Final Report of Granted Project
NSC98 - 2221 - E - 009 - 045 - MY3
Hsiao-feng Francis Lu
Department of Electrical and Computer Engineering
National Chiao Tung University
[email protected]
Tuesday 25
thSeptember, 2012
Abstract
This report provides an overview of major results we have obtained in the research project “Con-structions of Diversity-Multiplexing Tradeoff Optimal Codes for Multiuser MIMO Systems with Applications to MIMO Mobile Communications” supported by National Science Council under contract number NSC98 - 2221 - E - 009 - 045 - MY3 during August 2009 - July 2012.
In this report, we are concentrating explicit code constructions for input multiple-output (MIMO) multiple-access channels (MAC) with K users. The first construction is dedicated to the case of symmetric MIMO-MAC where all the users have the same number of transmit anten-nas ntand transmit at the same level of per-user multiplexing gain r. Furthermore, we assume that
the users transmit in an independent fashion and do not cooperate. The construction is systematic for any values of K, ntand r. It is proved that this newly proposed construction achieves the
opti-mal MIMO-MAC diversity-multiplexing gain tradeoff (DMT) provided by Tse et al. at high-SNR regime. We next take a further step to investigate the MAC-DMT of a general MIMO-MAC where the users are allowed to have different numbers of transmit antennas and can transmit at different levels of multiplexing gain. The exact optimal MAC-DMT of such channel is explicitly charac-terized in this report. Interestingly, in the general MAC-DMT, some users might not be able to achieve their single-user DMT performance as in the symmetric case, even when the multiplexing gains of the other users are close to 0. Detailed explanations of such unexpected result are provided in this report. Finally, by generalizing the code construction for the symmetric MIMO-MAC, ex-plicit code constructions are provided for the general MIMO-MAC and are proved to be optimal in terms of the general MAC-DMT.
We also answer several open questions related to diversity-multiplexing tradeoffs (DMTs) for point-to-point and multiple-access (MAC) MIMO channels. By analyzing the DMT performance of a simple code, we show that the optimal MAC-DMT holds even when the channel remains fixed for less than Knt+ nr− 1 channel uses, where K is the number of users, ntis the number
of transmit antennas of each user, and nr is the number of receive antennas at receiver. We also
prove that the simple code is MAC-DMT optimal. A general code design criterion for constructing MAC-DMT optimal codes that is much more relaxed than the previously known design criterion is provided. Finally, by changing some design parameters, the simple code is modified for use in point-to-point MIMO channels. We show the modified code achieves the same DMT performance as the Gaussian random code.
Keywords:Diversity-multiplexing gain tradeoff (DMT), multiple access channel (MAC), cyclic division algebras (CDAs), multiple-input multiple-output (MIMO) channel, space-time block codes (STBCs).
Referred Papers Supported by Granted
Project
Under the support of this three-years project, we have so-far successfully produced the following 17 papers ( 6 Journal Papers, and 11 conference papers published in the highest quality confer-ences):
1. 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.
2. H. F. Lu, R. Vehkalahti, C. Hollanti, J. Lahtonen, Y. Hong, and E. Viterbo, “New space-time code constructions for two-user multiple access channels,” IEEE Journal of Selected Topics in Signal Processing, vol. 3, nol. 6, pp. 939-957, Dec. 2009.
3. H. F. Lu and C. Hollanti, “Optimal diversity multiplexing tradeoff and code constructions of constrained asymmetric MIMO systems,” IEEE Trans. Inform. Theory, vol. 56, no. 5, pp.2121-2129, May 2010.
4. H. F. Lu, “Constructions of diversity-multiplexing tradeoff optimal vector codes for asyn-chronous cooperative networks using decode-and-forward protocols,” IEEE Trans. Wireless Communications, May 2010
5. H. F. Lu, C. Hollanti, R. Vehkalahti, and J. Lahtonen, “DMT optimal codes constructions for multiple-access MIMO channel,” IEEE Trans. Inform. Theory, vol. 57, no. 6, Jun. 2011. 6. H. F. Lu, ”Remarks on diversity-multiplexing tradeoffs for multiple-access and
point-to-point MIMO channels,” IEEE Trans. Inform. Theory, vol. 58, no. 2, Feb. 2012.
7. H. F. Lu and C. Hollanti “Diversity-multiplexing tradeoff-optimal code constructions for symmetric MIMO multiple access channels,” Proc. 2009 IEEE Int. Symp. on Inform. Theory (ISIT), Seoul, Korea.
8. C. Hollanti, H.F. Lu, and R. Vehkalahti, “An algebraic tool for obtaining conditional non-vanishing determinants,” Proc. 2009 IEEE Int. Symp. on Inform. Theory (ISIT), Seoul, Korea.
9. J. Lahtonen, R. Vehkalahti, H. F. Lu, C. Hollanti, and E. Vitero, “On the decay of the deter-minants of multiuser MIMO lattice codes,” Proc. ITW 2010, Cairo, Egypt, Jan. 2010. 10. H. F. Lu, J. Lahtonen, R. Vehkalahti, and Camilla Hollanti, “Remarks on the criteria of
constructing MIMO-MAC DMT optimal codes,” Proc. ITW 2010, Cairo, Egypt, Jan. 2010. 11. H. F. Lu, “Diversity-multiplexing tradeoff in MIMO Gaussian interference channels,” Proc.
2010 IEEE Int. Symp. Inf. Theory(ISIT 2010), Austin, TX, Jun. 2010.
12. H. F. Lu, “Approximately Universal MIMO Diversity Embedded Codes”, Proc. 2010 ISITA, Taichung, Taiwan, Oct. 2010.
13. R. Vehkalahti, C. Hollanti, J. Lahtonen, and H. F. Lu, ”Some Simple Observations on MISO Codes”, Proc. 2010 ISITA, Taichung, Taiwan, Oct. 2010.
14. R. Vehkalahti and H. F. Lu, “An algebraic look into MAC-DMT of lattice space-time codes,” Proc. 2011 IEEE Int. Symp. on Inform. Theory(ISIT), St. Petersburg, Russia.
15. T.W. Tang, M. K. Chen, and H. F. Lu, “Improving the DMT Performances of MIMO Linear Receivers,” Proc. 2011 IEEE Int. Symp. on Inform. Theory (ISIT), St. Petersburg, Russia. 16. R. Vehkalahti and H. F. Lu, “Diversity-multiplexing gain tradeoff: a tool in algebra?” in
Proc. ITW 2011, pp. 135 - 139 , Paraty, Brazil, Oct. 2011.
17. S. M. Huang, H. F. Lu, and S. M. Moser, “Minimal-Rate Description for Multiple-Access Channels,” in Proc. ISITA 2012, Hawaii, Oct. 2012.
Chapter 1
Introduction
During the last decade extensive research has been carried out in the design of point-to-point space-time (ST) codes [1, 2] for multiple-input multiple-output (MIMO) communication systems. ST codes based on cyclic division algebras (CDAs) [3–7] that can also be regarded as a kind of algebraic lattice codes and/or as a kind of linear dispersion ST codes [8] have been shown to perform extremely well. The error performance of these codes have been shown to be very close to the outage bound not only for practical numbers of antennas but also at moderate SNR values.
