Recovery of a Lattice Generator from its 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 ) [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 from a specified Gram matrix had no available method in the literature until recently [11]. where N is an (with real elements and abq%N& ; .), and where J is a (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. random variables, each one with a certain variance qk . B. The Gram matrix In the remainder of the paper we will resort to the equivalent real model of complex lattices: Gram matrix is obtained from æ Õ æ Õæ Õ æ Õ E õv õE +öE õu õk D ; E E , (3) v ; E u ) k ≤ ø Œ; ø Œø Œ) ø Œ(8) o o o o øövŒ øöE õEŒøöuŒ øökŒ where H is the Hermitian operator (conjugate and ). ¿ø œŒ ¿ø œŒ¿ø œŒ ¿ø œŒ The elements of D correspond to all the possible inner IV. THE MATRIX CONVERSION METHOD products ef ÷ e gbetween all generating vectors and thus is Given a rational Gram matrix D it is possible to unique to a lattice subject to unitary transformations N diagonalise the quadratic form as (albeit not unique for unimodular transformations J ). It D ; I ÷ A ÷ IQ , (9) should be noticed that, by construction, D is always symmetric (because the inner product is commutative) and a where L is a rational n×n lower with ones definite positive matrix. This second property can be verified in the diagonal (i.e., is a unit matrix) and D is a n×n diagonal from the squared Euclidean norm of (using the Hermitian y matrix with rational diagonal entries agg< - . operator):

Presented at the Int. Symposium on Communications, Control and Signal Processing, Limassol, Cyprus, March 2010. (IEEE Signal Processing Society.)

V. CLOSED LOOP TECHNIQUE We propose to expand these aggß ; into a sum of a fixed number of squares of R rational numbers, that is, Using traditional singular value decomposition (SVD) [1][24][25], ! / / / agg ; wg ) wg ) ?wgO . (10) Q ). )/ ) v ; R8)PS)9u ) k E An exact expansion of these djj can be accomplished by Q Q Q Q Q (14) applying a naive greedy algorithm as proposed for the first R8)9v ; R RPS &Su' ) R k ; P ÷ u ) R k v time by Lenstra in [11]. Imposing an exact expansion for a where P is a and R and S are unitary each a gg often leads to a large number of terms in the sum Q (10) and for that reason Lenstra also proposed the use of a matrices (i.e., norm-preserving rotations, and therefore R randomized algorithm given in [23], which assures the preserves the statistics of the noise term). A SVD-based bound O Ä 1 . scheme implies one SVD decomposition (requiring at least D 0 This paper proposes to replace an exact conversion from %3k & flops [22]) at the Rx in addition to matrix D to E by an approximate conversion (leading to a E! ) multiplications both at Rx and Tx (each multiplication D 0 using a fixed complexity algorithm that can be applied in a requiring %k & flops). This technique achieves capacity real time communication system. Based on the results in [23], through water filling power allocation (according to the we use a simple greedy algorithm for the expansion and singular values) [26]. T truncate the number of terms to O Ä 1 leading to a In [27] it was shown that the LDL decomposition truncated Lenstra algorithm. One way of achieving this is by achieves better performance than standard SVD, while being using R equal terms in (10). One may notice that when R=1, slightly less complex. Most importantly, that new approach the algorithm resorts to an approximated Cholesky takes advantage of having a precoding matrix with decomposition. %k/ +k&,/ zero elements. In fact it requires a in the O÷k One starts by constructing a tall matrix ? with precoder instead of the unitary S in (14). That lower rows and n columns. For the case with R=4 terms for each of triangular matrix is fedback from the R to the T , saving the d , ? has the form x x jj bandwidth in the feedback channel. Q æw w w w ------Õ This paper proposes a technique that achieves the same ø.). .)/ .)0 .)1 Œ ø- - - - B ------Œ performance as [27] while removing most of the complexity ? ; ø Œ ø B Œ from the receiver side to the transmitter side, which is an ø- - - - wg). wg)/ wg)0 wg)1 - - Œ ø Œ important feature in scenarios where the transmitter is a base ø------w w Œ ¿ k)0 k)1 œ station and the receiver terminal should be made as simple as (11) possible, i.e., by avoiding expensive processing. where each row has one and only one non-zero entry. Now, Remark: as in [27], this paper is not assessing the one can re-construct the diagonal matrix A from ? using capacity-achieving regime and thus, for simplicity, uniform power is allocated to the transmit antennas. ! Q A ; ? ÷ ? ; Both this proposal and [27] have a pre-processing stage at We have made this explicit to Rx consisting of the generation of a definite positive matrix emphasise how this ensures that each djj is a sum of squares G, the Gram matrix of the lattice defined by the columns of as defined in (10). Finally, the approximated generator E . This Gram matrix is formed by the left multiplication (3) matrix can be seen to be In this proposal it is imperative CSIT so that Tx can E! ; ?÷ IQ , (12) construct the precoding matrix M. This paper shows that his because can be achieved with the feedback of a lower triangular ! ! Q Q Q matrix only. Given the symmetry of D, the Tx only needs to D ; I÷ A÷ I ; I÷ ? ÷ ?÷ I / Q (13) receive %k )k&,/ coefficients and from them is able to ; &?÷ IQ ' ÷ ?÷ IQ ; E! Q ÷ E!) reconstruct the entire matrix, achieving the same bandwidth which verifies (3). savings seen in [27] for the feedback channel. After The complexity of this technique is cubic in the reconstructing the entire D (by symmetry), the Tx can use dimension n, due to the LDLT steps, to which we should add the truncated Lenstra algorithm described in Section IV to D obtain an equivalent generator matrix (Section II) for the &k' for the rational approximation steps. The overall ! complexity is D &k 0 ) k' , however, as it will be shown lattice. This matrix, E , is not the same as E but rather an later in Table 1, as the LDLT decomposition will be reused, equivalent generator matrix for the lattice, holding the same only the n term will add to the complexity of the technique Gram matrix. However, it is possible to obtain from them to be presented. the same and unique generator matrix resulting from QR It should be noticed that, unlike Cholesky decomposition, decompositions remembering that a QR decomposition is this technique is applicable to both square and non-square unique when imposing the positiveness of elements in the matrixes, allowing us to retrieve a rectangular E from D . main diagonal. Thus

