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Final Report of Granted Project NSC
97-2219-E-009-014
Hsiao-feng Francis Lu
Department of Electrical Engineering
National Chiao Tung University
[email protected]
September 7, 2009
Abstract
This article provides an overview of some major results we have obtained in the research project “Optimal coding and modulation designs for the fourth generation mobile com-munication systems under various MIMO-OFDM channels (3/3)”supported by National Science Council under contract number NSC 97-2219-E-009-014 during academic year 2008 (August 2008 - July 2009). Results contained in this article will be published in IEEE Journal of Selected Topics in Signal Processing, Special Issue: Managing Complexity in Multiuser MIMO Systems, Dec. 2009.
In this article, we address the problem of constructing multiuser input multiple-output (MU-MIMO) codes for two users. The users are assumed to be equipped with nt
transmit antennas, and there are nr antennas available at the receiving end. A general
scheme is proposed and shown to achieve the optimal diversity-multiplexing gain tradeoff (DMT). Moreover, an explicit construction for the special case of nt = 2 and nr = 2 is
given, based on the optimization of the code shape and density. All the proposed construc-tions are based on cyclic division algebras and their orders and take advantage of the multi-block structure. Computer simulations show that both the proposed schemes yield codes with excellent performance improving upon the best previously known codes. Finally, it is shown that the previously proposed design criteria for DMT optimal MU-MIMO codes are sufficient but in general too strict and impossible to fulfill. Relaxed alternative design criteria are then proposed and shown to be still sufficient for achieving the multiple-access channel diversity-multiplexing tradeoff.
Referred Papers Supported by Granted
Project
Under the support of this three-years project, we have successfully produced the following Nineteen papers ( Eight Journal Papers, Seven in IEEE Trans. IT and One in IEEE Trans. TCOM, and Eleven conference papers published in the highest quality confer-ences):
1. H. F. Lu, “On constructions of algebraic space-time codes with AM-PSK constella-tions satisfying rate-diversity tradeoff,” IEEE Trans. Inform. Theory, vol. 52, no. 7, pp. 3198-3209, Jul. 2006.
2. P. Elia, S. A. Pawar, K. Raj Kumar, P. V. Kumar, and H. F. Lu, “Explicit construction of space-time block codes achieving the diversity-multiplexing gain tradeoff,” IEEE Trans. Inform. Theory, vol. 52, no. 9, pp. 3869-3884, Sep. 2006.
3. H. F. Lu and M. C. Chiu, “Constructions of asymptotically optimal space-frequency codes for MIMO-OFDM systems,” IEEE Trans. Inform. Theory, vol. 53, no. 5, pp. 1676-1688, May 2007.
4. O. Moreno, R. Omrani, P. V. Kumar, and H. F. Lu, “A generalized Bose-Chowla family of optical orthogonal codes and distinct difference sets,” IEEE Trans. Inform. Theory, vol. 53, no. 5, pp. 1907-1910, May 2007.
5. H. F. Lu, “Constructions of multi-block space-time coding schemes that achieve the diversity multiplexing tradeoff,” IEEE Trans. Inform. Theory, vol. 54, no. 8, pp. 3790-3796, Aug. 2008.
6. C. Hollanti, J. Lahtonen, and H. F. Lu, “Maximal orders in the design of dense space-time lattice codes,” IEEE Trans. Inform. Theory, vol. 54, no. 10, pp. 4493-4510, Oct. 2008.
7. M. C. Chiu and H. F. Lu, “Accumulate Codes Based on 1+D Convolutional Outer Codes,” IEEE Trans. Commun., vol. 57, no. 2, pp. 331-334, Feb. 2009.
8. C. Hollanti and H. F. Lu, “Construction methods for asymmetric and multi-block space-time codes,” IEEE Trans. Inform. Theory, vol. 55, no. 2, pp. 1086-1103, Mar. 2009.
9. H. F. Lu, “Optimal Code Constructions for SIMO-OFDM Frequency Selective Fad-ing Channels,” Proc. 2007 Information Theory Workshop, pp. 12-16, Bergen, Nor-way, Jul. 1-7, 2007.
10. H. F. Lu, “Binary Linear Network Codes,” Proc. 2007 Information Theory Work-shop, pp. 223-227 Bergen, Norway, Jul. 1-7, 2007.
11. H. F. Lu, “Low Complexity Constructions of Multi-Block Space-Time Codes Achiev-ing Diversity-MultiplexAchiev-ing Tradeoff,” Globecom 2007, pp. 1724 - 1728.
12. C. Hollanti and H. F. Lu, “Normalized Minimum Determinant Calculation for Multi-Block and Asymmetric Space-Time Codes,” The 17th Int. Conf. on Applied Algebra, Algebraic Computation, and Error Correcting Codes (AAECC-17), Bangalore, India, Dec. 2007.
13. H. F. Lu “Diversity-Multiplexing Tradeoff Optimal Codes for OFDM-Based Asyn-chronous Cooperative Networks,” Proc. 2008 IEEE Int. Symp. on Inform. Theory (ISIT).
14. H. F. Lu “Constructions of Fully-Diverse High-Rate Space-Frequency Codes for Asyn-chronous Cooperative Relay Networks,” Proc. 2008 IEEE Int. Symp. on Inform. Theory(ISIT).
15. H. F. Lu and C. Hollanti “On the Construction of DMT-Optimal AST Codes with Transmit Antenna Selection,” Proc. 2008 IEEE Int. Symp. on Inform. Theory (ISIT). 16. C. Hollanti and H. F. Lu “Constructing Asymmetric Space-Time Codes with the Smart Puncturing Method,” Proc. 2008 IEEE Int. Symp. on Inform. Theory (ISIT). 17. H. F. Lu, “Optimal Diversity Multiplexing Tradeoff of Constrained Asymmetric MIMO
Systems,” Globecom 2008.
18. H. F. Lu and C. Hollanti “Diversity-Multiplexing Tradeoff-Optimal Code Construc-tions for Symmetric MIMO Multiple Access Channels,” Proc. 2009 IEEE Int. Symp. on Inform. Theory(ISIT).
19. C. Hollanti, H. F. Lu, and R. Vehkalahti, “An Algebraic Tool for Obtaining Condi-tional Non-Vanishing Determinants,” Proc. 2009 IEEE Int. Symp. on Inform. Theory (ISIT).
Chapter 1
Introduction
During the past five years extensive research has been carried out on single-user (SU) multiple-input multiple-output (MIMO) space-time (ST) lattice codes based on cyclic di-vision algebras (CDAs) [1–5]. At its best, this research has resulted in codes that get very close to the outage bound for practical numbers of antennas. Motivated by the promis-ing outcome in the SU-MIMO scenario, the aim in this report is to adapt the machinery provided by CDAs to the multiuser (MU) MIMO scenario as well, with the ultimate goal of producing diversity-multiplexing tradeoff (DMT) achieving codes in mind. We will concentrate on the multiple-access channel (MAC), i.e., on the uplink transmission from multiple users to a single access point (AP). Both the transmitters (=users) and the receiver (=AP) may be occupied with multiple antennas.
In general, multiuser MIMO coding is a very challenging topic. When the 3GPP (=third generation partnership project) asked the participating companies (cell phone manufactur-ers, chipset manufacturmanufactur-ers, operators etc.) to list research topics that they find essential for the next release, MU-MIMO was mentioned in nearly all the lists. The area is made very challenging by the diversity of potential applications all requiring slightly different treatment and design goals.
The idea of extending the single-user ST codes to the multiuser case and the design criteria for such MU-MIMO codes were given in [6]. An explicit (2 × 2) two-user MIMO construction exploiting independent Alamouti blocks was also introduced in [6]. By swap-ping columns for one user they managed to achieve a minimum rank of three. In [7], Tse et al. extended the DMT results from [8] to the MAC. The codes in [6] do not achieve the optimal MAC DMT. Nam et al. [9] proposed the first explicit DMT achieving trans-mission scheme based on a class of structured multiple access lattice ST codes. However, their scheme was not constructive and no explicit examples were provided. Some explicit, algebraic code constructions for the MAC with nt > 1 were introduced in [10] and [11].
The authors of [11] state that their construction is DMT optimal, but do not provide an explicit proof. In [10] a somewhat different approach was taken as compared to [6]: the authors propose a design criteria based on a truncated union-bound approximation. With the aid of these criteria they manage to outperform in error performance the other known two-user codes for the (2 × 2) MAC [6, 12]. Another group of multiuser ST codes was proposed in [12], but these codes suffer from high peak-to-average power ratio (PAPR) as the codeword matrices contain zero entries.
