Algebraic reduction for space-time codes based on quaternion algebras
aa r X i v : . [ c s . I T ] S e p Algebraic reduction for space-time codesbased on quaternion algebras
L. Luzzi G. Rekaya-Ben Othman* J.-C. Belfiore
Abstract
In this paper we introduce a new right preprocessing method for the decoding of × algebraicSTBCs, called algebraic reduction , which exploits the multiplicative structure of the code. The principleof the new reduction is to absorb part of the channel into the code, by approximating the channel matrixwith an element of the maximal order of the algebra.We prove that algebraic reduction attains the receive diversity when followed by a simple ZF detection.Simulation results for the Golden Code show that using MMSE-GDFE left preprocessing, algebraicreduction with simple ZF detection has a loss of only with respect to ML decoding. Index Terms:
Algebraic reduction, right preprocessing, Golden Code
EDICS category:
MSP-DECD
I. I
NTRODUCTION
Space-time coding for multiple antenna systems is an efficient device to compensate the effects offading 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 divisionalgebras, has been recently developed [17]. The algebraic constructions guarantee that these codes arefull-rank, full-rate and information-lossless, and have the non-vanishing determinant property.Up to now, the decoding of algebraic space-time codes has been performed using their lattice pointrepresentation. 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
Jean-Claude Belfiore, Ghaya Rekaya-Ben Othman and Laura Luzzi are with TELECOM ParisTech, 46 Rue Barrault, 75013Paris, France. E-mail: { belfiore , rekaya , luzzi } @ telecom − paristech . fr . Tel: +33 (0)145817705, +33 (0)145817633,+33 (0)145817636. Fax: +33 (0)145804036 October 26, 2018 DRAFT practical implementation, especially for lattices of high dimension, arising from MIMO systems with alarge number of transmit and receive antennas.On the other side, suboptimal decoders like ZF, DFE, MMSE have low complexity but their performanceis poor; in particular they don’t preserve the diversity order of the system.The use of preprocessing before decoding improves the performance of suboptimal decoders, and reducesconsiderably the complexity of ML decoders [12]. 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 a quasi-orthogonal lattice. The most widelyused lattice reduction is the LLL reduction.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 thespace-time code in addition to the lattice structure. Up to now, algebraic tools have been used exclusivelyfor coding but never for decoding. Algebraic reduction consists in absorbing a part of the channel into thecode. This is done by approximating the channel matrix with a unit of a maximal order of the quaternionalgebra.The algebraic reduction has already been implemented by Rekaya et al. [16] for the fast fading channel,in the case of rotated constellations based on algebraic number fields. In this context, the units in the ringof integers of the field form an abelian multiplicative group whose generators are described by Dirichlet’sunit theorem [10]. The reduction algorithm then amounts to decoding in the logarithmic lattice of theunit group, which is fixed once and for all. In this case, one can show that the diversity of the channelis preserved.For quaternion skewfields, which are the object of this paper, the situation is more complicated becausethe 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.Supposing that a presentation is known, we describe an algorithm to find the best approximation of thechannel matrix as a product of the generators.As an example, we consider the Golden Code, and find a set of generators for the unit group of its maximalorder. Our simulation results for the Golden Code show that using MMSE-GDFE left preprocessing, theperformance of algebraic reduction with ZF decoding is within of the ML.The paper is organized as follows: in Section II we introduce the system model; in Section III weexplain the general method of algebraic reduction. In Section IV we present the search algorithm to
October 26, 2018 DRAFT approximate the channel matrix with a unit; in Section V we prove that our method yields diversityorder equal to when followed by a simple ZF decoder. We discuss its performance obtained throughsimulations in the case of the Golden Code, and compare algebraic reduction and LLL reduction usingvarious decoders (ZF, ZF-DFE), with and without MMSE-GDFE preprocessing. Finally, in Section VIwe describe a method to obtain the generators of the unit group for a maximal order in a quaternionalgebra. The computations are carried out in detail for the case of the algebra of the Golden Code.II. S YSTEM MODEL AND NOTATION
A. System model
We consider a quasi-static × MIMO system employing a space-time block code. The receivedsignal is given by Y = HX + W, X, H, Y, W ∈ M ( C ) (1)The entries of H are i.i.d. complex Gaussian random variables with zero mean and variance per realdimension equal to , and W is the Gaussian noise with i.i.d. entries of zero mean and variance N . X is the transmitted codeword. In this paper we are interested in STBCs that are subsets of a principalideal O α of a maximal order O in a cyclic division algebra A of index over Q ( i ) (a quaternion algebra).We refer to [17] for the necessary background about space-time codes from cyclic division algebras, andto [8] for a discussion of codes based on maximal orders. Example ( The Golden Code ) . The Golden Code falls into this category (see [1] and [8]). It is based onthe cyclic algebra A = ( Q ( i, θ ) / Q ( i ) , σ, i ) , where θ = √ and σ : x ¯ x is such that σ ( θ ) = ¯ θ = 1 − θ and σ leaves the elements of Q ( i ) fixed.It has been shown in [8] that O = x x i ¯ x ¯ x , x , x ∈ Z [ i, θ ] (2)is a maximal order of A . O can be written as O = Z [ i, θ ] ⊕ Z [ i, θ ] j , where j = i (3)Up to a scaling constant, the Golden Code is a subset of the two-sided ideal O α = α O , with α = 1 + i ¯ θ [11]. Every codeword of G has the form X = 1 √ αx αx ¯ αix ¯ αx October 26, 2018 DRAFT with x = s + s θ , x = s + s θ . The symbols s , s , s , s belong to a QAM constellation. B. Notation