For high-SNR regime, the same point-to-point CDA-based ST codes have been shown [4] to be optimal in terms of the diversity-multiplexing tradeoff (DMT) proposed by Zheng and Tse [9]. Specifically, let nt and nr be respectively the numbers of transmit and receive antennas at
transmitter and receiver ends. Let r, 0 ≤ r ≤ min{nt, nr}, denote the multiplexing gain such that
the actual transmission rate equals
R = r log2SNR (bits per channel use). (1.1) Assuming a MIMO Rayleigh block fading channel, it was shown [4] that at multiplexing gain r, the CDA-based ST codes achieve the optimal codeword error probability
Pcwe(r)
.
= SNR−d∗nt,nr(r) (1.2)
at high-SNR regime, where by = we mean the exponential equality defined in [9]. That is, we. write f (SNR) = SNR. b if
lim
SNR→∞
log f (SNR) log SNR = b. The notations of ˙≥ and ˙≤ are similarly defined. The exponent d∗
nt,nr(r) is commonly known as the
DMT [9] and is given by a piecewise linear function connecting the points (r, (nt− r)(nr − r))
for r = 0, 1, · · · , min{nt, nr}. Furthermore, d∗nt,nr(r) represents the largest diversity gain that can
be achieved by any point-to-point ST codes under Rayleigh block fading channel whenever the channel remains static for at least a block of ntchannel uses [4] and varies independently from one
block to another.
For other types of fading statistics, the CDA-based ST codes are also known [4] to be capable of achieving the optimal error performance in such channels that include Rician, Weibull and Nak-agami as special cases. ST codes that are optimal in all fading statistics are coined approximately universal codes[4, 10].
If coding across independent fading blocks is allowed, the multi-block CDA code [6] has been shown to be approximately universal as well. In particular, it achieves codeword error probability
Pcwe(r)
.
= SNR−m·d∗nt,nr(r), (1.3)
at multiplexing gain r, where m is the number of independent fading blocks occupied by the code. The exponent m · d∗nt,nr(r) is known as the multi-block DMT [6, 9] when coding is applied over
m independent fading blocks. Therefore, the multi-block CDA-based ST code is optimal in terms of the multi-block DMT at high-SNR regime. More important, (1.3) indicates that the code has error probability decreasing to zero as m approaches infinity whenever d∗nt,nr(r) > 0. Hence, the multi-block ST code could potentially achieve the MIMO ergodic channel capacity at high-SNR regime and simultaneously be optimal in terms of the multi-block DMT at every discrete value m. Motivated by the promising outcome in the point-to-point scenario, the aim of this report is to investigate the code construction for the multiple-access channel (MAC) scenario. We will concentrate on the uplink transmission from multiple mobile users to a common base station (or access point). Both the mobile users and the base station may be equipped with multiple antennas. Consider a MIMO-MAC with K mobile users. For simplicity, we first focus on the case of symmetric MIMO-MAC[11], where each user is equipped with nttransmit antennas and
commu-nicates independently to the base station that has nr receive antennas. Furthermore, we assume
that all the users transmit at the same level of multiplexing gain. With a slight abuse of notation, hereafter we will denote by r the per-user multiplexing gain in the symmetric MIMO-MAC. Let S0, · · · , SK−1, be respectively the ST codes used by the kth user, k = 0, 1, · · · , K − 1. Each code
Sk, k = 0, 1, · · · , K − 1, consists of (nt× T ) matrices and satisfies the following power constraint:
ES∈SkkSk
2
F ≤ T · SNR, (1.4)
where by kSkF we mean the Frobenius norm of matrix S. Furthermore, we require |Sk| = SNRrT
for all k such that every user transmits at the same multiplexing gain r. Let Hk be the (nr× nt)
channel matrix of the kth user. We assume Hkis fixed for a block of T channel uses. Hkis known
completely to the receiver at base station but unknown to all the users. Entries of Hkare modeled
as i.i.d. CN (0, 1) complex Gaussian random variables to model the MIMO Rayleigh block fading channel. Let Sk ∈ Sk be the signal matrix transmitted by the kth user; then the signal matrix
received at base station is given by
Y =
K−1
X
k=0
HkSk+ W, (1.5)
where W is the (nr× T ) noise matrix with i.i.d. CN (0, 1) entries. When each user’s information
is encoded independently, Tse et al. [11] proved that the tradeoff between the diversity gain d and multiplexing gain r in a symmetric MIMO-MAC is governed by the following theorem.
Theorem 1 (Symmetric MAC-DMT [11]). In a symmetric MIMO-MAC with K users, each having nttransmit antennas and transmitting independently at multiplexing gainr, the maximal possible
diversity gain is given by
d∗nt,nr,K(r) := min 1≤k≤Kd ∗ knt,nr(kr) = d∗nt,nr(r), ifr ∈ 0, min nt,K+1nr , d∗Knt,nr(Kr), ifr ∈ min nt,K+1nr , min nt,nKr , (1.6) where d∗knt,nr(kr) is the point-to-point DMT for knt transmit antennas, nr receive antennas and
multiplexing gainkr defined as before (or see [9,11]). Equation (1.6) is termed optimal symmetric MAC-DMT. The multiplexing gain r for nonnegative diversity gain is bounded between
0 ≤ r ≤ min n nt, nr K o = rmax. (1.7)
Compared with the point-to-point scenario, the decrease of maximal multiplexing gain by a factor of K (see nr
K in rmaxof (1.7)) is due to the sharing of nr receive antennas among K users
and the fact that d∗Knt,nr(Krmax) = 0. Equation (1.6) also shows that when the level of
multi-plexing gain is low such that r ∈ 0, min nt,K+1nr , each user is able to retain his single-user
performance, i.e., d∗nt,nr,K(r) = d∗n
t,nr(r), as if there were no other users in the channel. On the
other hand, when the level of multiplexing gain is high and r ∈ min nt,K+1nr , min nt,nKr ,
the MIMO-MAC system would operate in the antenna pooling region [11], and single-user perfor-mance can no longer be maintained. As a consequence, a much lower diversity gain d∗Knt,nr(Kr) dominates the error performance in this region.
In Fig. 1.1 we demonstrate the above facts of the symmetric MAC-DMT for the case of K = 3 users, nt = 2 and nr = 2. It can be clearly seen that the turning point between the
single-user and antenna pooling regions is at r = minnt,K+1nr
= 12 and the cut-off point of r is at min{nt,nKr} = 23. 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 Multiplexing Gain r Diversity gain d
Figure 1.1: The MAC DMT for K = 3 users with nt= 2 and nr = 2.
The construction of MAC-DMT optimal codes calls for a coding scheme that independently encodes, but simultaneously transmits, each mobile user’s information over the MIMO channel such that at receiver end, the decoding of all users’ signals achieves the best possible error per-formance dictated by the MAC-DMT. Thus, a coding scheme is called MAC-DMT optimal if it achieves the following error performance under joint decoding
Pcwe(r)
.
= SNR−d∗nt,nr,K(r).
1.1
Prior Work
Several works have been reported in this area. Nam et al. [12] presented the first MAC-DMT optimal scheme using a class of structured multiple-access random lattice ST codes. For the con-structions of deterministic codes, below we briefly review some relevant earlier papers. Almost all deal exclusively with the two-user symmetric MIMO-MAC case, i.e., K = 2.
1. [13] extended the pairwise-error-probability-based design criteria of point-to-point ST codes to the MAC case for K = 2 users and nt = 2, nr = 2. An explicit (4 × 4) two-user MIMO
code1, i.e., a (2 × 4) code for each user, based on independent Alamouti blocks [2] is also
introduced in [13]. Yet, we remark that such code does not achieve the optimal symmetric MAC-DMT (1.6).