In [13], the authors propose design criteria for designing MAC-DMT optimal codes, and further propose a code construction that is claimed to fulfill their criteria. The criteria proposed in [13] are indeed sufficient for achieving the optimal DMT, but it turns out that it is not necessary to fulfill these criteria in order to do so. It will be shown that more relaxed design criteria will still provide us with MAC-DMT optimal codes. Especially, we will prove that it is not possible to design DMT optimal multiuser codes having the full NVD property when we have two users using one antenna. The general proof for an arbitrary number of users and antennas is presented in [14].
Our main goals in this report are to
1. construct explicit, sphere-decodable codes for the (2 × 2) situation where both of the two users are equipped with two transmitting antennas, and two antennas are available at the receiving end. We will compare our codes with the best known codes for this situation [10].
2. design a general, DMT-achieving, sphere-decodable (nt× nr) MU-MIMO scheme
for two users, that would yield good performance also at the low SNR end. We will compare our explicit (2 × 4) codes with the best known codes for this situation [11]. For the use of matrix representations of cyclic division algebras and their orders as space-time codes, we refer the reader to [1, 5, 15].
The report is organized as follows. In Section II we provide the reader with algebraic preliminaries, concentrating only on the facts that will be needed in this report. Section III is devoted to designing a 2 × 2 two-user code, whereas Section IV gives us a general DMT optimal nt× nr construction for two users. In Appendix I we prove the claimed
non-existence result of full-NVD multiuser codes in the case of two users equipped with one antenna.
Chapter 2
Algebraic preliminaries
In this chapter we introduce some concepts and results from the theory of central simple algebras for later use. For the proofs of these results and for a proper introduction we refer the reader to [16].
In the rest of the report we assume that all the fields are finite extensions of the field of rational numbers Q.
Definition 1. Let K be an algebraic number field and assume that E/K is a cyclic Ga-lois extension of degree n with Galois group Gal(E/K) = hσi. We can now define an associativeK-algebra
A= (E/K, σ, γ) = E ⊕ uE ⊕ u2E ⊕ · · · ⊕ un−1E,
where u ∈ A is an auxiliary generating element subject to the relations xu = uσ(x) for allx ∈ E and un = γ ∈ K∗. We call this type of algebra a cyclic algebra and the field K the center of the algebra. The center is the set of elements of A that commute with all the elements of A. Throughout the report,K denotes the center, and F denotes its subfield F ⊆ K. The inclusion may also be trivial, i.e., we allow K = F .
Definition 2. A cyclic algebra is a division algebra if and only if all the non-zero elements of the algebra are invertible.
Proposition 1 (Norm condition). The cyclic algebra A = (E/K, σ, γ) of degree n is a division algebra if and only if the smallest factort ∈ Z+ ofn such that γt is the norm of some element ofE∗ isn.
Due to the above proposition, the element γ is often referred to as the non-norm element. Definition 3. Let D be a K-central division algebra. We then callp[D : K] the index of the algebra.
Definition 4. Suppose that E is a cyclic extension of an algebraic number field K. Let D = (E/K, σ, γ) be a cyclic division algebra and let γ ∈ K∗ to be an algebraic integer. We immediately see that theOK-module
whereOE is the ring of integers ofE, is a subring in the cyclic algebra (E/K, σ, γ). We
refer to this ring as the natural order. Note also that ifγ is not an algebraic integer, then Λ fails to be closed under multiplication.
Let K/F be a finite extension (could be also the trivial extension) of algebraic number fields and D a K-central division algebra of degree n.
Definition 5. An OF-orderΛ in D is a subring of D, having the same identity element as
D, and such that Λ is a finitely generated module over OF and generates D as a linear
space overF .
Proposition 2. Every OK-orderΛ ⊆ D is also an OF-order.
Definition 6. An OF-orderΛ is called maximal, if it is not properly contained in any other
OF-order.
Proposition 3. Any K-central division algebra D has a maximal OF-order and any order
inside D is contained in at least one maximal order.
Example 1. Suppose that E/K is a cyclic extension of algebraic number fields. Let D = (E/K, σ, γ) be a cyclic algebra.
We can consider D as a right vector space overE, and every element a = x0 + ux1+
· · · + un−1x
n−1 ∈ D has the following representation as a matrix
A = x0 γσ(xn−1) γσ2(xn−2) · · · γσn−1(x1) x1 σ(x0) γσ2(xn−1) γσn−1(x2) x2 σ(x1) σ2(x0) γσn−1(x3) .. . ... xn−1 σ(xn−2) σ2(xn−3) · · · σn−1(x0) .
We call this representation theleft regular representation and denote A = ψ(a).
Definition 7. The determinant (resp. trace) of the matrix A above is called the reduced norm (resp. reduced trace) of the element a ∈ D and is denoted by nrD/K(a) (resp.
trD/K(a)).
Proposition 4. Let D be a K-central division algebra and a an element of D. Then nr(a) and tr(a) ∈ K.
Proposition 5. The norm and trace maps do not depend on the maximal representation, i.e., the left regular representation is not the only representation we can use. However, we stick toψ for simplicity.
Definition 8. We then define the reduced trace and norm of a to F by
trD/F(a) = trK/F(trD/K(a)) and nrD/F(a) = nrK/F(nrD/K(a)),
where nrK/F and trK/F are the usual relative norm and trace maps of a number field
Proposition 6. Let Λ be an OF-order in a K-central division algebra D. Then for any
elementa ∈ Λ its reduced norm nrD/F(a) and reduced trace trD/F(a) are elements of the
ring of integersOF of the fieldF . If a is non-zero, then so is nrD/F(a).
Now we are ready to define one of the main algebraic objects needed in this report. Definition 9. Let D be a K-central division algebra and m = dimFD. TheOF-discriminant
of theOF-orderΛ is the ideal d(Λ/OF) in OF generated by the set
{det(trD/F(xixj))mi,j=1| (x1, ..., xm) ∈ Λm}.
Here dimFD simply refers to the dimension of D as an F -linear vector space.
If Λ is a free OF-module, then
d(Λ/OF) = det(tr(xixj))mi,j=1,
where {x1, . . . , xm} is any OF-basis of Λ.
Proposition 7. All the maximal orders of a K-central division algebra share the same discriminant.
Now we can define the following.
Definition 10. Let D be a K-central division algebra and let Λ be some maximal order in D. Then we refer tod(Λ/OK) = dDas thediscriminant of the algebra D.
The following lemma connects the discriminants d(Λ/OK) and d(Λ/OF).
Lemma 8. Let D be a K-central division algebra of index n and let Λ be an OK-order. If
Λ is an OF-order in D, then
d(Λ/OF) = nrK/F(d(Λ/OK))d(OK/OF)n
2
Chapter 3
A Sphere decodable MU-MIMO code
for two users and two receive antennas
In this chapter we concentrate on designing a multiuser code for two users, both equipped with two transmit antennas, and for a receiver that has two antennas. This leads us to a situation where the single user must use a code that is sphere decodable with one receive antenna. Such MU-MIMO codes have been considered by Grtner and Blcskei [6] and by Hong and Viterbo in [10]. Our coding scheme is directly comparable to their codes.
In what follows, we first concentrate on the optimization of the single user code and then, in the very end of this section, we put our single-user codes into use in the multiuser scenario. The careful construction of the single-user code as a building block of the mul-tiuser code is crucial, as it will then guarantee good performance also when only one user is present.
3.1
Coding theoretic preliminaries of abstract multi-block
codes
In this chapter we consider abstract multi-block codes that are matrix lattices in the space Mn×nk(C). Particularly we are going to define the normalized minimum determinant and
normalized coding gainof such lattices and study the relation between these concepts. We can flatten the matrices A of Mn×nk(C) to real vectors α(A) ∈ R2kn
2
by first forming a vector of length kn2 out of the entries (e.g. row by row) and then replacing a complex number z with the pair of its real and imaginary parts Rez and Imz. This mapping α is clearly R-linear and maps t-dimensional Mn×nk(C) lattices to t-dimensional
R2kn2 lattices. We also have the equality ||A||F = ||α(A)||E, i.e., the Frobenius norm of the
matrix A coincides with the euclidean norm of the corresponding vector α(A). Therefore, α is also an isometry.
Definition 11. We say that a lattice L in Mn×nk(C) is orthogonal or rectangular if the
corresponding real latticeα(L) has a basis that is orthogonal with respect to the normal inner product of the space R2kn2.
We denote the measure (or hypervolume) of the fundamental parallelotope of the lattice α(L) by m(L) and we call it the volume of the fundamental parallelotope of the lattice L. If {x1, . . . , xt} is a basis of L, we can form a matrix M by using the vectors α(xi) as column
blocks. Then the Gram matrix of the lattice L is G(L) = M MT =
Retr(xix†j)
1≤i,j≤t,
where X† indicates the complex conjugate transpose of X. The Gram matrix then has a positive determinant equal to m(L)2.
