Algebraic reduction for the Golden Code

Transcription

1 Algebraic reduction for the Golden Code Ghaya Rekaya-Ben Othman, Laura Luzzi and Jean-Claude Belfiore TELECOM ParisTech, 46 rue Barrault, 753 Paris, France {rekaya, luzzi, paristech.fr Abstract In this paper we introduce a new right preprocessing method for the decoding of algebraic space-time codes, called algebraic reduction, which exploits the multiplicative structure of the code. The principle of the new reduction is to absorb part of the channel into the code, by approximating the channel matrix with an element of the maximal order of the code algebra. We prove that algebraic reduction attains the receive diversity when followed by a simple zero-forcing ZF detection. Simulation results for the Golden Code show that using minimum mean squared error generalized decision feedback equalization MMSE-GDFE left preprocessing, algebraic reduction with simple ZF detection has a loss of only 3dB with respect to optimal decoding. Index Terms Algebraic reduction, right preprocessing, Golden Code, space-time codes, decoding I. INTRODCTION Space-time coding for multiple antenna systems is an efficient device to compensate the effects of fading in wireless channels through diversity techniques, and allows for increased data rates. A new generation of space-time code designs for MIMO channels, based on suitable subsets of division algebras, has been recently developed []. The algebraic constructions guarantee that these codes are full-rank, fullrate and information-lossless, and have the non-vanishing determinant property. p to now, the decoding of algebraic space-time codes has been performed using their lattice point representation. In particular, maximum likelihood decoders such as the Sphere Decoder or the Schnorr-Euchner algorithm are currently employed. However, the complexity of these decoders is prohibitive for practical implementation. On the other side, suboptimal decoders like ZF, DFE, MMSE have low complexity but they don t preserve the diversity order of the system. The use of preprocessing before decoding improves the performance of suboptimal decoders, and reduces considerably the complexity of ML decoders [7]. Two types of preprocessing are possible: - Left preprocessing MMSE-GDFE to obtain a better conditioned channel matrix; - Right preprocessing lattice reduction in order to have aquasi-orthogonallattice.themostwidelyusedlattice reduction is the LLL reduction. The worst case complexity of sphere decoding for a space-time code is of the order of M 4,whereM is the size of the constellation. However, space-time codes that admit a reduced ML complexity of M have been recently proposed [3, 9]. We are interested here in the right preprocessing stage; we propose a new reduction method for space-time codes based on quaternion algebras which directly exploits the multiplicative structure of the space-time code. p to now, algebraic tools have been used exclusively for coding but never for decoding. Algebraic reduction consists in absorbing a part of the channel into the code. This is done by approximating the channel matrix with a unit of a maximal order of the quaternion algebra. The algebraic reduction has already been implemented by Rekaya et al. [] for the fast fading channel, in the case of rotated constellations based on algebraic number fields. In this context, the units in the ring of integers of the field form an abelian multiplicative group whose generators are described by Dirichlet s unit theorem [6]. For quaternion skewfields, which are the object of this paper, the situation is more complicated because the unit group is not commutative. However, it is still possible to find a finite presentation of the group, that is a finite set of generators and relations. As an example, we consider the Golden Code, and exhibit asetofgeneratorsfortheunitgroupofitsmaximalorder. Our simulation results for the Golden Code show that using MMSE-GDFE left preprocessing, the performance of algebraic reduction with ZF decoding is within 3dB of the ML. A. System model II. SYSTEM MODEL AND NOTATION We consider a quasi-static MIMO system employing aspace-timeblockcode.thereceivedsignalisgivenby Y = HX + W, X, H, Y, W M C The entries of H are i.i.d. complex Gaussian random variables with zero mean and variance per real dimension equal to, and W is the Gaussian noise with i.i.d. entries of zero mean and variance. X is the transmitted codeword. In this paper we are interested in STBCs that are subsets of a principal ideal Oα of a maximal order O in a cyclic division algebra A of index over Qi a quaternion algebra. We refer to [] for the necessary background about space-time codes from cyclic division algebras, and to [5] for a discussion of codes based on maximal orders. Example The Golden Code. The Golden Code falls into this category see [, 5]. It is based on the cyclic algebra A =Qi, θ/qi,σ,i, whereθ = 5+ and σ : x x is /9/$5. 9 IEEE