Presented at the Int. Symposium on Communications, Control and Signal Processing, Limassol, Cyprus, March 2010. (IEEE Signal Processing Society.)

E ; NO and E! ; N!O! (15) independent transmission channels and ii) geometrically it corresponds to a communication problem over a rectangular ! lead to O and O , which would be the same matrix (up signs lattice. Thus, it is convenient to think of a lattice as the result in main diagonal) if there was no distortion associated with of a linear transformation of the cubic lattice Dk . E! in the truncated Lenstra algorithm. A central aspect is that The diagonal matrix Abn corresponds to a set of the same Gram matrix is also obtainable from O alone, O! orthogonal generating vectors which span a rectangular alone or even a mixture of both, given their closeness: lattice (this lattice would even have the simplest trellis D ; OQ O >O! Q O! >OQ O! . (16) representation one may have for lattices, e.g., [16]). The As indicated in Section II.B and in Section IV, D would performance increase can be geometrically interpreted from have a LDLT decomposition, which can be calculated not this insight. A rectangular lattice is obtained from a k only given E or E! (in both cases applying (3)) but also deformation of a cubic lattice D by stretching each a given O , or O! , or both. However, this will not be the dimension according to each gg. Its decision regions are matrix to be LDLT decomposed. rectangles and thus even a zero-forcing detector experiences Taking advantage of these facts the mode in (7) and be no performance penalty. changed into The right multiplication of the channel matrix by O! Q in v ; N O u ) k (17) (17) changes the power at the transmitter. The geometric Q Q +. +.,/ interpretation is also useful on this matter. The “row lattice” and then, applying a precoding matrix M ; O! %I! & A! , A%EQ & has volume v~ ; N O %O! Q %I!Q &+. A! +.,/ &u ) k . (18) %((&((' SliA%O! Q & ; abqO! O! Q . (21) D~ & ' & ' The matrixes used in (18) will be presented and justified At the same time, because I! and I!Q are unit matrixes, on the fallowing. First, notice that this made a matrix ! ! ! ! !Q D~ ; OO! Q to appear (a may be needed Sli&A%A&' ; abq&A' ; abq&IAI ' together with O! to have a unique QR), which, despite not ! ! ! Q being the Gram matrix of the underlying lattice, it is the ; abq&D~' ; abq&OO ' . (22) (approximate) Gram matrix of the lattice spanned by the row Subsequently, one also needs the insertion of a diagonal lattice (as indicated in Section II). This matrix D~ also has an +.,/ T scaling A at the precoding stage so that the volume of LDL decomposition and thus (18) can equivalently be written as both lattices underlying the transmission scheme becomes the same. v~ ; N I! A! I!Q %I!Q &+. A! +.,/ u ) k . (19) Figure 1 depicts the overall transmission scheme that is %((&((' D~ proposed while in Figure 2 and in Figure 3 one can observe in detail the processing required respectively at the Tx and at Finally, after the detection filter at the receiver, the entire the Rx as well as the fluxes of CSI between both of them. chain becomes At the Rx it is important to highlight that there are two &I!+.N+. 'v~ ; &I!+.N+. 'N I! A! I!Q %I!Q &+. A! +.,/ u ) &I!+.N+. 'k parallel processing occurring at different stages and each one 8)))))9 associated with a different fading block: i) obtain G that will va~ ~ ! ! +.,/ !+. Q !+. Q be sent back to Tx in the form of a triangular matrix and ii) va ; AA ÷ u ) &I N 'k ; Abn÷ u ) &I N 'k %(((&(((' construct the receive filter from a received strictly upper k +. a triangular matrix and Q. In fact, I! is not only a lower (20) triangular but also a unit lower triangular (ones in the It should be noticed that, as Q is orthogonal (unitary if diagonal). This saves the transmission of the diagonal and / !+. considering complex models), then N+. ; NQ , which thus only the %k +k&,/ coefficients of I are needed to be forwarded to the Rx. These coefficients are denoted by further simplifies the computations at the receiver. The Rx I!+. . The process is summarised in Algorithm 1. (The will then just have to apply the filtering C ; I!+.NQ to the r channel is assumed to remain unchanged between adjacent incoming precoded signal, i.e., the unavoidable filtering symbols as it is common in the slow fading assumption.) multiplications that are present in all detectors. Besides that, +. Ir the Rx only needs to compute G and (given its symmetry) send back to the Tx only the lower or the upper parts of G, +. Q ~ ! u p v va which will be denoted by D.,/ . Moreover, both I and N ! Q !Q +. ! +.,/ !+. Q M ; O %I & A A! C ; I N are computed and sent from the Tx to the Rx. D k The resulting transmission chain (20) can be interpreted in .,/ two ways: i) algebraically it corresponds to a set of Figure 1: Proposed closed loop transmission scheme.

Presented at the Int. Symposium on Communications, Control and Signal Processing, Limassol, Cyprus, March 2010. (IEEE Signal Processing Society.)