2. In [14] Badr and Belfiore proposed an explicit algebraic code for K = 2 and nt = 1. The
idea can be extended to bigger values of K. The determinant of the code matrix is non-zero thanks to a ”twisting element.” However, the determinant is vanishing. The decay of determinants of this two-user MIMO-MAC code was carefully studied in [15]. It was shown that the code is MAC-DMT optimal, when r ≤ 15. Whether this code achieves the optimal MAC-DMT also when r > 1/5 remains an open question. In [15] it was shown, however, that the code fails to satisfy the criteria for achieving optimal MAC-DMT set forth in [16], when r > 1/5. This alone does not mean that their code could not be optimal, as the criteria in [16] is sufficient, but not necessary (see [17] for justification of this claim).
3. Some explicit, algebraic code constructions for nt > 1 and K = 2 were introduced by
Hong and Viterbo in [18]. A design criterion based on an approximation of truncated union bound was proposed. With such criterion they constructed a code that outperforms in error performance the aforementioned (4 × 4) two-user code [13].
4. Badr and Belfiore [19] proposed another (4 × 4) two-user MIMO-MAC code which is ob-tained by adding a twist matrix Γ to the (2 × 2) Golden ST code [20, 21] such that the overall code matrix is nonsingular whenever all the submatrices associated with each user are nonzero. However, because of this additional Γ matrix, the overall code matrix, though nonsingular, could be ill-conditioned at high-SNR regime, thereby resulting in a vanishing determinant, similarly as did their earlier one-antenna code [14] already discussed above. 5. [22] addressed the problem of whether there exists a two-user MIMO-MAC code satisfying
the non-vanishing determinant (NVD) property. This problem concerns whether the twisted Golden MIMO-MAC code [19] can be further improved to avoid the disadvantage of having a vanishing determinant. The answer is negative. [22] shows that if all the overall code matrices are nonsingular whenever the submatrices from each user are nonzero, then some of them must have determinant arbitrarily close to zero, i.e., have vanishing determinants. 6. By removing the Γ matrix, [22] reported another code construction and proved its
MAC-DMT optimality for K = 2 and for any values of ntand nr. Computer simulations showed
that this code outperforms the (4 × 4) code of [19] at all SNR values. Another important contribution reported in [22] was that, for the two-user MAC case, one does not need the whole code matrix to be nonsingular, and hence introducing the additional Γ rotation matrix is not necessary from the MAC-DMT point of view.
7. In [16], Coronel et al. studied the optimal DMT performance of a selective fading MIMO-MAC and provided a sufficient criterion for designing MIMO-MAC-DMT optimal codes for any K and nt. Noting that the Rayleigh block fading channel is a flat fading channel, a
simpli-fication of their criterion requires the product concatenation of codes from any subsets of K users to satisfy the NVD property such that the error probabilities associated with these subcodes do not exceed the corresponding outage probability. However, as already pointed out in [22], such codes do not exist for the case of K = 2 . A further investigation of their criterion can be found in [17].
1In this report, by an (m × n) code we mean a code consisting of (m × n) code matrices, where m is the number
of transmit antennas required for transmission, and n is the number of channel uses. The number m can be either nt
or Knt, depending on the discussion. When m = nt, the code is for each user’s use. When m = Knt, we mean
the vertical concatenation of all users’ codes as an overall code. Notation nt× nrwithout parenthesis is used for the
1.2
Complete Construction of DMT-Optimal Multiuser MIMO
Codes
A complete solution to the problem of constructing MIMO-MAC codes for K users that are MAC-DMT optimal in Rayleigh MIMO-MAC is presented in this report.
We first provide the constructions of MAC-DMT optimal codes for the symmetric MIMO-MAC. Later, we will give the code construction for the general MIMO-MAC where the users are allowed to have different numbers of transmit antennas and can transmit at different levels of multiplexing gain.
A general result on the nonexistence of NVD MIMO-MAC codes is presented in Chapter 2. This result suggests that the design criterion proposed by Coronel et al. [16] might be too strict to yield any MAC-DMT optimal codes. A relaxed design criterion is then provided in this section.
In Chapter 3, we present a new code construction for the symmetric MIMO-MAC for any K, nt and nr. Several nice properties of the proposed code are presented in this section. We prove
that this newly proposed construction is MAC-DMT optimal and meets the relaxed design criterion given in Chapter 2. For ease of reading, the proof of MAC-DMT optimality is relegated to Chapter 6.
In Chapter 4 we investigate the MAC-DMT in a general MIMO-MAC where the users are allowed to have different numbers of transmit antennas and transmit at different levels of multi-plexing gain. The exact general MAC-DMT in such channel will be given in Section 4.2, and it will be seen that unlike the symmetric case, some users in the general MIMO-MAC are no longer able to achieve their single-user performance even if the multiplexing gains of other users are extremely close to zero. The reasons for such unexpected result will be carefully explained therein. Finally, in Section 4.4 the newly proposed code construction for symmetric channels will be extended to cater to the general MIMO-MAC. The MAC-DMT optimality of the generalized construction will be presented in Chapter 7.
1.3
Several Open Problems in Multiuser MIMO
Communica-tion
It is known that using multiple antennas at both transmitting and receiving ends in a point-to-point multiple-input-multiple-output (MIMO) channel can increase the transmission rate and simulta-neously provide higher diversity gain. Assuming there are nt transmit antennas and nr receive
antennas, it has been shown that the ergodic channel capacity of such MIMO Rayleigh block fad-ing channel is approximately min{nt, nr} log2SNR in bits per channel use [23], and the maximal
achievable diversity gain is ntnr [1, 24], provided that the channel remains fixed for at least nt
channel uses. Let R = r log2SNR be the actual transmission rate, where r is termed the mul-tiplexing gain. Zheng and Tse [9] showed there is a fundamental tradeoff between mulmul-tiplexing gain r and diversity value d. Such tradeoff is commonly known as the diversity-multiplexing gain tradeoff (DMT) and is reproduced below.
Theorem 2 ( [9]). In a MIMO Rayleigh block fading channel with nt transmit and nr receive
antennas, assuming the transmitter transmits at multiplexing gain r, the maximal diversity gain d∗(r) can be achieved by any coding schemes is a piecewise linear function connecting the points (r, (nt− r)(nr− r)) for r = 0, 1, · · · , min{nt, nr}, when the channel is fixed for at least T ≥
nt+ nr− 1 channel uses.
If the MIMO channel cannot hold static for at least nt + nr − 1 channel uses, some lower
bounds on DMT based on Gaussian random coding schemes are provided in [9]. By using space-time codes constructed from cyclic division algebra (CDA) [25], Elia et al. [4] proved that the
same DMT d∗(r) holds whenever the channel is static for at least T ≥ ntchannel uses. However,
such result cannot be further improved, and the exact DMT for T < ntis still uncertain.