Any lattice L ⊆ Mn×nk(C) can be scaled (i.e. multiplied by a real constant s) to satisfy
m(sL) = 1.
If A is an element in the space Mn×nk(C) it can be written as (A1, . . . , Ak) where all
the matrices Aiare elements in Mn×n. We can then define the product determinant
pdet(A) = k Y i=1 det(Ai) of the matrix A.
Definition 12. The minimum determinant detmin(L) of a multi-block code L ⊆ Mn×nk(C)
is defined to be the infimum of the absolute valuespdet(A) of all the non-zero elements of the latticeL.
Thenormalized minimum determinant δ(L) of a lattice L is obtained by multiplying the lattice with a real constant such that the resulting latticeL0has fundamental parallelotope of volume 1 and then setting
δ(L) = detmin(L0) .
Definition 13. The coding gain CG(L) of the lattice L ⊆ Mn×nk(C), k ≥ n, is defined to
be the infimum of the absolute values of the determinants of matricesAA†of all non-zero matricesA in the lattice.
Thenormalized coding gain N CG(L) of a lattice L ⊆ Mn×nk(C) is obtained by
mul-tiplying the lattice by a real constant such that the resulting lattice L0 has a fundamental parallelotope of volume 1 and then set
N CG(L) = CG(L0).
Lemma 9. Let us suppose that A1, . . . , Ak are complexn × n matrices. We consider the
n × nk matrix (A1, A2, . . . , Ak) = A. We then have det(AA†) ≥ kn· (Qki=1|det(Ai)|)2/k.
Proof. First the Minkwoski determinant inequality states that (det(AA†))(1/n) ≥Pk
i=1|det(Ai)|2/n.
The AM-GM inequality on the arithmetic and geometric means then transforms this result into det(AA†)1/n ≥ k X i=1 |det(Ai)|2/n ≥ k · ( k Y i=1 |det(Ai)|2/n)1/k.
In the following corollary we use the notation of the previous lemma. Corollary 10. Let us suppose that L is a multi-block code in Mn×nk(C). Then
CG(L) ≥ kn(detmin(L))2/k and N CG(L) ≥ kn(δ(L))2/k.
Particularly the following will be of great interest for us.
Corollary 11. Let us suppose that L is a lattice in M2×4(C). Then NCG(L) ≥ 22δ(L).
Remark 1. The concept of the normalized minimum determinant of a multi-block code is related to the performance of the code when eachn × n block faces independent fading. On the other hand, the normalized coding gain is a relevant code design criterion when the channel stays stable during the transmission of the whole n × nk block. It is not a great surprise that these two concepts are so closely related.
3.2
Constructing the single user code
In this chapter we study the achievable normalized minimum determinant of 8-dimensional multi-block codes in the space M2×4(C). Notice that as we want to receive with only two
antennas (equipped with sphere decoders), we cannot use full lattices that would have dimension 16. In order to get well behaving 8-dimensional lattices we use real quadratic field as a center in the multi-block construction. We remark that while we came up with the idea independently it was discovered already in [17].
We begin by considering maximal order codes from division algebras. By discriminant analysis we are able to find the optimal algebras. In Section 3.4 we concentrate on rect-angular codes and derive a bound for normalized minimum determinant of such codes and give an example code achieving this bound. The minimum determinant analysis we are using is similar to that used in [18].
We will take advantage of multi-block constructions from division algebras. In Section 4 to follow the same trick will be used. The exception is that now the base field F is Q and the center K is some quadratic field, whereas in Section 4 we need full lattices; hence F = Q(i) and the center K is some suitable extension of F .
Let us consider the field E = KL that is a compositum of two quadratic fields K and L. We suppose that K ∩ L = Q and that Gal(K/Q) =< τ > and Gal(L/Q) =< σ >. We can then write that Gal(E/Q) =< σ > ⊗ < τ >.
Let us now consider the cyclic division algebra D = (E/K, σ, γ). As usually, we have the left regular representation ψ of the algebra D so that an element a maps to a 2 × 2 matrix ψ(a) ∈ M2(E), and the multi-block representation φ;
φ(a) 7→ (ψ(a), τ (ψ(a))). (3.1) Let us suppose that Λ is a Z-order in D. We call the φ(Λ) an order code. In the rest of this section, we suppose that the division algebras under consideration are of the previous type.
Lemma 12. Let a be an element of D. Then
det(ψ(a))det((τ (ψ(a))) = nrD/Q(a) and
Tr(ψ(a) + τ (ψ(a)) = trD/Q(a), whereT r is the usual matrix trace.
Proof. These results follow directly from Definition 8.
Proposition 13. Let us suppose that Λ is a Z-order of a division algebra D and that φ is a multi-block representation. The order codeφ(Λ) is an 8-dimensional lattice in the space M2×4(C) and
detmin(φ(Λ)) = 1.
Proof. The claim about the dimension of the lattice is easily seen. The second claim fol-lows directly from Proposition 6.
Remark 2. For every non-zero element (ψ(a), τ (ψ(a))) of an order code the rows are linearly independent over C. This follows as det(ψ(a)) 6= 0 and therefore the first two columns are linearly independent and generally in a matrix the number of linearly inde-pendent rows and columns is equal.
Corollary 14. With the previous notation we have δ(φ(Λ)) = m(φ(Λ))1 1/2.
The previous proposition reveals that the minimum determinant of an order code de-pends only on the volume of the fundamental parallelotope. The following lemma connects the volume of the fundamental parallelotope and the discriminant of the algebra. Here we identify the ideal discriminant and the element generating it. This allows us to discuss the absolute value of the Z-discriminant. In the following we identify the order of the algebra and its image in M2×4(C). If the regular representation ψ of the algebra fulfills the
fol-lowing conditions, then the discriminant and the fundamental parallelotope of an order are tightly connected.
In the case of a real center we must assume that the regular representation ψ gives us matrices of the following Alamouti-like type
a −b∗
b a∗
, (3.2)
where∗ is the complex conjugation. In the case of a complex center we must assume that the automorphism τ is the complex conjugation. It is an easy task to check that, with these assumptions,
Tr(ψ(a)ψ(b)†+ τ (ψ(a))τ (ψ(b))†) ∈ R, when a, b ∈ D.
Lemma 15. Let us suppose that D is a division algebra and Λ is an order in D. Then
m(Λ) =p|d(Λ/Z)| and δ(Λ) = 1 |d(Λ/Z)|1/4.
Proof. Let us suppose that Λ has a Z-basis B = {(A1, τ (A1)), . . . , (A8, τ (A8))}, where
Ai = ψ(ai), ai ∈ Λ. We can now flatten the matrix (Ai, τ (Ai)) into an 8-tuple L(Ai, τ (Ai))
by first forming a vector of length 4 out of the entries of Ai (e.g. row by row) and then
concatenating this with the 4-tuple similarly made out of the entries of the matrix τ (Ai).
We can now easily see the identities
L(Ai, τ (Ai))L(Aj, τ (Aj))T = Tr(AiATj + τ (Ai)τ (Aj)T) (3.3)
and
L(Ai, τ (Ai))L(ATj, τ (Aj)T)T = Tr(AiAj + τ (Ai)τ (Aj)). (3.4)
The Gram matrix of the lattice Λ is G = (Re(Tr(AiA
†
j+ τ (Ai)τ (Aj)†)))8i,j=1.
Due to the limitations we set above on the form of the matrices Ai, Tr(AiA †
j+τ (Ai)τ (Aj)†)
is already real and we can ignore taking the real part from the traces. According to Equation (3.3) we can write
G = (L(Ai, τ (Ai))L(A∗j, τ (Aj)∗)T))8i,j=1= L(B)L(B) †
,
where the rows of the 8 × 8 matrix L(B) consist of vectors L(Ai, τ (Ai)). A simple
permu-tation of the columns and elementary properties of determinants give us that
|det(L(B))det(L(B)†)| = |det(L(B))det(L(B)T)| = |det(L(B))det(L(B0)T)|, where L(B0) is a matrix with the rows L((Ai)T, τ (Ai)T). According to Equation (3.4) and
Lemma 12
L(B)L(B0)T = (Tr(AiAj + τ (Ai)τ (Aj))8i,j=1= d(Λ/Z).
Proposition 16. Of all the orders in a K-central division algebra, the maximal orders have the smallest Z-discriminant.
Lemma 17. Let us suppose that Λ is an order in a division algebra D. Then N CG(Λ) = 22(δ(Λ))2.