In the following paragraphs we will often pass from the × matrix notation for the transmitted andreceived signals to their lattice point representation as complex vectors of length . To avoid confusion, × matrices and vectors of length are written in boldface (using capital letters and small lettersrespectively), while × matrices are not in bold. Notation ( Vectorization of matrices ) . Let φ be the function M ( C ) → C that vectorizes matrices: φ : a cb d ( a, b, c, d ) t (4) 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 → C . That is, φ ( AB ) = A l φ ( B ) ∀ A, B ∈ M ( C ) A l is the block diagonal matrix A l = A A (5) Notation ( Lattice point representation ) . Let { w , w , w , w } be a basis of α O as a Z [ i ] -module. Everycodeword X can be written as X = X i =1 s i w i , s = ( s , s , s , s ) t ∈ Z [ i ] Let Φ be the matrix whose columns are φ ( w ) , φ ( w ) , φ ( w ) , φ ( w ) (6) Then the lattice point corresponding to X is x = φ ( X ) = X i =1 s i φ ( w i ) = Φs We denote by Λ the Z [ i ] -lattice with generator matrix Φ . A complex matrix T is called unimodular if the elements of T belong to Z [ i ] and det( T ) ∈ { , − , i, − i } .Recall that two generator matrices Φ and Φ ′ span the same Z [ i ] -lattice if Φ ′ = ΦT with T unimodular.The following remark explains the relation between the units of the maximal order O of the code algebraand unimodular transformations of the code lattice. This property is fundamental for algebraic reduction. October 26, 2018 DRAFT
Remark 1 ( Units and unimodular transformations ) . Suppose that U ∈ O ∗ is an invertible element:then { U w , U w , U w , U w } is still a basis of α O seen as a Z [ i ] -lattice. The codeword X can also beexpressed in the new basis: X = X i =1 s ′ i ( U w i ) , s ′ = ( s ′ , s ′ , s ′ , s ′ ) t ∈ Z [ i ] The vectorized signal is Φs = φ ( X ) = X i =1 φ ( U w i ) = X i =1 s ′ i U l φ ( w i ) = U l X i =1 s ′ i φ ( w i ) = U l Φs ′ Now consider the change of coordinates matrix T U = Φ − U l Φ ∈ M ( C ) from the basis { φ ( w i ) } i =1 ,..., to { φ ( U w i ) } i =1 ,..., . We have det( T U ) = det( U l ) = det( U ) = ± , see equation (5). Moreover, wehave seen that ∀ s ∈ Z [ i ] , s ′ = T U s ∈ Z [ i ] . Then T U is unimodular, and the lattice generated by ΦT U is still Λ . III. A LGEBRAIC REDUCTION
In this section we introduce the principle of algebraic reduction. First of all, we consider a normalizationof the received signal. In the system model (1), the channel matrix H has nonzero determinant withprobability , and so it can be rewritten as H = p det( H ) H , H ∈ SL ( C ) Therefore the system is equivalent to Y = Y p det( H ) = H X + W Algebraic reduction consists in approximating the normalized channel matrix H with a unit U of norm of the maximal order O of the algebra of the considered STBC, that is an element U of O such that det( U ) = 1 . A. Perfect approximation
In order to simplify the exposition, we first consider the ideal case where we have a perfect approxi-mation: H = U . Of course this is extremely unlikely in practice; the general case will be described inthe next paragraph.The received signal can be written: Y = U X + W (7) October 26, 2018 DRAFT and
U X is still a codeword. In fact, since U is invertible, { U X | X ∈ O α } = O α Applying φ to both sides of equation (7), we find that the equivalent system in vectorized form is y = U l Φs + w where Φ is the matrix defined in (6), s ∈ Z [ i ] , y = φ ( Y ) , w = φ ( W ) .We have seen in Remark 1 that since U is a unit, U l Φ = ΦT U , with T U unimodular. So y = ΦT U s + w = Φs + w , s ∈ Z [ i ] In order to decode, we can simply consider ZF detection: ˆs = (cid:2) Φ − y (cid:3) = " s + 1 p det( H ) Φ − w 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 givesoptimal (ML) performance. B. General case
In the general case, the approximation is not perfect with probability and we must take into accountthe approximation error E . We write H = EU , and the vectorized received signal is y = E l U l Φs + w = E l ΦT U s + w = E l Φs + w The estimated signal after ZF detection is ˆs = (cid:2) Φ − E − l y (cid:3) = " s + 1 p det( H ) Φ − E − l w = [ s + n ] (8)Finally, one can recover an estimate of the initial signal ˆs = T − U ˆs .Thus, the system is equivalent to a non-fading system where the noise n is no longer white Gaussian. October 26, 2018 DRAFT
C. Choice of U for the ZF decoder We suppose here for simplicity that the generator matrix Φ is unitary, but a similar criterion can beestablished in a more general case. We have seen that ideally the error term E should be unitary in orderto have optimality for the ZF decoder, so we should choose the unit U in such a way that E = H U − is quasi-orthogonal. We require that the Frobenius norm k E k F should be minimized : U = argmin U ∈O , det( U )=1 (cid:13)(cid:13) U H − (cid:13)(cid:13) F (9)This criterion corresponds to minimizing the trace of the covariance matrix of the new noise n in (8): Cov( n ) = Cov p det( H ) Φ − E − l w ! = 1 | det( H ) | Φ − E − l Cov( w ) (cid:0) E − l (cid:1) H (cid:0) Φ − (cid:1) H == N | det( H ) | Φ − E − l (cid:0) E − l (cid:1) H (cid:0) Φ − (cid:1) H and tr(Cov( n )) = N | det( H ) | (cid:13)(cid:13) Φ − E − l (cid:13)(cid:13) F = N | det( H ) | (cid:13)(cid:13) E − l (cid:13)(cid:13) F = 2 N | det( H ) | (cid:13)(cid:13) E − (cid:13)(cid:13) F (10)IV. T HE APPROXIMATION ALGORITHM
In this section we describe an algorithm to find the nearest unit U to the normalized channel matrix H with respect to the criterion (9). To do this we need to understand the structure of the group of unitsof the maximal order O . Notation.