In both DMT results, Theorems 1 and 2, the proofs proceed by first establishing an upper bound on DMT based on an outage formulation, and then by using a Gaussian random coding scheme to show the converse based on a union bound argument. It should be noted that in both point-to-point and MAC cases the requirement on the channel coherence time T for the optimal DMT to hold actually comes from the union bound, not the outage. When T ≥ Knt+ nt− 1, Coronel et al. [16]
presented a criterion for constructing MAC-DMT optimal codes. For any coding schemes, let Ek
denote the error event that only the messages from k users are erroneously decoded. Coronel et al. showed that for any k-subsets of users, 1 ≤ k ≤ K, if Pr {Ek} is upper bounded by the probability
of the corresponding outage event formulated by these k users, i.e. if one can show
Pr {Ek} ˙≤ Pr n log detInr + SNRHkH † k ≤ kr log SNRo, (1.8)
where H = [Hi1 · · · Hik] is the overall channel matrix and Hij is the (nr× nt) channel matrix of
the jth user, then the code is MAC-DMT optimal. Notions of exponential inequalities ˙≥, ˙≤, ˙>, ˙<, and equality = are defined in [9]. Specifically, in terms of code design, the above criterion (1.8). means that the (knt× T ) matrix obtained by vertically concatenating the signal matrices from k
users must be of full row rank and should perhaps satisfy the nonvanishing determinant (NVD) criterion [4, 26]. This full NVD design criterion was explicitly given in [16].
The aim of this report is to answer the following questions.
1. Is it possible to achieve the optimal MAC-DMT d∗nt,nr,K(r) when T < Knt+ nr− 1?
2. Is design criterion (1.8) necessary? or is it only sufficient?
3. In order to be MAC-DMT optimal, is it necessary for a code to satisfy the NVD criterion for any (knt× T ) submatrix formed by any k-subsets of users?
4. In point-to-point MIMO channel, can one design a non-random DMT optimal code for T < nt? Also, will the resulting DMT be the same as d∗nt,nr(r)? In other words, when T < nt, it
relates to the question of whether the outage event will dominate the error performance. The major contribution of this report is not to provide constructions of codes having per-formance better than the previously known DMT optimal codes, for example, the CDA based codes [4], the Golden perfect codes [20], the max-order codes [7], or the multi-block codes [6]. Instead, we aim to address the above four questions that none of these codes can answer.
By analyzing the DMT performance of a very simple code, we will provide answers to all the above questions. We will consider a MIMO-MAC channel with K = 2 users, each having only nt = 1 transmit antenna, and we will assume there are nr = 2 receive antennas at receiving end.
While Theorem 1 holds for codes with T ≥ Knt+ nr − 1 = 3 channel uses, we will prove this
simple code achieves the same optimal MAC-DMT d∗1,2,2(r) with only T = 1 or 2 channel uses. Furthermore, from the DMT analysis of this code we will see that criterion (1.8) is only sufficient, not necessary, and one does not need full NVD in order to achieve the optimal MAC-DMT. By slightly modifying the parameters of this code, we will show in the point-to-point MIMO scenario this simple code achieves the same DMT performance as the Gaussian random code over the fast Rayleigh fading channel, i.e. the case when T = 1, which relates to the fourth question in the above list.
In Chapter 8 we will present the simple code as well as the corresponding DMT performance analysis. Inferences from the DMT analysis will be given in Chapter 9 and will answer all the above questions of interest.
Chapter 2
Relaxed Design Criterion of MAC-DMT
Optimal Codes
In this section, we first present a rigorous, yet negative, result on the nonexistence of a MIMO-MAC lattice code that has the NVD property. This result suggests that the design criteria proposed by Coronel et al. [16] might be too strict to yield any MAC-DMT optimal codes. Following this, a relaxed design criterion will be presented and will be met by all subsequent constructions of MIMO-MAC codes in this report.
Consider a symmetric MIMO-MAC with K users, each having nttransmit antennas and
com-municating independently to the base station at the same level of multiplexing gain r. Let S0, · · · ,
SK−1, be respectively the (nt× T ) space-time codes used by the kth user, k = 0, 1, · · · , K − 1,
all satisfying the power constraint (1.4). If independent Gaussian random codebooks were used, i.e., the entries of code matrices Sk ∈ Skare i.i.d. CN
0,SNRn
t
random variables for all k, Tse et al.[11] showed that the event Em of m users in error has probability upper bounded by
Pr {Em} ≤ Pr {Om}
.
= SNR−d∗mnt,nr(mr), (2.1)
where Omis the event of m users in outage. Note that the overall error event E = E1∪E2∪· · ·∪EK.
The union bound on E gives
Pr {E } ≤ K X m=1 Pr {Em} ˙≤ max m Pr {Om} . (2.2)
Since the right-hand-side of (2.2) has a negative SNR-exponent equal to d∗nt,nr,K(r) defined in (1.6), (2.2) proved the achievability of MAC-DMT claimed by Theorem 1 based on the argument of Gaussian random codebooks.
We next turn our attention to the deterministic ST codes. From the point-to-point perspective, it is known [4] that ST codes satisfying the NVD property have the same error probability as the outage events. Thus, for any MIMO-MAC code {S0, · · · , SK−1}, set
Ck = 1 κSk: Sk ∈ Sk where κ2 = SNR. 1−ntr .
To see how κ is chosen, we offer the following insight. For each k, the code Ck has size |Ck| =
|Sk| = SNRrT so that it is of multiplexing gain r. An explicit construction of Ck was given in [4]
where the code is seen as a real algebraic ST lattice code with dimension 2ntT . Hence there
are |Ck|
1
2ntT = SNR r
2nt PAM signals selected from each dimension and kCkk2
F ≤ SNR˙
r
Ck ∈ Ck. Thus, the constant κ is chosen such that the code Sk= κ Cksatisfies the power constraint
(1.4).
From [4], it is easy to prove the following theorem which in turn gives a sufficient criterion for designing MAC-DMT optimal codes. We remark that this theorem is an alternative statement of the result given in [16] under certain restrictions, and we refer the interested readers to [17] for the connections.
Theorem 3 ( [16]). Let C0, · · · , CK−1 be given as above. For any Im = {i0, i1, · · · , im−1} ⊆
{0, 1, · · · , K − 1}, let CIm be the product concatenation ofCi0, · · · , Cim−1, defined by
CIm = CIm = Ci0 .. . Cim−1 : Cij ∈ Cij, ij ∈ Im .
If for all pairs of distinct code matricesCij 6= C
0
ij ∈ Cij,j = 0, 1, · · · , m − 1, the difference matrix
∆CIm = Ci0 − C 0 i0 .. . Cim−1− C 0 im−1 , (2.3) satisfiesdet(∆CIm∆C † Im) ˙≥ 1, where by C
† we mean the Hermitian transpose of matrixC, then
the codesC0, · · · , CK−1are jointly MAC-DMT optimal.
Proof. Note that the imposed condition implies that the code CIm satisfies the NVD property for
any Im. Along similar lines as in [4], it can be shown that the error event E (Im) associated with
code CIm, i.e., the error event of users in Imin error, has probability upper bounded by
Pr {E (Im)} ˙≤ Pr {O(Im)}
.
= SNR−d∗mnt,nr(mr),
where O(Im) is the event of users in Im in outage. Now taking union bound over all possible Im
as in (2.2) completes the proof.
Remark 1. The condition of det(∆CIm∆C
†
Im) ˙≥ 1 for all Im is called the full NVD criterion
and is actually equivalent to the criterion given by Coronel et al. in [16] with certain restrictions, see [17] for details. It should be noted that this full NVD condition is only sufficient, not necessary. However, the following result suggests that this condition might be too strong and precludes the existence of codes meeting the criterion. We call the stronger conditiondet(∆CIm∆C
†
Im) ≥ 1
theexactly full NVD criterion.
Theorem 4. For any K > 1 and for any nt≥ 1, there do not exist any linear MIMO-MAC codes1
that satisfy the exactly full NVD criterion.