Proof. Let us consider the lattice Λ without the normalization. We then have CG(Λ) ≥ 22(detmin(Λ))2= 22. On the other hand, det(φ(1D)φ(1D)H) = 22and therefore CG(Λ) =
3.3
Minimizing the discriminant
As previously stated, if we consider orders inside a fixed algebra, the smallest discriminant belongs to the maximal orders of the algebra and all the maximal orders share the same discriminant. Among those algebras having a regular representation fulfilling the condi-tions stated before Lemma 15, minimizing the discriminant of the algebra is now seen to be equivalent to maximizing the coding gain of a code from a maximal order.
In the following we forget the restrictions on the form of the regular representation and simply concentrate on finding the division algebras with the smallest possible discrimi-nants. Only after this we shall discuss whether the algebras have such regular representa-tions that Lemma 15 would be at their disposal. Still the solution to the problem of choosing an optimal division algebra is not an obvious one. The first step is the following. In our special case, Lemma 8 transforms into
|d(Λ/Z)| = |nrK/Q(d(Λ/OK))|d(OK/Z)4.
Here we see that for a fixed center K the second term d(OK/Z)4 is independent on the
chosen algebra and we can concentrate on the term |nrK/Q(d(Λ/OK))|. This leads us
to discuss the size of the ideals of OK. By this we mean that ideals are ordered by the
absolute values of their norms to Q, so e.g. in the case OK = Z[i] we say that the prime
ideal generated by 2 + i is smaller than the prime ideal generated by 3 as they have norms 5 and 9, respectively.
We have divided this chapter into two parts depending on the type of the center. Propo-sitions 18 and 20 that consider discriminants of division algebras are straightforward corol-laries of well known results and the proofs can be found for example from [16]. The min-imization problems that will have rather simple solutions here become more complicated in the case where the index of the algebra is greater than two. This question is of major importance when we consider general MIMO codes. We refer the interested reader to [5].
A complex quadratic center
In this chapter we consider the situation where the center K is a complex quadratic field of degree 2.
Proposition 18. Let us suppose that D is a K-central division algebra of index 2 containing anOK-orderΛ ⊆ D. Then
d(Λ/OK) = (P1· · · P2n)2,
where all thePiare distinct prime ideals of the centerK and n ≥ 1.
On the other hand, if we have an even numbered set of prime idealsP1, . . . , P2k, then
there exists a uniqueK-central division algebra D0 of index2 having an OK-orderΛ with
the discriminant
d(Λ/OK) = (P1· · · P2k)2.
Corollary 19. Suppose that P1andP2are a pair of smallest primes in the complex quadratic
fieldK. Then the smallest Z-discriminant of all the index 2 K-central division algebras is |nrK/Q(P1P2)|2d(OK/Z)4.
Example 2. Let us consider the center Q(i). It is readily seen that (2 + i) and (1 + i) are a pair of the smallest primes in this field. Proposition 18 proves that there exists a Q(i)-central division algebra D of index 2 having a maximal order Λ with the discriminant
dD = d(Λ/Z) = |(1 + i)(2 + i)|244 = 295.
If this algebra also has a suitable regular representation, then Lemma 15 infers that
δ(Λ) = 1
(295)1/4 = 0.140....
Example 3. Let us next consider the center K = Q(√−3). The smallest prime ideals in this center are 2 and√−3. According to Proposition 18 there exists a Q(√−3)-central division algebra D of index2 having a maximal order with the discriminant
dD = d(Λ/Z) = |2
√
3|234 = 972.
If this algebra also has a suitable regular representation, then Lemma 15 gives us that
δ(Λ) = 1
(972)1/4 = 0.179....
The discriminant 972 is already the smallest possible value we can achieve with a com-plex quadratic center K. This can be proved by simply trying different centers. It is easily done because for a given discriminant there is only one complex quadratic field. In the dis-criminant formula for the maximal order of a division algebra the term d(OK/Z)4is always
a factor and we already have 64 = 1296. Therefore it is enough to check the remaining
dis-criminants −4 and −5 that are still possible. In the previous example we saw that the center corresponding to discriminant −4 is Q(i) and that with this center the discriminant cannot be smaller than 972. The discriminant of the field Q(√−5) is −40 and there does not exist a field with discriminant −5.
A real quadratic center
In this chapter we fix the center K to be a real quadratic field of degree 2.
Proposition 20. Let us suppose that D is a K-central division algebra of index 2 and that Λ is a maximal Z-order in D. Then
d(Λ/OK) = (P1· · · Pn)2,
wherePiare separate prime ideals ofK and n ≥ 0. Here we use the notation that if n = 0
thend(Λ/OK) = OK.
On the other hand if we have a set of prime ideal P1, . . . , Pk then there exists a
K-central division algebra D0 of index2 having a maximal order Λ0 with discriminant d(Λ0/OK) = (P1· · · Pk)2
Corollary 21. Let us suppose that we have a real quadratic field K. Then the smallest discriminant of all the index 2 division algebras with the centerK is
d(OK/Z)4.
Example 4. The smallest discriminant of all the real quadratic fields belongs to the field Q(
√
5) = K. The following algebra Dicos= (Q(i,
√
5)/Q(√5), σ, −1)
is called the Icosian algebra. It is a known fact that |dDicos| = 1. This reveals that this
division algebra has the smallest Z-discriminant of all the index two division algebras with a real quadratic center. Lemma 8 then gives us thatd(Λ/Z) = 54. We immediately see that the regular presentation attached to the cyclic presentation of Dicosfulfills the expectations
of Equation 3.2. According to Lemma 15 we then have thatm(Λ) = 25, and according to Lemma 14
δ(Λ) = 1
5 = 0.2.
A comparison to complex centers proves that this algebra has the smallest discriminant of all the index two algebras where the center is a quadratic field.
Remark 3. We remark that the order code promised to exist by the previous example actu-ally played part in the construction of the Icosian code in [19].
The previous example gave us an idea of the achievable coding gain with order theo-retic methods. Yet a simple modulation scheme can easily ruin the performance of such codes. For instance, if we use a Z-module basis together with a PAM scheme the promised minimum determinant advantage might never get realized. Therefore the next chapter is devoted for constructing a code with rectangular shaping.
3.4
A rectangular MISO code with the best achievable
min-imum determinant
In this chapter we concentrate on the question of achievable minimum determinant of rect-angular multi-block codes in the space M2×4(C).
Proposition 22. Let us suppose that L is a rectangular multi-block code in the space M2×4(C). We then have that
δ(L) ≤ 1 16.
Proof. We expect w.l.o.g. that L has a fundamental parallelotope of volume 1. Consider an orthogonal basis of L. Due to the orthogonal shape at least one of the basis vectors must have length less than or equal to one. Let us suppose that (A1, A2) = A is a matrix
corresponding to such vector. This means that ||A||F ≤ 1. Let us consider the matrix
B = diag(A1, A2). According to Hadamard inequality we have that
| det(A1) det(A2)| = | det(B)| ≤
(||B||F)4 16 = (||A||F)4 16 ≤ 1 16.
In the following we are going to build an orthogonal order code that reaches the bound of the previous proposition. Let us consider the following algebra
Dort= (Q(i,
√
2)/Q(√2), σ, −1), and the natural order Λort of this algebra. The field L = Q(i,
√
2) can be seen as a Z[i]-module with a basis {1, ζ8}. Now the natural order can be written as
Λort = Z[i] ⊕ Z[i]ζ8⊕ uZ[i] ⊕ uZ[i]ζ8.
The operation of the automorphism τ is defined as τ (ζ8) = −ζ8, τ (i) = i and σ is just the
usual complex conjugation. The multi-block representation φ now gives us that φ(a1+ a2ζ8+ ua3+ uζ8a4) =
(a1+ a2ζ8) −(a3 + a4ζ8) a1− a2ζ8 −(a3− a4ζ8)
(a3+ a4ζ8) (a1+ a2ζ8) a3− a4ζ8 a1− a2ζ8
. By simply checking we see that
{1, i, ζ8, ζ8i, u, ui, uζ8, uζ8i}
forms a rectangular basis for the code. A particularly nice feature of this code is that we can apply QAM-modulation here, although the general construction method did not promise this.
We could now just calculate the fundamental parallelotope of this code and then de-termine the normalized minimum determinant, but we take a more general approach that sheds more light to the question of how we first came up with this code.
Lemma 23. [5, Lemma 2.9] Let us suppose that K is such an algebraic number field that OK is a principal ideal domain. If D = (E/K, σ, γ) is a K-central division algebra of
indexn and Λ is a natural order in D, then
|d(Λ/Z)| = |d(E/Q)nγ2n(n−1)|.
We now return to our example algebra above and to the fixed natural order Λort in it.