2 such that σθ = θ = θ and σ leaves the elements of Qi fixed. It has been shown in [5] that { } x x O =,x i x x,x Z[i, θ] is a maximal order of A. O can be written as O = Z[i, θ] Z[i, θ]j, where j = 3 i p to a scaling constant, the Golden Code is a subset of the two-sided ideal Oα = αo, withα =+i θ. Every codeword of G has the form X = 5 αx αx ᾱi x ᾱ x with x = s + s θ, x = s 3 + s 4 θ.thesymbolss,s,s 3,s 4 belong to a QAM constellation. B. Notation Notation Vectorization of matrices. Let φ be the function M C C 4 that vectorizes matrices: a c φ : a, b, c, d t 4 b d The left multiplication function A l : M C M C that maps B to AB induces a linear mapping A l = φ A l φ : C 4 C 4.Thatis,φAB =A l φb A, B M C, A with A l =. A Notation Lattice point representation. Consider a basis {w,w,w 3,w 4 } of αo as a Z[i]-module. Every codeword X can be written as X = s i w i, s =s,s,s 3,s 4 t Z[i] 4 i= If Φ is the matrix whose columns are φw, φw, φw 3, φw 4,thelatticepointcorrespondingtoX is x = φx = s i φw i =Φs i= We denote by Λ the Z[i]-lattice with generator matrix Φ. The following remark explains the relation between the units of the maximal order O of the code algebra and unimodular transformations of the code lattice. This property is fundamental for algebraic reduction. Remark nits and unimodular transformations. Suppose that O is an invertible element: then {w,w,w 3,w 4 } is still a basis of the Z[i]-lattice αo. The codeword X can be expressed in the new basis: X = s iw i, s =s,s,s 3,s 4 t Z[i] 4 i= The vectorized signal is Φs = φx = φw i = i= = l s iφw i = l Φs i= s i l φw i = i= Now consider the change of coordinates matrix T = Φ l Φ M 4 C between the basis {φw i } i=,...,4 and {φw i } i=,...,4. We have dett = det l = det = ±. Moreover, s Z[i] 4, s = T s Z[i] 4. Then T is unimodular, and the lattice generated by ΦT is still Λ. III. ALGEBRAIC REDCTION In this section we introduce the principle of algebraic reduction. First of all, we consider a normalization of the received signal. In the system model, the channel matrix H has nonzero determinant with probability, and so it can be rewritten as H = dethh, Therefore the system is equivalent to Y = H SL C Y deth = H X + W Algebraic reduction consists in approximating the normalized channel matrix H with a unit of norm of the maximal order O of the algebra of the considered STBC, that is an element of O such that det =. A. Perfect approximation In order to simplify the exposition, we first consider the ideal case where we have a perfect approximation: H =. Of course this is extremely unlikely in practice; the general case will be treated in the next paragraph. The received signal can be written: Y = X + W 5 and X is still a codeword. In fact, since is invertible, {X X Oα} = Oα Applying φ to both sides of equation 5, we find that the equivalent system in vectorized form is y = l Φs + w where Φ is the generator matrix of the code, s Z[i] 4, y = φy, w = φw.since is a unit, l Φ = ΦT, with T unimodular see Remark, therefore y = ΦT s + w = Φs + w, s Z[i] 4 In order to decode, we can simply consider ZF detection: ŝ = [ Φ y ]

3 where []denotes the rounding of each vector component to the nearest Gaussian integer. If Φ is unitary, as in the case of the Golden Code, algebraic reduction followed by ZF detection gives optimal ML performance. B. General case In general, the approximation is not perfect with probability and we must take into account the approximation error E. We have H = E, andthevectorizedreceivedsignalis y = E l l Φs + w = E l ΦT s + w = E l Φs + w The estimated signal after ZF detection is ŝ = [ Φ E ] l y = [ ] = s + Φ E w =[s + n] 6 deth Finally, one can recover an estimate of the initial signal ŝ = T ŝ. Thus,thesystemisequivalenttoanon-fadingsystem where the noise n is no longer white Gaussian. C. Choice of for the ZF decoder We suppose here for simplicity that the generator matrix Φ is unitary, but a similar criterion can be established in a more general case. Ideally the error term E should be unitary in order to have optimality for the ZF decoder, so we should choose the unit in such a way that E = H is quasiorthogonal. We require that the Frobenius norm E F should be minimized : =argmin H 7 F O, det= This criterion corresponds to minimizing the trace of the covariance matrix of the new noise n in 6: trcovn = Φ E l deth = F = E l deth = F E deth F IV. THE APPROXIMATION ALGORITHM In this section we describe an algorithm to find the nearest unit to the normalized channel matrix H with respect to the criterion 7. To do this we need to understand the structure of the group of units of the maximal order O. Remark nits of norm. The set O = { O det =} is a multiplicative subgroup of O.Infact,if is