Proof. For ease of reading, the proof is relegated to Chapter 5.
1Here by linear codes we mean codes having linear dispersion forms [8] or having a lattice structure. Almost all
Roughly speaking, the proof of Theorem 4 shows that while it is possible to construct DMT-optimal codes C0, · · · , CK−1for each user, as the existing CDA-based ST codes [4] would do, it is
impossible for the product code C0× · · · × CK−1to have an exactly full NVD. Any such product
code would have difference matrices ∆CImsuch that det(∆CIm∆C
†
Im) is extremely close to zero
at high-SNR regime. In terms of conventional rank and coding gain design criteria of ST codes, this means that even if the code achieves full diversity gain, it necessarily loses significantly in coding gain. Therefore, it becomes meaningless to say that the code achieves full rank and full diversity. We may conclude that the exactly full NVD condition is in practice too strict to yield MAC-DMT optimal codes.
Another implication from the proof of Theorem 4 is that the exactly full NVD condition can be met only if the users cooperate in their transmission. Without cooperation, the exactly full NVD condition can never be met and the determinant must be vanishing.
On the other hand, we may relax the exactly full NVD condition without adversely affecting the DMT performance. To do so, we will partition the error events in a different manner. Given the set of users Im, let En(Im), 1 ≤ n ≤ m, denote the error event when the users in Im are in
error and the corresponding error matrix ∆CIm (cf. (2.3)) has rank exactly nnt. Clearly event
E(Im) defined in the proof of Theorem 3 is a disjoint union of E1(Im), · · · , Em(Im). Now the
codes C0, · · · , CK−1are jointly MAC-DMT optimal if the following holds.
Theorem 5 (Relaxed design criterion). Let C0, · · · , CK−1be defined as above. Then they are jointly
MAC-DMT optimal if the error eventsEn(Im) have probabilities upper bounded by
Pr {En(Im)} ˙≤ SNR−d
∗
nnt,nr(nr) (2.4)
for all 1 ≤ n ≤ m ≤ K and for all Im ⊆ {0, 1, · · · , K − 1}. Furthermore, as for design of
MAC-DMT optimal codes we require at least that Pr {En(Im)} ˙≤ SNR
− min{d∗
nt,nr(r),d∗Knt,nr(Kr)} (2.5)
for all1 ≤ n ≤ m ≤ K and for all Im.
While (2.5) might be the most relaxed condition for designing MAC-DMT optimal codes, in this report we will focus on condition (2.4). The rationale behind the above theorem is the observation that the error probabilities SNR−d∗mnt,nr(mr) with 1 < m < K are not dominant in the
overall DMT performance. Hence we could relax the conditions such that the event
E(Im) = m
[
n=1
En(Im)
has probability larger than the corresponding outage probability, but no larger than the dominant error probability. That is, we could allow
Pr {E (Im)} Pr{O(Im)}
.
= SNR−d∗mnt,nr(mr), (2.6)
but would still require
Pr {E (Im)} ˙≤ SNR
− min{d∗
nt,nr(r),d∗Knt,nr(Kr)}.
Relaxation (2.6) would not affect the overall DMT performance. Compared with the exactly full NVD condition required by Theorems 3, Theorem 5 relaxes greatly the code design criterion in the following ways.
1. We do not require the difference matrix ∆CIm to be nonsingular and to satisfy the NVD
property when all the component matrices Cij − C
0
ij are nonzero, which has been shown to
be impossible by Theorem 4.
2. Should the difference matrix ∆CIm happen to be singular, (2.4) requires the resulting error
performance must be no worse than SNR−d∗nnt,nr(nr) for some n, 1 ≤ n ≤ m, in order to
maintain the MAC-DMT optimality.
3. In Theorem 3, events En(Im) with n < m were required to have probability absolutely zero.
Chapter 3
MAC-DMT Optimal Code Construction for
Symmetric MIMO-MAC Channels
For the symmetric MIMO-MAC coded system with K users, each having nttransmit antennas and
transmitting at multiplexing gain r, in this section we will propose a systematic code construction that is MAC-DMT optimal for any combinations of K, nt, nr, and r. The construction does not
assume any cooperation among the users. Furthermore, compared with the MAC-DMT optimal two-user code proposed in [22] where a sign change is required in the code matrices, here in the proposed method each user encodes his own information using an identical encoder. This greatly simplifies the hardware implementation of these encoders.
3.1
Proposed Construction
Given the number of users K, let Kobe the smallest odd integer such that Ko ≥ K, i.e.,
Ko =
K + 1, if K even,
K, if K odd. (3.1)
The construction calls for the following number fields. Let Ko = F(ηo) be a number field that
is a cyclic Galois extension of F = Q( ı ) with degree Ko, where ı =
√
−1. Let L = F(θ) be another cyclic Galois extension of F with degree nt. Let σ and τo be the generators of Galois
groups Gal(L/F) and Gal(Ko/F) with degrees nt and Ko, respectively. The fields Ko and L are
chosen1 such that K
o∩ L = F. Let Eo = KoL = F(ηo, θ) be the compositum of Ko and L. See
Fig.3.1 for the relation among the required number fields. The readers are referred to [4, 22, 28] for the constructions of such number fields.
Let Do := (Eo/Ko, σ, ζ) be a cyclic division algebra with
Do = Eo⊕ zEo⊕ · · · ⊕ znt−1Eo, (3.2)
where
ζ = γ
γ∗ , (3.3)
xz = zσ(x) (3.4)
for x ∈ Eo. The element z is an indeterminate satisfying znt = ζ ∈ F∗, and 0 6= γ ∈ OF is some
suitable nonnorm element2. By γ∗ we mean the complex conjugate of γ and OF is the algebraic closure of Z in F [25, 29, 30]. Notice that kζk = 1 and ζ is unimodular. It has been shown [5] that with such unimodular ζ, Do is always a cyclic division algebra.
1
A more general condition on Koand L is that the automorphisms σ and τ0commute.
2A sufficient criterion for finding a suitable nonnorm element γ is given in [26, Theorem 1]. Also, we refer the
Eo = F(θ, ηo) Ko ntKo nt L = F(θ) nt Ko = F(ηo) Ko F = Q( ı )
Figure 3.1: Field extensions required by the proposed code constructions.
Remark 2. While in the above we have set ζ to be of form ζ = γγ∗ such that ζ is unimodular, it
might be possible that in some CDAs, the nonnorm element γ is actually an nth root of unity for some integer n and is already unimodular. See [31] for such example construction. Should it be the case, we could set ζ = γ, and the discussion below can be easily modified to show that the MAC-DMT optimality of the proposed constructions remains to hold. Therefore, for simplicity, here we will focus only on the case ofζ = γγ∗.
Remark 3. We note that by construction the Galois groups of the numbers fields are Gal(Eo/Ko) = hσi ,
Gal(Eo/L) = hτoi ,
Gal(Eo/F) = hτo, σi = hτoi × hσi ,
where in the last line hτoi × hσi denotes the direct product of the groups generated by τo andσ,
respectively. It should also be noted that the automorphismsτoandσ commute, i.e.,
τoσ = στo
due to the direct product of two groups.