We denote elements of SL ( C ) with capital letters (for example H , U ) when considering theirmatrix representation, and with small letters (for example h , u ) when we want to stress that they aregroup elements. Remark 2 ( Units of norm ) . The set O = { u ∈ O ∗ | det( u ) = 1 } is a subgroup of O .In fact, if u is a unit of the Z [ i ] -order O , then N A / Q ( i ) ( u ) = det( u ) is a unit in Z [ i ] , that is, det( u ) ∈{ , − , i, − i } . O is the kernel of the reduced norm mapping N = N A / Q ( i ) : O ∗ → { , − , i, − i } whichis a group homomorphism, thus it is a subgroup of O . Remark that since det( E ) = 1 , k E k F = ‚‚ E − ‚‚ F . October 26, 2018 DRAFT
Table IG
ENERATORS OF O . u = iθ i ¯ θ = iθ, u − = i ¯ θ iθ = i ¯ θu = i ii − i = i + (1 + i ) j u − = i − − i − i + 1 i = i − (1 + i ) ju = θ ii − θ = θ + (1 + i ) j u − = ¯ θ − − i − i + 1 θ = ¯ θ − (1 + i ) ju = θ − − i − i + 1 ¯ θ = θ − (1 + i ) j u − = ¯ θ ii − θ = ¯ θ + (1 + i ) ju = i i ¯ θi (1 + iθ ) 1 + i = (1 + i ) + (1 + i ¯ θ ) j u − = i − − i ¯ θ − i (1 + iθ ) 1 + i = (1 + i ) + (1 + i ¯ θ ) ju = i iθi (1 + i ¯ θ ) 1 + i = (1 + i ) + (1 + iθ ) j u − = i − − iθ − i (1 + i ¯ θ ) 1 + i = (1 + i ) − (1 + iθ ) ju = − i ¯ θ + ii ( θ + i ) 1 − i = (1 − i ) + (¯ θ + i ) j u − = − i − ¯ θ − i − i ( θ + i ) 1 − i = (1 − i ) − (¯ θ + i ) ju = − i θ + ii (¯ θ + i ) 1 − i = (1 − i ) + ( θ + i ) j u − = − i − θ − i − i (¯ θ + i ) 1 − i = (1 − i ) − ( θ + i ) j Example ( The Golden Code ) . In the case of the Golden Code, N is surjective since N(1) = 1 , N( θ ) = θ ¯ θ = − , N( j ) = − j = − i , N( jθ ) = i . So { , − , i, − i } ∼ = O ∗ / O , and O is a normal subgroup ofindex of O ∗ . In order to obtain a set of generators, it is then sufficient to study the structure 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 canshow that O is finitely presentable , that is it admits a presentation with S and R finite [9]. Example ( Generators and relations in the case of the Golden Code ) . The group O is generated by units, that are displayed in Table I. The corresponding relations are shown in Table II.The method for finding a presentation is based on the Swan algorithm [18]. As it is not well knownand is rather complex, we have chosen to expose it in detail in Section VI. October 26, 2018 DRAFT
Table IIF
UNDAMENTAL RELATIONS AMONG THE GENERATORS OF O . u = − u = − ( u u ) = ( u u − ) = u u u = − u u u − = − u u u = − u − u u = − u u − u u − = u − u u − u − u u u u = u u − u u u − u − u − u − = A. Action of the group on the hyperbolic space H 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 ) , and consider the action of SL ( C ) on the hyperbolic -space H (see for example [5] or [13] for a reference).We refer to the upper half-space model of H : H = { ( z, r ) | z ∈ C , r ∈ R , r > } (11) H can also be seen as a subset of the Hamilton quaternions H : a point P can be written as ( z, r ) = z + r j = x + i y + r j , where { , i , j , k } is the standard basis of H . We endow H with the hyperbolicdistance ρ such that if P = z + r j , P ′ = z ′ + r ′ j , cosh ρ ( P, P ′ ) = 1 + d ( P, P ′ ) rr ′ , where d ( P, P ′ ) = | z − z ′ | + ( r − r ′ ) is the squared Euclidean distance. The corresponding surfaceand volume forms on H are ([13], pp. 48–49) ds = dx + dy + dr r , (12) dv = dxdydrr (13) October 26, 2018 DRAFT0
The geodesics with respect to this metric are the (Euclidean) half-circles perpendicular to the plane { r = 0 } and with center on this plane, and the half-lines perpendicular to { r = 0 } . Given a matrix g = a bc d ∈ SL ( C ) , its action on a point P = ( z, r ) is defined as follows: g ( z, r ) = ( z ∗ , r ∗ ) , with z ∗ = ( az + b )(¯ c ¯ z + ¯ d )+ a ¯ cr | cz + d | + | c | r ,r ∗ = r | cz + d | + | c | r (14)(Here we denote by ¯ z the complex conjugate of z ).The action of g and − g is the same, so there is an induced action of P SL ( C ) = SL ( C ) / { , − } . P SL ( C ) can be identified with the group Isom + ( H ) of orientation-preserving isometries of H withrespect to the metric defined previously ([13], p. 48).All the information we will gain about the group O will thus be modulo the equivalence relation g ∼ − g ;we denote by P O its quotient with respect to this relation.Consider the action of P SL ( C ) on the special point J = (0 ,
1) = j (15)which has the following nice property ([5], Proposition 1.7): ∀ g ∈ SL ( C ) , k g k F = 2 cosh ρ ( J, g ( J )) (16) Remark 3. If g ∈ U (2) is unitary, then g leaves every point of H fixed ([5], Proposition 1.1). Then byconsidering for example the mapping P SL ( C ) → H that sends g to g ( J ) , one can identify H withthe quotient space P SL ( C ) /U (2) . B. The algorithm
We assume here the following fundamental properties, which will be proven in Section VI:1) { u ( J ) | u ∈ O } is a discrete set in H .2) Given a unit u ∈ O , the set P u = { P ∈ H | ρ ( P, u ( J )) ≤ ρ ( P, u ′ ( J )) ∀ u ′ = u } is a compact hyperbolic polyhedron with finite volume and finitely many faces. October 26, 2018 DRAFT1
3) Two distinct polyhedra P u , P u ′ can intersect at most in one face; all the polyhedra are isometric,and they cover the whole space H , forming a tiling . Moreover, if P = P is the polyhedroncontaining J , P u = u ( P )
4) The polyhedra adjacent to P are given by u ( P ) , . . . , u r ( P ) , u − ( P ) , . . . , u − r ( P ) (17)where { u , . . . , u r } is a minimal set of generators for O .As anticipated in Section III, given the normalized channel matrix h ∈ SL ( C ) we want to find ˆ u = argmin u ∈O (cid:13)(cid:13) uh − (cid:13)(cid:13) F (18)But we know from equation (16) that (cid:13)(cid:13) uh − (cid:13)(cid:13) F = 2 cosh( ρ ( J, uh − ( J ))) = 2 cosh( ρ ( u − ( J ) , h − ( J ))) , since u is an isometry. So the condition (18) is equivalent to ˆ u = argmin u ∈O ρ ( u − ( J ) , h − ( J )) The point h − ( J ) is contained in the image ¯ u ( P ) = P ¯ u of the polyhedron P for some ¯ u ∈ O . It followsfrom the definition of P ¯ u that h − ( J ) is closer to ¯ u ( J ) than to any other u ( J ) , u ∈ O . Since all thepolyhedra are isometric, ρ (¯ u ( J ) , h − ( J )) ≤ R max where R max is the radius of the smallest (hyperbolic) sphere containing P . Therefore we have thefollowing property: (cid:13)(cid:13) uh − (cid:13)(cid:13) F ≤ C O (19)We now go back to the problem of finding a unit ¯ u such that h − ( J ) ∈ ¯ u ( P ) , given a normalizedchannel matrix h ∈ SL ( C ) . Let u , . . . , u r be the generators of O in (17) and u r +1 = u − , . . . , u r = u − r their inverses. The neighboring polyhedra of P are all of the form u i ( P ) , i = 1 , . . . , r. The idea is to begin the search from P and the neighboring polyhedra, corresponding to the