a unit of the Z[i]-order O, thenn A/Qi =det is a unit in Z[i], that is, det {,,i, i}. O is the kernel of the reduced norm mapping N=N A/Qi : O {,,i, i}. Remark that since dete =, E F = E F. l 8 Table I GENERATORS OF O. iθ +i +i θ = i θ 5 = i + iθ +i i +i +i +iθ = i i 6 = i + i θ +i θ +i i θ + i 3 = i θ 7 = iθ + i i θ i i θ + i 4 = i + θ 8 = i θ + i i Example The Golden Code. In the case of the Golden Code, N is surjective since N =, Nθ = θ θ =, Nj = j = i, Njθ =i. So{,,i, i} = O /O, and O is a normal subgroup of index 4 of O. Its cosets can be obtained by multiplying for one of the coset leaders {,θ,j,θj}. Our problem is then reduced to studying the subgroup O. In particular, we need to find a presentation of this group: a set of generators S and a set of relations R among these generators. In fact, one can show that O is finitely presentable, that is it admits a presentation with S and R finite. Example Generators and relations in the case of the Golden Code. The group O is generated by 8 units, that are displayed in Table I. The proof of the previous fact is omitted due to lack of space. The method for finding a presentation is rather complex and is based on the Swan algorithm [4, ]. A. Action of the group on the hyperbolic space H 3 The search algorithm is based on the action of the group on a suitable space. We use the fact that O is a subgroup of the special linear group SL C, andconsidertheactionof SL C on the hyperbolic 3-space H 3 see [4, 8]. We refer to the upper half-space model H 3 = {z,r z C, r R,r >} 9 endowed with the hyperbolic distance ρ such that if P = z,r, P =z,r, cosh ρp, P =+ z z +r r rr a b Given a matrix M = SL c d C, its action on a point P =z,r is defined as follows: { Mz,r =z,r z az+b c z+ d+a cr =,, with cz+d + c r r = r cz+d + c r Here we denote by z the complex conjugate of z. The action of M and M is the same, so there is an induced action of PSL C =SL C/{, }. PSL C can be identified with the group Isom + H 3 of orientationpreserving isometries of H 3 with respect to the metric we

4 defined previously [4], Proposition.3. Consider the action of PSL C on the special point J =, which has the following nice property [4], Proposition.7: M SL C, M F =coshρj, MJ As anticipated in Section III, given the normalized channel matrix H SL C we want to find Û =argmin O =argmin O H ρj, H since is an isometry. F =argmin O J = argmin O coshρj, H J = ρ J,H J Proposition. There exists a constant R max > such that H SL C, Û O with ρû J,H J <R max. Consequently, there exists a fixed constant C O such that H F C O The proof of this Proposition is based on the existence of Dirichlet polyhedra for Kleinian groups see [4, ]. For the Golden Code, the methods of [] allow to compute the constant C O explicitly: C O =4.47. B. The algorithm Let,..., r be a set of generators of O,and r+ =,..., r = r their inverses. Suppose that the matrix form of the i has been stored in memory, together with the images J,..., r J of J. Let i J =x i,y i,r i, i =,...,r INPT: H SL C. Initialization: let H = H, =, i =. REPEAT Compute H J =x, y, r. Compute the distances d i =coshρh J, i J, d =coshρh J,J 3 Let i = argmin i {,,...,r} d i. If several indices i attain the minimum, choose the smallest. 4 pdate i, H H i. NTIL i =. OTPT: Û = is the chosen unit. Remark 3. If the channel varies slowly from one time block to the next, the point H J will also vary little in time. Thus, this method requires only a slight adjustment of the previous search at each step. On the contrary, the LLL reduction method requires a full lattice reduction at each time block. V. PERFORMANCE OF THE ALGEBRAIC REDCTION A. Diversity It has recently been proved [5] that MIMO decoding based on LLL reduction followed by zero-forcing achieves the receive diversity. The following Proposition shows that algebraic reduction is equivalent to LLL reduction in terms of diversity for the case of transmit and receive antennas: Proposition. The diversity order of the algebraic reduction method with ZF detection is. Proof: Suppose that the symbols s i, i =,...,4 belong to an M-QAM constellation, with M = m.lete av be the average energy per symbol, and γ = Eav the SNR. For a fixed realization of the channel matrix H, equation6 is equivalent to an additive channel without fading where the noise n is no longer white. With symbol by symbol ZF detection, P e γ is bounded by the error probability for each symbol. sing the classical expression of P e in a Gaussian channel, for square QAM constellations [], 5..9, we obtain 3E av P ŝ i s i 4erfc σ 4e 3Eav M σ i M where σi is the variance for complex dimension of the noise component n i.singthebound8,wefind σi trcovn C deth because E F = H C F O,seeequation. Finally, we find P e γ H 6e 3 M C deth E av =6e c deth γ In order to compute P e γ, we need the distribution of deth. Itisknown[3]thattherandomvariable4 deth is distributed as the product of two independent chi square random variables X χ,y χ 4. Thecumulative distribution function of Z = deth = XY is F Z z =P { XY z} = xy z 8 ye x y dxdy From the change of variables u = y, v = xy, weobtain z v F Z z = e v u u du dv = z 4 K z where K is the modified Bessel function of the second kind. Finally, [ ] P e γ E 6e c γz =6 =6 c γ + πc γ 4 Ψ k + 5 z k= K ze c γz dz = Γk + 5 c γ k Γk + Ψk + lnc γ