Given multiplexing gain r, let A(SNR) be the base alphabet defined as
A(SNR) = a + b ı : −SNR r 2nt ≤ a, b ≤ SNR r 2nt, a, b ∈ Z, a, b odd ;
then the corresponding information set is
Ao(SNR) = (n t−1 X i=0 zi Kont−1 X k=0
xi,kek: xi,k ∈ A(SNR)
)
, (3.5)
where {e0, · · · , eKont−1} is an integral basis of Eo/F. It should be noted that
PKont−1
k=0 xi,kek ∈ Eo
for xi,k ∈ A(SNR) ⊂ OF and that Ao(SNR) ⊂ Do. Let
ψo : Do → Mnt(Eo)
be the left-regular map that maps elements in Do into (nt× nt) square matrices with entries in Eo.
Specifically, given u ∈ Do with
u =
nt−1
X
i=0
ψo(u) is given by ψo(u) := u0 ζσ(unt−1) · · · ζσ nt−1(u 1) u1 σ(u0) · · · ζσnt−1(u2) .. . ... . .. ... unt−1 σ(unt−2) · · · σ nt−1(u 0) . (3.6)
Note that the field Ko is the center of the division algebra Do, meaning that uk = ku for any
u ∈ Do and k ∈ Ko. Equivalently we have
ψo(u)ψo(k) = ψo(k)ψo(u),
showing that the matrix-product commutes.
Proposition 6 ( [4, 25]). Let Doandψo be defined as above. Then
det (ψo(u)) ∈ K∗o
for all0 6= u ∈ Do, where K∗o = Ko\ {0}.
Having defined the above, the encoding of each user’s data stream proceeds as follows. Given the multiplexing gain r, the ith user first partitions his binary data steam into blocks of rKontlog2
SNR bits. Then using the integral basis {e0, · · · , eKont−1} and set Ao(SNR) defined above, each
block of binary bits is mapped in a one-one fashion to a symbol xi ∈ Ao(SNR) ⊂ Do. The
encoding is performed independently at each user’s end.
Given xi ∈ Ao(SNR), the ith user actually sends out the following (nt× Kont) signal matrix
Si through his nttransmit antenna array in Kontchannel uses
Si = κ Xi τo(Xi) · · · τoKo−1(Xi) , (3.7)
where Xi = ψo(xi) and where κ is a normalizing constant such that
E kSik2F = ntKoSNR
.
= SNR.
Hence we have
κ2 = SNR. 1−ntr (3.8)
Remark 4. The above construction of the MIMO-MAC codes is reminiscent of the multi-block ST code presented in [6]. Some key differences are highlighted below.
1. In the proposed construction we require the length of the code to bent· Ko whereKo must
be an odd integer.
2. The number fields Ko and L are required such that the automorphisms σ and τo commute.
This was not needed in [6].
3. The elementζ of the CDA Domust be unimodular, and we have setζ = γγ∗.
We use the following example to illustrate the proposed construction.
Example 1. We consider the case of K = 2 and nt = 2. By construction Ko = 3 is the smallest
odd integer such thatKo ≥ K. Then it can be shown that with θ = eı
π
8 andηo = 2 cos 2π
7 the
number fields L = F(θ) and Ko = F(ηo) meet the required conditions of [L : F] = 2, [Ko : F] = 3
and L ∩ Ko = F. Furthermore, we have η3o + ηo2− 2ηo− 1 = 0. The generators σ and τo for the
Galois groupsGal(L/F) and Gal(Ko/F) are given respectively by
σ : θ 7→ −θ andτo : ηo 7→ η2o− 2 = 2 cos
4π 7
The set{1, θ, ηo, θηo, η2o, θηo2} is an integral basis for Eo/F.
As the prime ideal(2 + ı ) of Z[ ı ] remains inert in OKo andOL, following from [4] this gives an appropriate nonnorm element γ = 2 + ı . Hence we have ζ = 2+ ı2− ı. With Eo = F(θ, ηo),
Do = (Eo/Ko, σ, ζ) is a CDA of index 2 which is also a central simple Ko-algebra [25]. Next let
ui = xi,0+ θxi,1+ ηoxi,2+ θηoxi,3+ η2oxi,4+ θηo2xi,5
fori = 0, 1 with xi,j ∈ A(SNR). The Galois conjugates of ui are for example given by
σ(ui) = xi,0− θxi,1+ ηoxi,2− θηoxi,3+ ηo2xi,4− θηo2xi,5,
τo(ui) = xi,0+ θxi,1+ ηo0xi,2+ θη0oxi,3+ ηo02xi,4+ θηo02xi,5
whereηo0 = η2o− 2 = 2 cos 4π
7 and τo(η 0
o) = 1 − ηo− ηo2 = 2 cos 8π7 . With the above, the signal
matrix of the first user is given byS0 = κ X0 τo(X0) τo2(X0) , where κ2 = SNR1−
r 2 and X0 = u0 ζσ(u1) u1 σ(u0) .
By vertically concatenating the signal matrices from all users, the overall MIMO-MAC code of the K users is S = S = κ X0 · · · τoKo−1(X0) .. . . .. ... XK−1 · · · τoKo−1(XK−1) : Xi = ψo(xi), xi ∈ Ao(SNR) . (3.9)
For ease of code performance analysis that comes later we set C = 1κS, i.e., C = C = X0 · · · τoKo−1(X0) .. . . .. ... XK−1 · · · τoKo−1(XK−1) : Xi = ψo(xi), xi ∈ Ao(SNR) . (3.10)
Remark 5. Below we briefly compare the proposed construction of S with another MAC-DMT optimal code constructed forK = 2 users in [22]. The latter MIMO-MAC code takes the following form S2 = S2 = κ X0 τ (X0) X1 −τ (X1) : Xi = ψ(xi), xi ∈ A(SNR) . (3.11)
The construction ofS2requires a number field K = F(η) with [K : F] = 2 and Gal(K/F) = {1, τ }
such that E = KL = F(θ, η), [E : F] = 2ntandGal(E /F) = Gal(L/F)×Gal(K/F). Here by “1”
ofGal(K/F) we mean the trivial automorphism. The field L and the element θ are defined as be-fore. The elementxiis taken from the cyclic division algebra D= E ⊕z0E for some indeterminate
z0. A(SNR) is the base-information set defined similarly as Ao(SNR) in (3.5). Thus, compared
with the present proposed construction, we see thatS2 requires an additional sign change at the
second block matrix of the second user’s code. This sign change is essential to ensure an NVD-like property. It also endows S2 with another nice property that the transmission of code matrices in
S2 takes only2ntchannel uses, less than that required byS. However, this additional sign change
might complicate system design as the system must constantly check which user requires a sign change and which user does not. Such disadvantage does not exist in the proposed construction of S. Everything works perfectly after patching an extra block of transmission when K is even. Another drawback ofS2is the difficulty of generalization to the cases ofK > 2.
Let Hi be the (nr× nt) channel matrix of the ith user. We assume Hi is fixed for a block of
ntKochannel uses. Following (1.5), given the overall transmitted code matrix S ∈ S, the received
signal matrix at receiver end is
Y0 · · · YKo−1
= H0 · · · HK−1 S + W. (3.12)
W is the noise matrix whose entries are i.i.d. CN (0, 1) random variables, and Yj is the jth block
received signal matrix given by
Yj := κ K−1 X i=0 Hiτoj(Xi) + Wj, j = 0, 1, · · · , Ko− 1, and W = W0 W1 · · · WKo−1 .
3.2
Properties of the Proposed Construction
To simplify the analysis of code performance, below we define the extended versions of S and C.
