A Construction of a Space Time Code Based on Number Theory

Transcription

1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 3, MARCH A Construction of a Space Time Code Based on Number Theory Mohamed Oussama Damen, Associate Member, IEEE, Ahmed Tewfik, Fellow, IEEE, and Jean-Claude Belfiore, Member, IEEE Abstract We construct a full data rate space time (ST) block code over = transmit antennas and = symbol periods, and we prove that it achieves a transmit diversity of over all constellations carved from [ ]. Further, we optimize the coding gain of the proposed code and then compare it to the Alamouti code. It is shown that the new code outperforms the Alamouti code at low and high signal-to-noise ratio (SNR) when the number of receive antennas. The performance improvement is further enhanced when or the size of the constellation increases. We relate the problem of ST diversity gain to algebraic number theory, and the coding gain optimization to the theory of simultaneous Diophantine approximation in the geometry of numbers. We find that the coding gain optimization is equivalent to finding irrational numbers the furthest from any simultaneous rational approximations. Index Terms Block codes, diversity methods, lattices, maximum-likelihood (ML) decoding. I. INTRODUCTION The large capacity of multiple-antenna systems and the desire to transmit at higher data rates with better performance over wireless channels have motivated much research on signal processing over multiple transmit receive antennas [] [4]. A higher data rate can be achieved by transmitting symbols simultaneously from M transmit antennas []. In a fading environment, where the receiver cannot recover an information symbol affected by a deep fade, one improves the performance by providing the receiver with different replicas of the transmitted symbols. This can be done with multireceive antennas with sufficient spacing so that the fades over each receiver are independent from the fades on the other receivers. The diversity can also be obtained at the transmitter by spacing the transmit antennas sufficiently and introducing a code between the transmitted symbols over M transmit antennas (space) and T symbol periods (time) [], [3]. Improving the performance over wireless channels can be done by transmit and/or receive diversity techniques [], [3], [5]; in a system of M transmit and N receive antennas the maximum achievable transmit diversity is M and the total achievable diversity is MN. In [4], a multiantenna prototype, called Vertical Bell Labs Layered Space Time (V-BLAST), with a very high spectral efficiency is proposed, but the detection algorithm proposed in [4] does not fully exploit the receive diversity. In [6], Damen et al. exploited the lattice representation of a multiantenna system in order to apply the sphere decoder and fully exploit the receive diversity giving a large improvement over [4]. Alamouti [] proposed a space time (ST) block code of rate symbol per channel use (PCU) that achieves Manuscript received February, 00; revised September 5, 00. M. O. Damen is with the Department of Electrical and Computer Engineering, University of Alberta, ECERF W-073, Edmonton, AB T6G V4, Canada ( damen@ee.ualberta.ca). A. Tewfik is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN USA ( tewfik@ece.umn. edu). J.-C. Belfiore is with the COMELEC Department, Ecole Nationale Supérieure des Télécommunications (ENST) de Paris, 75634, Paris Cedex 3, France ( belfiore@com.enst.fr). Communicated by G. Caire, Associate Editor for Communications. Publisher Item Identifier S (0) a N diversity for M = transmit and N receive antennas, and has a linear processing optimal decoder. The latter scheme has been generalized for M in [3], where it has also been proved to be the unique scheme of rate one symbol PCU using complex constellations and linear decoder; for some other values of M >, ST codes have been proposed with rates of = and 3=4 symbol PCU. In [5], Damen et al. proposed a different approach to exploit the transmit diversity using rotated constellations and the Hadamard transform. The latter codes achieve the diversity MN and have a rate of one symbol PCU over real or complex constellations for M =or multiple of 4. The decoder used in [5] is the sphere decoder. However, for most values of M and N, the codes proposed in [], [3], [5] have small rates compared to the actual capacity of multiantenna systems []. In [7], ST codes with higher data rates than the ST codes in [], [3], [5] were proposed. However, the so-called linear and dispersive (LD) ST codes [7] do not necessarily satisfy the construction criteria in [8], [3]. In this correspondence, we relax the constraint of linear processing decoder and we propose an ST code for two transmit antennas which achieves the maximum transmit diversity [8], [3], and has a transmission rate of two symbols PCU. We present a comprehensive study of the proposed ST code properties based on number theory. It is shown that due to the lattice structure of the code, the maximum-likelihood (ML) decoder has a moderate complexity. The correspondence is organized as follows. In Section II, we give the system