antennas. algebra that commute with every other element in The presence of obstacles in the environment, such the algebra; the center is itself just a commutative as buildings, causes attenuation of the signals; in field, and the division algebra is naturally a vector addition, the signals are reflected several times space over its center. We consider only division and interfere with one another. The combined degradation of the signals is commonly referred to B. A. Sethuraman is professor of mathematics at Cali- as fading, and achieving reliable communication fornia State University Northridge. His email address is in the presence of fading has been the most [email protected]. challenging aspect of wireless communication. The author is supported in part by NSF grant DMS- The received and transmitted signals are modeled 0700904. The author wishes to thank P. Vijay Kumar by the relation for innumerable discussions during the preparation of Yr 1 θHr t Xt 1 Wr 1 this article; his counsel was invaluable, his patience mon- × = × × + × where X is a t 1 vector of information signals, umental. The author also wishes to thank Frederique × Oggier for her careful reading of a preliminary version of Y is an r 1 vector of received signals, W is an × this article. r 1 vector of additive noise, H is an r t matrix × × 1432 Notices of the AMS Volume 57, Number 11 that models the fading, and θ is a real number in Sk. Often itself is referred to as the space- chosen to multiply the information signals so as time code. ItX is typically assumed that the map to fit the power available for transmission. Under X is “linear in Sk”, that is, it is the restriction k k the most commonly adopted model, the entries to S of a group homomorphism S Mn(C), of the noise vector W and the channel matrix where S is the additive subgroup of C generated→ H are assumed to be Gaussian complex random by S. (The term “space-time” refers to the fact variables that are independent and identically that information (s1, s2, . , sk) is packaged in the distributed with zero mean. (A Gaussian complex spatial direction by sending it out through several random variable is one of the form w x ıy, physically separated transmit antennas and in the where x and y are real Gaussian random= variables+ time direction by sending it out in n consecutive that are independent and have the same mean and transmissions.) variance. The modulus of such a random variable, Under the information-theoretic framework de- and in particular the magnitude of each fading veloped by Shannon in 1948 ([18]) and adopted coefficient hij , is then Rayleigh distributed. This ever since within the telecommunication commu- model is hence also known as the Rayleigh fading nity, the amount of information conveyed by a k channel model.) It is the presence of fading in the message in this setting is equal to log2(q ) “bits”. channel that distinguishes this model from more Since this amount of information is conveyed classical channels, where the primary source of in n transmissions over the MIMO channel, the disturbance is the additive Gaussian noise W . rate of information transmission is then given by A common engineering model is to assume k n log2(q) bits per channel use. When q and n are that the channel characteristics (i.e., the fading fixed a priori, the quantity k serves as a measure coefficients hij ) stay constant in some fixed but of information rate. small time interval and that these characteristics Reliability of communication is commonly mea- are known to the receiver but not the transmit- sured by the probability Pe of incorrectly decoding ter. (This is known as coherent transmission.) If the transmitted message at the receiver. The pair- each antenna can transmit n times during such an wise error probability Pe(i, j) (for i ≠ j) is the interval, then the transmissionprocess is compart- probability that message i is transmitted and mes- mentalized into blocks of length n: each antenna sage j is decoded. Performance analysis of MIMO transmits n times, and each receiver waits to re- communication systems typically focuses on the ceive all n transmissions before processing them. pairwise error probability, as it is easier to es- A common simplifying assumption is to take timate and also because the error probability Pe r t n, and the equation above is accordingly can be upper and lower bounded in terms of the = = modified to read pairwise error probability. It was shown in [21, 11] that for a fixed SNR (1) Yn n θHn nXn n Wn n. × = × × + × (i.e., power) ρ, in order to keep the pairwise error Thus the ith column of Y , θX, and W represent probability low, the space-time code must meet (respectively) the received vectors, the transmitted the two criteria below, of which the firstX is primary: information, and the additive noise from the ith (1) Rank Criterion: For s : (s1, s2, . , sk) and transmission. A measure of the power available = s′ : (s1′ , s2′ , . , sk′ ) with s ≠ s′, the differ- during a single transmission from all n antennas, ence= matrix i.e., a single use of the telecommunication chan- X(s) X(s′) nel, is the signal-to-noise ratio (SNR) ρ. Recall that − must have full rank n, i.e., it must be the Frobenius norm X F of X (xi,j ) equals || || = 2 invertible. i,j xi,j . Since the power required to send a | | (2) Coding Gain Criterion: For s and s′ as complex number varies as the square of its mod- above, s ≠ s′, the modulus of the determi- ulus, the normalization constant θ must satisfy 2 2 nant of difference θ X F nρ. || || ≤ det(θX(s) θX(s′)) A subset S of the nonzero complex numbers | − | known as the signal set is selected as the alphabet must be as large as possible. (a common situation is that S is a finite subset of Clearly, the second criterion comes into play only size q of the nonzero Gaussian integers Z[ı] 0 ), when the first criterion has been met but then −{ } and a k-tuple (s1, s2, . , sk), si S, constitutes the subsumes it. Each criterion impacts a different message that the transmitter wishes∈ to convey to communication parameter, and the two are hence the receiver. Thus there are qk messages in all, and stated independently. Note that one cannot arbi- it is assumed that each message is equally likely trarily scale the matrices X to increase the coding to be transmitted. A space-time code is then a one- gain because the assumption of fixed ρ, along with k k 2 2 to-one map X : S Mn(C); we write for X(S ). the relation θ X nρ, would simply cause a → X || ||F ≤ The transmitted matrix θXn n in Equation (1) is corresponding decrease in θ. Note too that one × thus drawn from the set θ as (s1, s2, . , sk) vary cannot increase the quantity k (a proxy for the rate X December 2010 Notices of the AMS 1433 n of information) arbitrarily, as this would create k 2 for n > 4. It follows that these general- a larger set of matrices θX all circumscribed to izations≤ of the Alamouti code transmit too few lie within a sphere of radius √nρ, which would information symbols for more than two transmit then cause the determinant of their differences antennas.
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