The discriminant of the extension Q(i,√2)/Q has absolute value 256. Lemma 23 now states that
|d(Λort/Z)| = 2562
and because the left regular representation in this case is suitable Lemma 15 gives us that
δ(Λort) =
1 16.
Remark 4. The code Λort appeared in [20] as a4 × 1 MISO code. It was noted that Λort
3.5
A multiuser coding scheme
In this chapter we propose a simple multiuser coding scheme that is based on our previous work on MISO codes. The scheme is based on the criteria presented in [6].
As an example we apply the code of Section 3.4 and compare its performance to the corresponding codes in [6] and [10].
Let us assume that
Γ =ζ 0 0 ζ
,
where ζ is some primitive mth root of unity, m being sufficiently large so that ζ cannot possibly be a root for the determinant polynomial, meaning that our 2-user code matrix will end up having rank 4.
If only one user is transmitting the situation is equal to delay four 2 × 2 single-user MIMO transmission.
The infinite code lattice for the first user is α(Λ) where α(a) = Γψ(a) τ (ψ(a)) ,
where a ∈ Λ. The single user code lattice for the second user is β(Λ), where β(b) = ψ(b) Γτ (ψ(b)) ,
and b ∈ Λ.
If the users are independent yet synchronized the signal sent by the two users is
C = α(a) β(b)
.
If we suppose that neither a or b is zero, then the determinant of the matrix C is a polyno-mial of ζ and the term attached to its highest power is ψ(a)τ (ψ(b)). By our assumption this term is non-zero. If ζ is now a suitable primitive mthroot of unity, we see that as long as a
and b are non-zero elements, matrix C has rank 4. If only one user is transmitting, then by Remark 2 the matrix has rank 2.
Let us now consider a sample code based on our orthogonal code of Section 3.4. The code for the first user is
ζ7(a1+ a2ζ8) ζ7(−(a3+ a4ζ8)) a1− a2ζ8 −(a3− a4ζ8)
ζ7(a3+ a4ζ8) ζ7(a1+ a2ζ8) a3− a4ζ8 a1− a2ζ8
and for the second user
b1+ b2ζ8 −(b3+ b4ζ8) ζ7(b1− b2ζ8) ζ7(−(b3− b4ζ8))
b3+ b4ζ8 b1+ b2ζ8 ζ7(b3− b4ζ8) ζ7(b1− b2ζ8)
,
3.6
Simulations
In this chapter we compare our code construction to two previously proposed codes [10] (HV) and [6] (GB).
In [6] the coding scheme consist of two single user codes
U1 = x1(1) x1(2) x1(3) x2(4) x1(2)∗ x1(1) x2(4)∗ x1(3)∗ and U2 = x2(1) x2(3) x2(2) x2(4) x∗2(1) x2(4)∗ x2(1) x2(3)∗ ,
where in both cases the symbols xi(j) are independently chosen from some QAM-constellation.
When both users are transmitting the combined matrix has rank 3 (see [6]).
In [10] the HV code is based on the number field code used in the construction of the 4 × 4 Perfect code [1]. The key parts are the field extension L/K = Q(i, ζ15+ ζ15)/Q(i),
its cyclic Galois group G(L/K) =< σ >, and an ideal I of the ring of algebraic integers OL. Here the single user codes are
U1 =
a σ(a) σ2(a) σ3(a)
iσ3(a) σ(a)2 σ(a) a
and
U2 =
ib iσ(b) σ2(b) σ3(b)
iσ3(a) iσ(a)2 iσ(a) a
,
where a and b are elements of the ideal I corresponding to a given QAM constellation. When both users are transmitting the combined 4 × 4 matrix has rank 4, and when only one user is transmitting the rank is 2 (see [10]).
In Figures 3.1 and 3.2 we compare our new code (NC) to the codes in [10] (HV) and [6] (GB) in a slow fading situation where the channel remains fixed for four channel uses. We see a considerable gain compared to the previous code constructions. When compared to the GB code the performance advantage is explained by the fact that when both users are transmitting, the combined matrix of the NC code has rank 4, whereas the GB code has rank 3 only. Both codes are taking full advantage of the delay four, but encoding of the GB code is perhaps simpler. The decoding of both the GB code and the NC code can be simply done using a sphere decoder. Both the GB code and the NC code involve an Alamouti-like structure which can be taken advantage of in the decoding process.
When comparing the HV code and the NC code we have tie on ranks, but the optimality of our single user codes (see Proposition 22) expectedly gives us an edge in coding gain. In this case the encoding and decoding processes have similar complexity.
0 2 4 6 8 10 12 14 16 10−4 10−3 10−2 10−1 100 SNR P(e) HV GB New Codes
12 14 16 18 20 22 24 26 10−4 10−3 10−2 10−1 100 SNR CER HV−16QAM GB−16QAM New Code−16QAM
Chapter 4
DMT optimal code construction for two
users
In this chapter we will focus on the construction of DMT optimal multiuser codes when there are two users in the system, communicating simultaneously to a common base station. We assume that each user has nttransmit antennas and there are nrreceive antennas at the
receiving end. Further, we will assume a symmetric MAC channel [7], meaning the users transmit at same multiplexing gain r, or equivalently, both transmit at rate R = r log2SNR in bits per channel use.
4.1
DMT for MIMO-MAC Channels
Considering a MIMO Rayleigh block fading channel, Tse et al. [7] showed that the code-word error probability of any such multiuser codes is lower bounded by
Pcwe(SNR) ˙≥ max
n
SNR−d∗nt,nr(r), SNR−d∗2nt,nr(2r)
o
, (4.1)
where by ˙≥ we mean the exponential inequality defined in [8], i.e. f (SNR) ˙≥g(SNR) if
lim SNR→∞ log f (SNR) log SNR ≥SNR→∞lim log g(SNR) log SNR . Notions of= and ˙. ≤ are defined similarly.
The negative exponent d∗nt,nr(r) is the point-to-point DMT [8] for the case when there is only one user with nttransmit antennas communicating at multiplexing gain r to the base
station that has nr receive antennas. d∗nt,nr(r) is a piecewise linear function connecting
the points (r, (nt− r)(nr− r)) for r = 0, 1, · · · , min{nt, nr}. From this, in the two-user
symmetric MIMO-MAC scenario, the maximal multiplexing gain can be achieved by the users is upper bounded by rmax= min{nt,n2r} since d∗2nt,nr(2rmax) = 0.
The terms SNR−d∗nt,nr(r) and SNR−d ∗
2nt,nr(2r) are respectively the probabilities when
one or both users are in outage, i.e. the probabilities that the channel is not good enough to support the targeted rate. In particular, due to the behaviors of d∗nt,nr(r) and d∗2nt,nr(2r),
Tse et al. showed that SNR−d∗nt,nr(r) ≥ SNR−d ∗ 2nt,nr(2r), r ∈ h 0, min{nt, nr 3 } i .
That is, when r ∈ 0, min{nt,n3r}, each user can achieve his/her best possible error
per-formance as if the other user is not present in the channel. This is called the single-user performanceregime. For min{nt,n3r} ≤ r ≤ min{nt,n2r}, the lower bound (4.1) is
domi-nated by the second term, corresponding to the event of both users in outage. This is termed the antenna pooling regime [7]. These show a fundamental difference between single-user (or equivalently point-to-point) DMT and multiuser DMT.
By using independent Gaussian random codebooks for each user, the converse of (4.1) was proved by Tse et al. [7]. They partitioned the error events into two kinds, the kind when one of the two users is in error, denoted by E1, and the other kind when both users are in
error, denoted by E2. They showed that when only one user is in error, the Gaussian random
code is able to achieve an error performance with Pr{E1} ˙≤SNR−d
∗
nt,nr(r), and similarly
Pr{E2} ˙≤SNR
−d∗
2nt,nr(2r) for the case when both users are in error. The above amounts to
that given the multiplexing gain r, the maximal possible diversity gain can be achieved by any multiuser codes is min{d∗nt,nr(r), d∗2nt,nr(2r)}. This is commonly referred to as the optimal MAC-DMT. Codes achieving this optimality are thus termed MAC-DMT optimal codes.
On the other hand, if deterministic codes were used; say code S1for the first user and S2
for the second. Both codes consist of (nt× T ) code matrices for some T that corresponds
to the channel coherence time, meaning the MIMO channel remains fixed during T symbol time. Further, the code matrices in S1 and S2 are required to satisfy the following power
constraint: ES1∈S1kS1k 2 F ≤ T · SNR and ES2∈S2kS2k 2 F ≤ T · SNR. (4.2)
By kAkF we mean the Frobenius norm of matrix A. Coronel et al. studied the op-timal DMT performance of a selective fading MIMO multiple-access channel [13] and gave a sufficient criterion for designing MAC-DMT optimal multiuser codes. Noting that Rayleigh block fading channel can be regarded as a frequency selective fading channel with only one multipath, to our present interest, the criterion shown in [13] is equivalent to the following.