5 3 FER FER ML MMSE GDFE + AR + ZF DFE MMSE GDFE+AR+ZF AR+ZF ZF 3 ML MMSE GDFE + LLL + ZF MMSE GDFE + LLL + ZF DFE MMSE GDFE + AR + ZF MMSE GDFE + AR + ZF DFE SNR SNR Figure. Performance of algebraic reduction followed by ZF or ZF-DFE decoders using 4-QAM constellations. Figure. Comparison of algebraic reduction and LLL reduction using MMSE-GDFE preprocessing combined with ZF or ZF-DFE decoding with 6-QAM constellations. where Ψ is the Digamma function. The series in the last expression being uniformly bounded for large γ, the leading term is of the order of γ. B. Simulation results Figure shows the performance of algebraic reduction followed by ZF and ZF-DFE decoding compared with ML decoding using 4-QAM constellations; the slope of the probability of error in the case of algebraic reduction with ZF detection without preprocessing is very close to, confirming the result of Proposition concerning the diversity order. One can add MMSE-GDFE left preprocessing to solve the shaping problem for finite constellations [7] in order to improve this performance. With MMSE-GDFE preprocessing, algebraic reduction is within 4.dB and 3.dB from the ML using ZF and ZF-DFE decoding, at the FER of 4. In the 6-QAM case, the loss is of 3.4dB and.6db respectively for ZF and ZF-DFE decoding at the FER of 3 Figure. In the same figure we compare algebraic reduction and LLL reduction. The two performances are very close; with ZF-DFE decoding, algebraic reduction has a slight loss.3db. On the contrary, with ZF decoding, algebraic reduction is slightly better.4db gain, showing that the criterion 7 is indeed appropriate for this decoder. Numerical simulations also evidence that the average complexity of algebraic reduction is low: in fact the average number of steps of the unit search algorithm is only.93. VI. CONCLSIONS In this paper we have introduced a right preprocessing method for the decoding of space-time block codes based on quaternion algebras, which allows to improve the performance of suboptimal decoders and reduces the complexity of ML decoders. The new method exploits the algebraic structure of the code, by approximating the channel matrix with a unit in the maximal order of the quaternion algebra. Our simulations show that algebraic reduction and LLL reduction have similar performance. Future work will deal with the generalization of algebraic reduction to higher-dimensional space time codes based on division algebras. REFERENCES [] J-C. Belfiore, G. Rekaya, E. Viterbo, The Golden Code: a full-rate Space-Time Code with non-vanishing determinants, IEEE Trans. Inform. Theory, vol5n.4,5 [] C. Corrales, E. Jespers, G. Leal, A. del Rìo, Presentations of the unit group of an order in a non-split quaternion algebra, Advances in Mathematics, 86n [3] A. Edelman, Eigenvalues and condition numbers of random matrices, Ph.D. Thesis, MIT 989 [4] J. Elstrodt, F. Grunewald, J. Mennicke, Groups acting on Hyperbolic Space, Springer Monographs in Mathematics, 998 [5] C. Hollanti, J. Lahtonen, K. Ranto, R. Vehkalahti, On the densest MIMO lattices from cyclic division algebras, submitted to IEEE Trans. Inform. Theory [6] S. Lang, Algebraic number theory, Springer-Verlag [7] A. D. Murugan, H. El Gamal, M. O. Damen, G. Caire, A unified framework for tree search decoding: rediscovering the sequential decoder, IEEE Trans. Inform. Theory, vol 5 n. 3, 6 [8] C. Maclachlan, A. W. Reid, The arithmetic of hyperbolic 3- manifolds, Graduate texts in Mathematics, Springer, 3 [9] J. Paredes, A.B. Gershman, M. Gharavi-Alkhansari, A space-time code with non-vanishing determinants and fast maximum likelihood decoding, ICASSP 7 [] J. Proakis, Digital communications, McGraw-Hill [] G. Rekaya, J-C. Belfiore, E. Viterbo, A very efficient lattice reduction tool on fast fading channels, Proceedings of ISITA 4, Parma, Italy [] B. A. Sethuraman, B. Sundar Rajan, V. Shashidar, Fulldiversity, high-rate space-time block codes from division algebras, IEEE Trans. Inform. Theory, vol49n., 3, pp [3] S. Sezginer, H. Sari, Full-rate full-diversity space-time codes of reduced decoder complexity, IEEE Commun. Letters vol n., 7 [4] R. G. Swan, Generators and relations for certain special linear groups, Adv. Math. 6, 77 [5] M. Taherzadeh, A. Mobasher, A. K. Khandani, LLL reduction achieves the receive diversity in MIMO decoding, IEEE Trans. Inform. Theory, vol53n.,7,pp48 485