model and recall the construction criteria of block ST codes in [8], [3]. In Section III, we give preliminary definitions and results from number theory, which we will use to construct our ST code. Section IV describes the properties of the proposed ST code, where we prove it to satisfy the construction criteria in [8], [3] without data rate reduction. We consider the decoding in Section V. In Section VI, we give further results from number theory, where we relate the optimization of the coding gain of the proposed code to the simultaneous Diophantine approximation. Before concluding the correspondence, we give some simulation results in Section VII. II. PROBLEM FORMULATION Consider an information symbol vector s =(s;...;s q ) T ;q, where s j ;j=;...;q, belongs to a given constellation (pulse-amplitude modulation (PAM), quadrature-amplitude modulation (QAM), or phase-shift keying (PSK)). A block ST code associates with each information symbol vector s an M T matrix B(s) where M encoded symbols b mt (m = ;...;M) are transmitted simultaneously from all transmit antennas at time t (t =;...;T). There is a one-to-one correspondence between the information symbol vector s and B(s). When there is no confusion, we denote the block ST code by B. We normalize the transmitted power by M so that the total transmitted power is independent of M. At time t and over the nth receive antenna, one has M x nt = h nmb mt + wnt () m= where h nm are the fades between transmit antenna m and receive antenna n, and are assumed to be independent and identically distributed (i.i.d.) zero-mean complex Gaussian random variables with variance 0:5 per real dimension, and fixed during T time periods (quasi-static fading). w nt are assumed to be independent samples of a zero-mean complex Gaussian random variables with variance per dimension. Let X be the N T received signal matrix; H the N M channel matrix; and W the N T noise matrix. One has X = HB + W : () /0$ IEEE

2 754 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 3, MARCH 00 Over a quasi-static fading, minimizing the pairwise error probability (PEP) of the ML detection of s given that s 6= s has been sent is equivalent to satisfying [8], [3]. The Rank Criterion: The minimum rank r of B(s )0B(s ) taken over all distinct pairs (s ;s ) is the diversity gain and should be maximized. The Determinant Criterion: Let A = B(s ) 0 B(s ), then r the minimum of ( j= j )=r, taken over all distinct codeword pairs, is the coding gain and must be maximized, where j ;j=r; are the nonzero eigenvalues of AA H, with A H denoting the conjugate transpose of matrix A. For example, the Alamouti scheme is given by [] s 0s 3 B = G = s s 3 (3) where s ;s are complex information symbols (PSK, QAM), and s 3 denotes the complex conjugate of s. The code (3) satisfies the criteria above; in fact, its coding gain is A = js j + js j > 0: (4) min (s ; s )6=(0; 0) The ST code (3) transmits at a rate of one symbol PCU, and due to the orthogonality of the columns of G, The metric for ML decoding can be produced by linear processing []. III. AN ST CODE BASED ON NUMBER THEORY Before the construction of our ST code we give hereafter some necessary definitions from number theory [9] []. A. Preliminary Definitions From Number Theory Let ; ; ; and denote, respectively, the ring of rational integers, the field of rationals, the field of reals, and the field of complex numbers, and let i = p 0. Also, let [i] denote the ring of complex integers and let (i) denote the field of complex rational. Definition : Let. Then is called an algebraic number if there exists a polynomial P [X] such that P () = 0. We call an algebraic integer if, in addition, one can choose P to be monic (i.e., with leading coefficient equal to ). For algebraic, there exists a unique irreducible polynomial (X) [X] with leading coefficient which has as a root. We call (X) the minimal polynomial of. The degree of ; n is defined to be the degree of (X).If is complex, one can also define (X) over [ix]. Example : Let = e i=4. Then is a root of the irreducible polynomial (X) =X 4 + [X]. The degree of over is 4 and one has 4 = 0. The minimal polynomial of over [ix] is given by (X) =X 0 i, and has a degree over (i). Definition : We define the height of a polynomial P (X) [X] as the maximum modulus of its coefficients. Suppose that (X) [X] is the minimal polynomial of the algebraic number. If we multiply (X) by the least common multiple of the denominators of its coefficients, we obtain a primitive polynomial 0 (X) [X] such that 0 () =0(a polynomial in [X] is called primitive if there is no integer greater than which divides all of its coefficients). Then the height of is defined to be the height of 0 (X). It is easily seen that for algebraic, there exists an integer r>0such that r is an algebraic integer. Example : Take = e i=4 with (X) =X 4 + [X]. (X) [X] and it is primitive, so the height of equals. Definition 3: An algebraic number field is a field containing which, considered as a -vector space, is of a finite dimension. The dimension of over is called the degree of over and is denoted by [ : ]. It can be proved that given an algebraic number field of