generatorsof the group and their inverses, and choose the U i such that u i ( J ) is the closest to h − ( J ) . Since u i isan isometry of H , at the next step we can apply u − i and start again the search of the u i ′ that gives the October 26, 2018 DRAFT2 v ( J ) v ( P ) J P v − J v − h − ( J ) P h − ( J ) v − ( P ) v − ( J ) Figure 1. A step of the algorithm. The polyhedra are represented as two-dimensional polygons for simplicity. closest point to u i − h − ( J ) . With this strategy we only need to update a single point and perform r comparisons at each step of the search.The algorithm is illustrated in Figure 1.Suppose that the matrix form of the u i has been stored in memory at the beginning of the program,together with the images u ( J ) , . . . , u r ( J ) of J , for example using the coordinates in the upper half-space model (11). Let u i ( J ) = ( x i , y i , r i ) , i = 1 , . . . , r INPUT: h ∈ SL ( C ) .Initialization: let h = h , ¯ u = , i = 0 . REPEAT Compute h − ( J ) = ( x, y, r ) . Compute the distances d i = 2 cosh ρ ( h − ( J ) , u i ( J )) = 1 + ( x − x i ) + ( y − y i ) + ( r − r i ) rr i , i = 1 , . . . , r,d = 2 cosh ρ ( h − ( J ) , J ) Let i = argmin i ∈{ , ,..., r } d i . (If several indices i attain the minimum, choose the smallest.) Update ¯ u ← ¯ uu i , h ← hu i . UNTIL i = 0 . OUTPUT: ˆ u = ¯ u − is the chosen unit. October 26, 2018 DRAFT3
Remark 4 ( Advantage of the algebraic reduction in the case of slow fading channels ) . If thechannel varies slowly from one time block to the next, it is reasonable to expect that the polyhedron ¯ u ( P ) containing h − ( J ) at the time t will be the same, or will be adjacent, to the polyhedron chosen atthe time t − . 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. P ERFORMANCE OF THE ALGEBRAIC REDUCTION
A. Diversity
It has recently been proved [19] that MIMO decoding based on LLL reduction followed by zero-forcingachieves the receive diversity. The following Proposition shows that algebraic reduction is equivalent toLLL reduction in terms of diversity for the case of transmit and receive antennas: Proposition 1.
The diversity order of the algebraic reduction method with ZF detection is .Proof: We suppose that the symbols s i , i = 1 , . . . , belong to an M -QAM constellation, with M = 2 m . Let E av be the average energy per symbol, and γ = E av N the SNR.For a fixed realization of the channel matrix H , equation (8) is equivalent to an additive channel withoutfading where the noise n is no longer white.We can compute the error probability using ZF detection conditioned to a certain value of H , and thenaverage over the distribution of H : P e ( γ ) = Z P e ( γ | H ) dH (20)With symbol by symbol ZF detection, P e ( γ ) is bounded by the error probability for each symbol: P e ( γ ) ≤ X i =1 P (( ˆs ) i = ( s ) i ) , Using the classical expression of P e in a Gaussian channel, for square QAM constellations ([15], § P (( ˆs ) i = ( s ) i ) ≤ s E av σ ( M − ! ≤ e − E av2( M − σ i where σ i is the variance for complex dimension of the noise component n i .We have seen in (10) that the trace of the covariance matrix of the new noise n is bounded by N | det( H ) | (cid:13)(cid:13) Φ − (cid:13)(cid:13) F (cid:13)(cid:13) E − l (cid:13)(cid:13) F , recalling that the Frobenius norm is submultiplicative. Thus σ i ≤ Cov( n ) ≤ CN | det( H ) | October 26, 2018 DRAFT4 because (cid:13)(cid:13) E − (cid:13)(cid:13) F = (cid:13)(cid:13) U H − (cid:13)(cid:13) F ≤ C O , see equation (19).Indeed if Φ is unitary, as in the case of the Golden Code, (cid:13)(cid:13) Φ − E − l (cid:13)(cid:13) F = (cid:13)(cid:13) E − l (cid:13)(cid:13) F ≤ C O .Finally, P e ( γ | H ) ≤ e − “ M − C ” | det( H ) | E av N = 16 e − c | det( H ) | γ In order to compute the error probability in equation (20), we need the distribution of | det( H ) | . It isknown [7, 4] that if H is gaussian with i.i.d. N (0 , entries (variance per real dimension ), the randomvariable | det( H ) | , corresponding to the determinant of the Wishart matrix HH H , is distributed as theproduct of two independent chi square random variables with and degrees of freedom respectively.Consider two random variables X ∼ χ (2) , Y ∼ χ (4) : their joint probability distribution function is p X,Y ( x, y ) = 18 ye − x − y x, y > Then the cumulative distribution function of Z = 2 | det( H ) | = √ XY is F Z ( z ) = P {√ XY ≤ z } = Z Z √ xy ≤ z p X,Y ( x, y ) dxdy From the invertible change of variables u = y , v = √ xy with Jacobian J = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − v u vu (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − vu we obtain F Z ( z ) = Z z Z ∞ p X,Y (cid:18) v u , u (cid:19) | J | dudv = Z z v (cid:18)Z ∞ e − v u − u du (cid:19) dv,p Z ( z ) = ∂F Z ( z ) ∂z = z Z ∞ e − z u − u du = z K ( z ) , where K is the modified Bessel function of the second kind. Finally, P e ( γ ) ≤ E h e − c ′ γZ i = 16 Z ∞ z K ( z ) e − c ′ γz dz == 16 c ′ γ ) + 2 π ( c ′ γ ) ∞ X k =0 c ′ γ ) k Γ( k + )Γ( k + 1) (cid:18) Ψ (cid:18) k + 52 (cid:19) − Ψ( k + 1) − c ′ γ ) (cid:19)!! , 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 . October 26, 2018 DRAFT5
B. Some remarks about complexity
The length of the algorithm described in Section IV-B is related to the initial distance ρ ( h − ( J ) , J ) = (cid:13)(cid:13) h − (cid:13)(cid:13) F = k h k F = (cid:13)(cid:13)(cid:13)(cid:13) H √ det H (cid:13)(cid:13)(cid:13)(cid:13) F = k H k F | det( H ) | In order to have more information about the distribution of this distance, one has to find the distributionof the random variable k H k F | det( H ) | . From [4], we learn that H is unitarily similar to e H = 12 X Y Z , where X ∼ χ (4) , Y , Z ∼ χ (2) and X, Y, Z are independent. Therefore k H k F | det( H ) | = (cid:13)(cid:13)(cid:13) e H (cid:13)(cid:13)(cid:13) F (cid:12)(cid:12)(cid:12) det( e H ) (cid:12)(cid:12)(cid:12) = X + Y + Z XZ We want to find the distribution of the random variable T = X + Y + Z XZ knowing the distributions of X, Y, Z : p X ( x ) = x e − x , p Y ( y ) = ye − y , p Z ( z ) = ze − z Their joint probability distribution is p X,Y,Z ( x, y, z ) = 12 x yze − ( x + y + z ) , and the distribution of T is given by p T ( t ) = ∂∂t F T ( t ) = ∂∂t Z Z x y z xz ≤ t p X,Y,Z ( x, y, z ) dxdydz == ∂∂t Z ∞ Z x ( t + √ t − x ( t − √ t − Z √ txz − x − z x yz e − x y z dydzdx With the change of variables t = cosh( u ) , w = y this integral becomes ∂∂u Z ∞ Z x (cosh( u )+sinh( u )) x (cosh( u ) − sinh( u )) Z u ) xz − x − z x z e − x z w dwdzdx ! ∂u∂t == ∂∂u tanh ( u ) ∂u∂t = 12 √ t − t The following example shows that the distance ρ ( h − ( J ) , J ) is mostly concentrated near the origin: Example.