Co := Co = X0 · · · τoKo−1(X0) .. . . .. ... XKo−1 · · · τ Ko−1 o (XKo−1) : Xi = ψo(xi), xi ∈ Ao(SNR) , (3.13) So := {So= κCo : Co ∈ Co} . (3.14)
Given the overall signal matrix S ∈ S, let So ∈ So be any signal matrix such that the upper
(Knt× Kont) submatrix of Soequals S. Then we can rewrite (3.12) as
Y0 · · · YKo−1 = H0 · · · HKo−1 So+ W, (3.15) where HKo−1 = HK−1, if K odd, 0, if K even.
By 0 we mean the all-zero matrix of proper size. Noting (3.12) and (3.15) are equivalent, hence-forth we will work only with the extended codes So and Co, rather than S and C. We next show
Property 1. For any Co ∈ Co, we have (γ∗)Ko(nt−1) det(Co) ∈ Z[ ı ]. (3.16)
Proof. We first claim
τo(det(Co)) = det(Co). (3.17)
To see this, notice that
τo(det(Co)) = det τo(X0) · · · τoKo(X0) .. . . .. ... τo(XKo−1) · · · τ Ko o (XKo−1) = det τo(X0) · · · X0 .. . . .. ... τo(XKo−1) · · · XKo−1 = (−1)nt(Ko−1)det(C o) = det(Co),
where the last equality follows from the fact that Ko− 1 is even, hence the claim (3.17) is proved.
Next, we show
σ(det(Co)) = det(Co). (3.18)
To this end, define
Z = ψo(z), (3.19)
where z is the indeterminate defined as in (3.2). Since from (3.4) xz = zσ(x) for all x ∈ Eo, it is
clear that σ(X) = Z−1XZ, where X = ψo(x). Now we have
σ(det(Co)) = Z−1X0Z · · · τoKo−1(Z −1X 0Z) .. . . .. ... Z−1XKo−1Z · · · τ Ko−1 o (Z −1X Ko−1Z) = Z−1X0Z · · · Z−1τoKo−1(X0)Z .. . . .. ... Z−1XKo−1Z · · · Z −1τKo−1 o (XKo−1)Z = Z−1 . .. Z−1 × X0 · · · τoKo−1(X0) .. . . .. ... XKo−1 · · · τ Ko−1 o (XKo−1) Z . .. Z = det(Co),
where we have used the fact that τo(Z) = Z since 0 6= ζ ∈ F by construction. Thus, as det(Co) is
fixed by both τoand σ, we see that det(Co) ∈ F = Q( ı ).
Finally, from the definition of ψo (3.6), the matrix
τoj(Xi) 1 γ∗ . .. γ∗
has entries in OEo for all i = 0, 1, · · · , Ko− 1 and j = 0, 1, · · · , nt− 1 since
Ao(SNR) ⊂ OEo ⊕ zOEo ⊕ · · · ⊕ z
nt−1O
Eo.
OEo is the ring of algebraic integers in number field Eo. It then follows that
(γ∗)Ko(nt−1)
det(Co)
∈ OEo.
Summarizing the above results, we conclude that
(γ∗)Ko(nt−1)
det(C)
∈ OEo ∩ Q( ı ) = Z[ ı ],
and this completes the proof.
Property 2. Let C= x>0 .. . x>K o−1 = x0 · · · τoKo−1(x0) .. . . .. ... xKo−1 · · · τ Ko−1 o (xKo−1) (3.20) and Co = X0 · · · τoKo−1(X0) .. . . .. ... XKo−1 · · · τ Ko−1 o (XKo−1) ∈ Co
withXi = ψo(xi), xi ∈ Ao(SNR), where by x>we mean the transpose of vectorx. Let m be the
maximal number of rows in C that are linearly independent as a left Do-module; then
rank(Co) = mnt (3.21)
where the rank is measured in the complex number field C.
Proof. To find out the rank of matrix Co, we use the elementary row operations from Gaussian
elimination method. Note that the same row operations can be performed on C whose entries are in Do. Extra care must be taken because multiplication in Do is non-commutative. Further, we
note that elementary row operations on C are equivalent to the block elementary row operations on Co. By this we mean that, say P is a (Ko× Ko) elementary matrix with entries in Do; then it is
clear
Ψo(P C) = Ψo(P )Co,
where Ψois the natural extension of ψo to the (Ko× Ko) central simple matrix algebra MKo(Do)
over Do[25], i.e.,
Ψo(P ) = [ψo(Pi,j)] . (3.22)
From hypothesis, assume {x>i0, · · · , x>im−1} is the maximal subset of the rows of C that are linearly independent over Do. Then it follows that there are m leading ones in the row-reduced matrix of C.
Equivalently, the same block elementary operations Ψo(P ) would reduce matrix Co into a matrix
whose main diagonal consists of m identity matrices, each of size (nt× nt), after permuting the
Property 2 shows that the overall code matrix Co ∈ Co might not always have full rank Kont,
and the rank of Co is always a multiple of nt. This is not too much of a surprise as it is
straight-forward to see that in (3.13) if some Xi’s are identical, then the overall code matrix Co cannot be
nonsingular.
Compared with the constructions proposed in [18,19], the matrix Coof the present construction
could be singular even when the component matrices Xi are all distinct and nonzero as shown
by Property 2. Nevertheless, we will prove in Chapter 6 that in order to achieve the optimal MAC-DMT performance at high-SNR regime, it is unnecessary to construct codes such that Cois
nonsingular whenever all the component matrices Xiare distinct and nonzero.
Before rigorously proving the above statement, a heuristic way to see this is the following. Since the users communicate independently to the base station, for any overall MIMO-MAC code C it is impossible for all the code matrices C ∈ C to be nonsingular as some component matrices Ck of the kth user could be zero. Also, from the pairwise error probability point of view, for any
C 6= C0 ∈ C, C − C0
can be singular at least when the information symbols transmitted by some users are the same. The rank of overall code matrices Co is at best a multiple of nt. Therefore,
intuitively speaking, perhaps it would not hurt to make things a bit worse in the sense that the difference matrix C − C0 can be singular in other cases. By this we mean that if there are m distinct information symbols in the difference matrix C − C0, the maximal possible rank of C − C0 is mnt. We claim that it would not hurt in the DMT sense if the construction can provide only
rank nnt for some n with 1 < n < m. The reason for this actually follows from Theorem 5 that
the error events En(Im) of m users in error but getting only rank distance nnt do not dominate
the error performance in the final DMT performance. Therefore, we strongly speculate that such difference matrices C − C0do not have to achieve the same rank mntas the Gaussian random code
does. The rank can be less, as long as the resulting error performance is not worse than those of m = 1 and m = Ko.
Although we do not need the whole code Coto satisfy the full NVD property as in the
point-to-point scenario, an alternative NVD-like property is preferred and is given as below.
Property 3. Let C be defined as in (3.20) and assume that {x>i0, · · · , x>im−1} is a subset of rows of C that are linearly independent as a left Do-module. Define
Cs := x>i0 .. . x>im−1 andCs:= Ψo(Cs) , (3.23) i.e., Cs is the submatrix of Co consisting of the corresponding linearly independent mnt rows,
whereΨois the natural extension ofψo. Then
1 ≤ |γ|2mnt · det C sCs† ∈ Z, (3.24)
where byA†we mean the hermitian transpose of matrixA. Proof. First, it follows from Property 2 that
|γ|2mnt · det C sCs† > 0
since Cs has full row rank mnt and γ 6= 0 by assumption. To show |γ|
2mnt · det C
sCs† ∈ Z,
we shall first verify that det CsCs† is fixed under automorphisms τoand σ. For τo, it can be seen
from the proof of Property 1 that
τo det CsCs† = det
τo(Cs) [τ0(Cs)] †
and τo(Cs) = τo(Xi0) · · · τ Ko o (Xi0) .. . . .. ... τo(Xim−1) · · · τ Ko o Xim−1 = τo(Xi0) · · · Xi0 .. . . .. ... τo(Xim−1) · · · Xim−1 = CsP
for some column permutation matrix P of size (Kont× Kont), where Xij = ψo(xij,0) and x
> ij =
xij,0, · · · , xij,Ko−1, j = 0, 1, · · · , m − 1. Now it follows that
detτo(Cs) [τ0(Cs)] †
= det CsP P†Cs† = det CsCs†
as P P† = IKont, and we have proved det CsC
†
s is fixed by τo.