degree [ : ] over, then there exists an algebraic number of degree n = [ : ] such that = (). Such a number is called a primitive element. The set f; ;...; n0 g generates () and is called a basis of (). Each element (), has a unique representation n0 = r() = a l l with a l ;l=0n0. The roots of (X); = ; ;...; n are called the conjugates of. We define the size of = r() () as jj = max jn jr( j )j. Example 3: The roots of X 4 + are = e i=4 ; = 0e i=4 ; 3 = ie i=4 ; and 4 = 0ie i=4. The roots of X 0 i are = e i=4 and = 0e i=4. The algebraic number field generated by is l=0 () =fa + b + c + d 3 ;a;b;c;d when is considered as an extension of. = (i; ) =fe + f; e; f (i)g when is considered as an extension of (i). The set f; ; ; 3 g is a basis of = () over and the set f; g is a basis of = (i; ) over (i). Definition 4: If is not algebraic it is called transcendental. Example 4: = e; = e i ; and = are transcendental (Hermite, Lindemann) [9, p. 47]; but = e ik=n ;k;n is algebraic (Hermite) [9, Ch. ]. B. Code Construction In order to transmit at a full data rate and also satisfy the construction criteria [8], [3], we propose the following ST code: s + s (s 3 + s 4 ) B ; = p (5) (s 3 0 s 4 ) s 0 s with = and = e i, such that is a real parameter to be optimized. We suppose that the information symbol vector s = (s ;...;s 4 ) T belongs to a (TM = 4)-dimensional constellation C carved from [i] 4 (QAM, or PAM). We note that hereafter we use the Cartesian product sign in order to denote multidimensional constellations, while when using a specific constellation of size q, we put q before the constellation name; for example, if we have four symbols s ;...;s 4, such that each one belongs to the constellation S = q-qam, then s = (s ;...;s 4) T C = S 4 = q-qam 4. Independent of C, the coding gain over [i] 4 of our ST code (5) is B () = inf s6=(0;0;0;0) [i] = = inf s6=(0;0;0;0) [i] inf s6=(0;0;0;0) [i] det B ; B H ; = g s 0 s 3 0 s + s 4 3 T ~s (6) where ~s =(s ; 0s3; 0s;s 4) T ; =(;; ; 3 ) T. Over a finite constellation C [i] 4, the coding gain of B ; is C B() = min s=(=)(s 0s );s 6=s C T ~s B () (7) where multiplying the vector (s 0 s ) by = denotes the multiplication of each component of the vector by =. This is performed for normalization purposes. The inequality C B() B() is due to

3 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 3, MARCH C [i] 4. Our aim is to maximize B() C (maximum transmit diversity and coding gains) for every C [i] 4. IV. CODE PROPERTIES A. Primary Results on Diversity and Coding Gain We discuss in this section the choice of such that the ST code B ; has a maximum transmit diversity over all constellations carved from [i] 4. Proposition : ) If is an algebraic number of degree 4 over (i) then one guarantees the maximum transmit diversity over all constellations carved from [i] 4. ) If is an algebraic number of degree < over (i) then there exists a constellation C [i] 4 such that B() C =0. Proof: Let be an algebraic number of degree 4 over (i), then f; ; ; 3 g is a basis of (i; ) and, therefore, it is a free set, i.e., if 3 j=0 a j j =0; for a j (i); j=0;...; 3 then a 0 = a = a = a 3 =0. This guarantees that B() C 6= 0for all constellations carved from [i] 4. Now, suppose that the degree of <. Then there exists a polynomial of degree over (i) which has as a zero. Let (X) =X + p q [ix] be such a polynomial. Then = 0 p q (i), and hence = p q. There exist a finite constellation C [i] 4 that contains two vectors s =(p; q; 0; 0) 6= s (0p; 0q; 0; 0) such that (=)(s 0 s )= (p; q; 0; 0) T. Substituting in (7) gives C B() jp 0 q j =0: Example 5: Take = e i=8 then the minimal polynomial over (i) equals (X) =X 4 0 i, and is of degree 4. Hence, one guarantees that there are no linear combinations from (i) among the set of numbers f; e i=8 ;e i=4 ;e 3i=8 g, since this set is a basis of (i; ). The coding gain obtained for this value of equals 0:304 for C = 4-QAM 4. Example 6: Let =, and let C =4-QAM 4 be a constellation that contains the two vectors and Substituting s =(+i; +i; +i; +i) T s =(0+i; 0 +i; +i; +i) T : s =(=)(s 0 s )=(; ; 0; 0) T in (7) yields B() C j0 j =0. The first condition in Proposition is sufficient for the maximum transmit diversity, while the second condition is a necessary one. One cannot say much about algebraic numbers of degrees and 3. However, since in (6) one has only a linear combination of f; ; ; 3 g by sj ;j=;...; 4, with s j [i], one has the following result. Proposition : If is an algebraic number of degree over (i) and if (i), then one guarantees the maximum transmit diversity over all constellations carved from [i] 4. Proof: Let us suppose that there exists s 6= (0; 0; 0; 0) T [i] 4 such that ~s T =0 =s 0 s 3 0 s + s 4 3 0= s 0 s 0 s3 0 s4 : (8) Sincef; g is a free set over (i) and since (i), one has s 0 s =0 (9) and s3 0 s4 =0: (0) Since is of degree > ; is not a square in (i), i.e., there is no z z (i) such that = z z. Thus, in (9) one has s = s = s3 = s4 = 0 and therefore s = (0; 0; 0; 0) T which proves our proposition. Example 7: Let = e i=4 of degree over (i). One has = i (i), which is not a square in (i) since p i = p +i p does not belong to (i). It yields that B ; has a