In Section VI, we will see that the minimum of the distances between J and the vertices of P for the Golden Code is R min = arccosh(1 . · · · ) = 1 , · · · . From the distribution of T , we findthat the probability that ρ ( J, h − ( J )) > R min is approximately . : in most cases h − ( J ) is already October 26, 2018 DRAFT6 −7 −6 −5 −4 −3 −2 −1 SNR F E R MLMMSE−GDFE + AR + ZF−DFEMMSE−GDFE+AR+ZFAR+ZFZF
Figure 2. Performance of algebraic reduction followed by ZF or ZF-DFE decoders using -QAM constellations. contained in P or in one of the neighboring polyhedra, and the algorithm stops after one step! Moreover,the probability that ρ ( J, h − ( J )) > R min is of the order of − , which is negligible since for practicalvalues of the SNR the probabilities of error for ML detection are typically of the order of − at best. C. Simulation results
Figure 2 shows the performance of algebraic reduction followed by ZF and ZF-DFE decoding comparedwith ML decoding using -QAM constellations. One can verify that the slope of the probability of error inthe case of algebraic reduction with ZF detection (without preprocessing) is very close to − , confirmingthe result of Proposition 1 concerning the diversity order.One can add MMSE-GDFE left preprocessing to solve the shaping problem for finite constellations [12]in order to improve this performance. With MMSE-GDFE preprocessing, algebraic reduction is within . and . from the ML using ZF and ZF-DFE decoding, at the FER of − .In the -QAM case, the loss is of . and . respectively for ZF and ZF-DFE decoding at theFER of − (Figure 3). In the same figure we compare algebraic reduction to LLL reduction usingMMSE-GDFE preprocessing. The two performances are very close; with ZF-DFE decoding, algebraicreduction has a slight loss ( . ). On the contrary, with ZF decoding, algebraic reduction is slightlybetter ( . gain), showing that the criterion (9) is indeed appropriate for this decoder.Numerical simulations also evidence that the average complexity of algebraic reduction is low. InSection IV-B we have seen that each step of the unit search algorithm requires only a few operations.Table III shows the actual distribution of the number of steps in the unit search algorithm. The data October 26, 2018 DRAFT7 −4 −3 −2 −1 SNR F E R MLMMSE−GDFE + LLL + ZFMMSE−GDFE + LLL + ZF−DFEMMSE−GDFE + AR + ZFMMSE−GDFE + AR + ZF−DFE
Figure 3. Comparison of algebraic reduction and LLL reduction using MMSE-GDFE preprocessing combined with ZF orZF-DFE decoding with -QAM constellations. Table IIIN UMBER OF STEPS OF THE SEARCH ALGORITHM . average number of steps distribution of the number of steps > . .
2% 39 .
4% 16 .
0% 4 .
8% 1 .
2% 0 .
2% 3 . · − % 7 . · − % 2 . · − % 10 − % 0 refers to a computer simulation for the Golden Code using a ZF decoder, for 16-QAM constellations, forthe transmission of codewords. (Clearly this distribution does not depend on the SNR.) The averagelength of the algorithm is less than .VI. F INDING THE GENERATORS
In this Section, we describe a method to find a presentation for the group O of units of norm , andthe corresponding polyhedron P . The computations are carried out in detail for the Golden Code. A. Kleinian groups and Dirichlet polyhedra
We introduce some terminology that will be useful later:
Definition 1 ( Kleinian groups ) . Let Γ be a subgroup of the projective special linear group P SL ( C ) acting on the hyperbolic space H .- If Γ is discrete , that is if the subspace topology on Γ is the discrete topology, Γ is called a Kleiniangroup . Remark that then Γ is countable. October 26, 2018 DRAFT8 - The orbit of a point x ∈ H is the set { g ( x ) | g ∈ Γ } .- A fundamental set for the action of Γ is a subset of H containing exactly one point for every orbit.- A fundamental domain for Γ is a closed subset D of H such thata) S g ∈ Γ g ( D ) = H ,b) If g ∈ Γ \ { } , the interior of D is disjoint from the interior of g ( D ) .c) The boundary of D has measure .- Γ is called cocompact if it admits a compact fundamental domain; we say that Γ has finite covolume if it admits a fundamental domain with finite volume.- If Γ has finite covolume, and D and D are fundamental domains for Γ , then Vol( D ) = Vol( D ) < ∞ ([13], Lemma 1.2.9).In the case of a Kleinian group Γ , one can obtain a fundamental domain that is a hyperbolic polyhedron[13, 2]. This polyhedron can be obtained as an intersection of hyperbolic half-spaces.For any pair of distinct points Q, Q ′ ∈ H , the set of points equidistant to Q and Q ′ with respect to ρ is a hyperbolic plane, called the bisector between Q and Q ′ , which divides H into two open convexhalf-spaces, one containing Q and the other containing Q ′ . Given g ∈ Γ , let D g ( Q ) = { P ∈ H | ρ ( Q, P ) ≤ ρ ( g ( Q ) , P ) } (21)the closed half-space of the points that are closer to Q than to g ( Q ) . If Q is not fixed by any nontrivialelement of Γ , the Dirichlet fundamental polyhedron of Γ with center Q is defined as the intersection ofall the bisectors corresponding to nontrivial elements: P Γ = \ g ∈ Γ ,g = D g ( Q ) (22)The definition (22) cannot be used directly to compute the polyhedron, since we ought to intersect aninfinite number of bisectors. Let B ( Q, R ) denote the closed ball with center Q and radius R , and let D R ( Q ) = \ { D g ( Q ) | g = , g ( Q ) ∈ B ( Q, R ) } (23)If P Γ is compact, it has finite diameter, so there exists R > such that P Γ = D R ( Q ) . B. Poincar´e’s theorem