!+. Qork`^qba„Ibkpqo^„ Ir coefficients flowing in both the uplink and downlink. The FksboqI! >idlofqej !Q number of operations in Table 1 is presented in a way that D E! FksboqI Mob`labo p Ob`lkpqor`qD shows the contribution of each individual processing stage to ! Q !Q +. ! +.,/ %`lmv„bibjbkqp& M ; O %I & A D! ~ ; O!O! Q the total number of operations of the Rx or Tx (matrix E! ; N! O! 0 ! ! Q ! ! !Q D%k & OO ; IAI u multiplications are counted as only one though). The complexity at the receiver comes from a QR decomposition D.,/ and two matrix multiplications: one to initially obtain D (similar to [27]) and then the unavoidable filtering Figure 2: Processing at the transmitter. multiplication by C . One should remember that this last multiplication is common to all types of receivers in both D.,/ closed or open-loop configurations. Q D ; E E E ; N O TABLE 1 E SVD [27] Proposal NQ D 0 0 D 0 0 0 0 0 # flops at Rx &1k ) k ' &k ) k 0 ) k ' L &/k ) k ' !+. !+. v Ir I a !+. Q ª 0 C ; I N ºk ) /k )À D k0 D k 0 Lº À # flops at Tx & ' & ' º 0 0 À v~ Ωºk 0 ) k ÀÃ / Coefficients k k/ ) k / k/ ) k / Figure 3: Processing at the receiver. in feedback & ' & ' Coefficients − − k/ + k / in downlink & ' ALGORITHM 1: CLOSED LOOP TECHNIQUE / / Total of k k/ ) k / k coefficients & ' 1: Channel estimation at Rx : H Q 2: Gram matrix of the lattice at Rx : D ; E E VI. ASSESSMENT OF THE APPROXIMATION 3: Lower triangular matrix is feedback: D.,/ (Rx→ Tx) To access the approximation one first computes the error 4: Gram matrix is reconstructed at Tx: Q matrix of the Gram matrix involved (i.e., the Gram matrix D ; D.,/ )&D.,/ ' + af^d&D.,/ ' associated with the “row lattice”, as indicated in Section V)

! ! 5: Obtain approximate E from D using the algorithm B; D~ + D~ (23) in Section IV (which encompasses a IAIQ for D ) and one applies to it the squared norm [22] ! ! ! ! / / 6: Compute QR decomposition of E : E ; NO B ; b ; Qo^`bBE B (24) C ∫ f)g & ' 7: Compute the Gram matrix of the “row lattice” at Tx: f)g ! ! ! Q D~ ; OO as the evaluation metric. ! ! ! !Q Figure 4 shows the distribution of this error for three 8: Decomposition of D~ at Tx: D~ ; IAI example cases having the number of real dimensions most 9: Precoding at T : M ; O! Q %I!Q &+. A! +.,/ ; p is sent x common in MIMO wireless communications (and with (note: for a non-squared O! the non-zero rows must be variance 0.5 per real component). deleted) Notice that despite the Gram matrix of the “row lattice” !+. and the one of “column lattice” being different, they hold the 10: Strictly lower triangular matrix is sent: Ir (Tx→ Rx) same distribution because E and E E exhibit the same 11: Compute QR decomposition of E : E ; NO statistics and consequently they are interchangeable in (3). (note: can be computed at the same time as steps 4-10) For a NT =4, NR =4 configuration (i.e., n=8 dimensions) !+. !+. !+. 12: I is reconstructed at Rx: I ; Ir ) Fkèk under a Rayleigh fading channel and using 16-QAM T +. Q modulation, it was verified that with the 13: Receiver filter C ; I! N multiplies the received LDL decomposition proposed in this paper the error shown in ! .,/ chain and the received vector becomes va ; A u ) ka Figure 4 leads to a negligible performance penalty in terms of symbol error rate (SER) in respect to the results presented The number of flops required by the LDLT decomposition in [27] for the same configuration when using the same is D%k0 ,0&, which is half of the number of flops needed in minimum mean squared error (MMSE) receiver. Gaussian elimination, the number of flops of QR VII. CONCLUSIONS decomposition is D%/k0& and for the standard matrix ,0 This paper shows how to reconstruct (with very low multiplication one has D%k &[21][22][28](though there are distortion and fixed complexity) a generator matrix of a more efficient algorithms for matrix multiplication). Table 1 lattice from one given Gram matrix of the same lattice for contains a comparison of the proposed technique with SVD non-square matrixes. This opens new possibilities in several and with [27] in terms of the number of flops and number of problems of engineering and computer science that rely on

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Presented at the Int. Symposium on Communications, Control and Signal Processing, Limassol, Cyprus, March 2010. (IEEE Signal Processing Society.)