maximum transmit diversity over all constellations carved from [i] 4. For C = 4-QAM 4 and C = 6-QAM 4, we maximize C B() by computer search over varying from 0 to = with a step size of 0:00. The maximum coding gain for C = 4-QAM 4 is attained for = e i= and equals 0:369.ForC = 6-QAM 4 the maximum coding gain is attained for = e 0:5i and equals 0:059. Note that C B decreases when the size of the constellation increases (see Section VI). For example, C B(e i= )=0:037, and C B(e i=8 )=0:006 for C = 6-QAM 4.Itis noted that = e i=4 presents a local maximum of the coding gain for C = 4-QAM 4 and C = 6-QAM 4, with coding gain values of 0:0858 and 0:07, respectively. We note that for a four-dimensional constellation C with a very high spectral efficiency (where the size of the constellation #(C) ) the maximization of C B() by computer search is very complex. This, along with the perspective of the generalization of our scheme to more than two transmitters, motivates the research of algebraic techniques for choosing (see Section VI). B. Mutual Information We write the received signal () when using (5) with SNR equal as X = M HB ; + W x = vec (X) = = M M H 0 NM 0 NM H 8s+ vec (W ) Hs + w () where vec(x) arranges the matrix X in one column vector by putting its columns one after the other, 0 NM is an N M matrix with entries 0, and w = vec(w ) is a N column vector which represents the additive white Gaussian noise (AWGN). The noise vector w is component-wise independent with variance =0:5 per real dimension. The channel H is modeled as in () and H is a new N M channel matrix with 88 H = I M. 8 = p Definition 5: We say that the ST code B is information lossless if the capacity of the new precoded channel, H, obtained by considering

4 756 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 3, MARCH 00 the ST code as a part of the channel, has the same capacity of the original channel H. The difference between the capacities of the new and original channels represents the information loss of the ST code. The information loss can be measured either in bits PCU at a given SNR (decibels) or in decibels at the given spectral efficiency (bits PCU). The capacity of the new channel () is [], [7] C B (; N) = max E trr =M log det I N + M HR s H H () where R s is the covariance matrix of s, and the multiplicative factor = is for the normalization of the capacity PCU (T =). Proposition 3: The code B ; (5) is information lossless, where the capacity of the new channel in () equals C B (; N )=E log det IN + M HHH : (3) Proof: The proof is direct when computing () with R s = I M which maximizes the mutual information [7], [] C B (; N )= E log det I N + M HHH = E log det I N + M HHH where one uses the equality det diag I N + M HHH ;I N + to justify the second equation in (4). = E log det I N + M HHH (4) M HHH = det I N + M HHH The capacity of the new precoded channel of the code G equal to the capacity of an equivalent multiantenna system with M =N transmit and one receive antenna with an SNR scaled by N [7], [] C G (; N )=C(N; N; ) C(; ; N) (5) where C(; M; N ) is the channel capacity for SNR =, over M transmit and N receive antennas. The inequality in (5) becomes strict when N > [7]. When N =, both ST codes B ; and G are information lossless, but G outperforms B ; since the former ST code has a better coding gain ( compared to 0:369 for 4-QAM). When N>, Proposition 3 states that the Alamouti code has an information loss compared to the ST code B ; by virtue of C(N; N; ) <C(; ; N) (5). This also allows us to state that the information loss increases with N. Computing the quantities in (4) and (5), one obtains that the Alamouti code has an information loss of about.6 db compared to the code B ; at a spectral efficiency of 4 bits PCU when N =receive antennas. When N =0, the information loss of the Alamouti code is approximately 3.7 db at a spectral efficiency of 4 bits PCU, and it is more than 0 db at a spectral efficiency of 6 bits PCU. It is worth noting that the information loss gives a good approximation of the difference in performance between G and B ; obtained in Section VII, since both codes achieve the maximum transmit diversity. Note that when considering the information loss, an ST code can only worsen things or at most be harmless. The role of ST codes is to provide transmit diversity, and/or to simplify the ML decoding by introducing structure in the new channel matrix as done by the Alamouti code or the codes from orthogonal design [3]. When N =,we are in the lucky situation where the Alamouti code is information lossless, while achieving the maximum transmit diversity and coding gains. However, as it is impossible to have information lossless ST codes from Fig.. The ST code B versus the Alamouti scheme and the code LD with N = receive antennas. orthogonal design when N> [3], [7], the ST code B ; is an advantageous alternative to the Alamouti code since it is information lossless and it achieves the maximum transmit diversity with optimized coding gain. The drawback of using the code B ; is the increase of the decoding complexity. Remark : Note that the ST code B ; can be obtained from the code LD [7] (and vice versa) by means of a unitary transformation s 0! 