From the Dirichlet polyhedron of a Kleinian group one can obtain a complete description of the latter,including generators and relations. In fact, a famous theorem due to Poincar´e establishes a correspondence
October 26, 2018 DRAFT9 between a set of generators of the group and the isometries which map a face of the polyhedroninto another face, called side-pairings . The sequences of side-pairings which send an edge into itselfcorrespond to a complete set of relations among the generators.A complete exposition of Poincar´e’s theorem in the general case can be found in [6]. We only need arather weak version of the theorem that we state as follows:
Theorem 2 ( Poincar´e’s polyhedron theorem ) . Let P be a hyperbolic polyhedron in H with finitelymany faces. Let F denote the set of faces of P , and suppose that: a) [ Metric condition ] For every pair of disjoint faces of P , the corresponding geodesic planes haveno common point at infinity. b) [ Side-pairings ] There exist two maps R : F → F , U : F →
Isom( H ) such that: - ∀ F ∈ F , R ( F ) = F - If R ( F ) = F ′ , U ( F ) = u F maps F ′ onto F , sending distinct vertices into distinct vertices,and distinct faces into distinct faces, and maps the interior of P outside of P . Moreover u F ′ = ( u F ) − R is called a side-pairing for P . c) [ Cycles ] For each edge E of P , there is a cycle starting with E , that is a sequence of the form [ E , . . . , E n +1 ] , where E i , i = 1 , . . . , n + 1 are edges of P , and ∀ i ∈ { , . . . , n } there exists agenerator u ( i ) ∈ U ( F ) such that u ( i ) ( E i ) = E i +1 , and E n +1 = E . Moreover, we suppose that u = u ( n ) ◦ · · · ◦ u (1) is a rotation through an angle πm , m ∈ Z + , and that its restriction to E isthe identity.Consider the group Γ generated by U ( F ) . Then P is a fundamental domain for the action of Γ on H . The proof of this theorem is a special case of the proof of Theorem 4.14 in [6].
C. The structure of O We now have all the necessary background to find a fundamental domain, and thus a set of generators,for P O = O / { , − } . The following theorem shows that P O is a Kleinian group, and describes itsDirichlet polyhedron (see [5] or [20]): Theorem 3.
Let A be a quaternion algebra over a number field K such that a) K has exactly one pair of complex embeddings b) A is ramified at all the real places, that is A ⊗ Q K ν is a division ring for every real place ν of K . October 26, 2018 DRAFT0
Let O be an order of A . Then: - P O is a Kleinian group. - P O has finite covolume and its Dirichlet polyhedron has finitely many faces. - P O is cocompact if and only if A is a division ring. Remark that conditions (a) and (b) of the theorem are verified since K = Q ( i ) is an imaginary quadraticnumber field and thus has a pair of complex embeddings and no real embeddings.Thus P O admits a compact fundamental polyhedron P O of the form (22), with finitely many facesand finite volume. This volume is known a priori and only depends on the choice of the algebra A (see[13], p.336): Theorem 4 ( Tamagawa Volume Formula ) . Let A be a quaternion algebra over K such that A ⊗ Q R ∼ = M ( C ) . Let O be a maximal order of A . Then the hyperbolic volume Vol( P O ) = 14 π ζ K (2) | D K | Y p | δ O ( N p − In the previous formula, ζ K denotes the Dedekind zeta function relative to the field K , D K is thediscriminant of K , δ O is the discriminant of O , p varies among the primes of O K , and N p = [ O K : pO K ] ,where O K is the ring of integers of K . Example ( The Golden Code ) . In the case of the Golden Code algebra, D Q ( i ) = − , and δ ( O ) = 5 Z [ i ] .The only primes that divide the discriminant of the maximal order are (2 + i ) and (2 − i ) , both withalgebraic norm . In conclusion, Vol( P O ) = 8 ζ Q ( i ) (2)164 π = 32 ζ Q ( i ) (2) π = 4 , · · · (24)since ζ Q ( i ) (2) = 1 . · · · . Remark 5.
We have seen in section IV-B that a smaller polyhedron P results in a better average distancebetween ˆ u ( J ) and h − ( J ) and a better approximation. So the algebraic codes such that Vol( P ) is smallare better suited for the method of algebraic reduction. The Dedekind zeta function is defined as ζ K ( s ) = P I ([ O K : I ]) − s , where I varies among the proper ideals of O K . October 26, 2018 DRAFT1
D. Computing the Dirichlet polyhedron for O We suppose here that we already have an estimate of the volume of P O , given by Theorem 4. Forthis reason our strategy to find the Dirichlet polyhedron differs slightly from that described in [2]. Theidea is to compute the sequence D R ( Q ) defined by (23) for an increasing sequence of values of R , untilwe find R such that D R ( Q ) is compact. If the hypotheses of Poincar´e’s Theorem are verified for a setof side-pairings belonging to O , D R ( Q ) is a Dirichlet polyhedron for some subgroup of O . To checkwhether this subgroup coincides with O it is sufficient to estimate of the volume of D R ( Q ) .Following [2], we take as our base point Q the point J defined in (15). One needs to check that J isnot fixed by any nontrivial element of P O . Remark 6.
