Recovery of a Lattice Generator Matrix from its Gram Matrix for Feedback and Precoding in MIMO Francisco A. Monteiro, Student Member, IEEE, and Ian J. Wassell Abstract— Either in communication or in control applications, coding, MIMO spatial multiplexing, and even OFDM) multiple-input multiple-output systems often assume the unified from a lattice perspective as a general equalization knowledge of a matrix that relates the input and output vectors. problem (e.g., [14]). Advances in lattice theory are therefore For discrete inputs, this linear transformation generates a of great interest for MIMO engineering. multidimensional lattice. The same lattice may be described by There are several ways of describing a lattice (e.g., via an infinite number of generator matrixes, even if the rotated versions of a lattice are not considered. While obtaining the modular equations [15] or trellis structures [16]), however, Gram matrix from a given generator matrix is a trivial the two most popular ones in engineering applications are i) operation, the converse is not obvious for non-square matrixes the generator matrix and ii) the Gram matrix. The and is a research topic in algorithmic number theory. This computation of the latter given the former is trivial. The paper proposes a method to execute such a conversion and reverse is not, and an efficient algorithm for this conversion applies it in a novel MIMO system implementation where some remains an open problem in the theory of lattices. of the complexity is taken from the receiver to the transmitter. It should be noticed that an efficient algorithm for this Additionally, given the symmetry of the Gram matrix, the reverse operation can allow a lattice to be described using number of elements required in the feedback channel is nearly only about half the number of elements usually required halved. when the dimensionality of the space is sufficiently high, I. INTRODUCTION provided that the Gram matrix is always symmetric. For example, in MIMO communications with channel state Any multiple-input multiple-output (MIMO) system is information at the transmitter (CSIT), this means that about traditionally described by a generator matrix. In the wireless half the number of coefficients would need to be sent to the (and recently also in wired [1],[2]) communications systems transmitter (T ) when compared with that when using context, the matrix storing the fading coefficients between x traditional feedback [17]. Using the traditional example in transmit and receive antennas is known as the channel [17] , while in a single-input single-output configuration matrix, however in other contexts which operate with (with BPSK modulation) the channel state information is vectorial spaces, the matrix receives other names. conveyed by one coefficient only, in a 4×4 antenna system Considering that the inputs are restricted to a set of discrete one has 16 complex variables describing the channel, or inputs isomorphic to D , these systems can be framed in the equivalently 32 real coefficients, that need to be periodically general theory of lattices. fedback to the transmitter. In fact, the number of coefficients The regularity of a lattice lends itself to the representation to be fedback is the product of the number of antennas at the of problems where different signals are interpreted as a point transmitter, at the receiver (R ), the delay spread and, in in a multidimensional space. They appear in many areas of x multi-user environments, also proportional to the product signal processing such as quantization[3][4][5] or image with the number of users. processing [6]. Recently, lattices have also become a central This paper shows how an approximate solution to an open tool in cryptography [7] [8]; they are also used in numerical problem in algorithmic number theory may lead to a more integration (i.e., quadrature) of multi-dimensional functions efficient CSIT mechanism. The paper proposes an algorithm constituting lattice rules [9][10], and have a long history in to obtain a close approximation for a generator matrix given the fields of geometry of numbers, algorithmic number a Gram matrix of a lattice. The algorithm is based on an theory [11], multidimensional sphere packing (important in exact technique recently proposed by Lenstra [11] (an coding theory) [12] and also in integer programming [13]. historical figure in the fields of algorithms for lattices). This The communication theory community has recently seen paper uses the proposed algorithm as a constituent block in a topics that were thought to be distinct (such as the multiple novel strategy for closed-loop MIMO communication. access channel, the broadcast channel, precoding, space-time