9s with 9 = diag(; ;;) (6) with and as in (5). Both ST codes B ; and LD are information lossless. The difference with [7] is that our ST code satisfies also the rank and determinant criteria [8], [3] which boosts the performance of B ; compared to the code LD [7] (see Fig. ). Note that the code LD equal to B ;, with =. As noticed in Example 6, the code LD does not satisfy the rank and determinant criteria. C. Constellation Shape We study here some practical properties of the ST code (5). Over the first transmit antenna and the first time interval and also the second transmit antenna and the second time interval, the code B ; transmits signal points from Q = p fs + s ;s ;s Sg (7) where S can be QAM or PAM modulation. Over the first transmit antenna and the second time interval and also the second transmit antenna and the first time interval, the code B ; transmits signal points from Q = fs; s Qg, with = = e i. While the average transmitted energy of the original constellation S is preserved when using the code B ;, the dynamic range of each transmitted symbol increases considerably. A high dynamic range may cause operating in the nonlinear region of power amplifiers due to a high peak-to-average power ratio (PAPR). We note the similarity of our problem with that of multicarrier transmission (orthogonal frequency-division multiplexing; OFDM) where one reduces the PAPR by the use of concatenating error

5 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 3, MARCH correcting codes (see for example [3] and [4] and references therein). The use of similar techniques as in OFDM to reduce the PAPR for the code B ; is under investigation. V. DECODING Consider the received signal (). Assuming that there is sufficient spacing between each of the M transmit antennas (and likewise between each of the N receive antennas), it can be proved that the new channel matrix H is almost surely (a.s.) full rank. The latter property allows to apply the sphere decoder on the real representation of () for N, which can be seen as a lattice point in TN [6]. The principle of the sphere decoder is to limit the minimization of the ML metric to the lattice points enclosed inside a sphere of radius C. It takes advantage of the regular lattice structure in order to enumerate the lattice points enclosed by the sphere. Each time a lattice point is found inside the sphere, one decreases the sphere radius accordingly [6], [5]. The sphere decoder reaches ML performance and has a polynomial complexity in TM, linear in N and nearly independent of the rate [6], [5], [6]. For N =, one can apply the generalized sphere decoder and reaches near ML performance, but with an additional complexity proportional to the square of the constellation size [7]. When the number of receive antennas N M =, one can use a suboptimal scheme based on successive interference cancellation (SIC) such as [4], [8] to perform the decoding of (). Because in such a situation, the performance of the suboptimal scheme approaches the ML decoding performance with a lower complexity [4], [8]. VI. RELATING THE PROBLEM OF MAXIMIZING THE CODING GAIN TO NUMBER THEORY AND LATTICES Here, we relate the maximization of B() C (7) to the simultaneous Diophantine approximation of complex numbers by algebraic numbers. This will allow for the derivation of lower and upper bounds on B(). C Next, we relate the considered problem to lattices, and propose a heuristic approach to choose when the size of the constellation is very large. The interested reader might consult [9, Ch. ], [0, Ch. 5], or [, Ch. ]. First, let us define the quantity = s 0 s3 0 s + s4 3 (8) for s j [i]; j=;...; 4 and algebraic of degree n. By definition, B() C is the minimum of (=) over s =(s ;...;s 4) T =(=)(s 0 s ); s 6= s C: Number theory allows us to determine how small we can make in terms of and the amplitudes and degrees of s ;...;s 4. Thus, one expects to determine the order of magnitude of C B() in terms of and C. Let us recall that our aim is to choose that maximizes the minimum of over a given four-dimensional constellation C. A. Number Theory Definitions and Results Definition 6: The Simultaneous Diophantine Approximation: Given rationals ;...; n ;> 0, and an integer > 0, decide if there exist integers p ;...;p n and an integer q> 0 such that q and jq j 0 p jj <for j =;...;n. Since is everywhere dense in, it follows that the above approximation can be made as close as necessary. Thus, it makes sense to consider its relative closeness, and especially to answer the question how small can we make the integer q. A closely related problem is the following [0]. Definition 7: The Small Linear Form Problem: Given rationals ;...; n; > 0, and an integer > 0, find integers p ;...;p n, not all 0, such that jp jj for j = ;...;nand j p + + n p n j <. Similarly, one can consider the approximation of complex numbers by algebraic numbers, which is our case of interest in this correspondence. Given z, we consider jz 0 j with algebraic. Since the field of algebraic numbers is everywhere dense in, it follows that z can be approximated arbitrarily closely by algebraic. The approximation