As pointed out in [2], if on the contrary J is fixed by some nontrivial element, one needsfirst to compute a fundamental domain for the stabilizer Γ J of J (the subgroup of elements that fix J ),and then intersect it with T g ∈ Γ \ Γ J D g ( J ) .Because of the property (16), in order to find D R ( J ) we only need to intersect the bisectors corre-sponding to elements of O with square Frobenius norm less or equal to R ) . Since O can beidentified with a discrete lattice in C using the map φ defined in (4), and the Frobenius norm correspondsto the Euclidean norm in C , clearly there is only a finite number of these elements.Since cosh is increasing on the positive half-line, in order to find the half-space D g ( J ) one can solvethe inequality cosh( ρ ( Q, J )) ≤ cosh( ρ ( Q, g ( J ))) . For a general g = a bc d , g ( J ) = (cid:18) b ¯ d + a ¯ c | d | + | c | , | d | + | c | (cid:19) , and the corresponding half-space has equation ( C − x + ( C − y + ( C − r − Ax − By + A + B + 1 C − ≥ , where A = ℜ ( b ¯ d + a ¯ c ) , B = ℑ ( b ¯ d + a ¯ c ) , C = | d | + | c | . Its boundary is a sphere of center (cid:16) AC − , BC − , (cid:17) and square radius C (cid:16) A + B ( C − + 1 (cid:17) . Remark that if we change the sign of the pair a, d or b, c , the radiusdoesn’t change, while the center is reflected with respect to the origin.If a ball or complementary of a ball (according to the sign of C − ) in the list (23) is already containedin the intersection, we can discard the corresponding element of the group. Since all the spheres havecenter on the plane { r = 0 } , in order to determine whether a sphere is contained in another we onlyneed to consider their intersections with this plane. October 26, 2018 DRAFT2
Figure 4. The projection of the bisectors on the plane { r = 0 } . Figure 5. The intersection of the spheres (the picture showsonly half of the space for better understanding). E. Computing the generators for the Golden Code
We now apply the method described in the previous Section to the maximal order O of the algebra ofthe Golden Code. One can easily verify that in this case J is not fixed by any nontrivial element of O .Considering the elements g ∈ O such that k g k F ≤ by computer search, we find that P = D R ( J ) iscompact with R = arccosh (cid:0) (cid:1) , since it doesn’t intersect the plane “at infinity” { r = 0 } . Table I liststhe elements { u i , u − i } , i = 1 , . . . , of the group that are necessary to obtain P . The equations of thecorresponding spheres, the bisectors S ( u i ) = D u i ( J ) , S ( u − i ) = D u − i ( J ) , i = 1 , . . . , (see definition (21)) can be found in Table IV.The vertices of P are the intersections of all the triples of spheres S ( u i ) : V i , V ′ i = π ′ ( V i ) , V ′′ i = π ′′ ( V i ) , V ′′′ i = π ′′′ ( V i ) , i = 1 . . . , Here π ′ , π ′′ and π ′′′ denote the reflections with respect to the plane { y = x } , the plane { y = − x } andthe line { x = 0 , y = 0 } respectively. V = √ , √ − , q
33 + 11 √ ! = S ( u ) ∩ S ( u ) ∩ S ( u ) ,V = (cid:18) θ , − , θ (cid:19) = S ( u ) ∩ S ( u ) ∩ S ( u ) , October 26, 2018 DRAFT3
Table IVB
ISECTORS unit center radius int/ext u (0 , , θ I u − (0 , ,
0) ¯ θ E u (1 , ,
0) 1 E u − ( − , − ,
0) 1 E u ( − θ, − θ, θ E u − (¯ θ, ¯ θ, − ¯ θ E u ( θ, θ, θ E u − ( − ¯ θ, − ¯ θ, − ¯ θ E u “ − √ , − − √ , ” √ (7 − √ E u − “ √ − , √ , ” √ (7 − √ E u “ √ , − √ , ” √ (7 + √ E u − “ − √ − , − √ , ” √ (7 + √ E u “ − − √ , − √ , ” √ (7 − √ E u − “ √ , √ − , ” √ (7 − √ E u “ − √ , √ , ” √ (7 + √ E u − “ − √ , − √ − , ” √ (7 + √ E –1–0.500.51–1 –0.5 0 0.5 10.40.60.8 Figure 6. A schematic representation of the polyhedron P (in the picture, the edges have been replaced by straightlines). Figure 7. The projection of the polyhedron P on the plane { r = 0 } . October 26, 2018 DRAFT4 V = (cid:18) θ , − ¯ θ , (cid:19) = S ( u ) ∩ S ( u ) ∩ S ( u − ) ,V = √
520 + 12 , √ − , r ! = S ( u ) ∩ S ( u ) ∩ S ( u − ) V = √ , √ − , q − √ ! = S ( u − ) ∩ S ( u − ) ∩ S ( u − ) ,V = (cid:18) , − ¯ θ , ¯ θ (cid:19) = S ( u − ) ∩ S ( u ) ∩ S ( u − ) The faces of P correspond to portions F ( u ) of the spheres S ( u ) , with u one of the units in Table I.The projection of the faces of P on the plane { r = 0 } is shown in Figure VI-E.As explained in Section VI-D, in order to prove that P is a Dirichlet polyhedron for O , we will firstshow that it is a fundamental domain for some subgroup Γ of O using Poincar´e’s Theorem. Comparingthe volume of P with the value (24), we will find that Γ = O .The metric condition in Poincar´e’s Theorem can be verified given the equations of the spheres (see alsoFigure 4).Define a side-pairing as follows: U ( F ( u )) = u , R ( F ( u )) = F ( u − ) for every u in Table I. The actionof the generators on the faces and vertices is summarized in Table VI, and it is not hard to see that itsatisfies all the conditions in the theorem. In fact every face F ( u ) = P ∩ u ( P ) ⊂ S ( u ) . Remark thatan isometry between polygons with the same number of vertices, sending distinct vertices in distinctvertices, must be onto.In order to check that the cycle condition of Theorem 2 holds, we need to compute the minimal relationsor “cycles” between the generators, by finding the sequences of edges of P of the form [ E , . . . , E n +1 ] ,such that u ( i ) ( E i ) = E i +1 , and E n +1 = E . As F (( u ( i ) ) − ) must contain E i , there are only two possiblechoices for u ( i ) , corresponding to the two faces containing the edge E i .Given such a sequence, u (1) · · · u ( n ) is an element of finite order in O , that is ( u (1) · · · u ( n ) ) k = forsome k . (Remark that every cyclic permutation of the sequence [ E , . . . , E n +1 ] gives rise to a new cycle.)Actually it is necessary to “lift” the relation from P SL ( C ) to SL ( C ) . We also require our sequencesto be irreducible , that is u ( i +1) = ( u ( i ) ) − for all i .In this way we obtain a decomposition of the set of edges of P into cycles. The action of the generatorson the faces is summarized in Table VI; the cycles are described in Table V.A complete set of relations is listed in Table II. Except for the first four, the products correspond to theidentity in P SL ( C ) (thus, a trivial rotation). By computing the eigenvalues of u , u , u u and u u − , October 26, 2018 DRAFT5