Theorem 24 ( [13]). Let S1 and S2 be defined as above with nr ≥ 2nt and T ≥ 2nt.
Then codesS1 andS2 achieve the optimal MAC-DMT if the following inequalities are all
satisfied: min S16=S10∈S1 det (S1− S10)(S1− S10) † ˙ ≥ SNRnt−r min S26=S20∈S2 det (S2− S20)(S2− S20) † ˙ ≥ SNRnt−r min S16=S10∈S1,S26=S20∈S2 det ∆S∆S† ≥ SNR˙ 2nt−2r, where ∆S := S1− S 0 1
and where byA†we mean the hermitian transpose of matrixA.
We remark that the actual result of [13] was stated in a form different from the above and we do require nr ≥ 2nt in Theorem 24. When nr ≥ 2nt, showing the two results are
equivalent is not hard, yet the derivation steps can be somewhat lengthy. For brevity, we do not elaborate on the details and we refer the interested readers to [21, Section II] for details. Next we set C1 = C1 = 1 κS1 : S1 ∈ S1
and similarly C2 = κ1S2 with κ2 = SNR
1− r
nt, then the three criteria in Theorem 24 are
equivalent to min C16=C10∈C1 det (C1− C10)(C1 − C10) † ˙ ≥ 1 (4.3) min C26=C20∈C2 det (C2− C20)(C2 − C20) † ˙ ≥ 1 (4.4) min C16=C10∈C1,C26=C20∈C2 det ∆C∆C† ≥ 1˙ (4.5)
where ∆C = 1κ∆S. We remark that the constant κ is a power scaling factor frequency used in [22–24] such that the approximate universal cyclic division algebra space-time codes given in [22–24] also satisfy the same power constraint as S1 and S2. In other words, here
the codes C1 and C2 are reminiscent of the cyclic division algebra space-time codes. Now
with such transformation, we immediately recognize these three conditions (4.3)-(4.5) are the well-known non-vanishing determinant (NVD) criteria [23–26] for constructing point-to-point DMT optimal space-time codes except that a normal inequality ≥ was actually used in these works, rather than the exponential inequality ˙≥. Nevertheless, we remark that results in these works hold the same under exponential inequality ˙≥. With the above observations, Theorem 24 is equivalent to the following. The proof can be regarded as an alternative proof to Theorem 24 in the flat fading case.
Theorem 25. Let C1 andC2 be defined as above, and let the codeC1× C2 be obtained by
vertically concatenating the code matrices fromC1andC2. IfC1,C2, andC1× C2all satisfy
NVD criterion, then the codes are MAC-DMT optimal.
Proof. Similar to [7], we partition the error event into E1 and E2 that correspond
respec-tively to the events when one or both users are in error. Then we have Pr{E1} ≤ Pcwe(C1) + Pcwe(C2) ˙≤ SNR−d
∗ nt,nr(r)
Pr{E2} = Pcwe(C1× C2) ˙≤ SNR−d
∗
2nt,nr(2r),
where it follows from the fact that C1, C2, and C1× C2are all DMT optimal in the
point-to-point MIMO scenario. The readers are referred to [24] for the details. Pcwe(C) denotes the
Henceforth, we will refer to the criteria (4.3)-(4.5) as the full NVD condition. We note that as stated earlier, the full NVD condition is only sufficient for constructing MAC-DMT optimal codes, not necessary. In fact, we report the following negative result.1
Theorem 26. When nt = 1, i.e., each user with only one transmit antenna, there does not
exist any multiuser codes that are full NVD when normal inequality≥ is used in (4.3)-(4.5).
2
Proof. For ease of reading, the proof is relegated to the Appendix A.
In a nutshell, the proof shows that while it is possible to construct DMT optimal codes C1 and C2for user 1 and 2 respectively, as the existing cyclic-division algebra-based
space-time codes [24] would do, it is impossible for the product code C1× C2to be NVD, i.e.
hav-ing minimum nonzero determinant ≥ 1. Any such product code would be ill-conditioned and have determinant extremely close to 0 at high SNR regime. Thus, Theorem 26 shows the nonexistence of codes satisfying the design criteria provided by Coronel et al. in [13] if we require the minimum determinant ≥ 1. A similar, but much stronger, result is later given in [21, Theorem 5] and shows such codes do not exist even when we replace the normal inequality by the exponential inequality, i.e. when exactly (4.3)-(4.5) are required. Therefore, we may conclude that the full NVD condition is in general too strict to yield any MAC-DMT optimal codes. Another implication can be made is the following. The full NVD condition can be met only if the two users cooperate in their transmission. Once without cooperation as it is in MIMO-MAC channel, the full NVD condition can never be met and the determinant must be vanishing.
However, we may relax the full NVD condition without affecting the DMT perfor-mance. To do so, we will use a different partition of error events. Let E1 denote again the
event when one of the two users is in error. But let E2,1 (resp. E2,2) denote the error event
when two users are in error and the error matrix is of rank nt (resp. 2nt.) Clearly E2 is
a disjoint union of E2,1 and E2,2. Now the codes C1 and C2 are MAC-DMT optimal if the
following holds.
Theorem 27. Let C1 andC2 be defined as above. Then they are MAC-DMT optimal if the
error events have probabilities upper bounded by Pr{E1} ≤ SNR˙ −d ∗ nt,nr(r), Pr{E2,1} ≤ SNR˙ −d ∗ nt,nr(r), Pr{E2,2} ≤ SNR˙ −d ∗ 2nt,nr(2r).
The rationale behind the above theorem is the observation that in the single-user perfor-mance regime, the error probability SNR−d∗2nt,nr(2r) is not dominant, hence we could relax
1A more general result of the nonexistence of full NVD multiuser codes that satisfy the criteria given
by Coronel et al. [13] for arbitrary number of transmit antennas and for arbitrary number of users has been proven by the authors, but it will be treated in a separate paper [14].
the condition such that event E2,1has larger probability SNR−d
∗
nt,nr(r)than the actual outage
probability SNR−d∗2nt,nr(2r). This will not affect the overall DMT performance. Compared
with the full NVD condition required in Theorems 24 and 25, Theorem 27 relaxes greatly the code design criterion. Specifically, the full NVD condition requires that whenever C1 6= C10 ∈ C1 and C2 6= C20 ∈ C2, the matrix ∆C must be nonsingular and be NVD, i.e.
having determinant det(∆C∆C†) ≥ 1. This has been shown to be impossible by Theorem 26. On the other hand, Theorem 27 says that the difference matrix ∆C can be singular, and the only condition is that should it happen, the resulting error performance cannot be worse than SNR−d∗nt,nr(r), in order to maintain the MAC-DMT optimality. In [13], event E2,1was
required to have probability absolutely zero, which is too strict and forbids the existence of MAC-DMT optimal codes.
4.2
Construction of MAC-DMT Optimal Codes
In this section, we will provide a systematic construction of multiuser codes for the two-user case. The proposed codes will not meet the full NVD criterion as such codes do not exist. In the next chapter we will analyze the DMT performance of these newly proposed codes and show that they actually achieve the relaxed criteria given in Theorem 27.
Let F = Q(i) be the base number field. The proposed construction calls for two addi-tional number fields L = F(θ) and K = F(η) that are cyclic Galois extension of F with [L : F] = ntand [K : F] = 2. We require further that L ∩ K = F. Let Gal(L/F) = hσi and
Gal(K/F) = hτ i, and let E = LK = F(θ, η) be the compositum of the fields L and K. The relation between these field extensions is shown in Fig. 4.1.
E = F(θ, η) hτ i oooooo ooooo hσi O O O O O O O O O O O L = F(θ) hσiOOOOO O O O O O O K = F(η) hτ i
ooooooooo oo
F = Q(i)
Figure 4.1: Field extensions required by the proposed code constructions.