theory provides the order of this closeness jz 0 j, which depends on the considered (). One has the following theorem which is a generalization of Liouville s theorem over real numbers [9, pp. 3 3]. Theorem : If is an algebraic number of degree n and height h, then for any 6= algebraic of degree k and height H, one has j 0 j > 3 k(n0) h kn (h +) k (h +3) k H n : (9) The generalized Liouville s theorem is a necessary condition for a number to be algebraic. It proves that the numbers in which are algebraic are not very well approximated by other algebraic numbers. The restriction of Theorem over allows one to show that the irrational quadratic numbers have the worst order of approximation by rationals p q. It is worth mentioning that in this case, a well-known approach to determine the order of approximation of irrational numbers by rationals is the theory of continued fractions [9]. Definition 8: A continued fraction expansion of is given by = a 0 + a + a + Another way of writing the above expansion of is : (0) =[a 0 ; a ;a ;...] () where a j ;j=0; ; ;...; are called the quotients of the continued fraction expansion of. The expansion of to the order k, i.e., [a 0; a ;...;a k ] is called the kth-order convergent of. It is clear that if then its continued fraction expansion is finite (i.e., a j = for some j). On the other hand, is not well approximated by a rational if a j;j, are all relatively small [0]. The generalized Liouville s theorem gives a lower bound on the approximation of algebraic by another 6= algebraic, which can be considered as one-dimensional Diophantine approximation. The following result, which is based on Liouville s theorem, gives a lower bound on the simultaneous approximation of algebraic numbers by other algebraic numbers [9, p. 34]. Theorem : Suppose that ;...; m are algebraic numbers, and n is the degree of the algebraic number field () that contains these numbers. Then for any polynomial P (X ;...;X m) [X;...;X m] with degree k and height H, either P ( ;...; m )=0or else c k jp ( ;...; m)j () H n0 with c constant depending only on j;j=;...;m c = r n c n0 (3) 0 with r> 0 as an integer such as r j is an algebraic integer for j = ;...;m;and c 0 =+ m j= j jj

6 758 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 3, MARCH 00 with j jj as the size of the algebraic number j;j=;...;m(see Definitions 3). We have only considered until now the behavior of algebraic numbers when approximated by other algebraic numbers. The following theorem concerning the behavior of transcendental numbers is due to Lindemann [9, pp. 5 7]. Theorem 3: If ;...; m are distinct algebraic numbers, and if c ;...;c m are algebraic numbers not all equal to zero, then m l= c l e 6=0: (4) Lindemann also showed that if is an algebraic number 6= 0, then e is transcendental. B. Further Results on Diversity and Coding Gain In view of Theorem 3, one obtains the following result. Proposition 4: A maximum transmit diversity of B ; can be guaranteed by choosing = e i with 6= 0algebraic ( is transcendental). In particular, = e i= and = e 0:5i,guarantee a maximum transmit diversity over all constellations carved from [i] 4. Proof: Let 6=0be an algebraic number, then f0; i; i; 3ig are distinct algebraic numbers. Applying the Theorem of Lindemann on f;; ; 3 g with = e i, yields that if 3 j=0 a j j =0 with a j ;j = 0;...; 3 algebraic, then a 0 = a = a = a 3 = 0. This yields that B() C 6= 0for every C [i] 4, which proves our proposition. Having a maximum transmit diversity does not prevent B C from vanishing when the size of the constellation #(C) increases since (i) is everywhere dense in. Hence, we want to find such that B() C decreases as slowly as possible when the size of the constellation #(C) increases. With respect to Theorem we obtain the following. Proposition 5: For s j [i]; j = ;...; 4, not all zero, and algebraic integer of degree n>3over (i) = s 0 s 3 0 s + s js j j j= 3(n0) : (5) Proof: Let s j [i]; j=;...; 4 and algebraic integer of degree n>3over (i) such that s ;...;s 4; (i; ) of degree n over (i). Note that due to Galois theorem [] (i; ) is of degree n over. Furthermore, let with c as a constant depending only on s ;...;s 4;. Note that = jp (s ;s ;s 3 ;s 4 ;)j. Due to Proposition, one guarantees that P (s ;...;s 4 ;) 6= 0since n>3. Hence c k = (8) r nk c k(n0) 0 where r > 0 an integer such that rs j is algebraic integer for j = ;...; 4 and r is algebraic integer. c 0 =+ 4 j= js jj + jj (see Definitions 3). Since s ;...;s 4 ; are algebraic integers, one obtains r =. Since s j [i], one has that the size of s j equals js jj = js jj for j = ;...; 4. The size of equals over the basis f; ; ; 3 g. Finally, substituting in (8) one proves the Proposition. As a corollary of Proposition 5 one has the following. Corollary : Given an algebraic integer of degree n > 3 over (i), and given a constellation C with One has = max s=(s ;...;s ) (js j j;j=;...; 4): C B C : (9) ( + 4) 3(n0) Corollary shows that we can make the vanishing of B C slow as the size of C increases by choosing algebraic with the smallest possible degree n. Proposition tells us that the degree of over (i) should be at least. Heuristically, by combining