Table VC
YCLES FOR THE EDGES OF P . V ′′ V ′′′ u −→ V ′′ V ′′′ V V ′ u −→ V V ′ V V ′′ u −→ V ′′′ V ′ u −→ V V ′′ V V ′′ u − −−−→ V ′′′ V ′ u −→ V V ′′ V V u − −−−→ V ′′′ V u − −−−→ V ′′′ V ′′′ u − −−−→ V V V V u − −−−→ V ′′′ V ′′′ u − −−−→ V V u −→ V V V ′ V ′ u −→ V ′′ V ′′ u −→ V ′′ V ′′ u −→ V ′ V ′ V ′ V ′ u −→ V ′ V ′ u − −−−→ V ′′ V ′′ u − −−−→ V ′ V ′ V V ′ u − −−−→ V V ′ u −→ V ′′′ V ′′ u − −−−→ V ′′′ V ′′ u −→ V V ′ V ′ V ′ u −→ V ′′ V ′′ u −→ V ′ V ′ u −→ V ′′ V ′′ u −→ V ′ V ′ u − −−−→ · · ·· · · V ′′ V ′′ u − −−−→ V ′ V ′ u −→ V ′′ V ′′ u − −−−→ V ′ V ′ V V u − −−−→ V ′′′ V ′′′ u − −−−→ V V u − −−−→ V ′′′ V ′′′ u − −−−→ · · · V V u −→ · · · V ′′′ V ′′′ u −→ V V u − −−−→ V ′′′ V ′′′ u −→ V V Table VIA
CTION OF THE GENERATORS ON THE VERTICES OF P . u ( S ( u − )) = S ( u ) u ( V ) = V ′′′ , u ( V ′ ) = V ′′ , u ( V ′′ ) = V ′ , u ( V ′′′ ) = V , u ( V ) = V ′′′ , u ( V ′ ) = V ′′ , u ( V ′′ ) = V ′ , u ( V ′′′ ) = V u ( S ( u − )) = S ( u ) u ( V ′ ) = V ′′ ) , u ( V ′ ) = V ′′ ) , u ( V ′ ) = V ′′ , u ( V ′′′ ) = V , u ( V ′′′ ) = V , u ( V ′′′ ) = V u ( S ( u − )) = S ( u ) u ( V ′′ ) = V ′′ , u ( V ′′′ ) = V ′′′ , u ( V ′′ ) = V ′′ , u ( V ′′′ ) = V ′′′ u ( S ( u − )) = S ( u ) u ( V ) = V , u ( V ′ ) = V ′ , u ( V ′ ) = V ′ , u ( V ) = V u ( S ( u − )) = S ( u ) u ( V ′ ) = V ′′ , u ( V ′ ) = V ′′ , u ( V ′ ) = V ′′ , u ( V ′ ) = V ′′ u ( S ( u − )) = S ( u ) u ( V ′′′ ) = V , u ( V ′′′ ) = V , u ( V ′′′ ) = V , u ( V ′′′ ) = V u ( S ( u − )) = S ( u ) u ( V ) = V ′′′ , u ( V ) = V ′′′ , u ( V ) = V ′′′ , u ( V ) = V ′′′ u ( S ( u − )) = S ( u ) u ( V ′′ ) = V ′ , u ( V ′′ ) = V ′ , u ( V ′′ ) = V ′ , u ( V ′′ ) = V ′′ October 26, 2018 DRAFT6 we find that they are indeed conjugated to rotations of an angle π around the axis { x = 0 , y = 0 } . We have thus shown that P is a Dirichlet polyhedron for some subgroup Γ of O . But if Γ were aproper subgroup, the volume of P would be a multiple of the volume of the fundamental polyhedron for O that we computed in (24), that is it should be at least · . · · · = 9 . · · · .So the last step of the proof that P is a fundamental polyhedron for O is the following: Lemma 5.
Vol( P ) < . · · · . The proof of this fact is rather tedious and is reported in the Appendix.
Remark 7.
From the coordinates of the vertices of P , one finds that the radius of the smallest hyperbolicsphere containing P is R max = arccosh(2 . · · · ) = 1 . · · · , while the minimum of the distances between J and the vertices of P is R min = arccosh(1 . · · · ) = 1 . · · · VII. C
ONCLUSIONS
In this paper we have introduced a right preprocessing method for the decoding of space-time blockcodes based on quaternion algebras, which allows to improve the performance of suboptimal decodersand reduces the complexity of ML decoders.The new method exploits the algebraic structure of the code, by approximating the channel matrix witha unit in the maximal order of the quaternion algebra. Our simulations show that algebraic reductionand LLL reduction have similar performance. However in the case of slow fading, unlike LLL reduction,algebraic reduction requires only a slight adjustment of the previous approximation at each time block,without needing to perform a full reduction.In future work we will deal with the generalization of algebraic reduction to higher-dimensional spacetime codes based on cyclic division algebras. g ∈ SL ( C ) is called elliptic , and is a rotation around a fixed geodesic, if and only if tr( g ) ∈ R and | tr( g ) | < , see [5],Prop. 1.4. If its eigenvalues are e iβ , e − iβ , then the angle of rotation is β . October 26, 2018 DRAFT7 A PPENDIX
A. Proof of Lemma 5
In order to prove that the volume of P is smaller than the required constant, we can compute thevolume of the hyperbolic polyhedron Q enclosed by S ( u ) , the plane { r = − ¯ θ } , and the spheres S ( u ) , S ( u − ) , S ( u − ) , S ( u ) , S ( u ) , S ( u − ) , S ( u − ) . Clearly Q ⊃ P . Recalling the definition of thehyperbolic volume in (13), the volume of the spherical sector T enclosed by S ( u ) and n r = − ¯ θ o is Z θ − ¯ θ π ( θ − r ) r dr = π (cid:18) − − ln( θ ) + 2 θ + ln (cid:18) − ¯ θ (cid:19)(cid:19) = 36 . · · · To this volume we must subtract the volume of the intersection of T with the chosen spheres.From the expression for the area of the intersection of two circles of radii R and R whose centershave distance d [21] A ( R , R , d ) = R arccos (cid:18) d + R − R dR (cid:19) + R arccos (cid:18) d + R − R dR (cid:19) ++ 12 p ( − d + R − R ) p ( d + R − R )( d − R + R )( d + R + R ) , we obtain the area of the horizontal sections of T ∩ S ( u ) . Since R = √ θ − ¯ r , R = √ − ¯ r are the radii of S ( u ) ∩ { r = ¯ r } , S ( u ) ∩ { r = ¯ r } respectively, and the distance between the centers is d = √ , we find A ( R , R , d ) = π (1 − ¯ r ) + (¯ r −
1) arccos √ θ − √ − ¯ r ! ++ ( θ − ¯ r ) arccos √
24 1 + θ √ θ − ¯ r ! − p − θ − θ − v , which is defined for ¯ r ≤ √ √ . In conclusion, Vol(
T ∩ S ( u )) = Vol( T ∩ S ( u − )) = Z √ √ − ¯ θ A ( R , R , d ) r dr = 5 . · · · Proceeding in the same way, one can compute
Vol(
T ∩ S ( u )) = Vol( T ∩ S ( u )) = 5 . · · · , Vol (cid:0) ( T ∩ S ( u − )) \ ( S ( u − ) ∩ ( S ( u ) ∪ S ( u − )) (cid:1) = 2 . · · · , Vol (cid:0) ( T ∩ S ( u − )) \ ( S ( u − ) ∩ ( S ( u ) ∪ S ( u − )) (cid:1) = 0 . · · · October 26, 2018 DRAFT8
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