Clearly, L/F is cyclic Galois; so is E/K. Moreover, we have Gal(E/K) = hσi. Hence there exists some suitable non-norm element γ ∈ OFsuch that
D= (E/K, σ, γ) = E ⊕ uE ⊕ · · · ⊕ unt−1
E
is a division algebra, where by OF we mean the ring of algebraic integers in F and u is an indeterminate satisfying unt = γ and xu = uσ(x) for every x ∈ E. Similarly as in
x =Pnt−1 i=0 u ix i ∈ D, xi ∈ E, as an nt× ntmatrix given by ψ(x) := x0 γσ(xnt−1) · · · γσ nt−1(x 1) x1 σ(x0) · · · γσnt−1(x2) .. . ... . .. ... xnt−1 σ(xnt−2) · · · σ nt−1(x 0) . (4.6) According to Definition 7 and Proposition 4 det(ψ(x)) ∈ K for every x ∈ D, and hence clearly
nrK/F(det(ψ(x))) = det(ψ(x))τ (det(ψ(x))) ∈ F (4.7) where nrK/F(a) is the algebraic norm of a from K to F. Note that when the element x is taken from the natural order OD := OE⊕ · · · ⊕ u
nt−1O
E, it can be further shown that
nrK/F(det(ψ(x))) = det(ψ(x))τ (det(ψ(x))) ∈ OF (4.8) and OF = Z[i]. It in turn implies that the absolute |nrK/F(det(ψ(x))) | is bounded from
below by 1 whenever 0 6= x ∈ OD. This property is termed generalized non-vanishing
determinant condition in [23] (also cf. Definition 12) and is required in constructing the DMT optimal multi-block space-time codes.
Having said the above, the proposed construction is the following. Given the multiplex-ing gain r, let
A(SNR) = na + bi : −SNR2ntr ≤ a, b ≤ SNR r
2nt, a, b odd
o
(4.9) and let {e0, · · · , e2nt−1} be an integral basis of E/F. Given A(SNR) we define the
infor-mation set A(SNR) = (n t−1 X i=0 ui 2nt−1 X j=0
ai,jei : ai,j ∈ A(SNR)
)
. (4.10)
It is clear that A(SNR) ⊂ OD.
If the first user wishes to transmit information x ∈ A(SNR), the transmitter actually sends in 2ntchannel uses the (nt× 2nt) code matrix
Sx = κ ψ(x) τ (ψ(x)) , (4.11)
where κ is a constant given by
κ2 = SNR. 1−ntr (4.12)
and is set such that E kSxk 2
F = 2nt· SNR.
On the other hand, if the second user wishes to transmit information y ∈ A(SNR), the resulting code matrix associated with y is
Sy = κ ψ(y) −τ (ψ(y)) . (4.13)
matrices associated with the first and the second users. The overall received signal matrix Ro is given by Ro = H1Sx+ H2Sy+ W = κ H1 H2 X τ (X) Y −τ (Y ) + W (4.14) where X := ψ(x), Y := ψ(y), and where W is the (nr× 2nt) noise matrix whose entries
are i.i.d. CN (0, 1) random variables. Therefore, our proposed multiuser code may be described as follows S = κ X τ (X) Y −τ (Y ) : X = ψ(x), Y = ψ(y), x, y ∈ A(SNR) . (4.15) For every code matrix S ∈ S, the upper half submatrix corresponds to the information sent by the first user and the lower half comes from the second user. Clearly the two submatrices are coded independently, and there is no cooperation between these two users. As κ is a normalizing constant for power constraint, below we will pay our attention only to the set of unnormalized code matrices, i.e.
C = X τ (X) Y −τ (Y ) : X = ψ(x), Y = ψ(y), x, y ∈ A(SNR) . (4.16) First, we show that every code matrix C ∈ C has determinant in Z[i].
Lemma 28. Let C be defined as above; then for every C ∈ C, det(C) ∈ Z[i].
Proof. Clearly, the entries of C lie in OE, the ring of algebraic integers in E; hence det(C) ∈ OE. It suffices to show that the determinant is fixed by the automorphisms τ and σ. To this end, given any C ∈ C, we simply check
τ (det(C)) = det τ (X) X τ (Y ) −Y = (−1)ntdet X τ (X) −Y τ (Y ) = (−1)ntdet Int −Int X τ (X) Y −τ (Y ) = (−1)2ntdet(C) = det(C) and σ(det(C)) = det Z −1XZ τ (Z−1XZ) Z−1Y Z −τ (Z−1Y Z) = det Z −1XZ Z−1τ (X)Z Z−1Y Z −Z−1τ (Y )Z = det Z −1 Z−1 X τ (X) Y −τ (Y ) Z Z = det(C)
where Z := ψ(u) and where we have used the fact that τ (Z) = Z as γ ∈ OF. Overall, these show det(C) ∈ Z[i].
While the above lemma shows that the determinant of the matrix C lies in Z[i], it does not necessarily mean that the code satisfies the NVD property. For example, if τ : η → −η, then setting y = ηx ∈ A(SNR) makes the resulting code matrix C singular as the lower half can be obtained by multiplying from the left the upper half by matrix ψ(η). In particular, whether the code matrix C is singular or not, is completely characterized by the following lemma. Lemma 29. Given C = X τ (X) Y −τ (Y ) ∈ C withX = ψ(x) and Y = ψ(y), x, y ∈ A(SNR), if x 6= 0, then
rank(C) = nt, ifyx
−1+ τ (yx−1) = 0
2nt, otherwise.
(4.17)
Moreover, ifτ : η → −η then rank(C) = ntif and only if
yx−1 ∈
nt−1
M
i=0
uiηL := L. (4.18)
Proof. To find out the rank of matrix C, we follow the conventional Gaussian eliminant procedure with elementary row operations. In particular, we remark that such operations would be easier to carry out if we change our focus to the matrix
˜
C = x τ (x) y −τ (y)
∈ M2(D).
This is because elementary row operations in M2(D) correspond exactly to block
elemen-tary row operations in C. Specifically, we mean following ψ p q C˜= ψ(p) ψ(q) C.
Thus, if x 6= 0 by assumption we see that rank(ψ(x)) = ntas D is a division algebra, and
secondly that there must exist p ∈ D such that y = px since yx−1 ∈ D. Then we can rewrite ˜C as ˜ C = x τ (x) px −τ (p)τ (x) .
Multiplying from the left the first row of ˜C by −p and adding to the second row yields x τ (x)
0 − (τ (p) + p) τ (x)
.
It is clear that ˜C is left- and right- invertible in Mnt(D) if and only if τ (p) + p 6= 0. In other
To prove the second claim, we first note that {1, θ, · · · , θnt−1} is a basis of L/F and
similarly {1, η} a basis for K/F. p = yx−1 can be uniquely represented as
p = nt−1 X i=0 ui nt−1 X j=0 p1,i,jθj+ nt−1 X i=0 uiη nt−1 X j=0 p2,i,jθj
for some p1,i,j, p2,i,j ∈ F. Hence
τ (p) = nt−1 X i=0 ui nt−1 X j=0 p1,i,jθj − nt−1 X i=0 uiη nt−1 X j=0 p2,i,jθj.
Now we see p = −τ (p) if and only if p1,i,j = 0 for all i and j. This proves the claim.
Remark 5. The above lemma shows that the proposed construction does not satisfy the full NVD criterion. This is not surprising as already pointed out in Theorem 26 that codes satisfying full NVD criterion do not exist. Yet, as suggested by the reviewers, it is some-times interesting to see how often the code violates the full NVD criterion. That is, we are interested in knowingPr{p + τ (p) = 0}. Although such probability depends closely upon the underlying set of base alphabetA(SNR), we can argue heuristically to show such probability is extremely small. Furthermore, our estimate of Pr{p + τ (p) = 0} will be asymptotically tight at high SNR regime, i.e. when the transmission rate R (in bits per channel use) gets larger and larger.
To see the above, let us fixx, the symbol sent by the first user and consider all possible choices ofy sent by the second user. Clearly, as p = yx−1 ∈ D we have p = p0 + up1+
· · · + unt−1p
nt−1withpi ∈ E. Define
P := p = yx−1: y ∈ A(SNR), p + τ (p) = 0 .
Note that from(4.18) we have |P| =
p = yx−1
: y ∈ A(SNR), p ∈ L
≤ |{z ∈ A(SNR) : z ∈ L}| = |A(SNR)|n2t .
The inequality ˙≤ is because of the following. Given any p =Pnt−1
i=0 u ip iwithp + τ (p) = 0, the element y = px = nt−1 X i=0 ui 2nt−1 X j=0 yi,jej
might not be in A(SNR), since
1. the elementyi,j might not be a Gaussian integer, and
2. yi,j might not be inA(SNR), especially when A(SNR) is of small size.
Thus, the above estimate of |P| is generally loose for small A(SNR). However, when A(SNR) becomes larger, px is likely to be in A(SNR) and the proposed estimate becomes more accurate. Overall, as|A(SNR)| = SNR2ntrwe see
Pr {p + τ (p) = 0} ≤ |P| |A(SNR)| =
1
p|A(SNR)| = SNR
Whennt = 2, we numerically simulated the probability Pr {p + τ (p) = 0} at different
rates.
• At R = 4 and A(SNR) being QPSK, the probability Pr{p+τ (p) = 0} ≈ 5.15×10−5,
while(4.19) gives 4−4 ≈ 4 × 10−3.