Proposition and Corollary, one prevents B C 6= 0from vanishing rapidly when #(C) increases by choosing algebraic with the smallest degree such that B C 6= 0. It is somewhat surprising that the maximum of B C over the constellation C = 4-QAM 4 is achieved by = e i= transcendental. Nevertheless, number theory guarantees a lower bound on the shrinking of B() C for algebraic; while there is no such guarantee in the case when is transcendental. The latter observation is confirmed by computer search where B(e C i= ) drops from 0:369 for C = 4-QAM 4 to 0:037 for C = 6-QAM 4 (7.3 times); B(e C i=8 ) drops from 0:304 for C = 4-QAM 4 to 0:006 for C = 6-QAM 4 (.3 times); finally, B(e C i=4 ) drops from 0:0858 for C = 4-QAM 4 to 0:07 C = 6-QAM 4 (3.5 times). C. Relation to Lattices We note briefly here the relation between maximizing the coding gain of our ST code and the problem of finding nonzero short vectors in a given lattice []. First, let us define the quantity C B = min j T sj: (30) s=(=)(s 0s );s 6=s C It is easily seen that for a given constellation C 0 [i] 4 there exists CC 0 [ s ;...;s 4 T ; (s ;...;s 4 ) T C 0 P (X ;...;X 5 ) = X 0 X 3 X 5 0 X X 5 + X 4 X 3 5 [X ;...;X 5]: (6) such that C B C B : (3) The degree of P (X ;...;X 5 ) is k 3 and its height H =.By Theorem, either P (s ;...;s 4;)=0, or else jp (s ;...;s 4 ;)j ck H n0 (7) Hence, finding that guarantees an acceptable value of C B ensures a good value of C B. Based on the above discussion we propose a heuristic approach to find with C B() not too small for #C, or for an eventual generalization of our scheme to more than two transmit antennas.

7 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 3, MARCH ) As a necessary condition, should be chosen such that f ;; g are not well approximated by rational complex numbers with small denominators. Because from (6) we develop the following upper bounds on B : B min (s ;...;s )6=(0;0;0;0) [i] min (s ;...;s )6=(0;0;0;0) [i] min (s ;...;s )6=(0;0;0;0) [i] s 0 s 3 (s = s 4 =0) s 0 s (s 3 = s 4 =0) s + s 4 3 (s = s 3 =0) (3) so if any of f ;; g can be well approximated by a rational complex number, the coding gain vanishes rapidly. ) Given =(;; ; 3 ), test the value of C B using the sphere decoder to search the shortest nonzero vector in the lattice generated by A = I 4 p (; ; ; 3 ) with being the maximum absolute value taken by the symbols from the considered constellation. I 4 is the identity matrix of dimension 4. This step gives a good approximation of C B() []. In fact, the lattice generated by A is 3 = fu = As; s [i] 4 g 5 which can also be seen as a subset of A 54 as <(A) =(A) 0=(A) <(A) 0 upon representing and representing u and s as (<(u); =(u)) T and (<(s); =(s)) T, respectively, where <( ) and =( ) are the real and imaginary operators. The norm square of a vector u 3 equals kuk = 4 j= js j j + 4 j= s j j Hence, the shortest nonzero vector in the lattice 3 guarantees a small value of 4 s j j j= which gives a good approximation of C B() (30). This simplifies slightly the problem of maximizing the coding gain when the size of the constellation increases. Because we know that step ) can be approached in two ways. One may choose algebraic with small degree and small height (Liouville s thereom). Some interesting numbers satisfying the above constraints are the cyclotomic numbers which are the nth roots of unity n = e i=n (such as e i=8 or e i=4 ) []. The minimal polynomial of n over is (X) = j<n; GCD(n; j)= : X 0 j n with GCD(n; j) the greatest common divisor of n and j. The degree of n over equals (n), the Euler function [], which represents the number of integers j < n such that GCD(n; j) =. Or one may choose transcendental such as the quotients of its continued fraction expansion are bounded [9]. The advantage of step ) above is that the sphere decoder is an efficient algorithm nearly independent of the constellation size. Thereby, one can guarantee acceptable values of C B() for C with very large size. We used step ) above to compute C B for C with different values of. We have found that C B(), with = e i=8, is always a local maximum for = ; 4; 00, which confirms our results in Section VI-B. Note that e i=8 gives an acceptable value of B C for C = 4-QAM 4 and C = 6-QAM 4. We note also that C B(e i=4 )=0, because ie i=4 = 0i. It is an interesting approach to see the design of ST encoding as searching irrational numbers the furthest from rational approximations. On the other hand, the decoding process is equivalent to searching rational integers the closest to irrational numbers; and both encoding and decoding can be approached by the same algorithm (sphere decoder) of searching nonzero short vectors in a given lattice. Remark : The real rotation obtained in [0] in order to maximize the minimum product distance defined as d p; min = min (s ;s )6=(0;0) fjy ky j; y =sg can also be considered as a one-dimensional Diophantine approximation. One has Hence, d p; min = = min (s ;s )6=(0;0) : (33) fjs + s k0s + s jg: Note that by choosing not to be well approximated by a rational number s one maximizes d s p; min. It has been shown by Shokrollahi in [] that when using the rotation (33) with a