II. LATTICE BASICS Manuscript received November 23, 2009. This work was supported in A lattice is a discrete subgroup (of maximal rank) in a part by a scholarship from the Foundation for Science and Technology, Euclidean space and can be defined in a number of ways, as Portugal. listed in Section I. We summarise here the most common Francisco A. Monteiro is with the Department of Engineering and with ones. The Computer Laboratory, University of Cambridge, UK, and also with the Telecommunications Institute at the Lisbon University Institute, Portugal A. The generator matrix (e-mail: [email protected]). A n-dimensional lattice J may be defined by Ian J. Wassell is with The Computer Laboratory, University of Cambridge, UK (e-mail: [email protected]). ƒ¡ k ƒ– Revised version of the paper in the ISCCSP 2010 proceedings: a QR is J ; ¬ƒv ß yk 7v ; e u; E ÷ u) uß zŸƒ (1) moved into the receiver (Fig.3) and the implications from that. ƒ ∫ f f f ƒ √ƒ f;. ⁄ƒ Presented at the Int. Symposium on Communications, Control and Signal Processing, Limassol, Cyprus, March 2010. (IEEE Signal Processing Society.) E where are the points of the lattice, are the generating / E E E E y hi v ; v ÷ v ; Eu ÷ Eu ; u E Eu ; u Du . (4) vectors where each corresponds to the ith column of the n×n & ' & ' generator matrix H (considering full-rank lattices only). Consequently, one can state that G induces a quadratic Each integer xi is an element of the column vectors x. Thus, form and is definite positive because v < - for any u î -. a lattice defined as in (1) is the span of the column space of This permits us to say that D always has a LDLT the generator matrix H, when we restrict the input to integers. decomposition [21][20][22]. Note that the prevalent notation in MIMO literature Obtaining a valid E from D is not simple. D defines an considers column vectors while in channel coding or in other abstract lattice, however, two versions of a lattice will have fields in mathematics lattices are traditionally represented by the same Gram matrix and in general, for a given D , the span of the row space of a generator matrix. Notice that obtaining a possible E is named the Gram matrix rows and columns of a given H span different lattices. factorization problem. When E is square, the Cholesky Any generator matrix E can be transformed into one decomposition offers a good solution as it applies to representing an equivalent lattice defined by symmetric definite positive matrices [21]. For the general Ebn :N E J (2) m×n case, obtaining a basis from a specified Gram matrix had no available method in the literature until recently [11]. where N is an unitary matrix (with real elements and abq%N& ; .), and where J is a unimodular matrix (with C. The volume of lattices Full rank lattices are specified by a full-rank generator all elements integers and abq%J& ; .). matrix and the volume of a lattice (e.g., [8]) is given by With this in mind, the unitary transformation N performs a rigid rotation of the lattice structure (i.e., of its generating Sli%J& ; abq&E' . (5) vectors), while the unimodular matrix replaces a set of When E is rank-deficient (which is the case when E is generating vectors by a different set that still generates the non-square), the volume is same lattice. Essentially, J finds an equivalent basis for E the same structure and N rotates the entire structure in Sli&J' ; abq&E E' ; abq&D' . (6) space. One of the hardest lattice problems is the lattice III. MIMO MODELS distinguishing problem, i.e., to discover if two lattices are For a traditional complex representation with NT inputs “the same”. Once N or J are fixed, answering the and NR antennas as outputs, the received signal is question becomes trivial. But when both transformations are v ; Eu ) k , with (7) unknown the question is difficult to solve in some particular Q K Q problems [18],[19]. This decision problem has a simple O è. æv v v Õ ß æu u u Õ v ; .) /)ó) K u ; ø.) /)ó) K Œ solution when the lattices are rational lattices (when all ¿ø O œŒ ¿ Q œ entries are in ; ). In that case two lattices are equivalent if K è. K èK ß Q and E ß O Q , where each entry e is a zero- and only if their generating matrixes have the same Hermite f)g Normal Form [13],[20]. However, the real lattices that arise mean circularly symmetric Gaussian random variable with in communication problems lead to numerical problems Q æk k k Õ unitary variance. The noise vector is k ; ø.) /)ó) K Œ given the large numerators and denominators in the fractions ¿ O œ representing fading coefficients. In that case the best K è. ß O approach is to use the fact that the QR decomposition is with independent circularly symmetric Gaussian / unique up to signs in the main diagonal.