• At R = 6 and A(SNR) being 8QAM, we get Pr{p + τ (p) = 0} ≈ 1.104 × 10−8,
while(4.19) gives 8−4 ≈ 2 × 10−4.
• At R = 8 and A(SNR) being 16QAM, we report Pr{p + τ (p) = 0} ≈ 1.194 × 10−9,
while(4.19) gives 16−4 ≈ 10−5.
Thus we see in general for high transmission rate, Pr{p + τ (p) = 0} is extremely close to 0, and the difference matrix ∆C is of full rank with probability close to 1. Further-more, from the simulations above we see that at small size of A(SNR), the probability Pr {p + τ (p) = 0} behaves more like
Pr {p + τ (p) = 0} ≈ |A(SNR)|
|A(SNR)| = |A(SNR)|
−(2n2
t−1)
since not ally = px belong to A(SNR) for a fixed x and a random p with p + τ (p) = 0. Armed with the two above lemmas, we are now ready to show that the proposed code S is MAC-DMT optimal. The proof will be given in the next subsection.
Theorem 30. Given the multiplexing gain r, the proposed code S achieves over quasi-static Rayleigh fading channel with coherence timeT ≥ 2ntthe DMT
d(r) = d ∗ nt,nr(r), if r ≤ minnt, nr 3 d∗2nt,nr(2r), if r ∈ minnt,n3r , min nt,n2r (4.20) meaning thatS is MAC-DMT optimal.
4.3
Proof of Theorem 30
For any S 6= S0 ∈ S with
S = κ ψ(x) τ (ψ(x)) ψ(y) −τ (ψ(y)) and S0 = κ ψ(x 0) τ (ψ(x0)) ψ(y0) −τ (ψ(y0)) , define dx := x − x0and dy := y − y0. Hence
∆S = S − S0 = κ ψ(dx) τ (ψ(dx)) ψ(dy) −τ (ψ(dy))
1. Event E1 corresponds to the case when either user one or user two is in error, but
not both. This means that the difference matrix ∆S of (4.21) has either dx = 0 or dy = 0.
2. Error event E2,1 concerns the case when both users are in error, but the overall error
matrix ∆S is not of full rank 2nt. That is, we have both dx and dy being nonzero,
but the error matrix ∆S is only of rank ntand dy(dx)−1+ τ (dy(dx)−1) = 0.
3. Error event E2,2is the case when both users are in error and the error matrix ∆S is of
full rank 2nt.
Clearly, whenever a decoding error occurs, the error event E is a union of the above three error events, namely, we have
E = E1 ∪ E2,1 ∪ E2,2
and the corresponding error probability achieved by S is
Pcwe(SNR) = Pr{E} ≤ Pr{E1} + Pr{E2,1} + Pr{E2,2}.
Thus, in the remaining of this chapter we will show Pr{E1} ≤ SNR˙ −d ∗ nt,nr(r), Pr{E2,1} ≤ SNR˙ −d ∗ nt,nr(r), Pr{E2,2} ≤ SNR˙ −d∗ 2nt,nr(2r).
Error Event E1 We first focus on analyzing the error event E1 that corresponds to the
case when either user one or user two is in error, but not both. Given the channel matrices H1and H2 we define the squared Euclidean distance between S and S0 as
d2E(S, S0) := kH∆Sk2F (4.22) where H = [H1H2]. Due to the structure of S, we can without loss of generality assume
that dx 6= 0 but dy = 0. The other case of dx = 0, dy 6= 0 can be analyzed in a similar fashion. Thus in this case we have
d2E(S, S0) = kH1ψ(dx)k2F + kH1τ (ψ(dx))k2F. (4.23)
To obtain a lower bound on d2E(S, S0), let λ1,1 ≥ · · · ≥ λ1,m be the set of ordered nonzero
eigenvalues of H1H †
1 where m = min{nt, nr} and let `1,1 ≤ · · · ≤ `1,nt and `2,1 ≤
· · · ≤ `2,nt be the ordered nonzero eigenvalues of ψ(dx)ψ(dx)
†and τ (ψ(dx))τ (ψ(dx))†,
respectively. Using the mismatch eigenvalue bound [23, 24, 27] we see d2E(S, S0) is lower bounded by
d2E(S, S0) ≥ κ2
m
X
i=1
Note that nt Y i=1 2 Y j=1 `j,i = nrK/F(det(ψ(dx))) 2 ≥ 1. (4.25)
Repeatedly using the arithmetic mean-geometric mean inequality and (4.25) along the same lines as in [23, 24], given k, k = 1, 2, · · · , m, it can be shown that
d2E(S, S0) ˙ ≥ κ2 " m Y i=m−k+1 λ1,i #1k kψ(dx)k2 F + kτ (ψ(dx))k 2 F −nt−kk ˙ ≥ SNR1−ntr " m Y i=m−k+1 λ1,i #1k SNR−ntr nt−k k .
Setting λ1,i = SNR−α1,i gives
d2E(S, S0) ˙≥ SNRδ1,k(α1) (4.26) where α1 = [α1,1· · · α1,m] t and δ1,k(α1) := 1 k " m X i=m−k+1 (1 − α1,i) # − r k. (4.27) Following the sphere bound argument as in [24], the probability of event E1 given the
channel matrices H1 and H2can be upper bounded by
Pr {E1|H1, H2} ≤ Pr kW k2F ≥ d 2 E(S, S0) 4 = exp −d 2 E(S, S0) 4 2nrnt−1 X j=0 (d2E(S, S0))j j! .
As d2E(S, S0) ˙≥SNRδ1,k(α1)for all k, we see from the above that Pr {E
1|H1, H2}
.
= 0 if there exists k such that δ1,k(α1) > 0. Since Pr {E1|H1, H2} ≤ 1, it follows that
Pr{E1} = EH1,H2Pr {E1|H1, H2} ≤ 2 Pr {α1 : δ1,k(α1) ≤ 0, all k} ,
where the extra factor of 2 shown above is due to the inclusion of the other case when user two is in error which has the same probability as the present case. Clearly, in terms of diversity analysis one can safely neglect this factor of 2.
Arguing similarly as [22, 23] it can be shown that
{α1 : δ1,k(α1) ≤ 0, k = 1, · · · , m} = ( α1 : m X i=1 (1 − α1,i) + ≤ r ) (4.28)
where (x)+ := max{x, 0}. Now we see Pr{E1} ≤ Pr ( α1 : m X i=1 (1 − α1,i) + ≤ r ) = Prnlog detInr + SNRH1H † 1 ≤ r log SNRo . = SNR−d∗nt,nr(r),
where the last exponential equality follows from [8].
Error Event E2,2 For simplicity, we will first analyze the event E2,2, and leave the most
tedious event E2,1 to the last. Recall that E2,2 is the event when both users are in error, and
the error matrix ∆S is of full rank 2nt. In other words, we have in (4.21) that dx, dy 6= 0
and dy (dx)−1+ τ dy (dx)−1 6= 0. Lemmas 28 and 29 then imply the matrix
∆C = ψ(dx) τ (ψ(dx)) ψ(dy) −τ (ψ(dy))
(4.29)
must have full rank 2ntand 1 ≤ | det(∆C)| ∈ Z. Let `1 ≤ `2 ≤ · · · ≤ `2nt be the ordered
eigenvalues of ∆C∆C†, and let λ2,1 ≥ · · · ≥ λ2,m0 be the ordered nonzero eigenvalues of
HH†with H = [H1 H2] and m0 = min{2nt, nt}.
Following arguments similar to E1, the squared Euclidean distance d2E(S, S
0) for the
pair (S, S0) falling in the category of E2,2 is lower bounded by d2E(S, S0) ˙≥SNR
δ2,k(α2), for
k = 1, 2, · · · , m0, where λ2,i = SNR−α2,i and
δ2,k(α2) := 1 k " m0 X i=m0−k+1 (1 − α2,i) # − 2r k . (4.30) Again along the same lines as in the previous case we can show that
Pr {E2,2} ≤ Pr {α˙ 2 : δ2,k(α2) ≤ 0, all k} = Pr ( α2 : m0 X i=1 (1 − α2,i) + ≤ 2r ) = Prlog det Inr + SNRHH † ≤ 2r log SNR . = SNR−d∗2nt,nr(2r),
proving that the code S satisfies the third condition required in Theorem 27.
Error Event E2,1 Finally we are left with the last type of error event, the event E2,1
oc-curring when both users are in error, but the error matrix does not have full rank. In other words, it is the case when dx, dy 6= 0, p = dy (dx)−1and p + τ (p) = 0 in (4.21). From the proof of Lemma 29 these conditions mean