differential diagonal double antennas ST code, one optimizes the so-called diversity product (d p; min ) by choosing opt = n g = + p 5. Indeed, opt = n g is found by computer search to maximize d p; min over the binary phase-shift keying (BPSK) constellation in [0] and over the 6-PAM constellation in []. We note that the Golden number n g is in a sense the furthest real number from any rational approximation. In fact, the continued fraction expansion quotients of n g are [; ; ;...], and hence n g is the irrational number the most poorly approximated by rational fractions [9, p. 36]. We also note that the rotation in (33) is used for the diagonal algebraic ST code of rate one symbol PCU over M =transmit antennas along with Hadamard transform in [5]. VII. SIMULATIONS RESULTS The channel is modeled as () (); we compare the ST code B ; (with = e i= for 4-QAM, = e 0:5i for 6-QAM, and = e i=4 for 56-QAM) with the Alamouti scheme and the code LD, at the same spectral efficiency and the same average transmitted power. The sphere decoder is used to decode both ST codes B ; and LD. The number of receive antennas equals N =in Fig. and N =0in Fig.. In both figures, we plot the average bit-error rate as a function of SNR. Fig. shows that the code B ; outperforms the Alamouti code at low and high SNR. For example, at a spectral efficiency of 8 bits PCU, ST code B ; shows about.5-db gain over the Alamouti code. At a spectral efficiency of 8 bits PCU, the code LD has a better performance than the Alamouti code for SNR 33 db. This is due to the information loss of the Alamouti code. Note that the slope of the

8 760 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 3, MARCH 00 Fig.. The ST code B versus the Alamouti scheme with N = 0 receive antennas. LD curves are different from the Alamouti code and B ;, because LD does not satisfy the rank and determinant criteria [8], [3], thus, for relatively high SNR, the Alamouti code becomes better. On the other hand, the code B ; satisfies all the desired criteria, and has a better performance than both the Alamouti code and the code LD at low and high SNR. In Fig., for N =0and at a spectral efficiency of 6 bits PCU our scheme has a gain of more than 0 db over the Alamouti code. We also notice the difference in performance is enhanced when N or the spectral efficiency increases, which is confirmed by the information loss of the Alamouti code (see Section IV). It has been observed that small changes of imply small changes of the error probability curves as long as one chooses to guarantee the maximum transmit diversity. Nonetheless, the problem of searching that maximizes the coding gain is very interesting from the theoretical point of view. As for optimizing the performance of the scheme, one notes that the main role is played by the sphere decoder which reaches the ML detection performance of the proposed code with a moderate complexity, and in most cases, has a dramatic improvement over suboptimal detection schemes [6]. This emphasis on the importance of the sphere decoder is true for many high-rate linear ST codes such as the codes in [5] [7]. VIII. CONCLUSION We have proposed a new ST code over two transmit and N receive antennas which maximizes the transmit diversity and the coding gain. The proposed code is information lossless, and is shown to largely outperform the Alamouti code when N. Due to the lattice structure of the proposed code, the ML decoding can be performed at a moderate complexity using the sphere decoder. Tools from algebraic number theory and the geometry of numbers were used to optimize the diversity and coding gains of the proposed code. REFERENCES [] S. M. Alamouti, A simple transmit diversity technique for wireless communications, IEEE J. Select. Areas Commun., vol. 6, pp , Oct [] I. E. Telatar, Capacity of multi-antenna Gaussian channels, Europ. Trans. Telecommun., vol. 0, pp , Nov [3] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space time block codes from orthogonal designs, IEEE Trans. Inform. Theory, vol. 45, pp , July 999. [4] G. D. Golden, G. J. Foschini, R. A. Valenzuela, and P. W. Wolniasky, Detection algorithm and initial laboratory results using V-BLAST space time communication architecture, IEE Electron. Lett., vol. 35, pp. 4 6, Jan [5] M. O. Damen, K. Abed-Meraim, and J.-C. Belfiore, Diagonal algebraic space time block codes, IEEE Trans. Inform. Theory, vol. 48, pp , Mar. 00. [6] M. O. Damen, A. Chkeif, and J.-C. 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Chicago Press, 964. [0] V. M. DaSilva and E. S. Sousa, Fading-resistant modulation using several transmitter antennas, IEEE Trans. Commun., pp , Oct [] A. Shokrollahi, Double antenna diagonal space time codes and continued fractions, in Proc. IEEE Int. Symp. Information Theory (ISIT 00), Washington, DC, June 00, p. 08. [] J. Boutros and E. Viterbo, Signal space diversity: A power and bandwidth efficient diversity technique for the Rayleigh fading channel, IEEE Trans. Inform. Theory, vol. 44, pp , July 998. ACKNOWLEDGMENT The authors are grateful to the anonymous reviewers for their helpful comments. M. O. Damen would like to thank Dr. B. Hassibi for pointing out Remark.