For σ, again from the proof of Property 1 we see that σ det CsCs† = det σ (Cs) [σ (Cs)] † and σ(Cs) = Z−1Xi0Z · · · Z −1τKo−1 o (Xi01)Z .. . . .. ... Z−1Xim−1Z · · · Z −1τKo−1 o (Xim−1)Z = Z−1 . .. Z−1 Cs Z . .. Z , where Z = ψo(z) = 0 0 0 · · · ζ 1 0 0 · · · 0 0 1 0 · · · 0 .. . ... . .. ... ... 0 0 · · · 1 0 . (3.25)
From (3.25) it is clear that ZZ†= Int since ζζ
∗ = 1 by construction. Therefore, we see that
σ(Cs) [σ(Cs)] †
= diag(Z−1, · · · , Z−1) CsCs†diag((Z −1
)†, · · · , (Z−1)†).
Taking into account that detZ−1(Z−1)†= 1 it follows that σ det(CsCs†) = det(CsCs†). So far,
we have proved that det(CsCs†) is fixed by both τo and σ. This in turn implies that det(CsCs†) ∈
Q ∩ R = Q. Finally, the proof is complete after noting that γ∗Cshas entries in OEo.
In Property 2 we have shown that the overall code matrix Co might not have full rank, and
when that happens, its rank always equals mntfor some m. The number m indicates the number
of users whose transmitted signal vectors, when regarded as rows of matrix C in (3.20), are linearly independent over Do. Further, Property 3 shows that even when Co is singular and fails to have
NVD, i.e., fails to satisfy det(CoCo†) ≥ 1, the submatrix Cs formed by the transmitted signal
matrices of those m users still satisfies the NVD property. Such result can be further extended to yield the following property on the nonzero eigenvalues of CoCo†.
Property 4. Let CoandCsbe defined as above withrank(Co) = rank(Cs) = mnt. Letβ1, · · · , βmnt
be the nonzero eigenvalues ofCoCo†. Then mnt
Y
i=1
βi ≥ det (CsCs) ˙≥ 1. (3.26)
Proof. Here we take an information theoretic approach to prove the first inequality. To this end, let N = [N1, · · · , NKont]
>
be a complex Gaussian random vector of length Kontwith zero mean
and covariance matrix
E N N† = CoCo†.
Without loss of generality we can assume that m linearly independent users are the first m users and ij corresponds to the jth user, j = 0, 1, · · · , m − 1. Hence the covariance matrix of the
sub-vector Ns = [N1, · · · , Nmnt] > equals E NsN † s = CsCs†.
We have the following inequality for the differential entropies of N and Ns h (N1, · · · , NKont) ≥ h (N1, · · · , Nmnt)
= log det CsCs† + mntlog(2πe)
(3.27) Notice that the covariance matrix of N can be decomposed as
CoCo† = U ΣU †
for some (Kont× Kont) unitary matrix U . Σ is a diagonal matrix whose nonzero entries are the
βi’s. Thus setting N0 = U N we have
h(N ) = h(U†N0) =
mnt
X
i=1
log βi+ mntlog(2πe).
Now combining the above results proves the first inequality in (3.26). The second inequality in (3.26) follows directly from Property 3 and from |γ|= 1..
Remark 6. The above property shows that despite Cocan be singular, the product of the nonzero
eigenvalues ofCoCo†is always bounded from below by1. This can be regarded as a relaxation of the
conventional NVD property. In the design of ST codes, satisfying the NVD criterion is a sufficient condition to achieve the optimal to-point DMT performance. To guarantee NVD in the point-to-point MIMO, we require all the users to cooperate fully as already seen in Theorem 4. However, it is not allowed in MIMO-MAC where users transmit independently their own information to the common receiver. Thus, in MIMO-MAC we do not demand full NVD, and only partial NVD is
required as shown in(3.26).
3.3
MAC-DMT Optimality of the Proposed Construction
Armed with the properties discussed in the previous section, below we are able to show the pro-posed code is MAC-DMT optimal.
Theorem 7. Given multiplexing gain r, the proposed code S defined as in (3.9) achieves the following diversity gain
d(r) = min
1≤k≤Kd ∗
knt,nr(kr) (3.28)
over Rayleigh block fading channel with channel coherence timeT ≥ Kontchannel uses. Thus,S
is MAC-DMT optimal.
Chapter 4
MAC-DMT Optimal Codes for General
MIMO-MAC Systems
In [11], Tse et al. focused on analyzing the DMT in a symmetric MIMO-MAC system. By sym-metric we mean that every mobile user in the system has the same number of transmit antennas and transmits at the same level of multiplexing gain. However, the symmetric MIMO-MAC might not be practical enough. In the near future, the mobile communication is likely to be at a transi-tion stage, migrating from conventransi-tional SISO (single-input single-output) to MIMO. In fact, such transition already takes place in wireless local area networks where some old laptops have single transmit antenna while the latest ones could have more than two transmit antennas. In the mixture of SISO and MIMO communication environment, one would expect the mobile users having dif-ferent numbers of transmit antennas. Furthermore, in practice it is often possible that mobile users transmit at different rates because of the different plans they purchase from the service provider. The different rate implies a different level of multiplexing gain in the DMT sense. It is then of fundamental importance that we must have a general code construction that works for any MIMO-MAC systems where the mobile users are allowed to have different numbers of transmit antennas and can transmit at different levels of multiplexing gains. In the previous sections we have pro-vided a systematic construction for the symmetric MIMO-MAC and have proved that it achieves the optimal MAC-DMT. Below we will extend these results to the general channel.
4.1
Decoding in General MIMO-MAC
There can be at least two decoding methods in the general MIMO-MAC, depending on how much computational complexity one can afford. The first decoder is the joint ML decoder, by which we mean the following. Assuming there are K users, each transmitting using a codebook Si that
consists of (ni × T ) ST code matrices, for i = 0, 1, · · · , K − 1. Let Si ∈ Si be the signal matrix
transmitted by the ith user, and let
Y =
K−1
X
i=0
HiSi + W
be the received signal matrix; then the joint ML decoder seeks the optimal joint ML estimate ˆS0, · · · , ˆSK−1 by ˆS0, · · · , ˆSK−1 = arg max S∈S Pr {S = (S0, · · · , SK−1) |Y } = arg min S∈S Y − K−1 X i=0 HiSi F , (4.1)
where S = S0 × S1 × · · · × SK−1. This joint ML decoder was used in [11] for analyzing the
MAC-DMT performance in symmetric MIMO-MAC.
However, the above joint ML decoder might not be optimal in terms of the error performance of each user. For the ith user, the truly optimal decoder, though having extremely high computational complexity, is the individual ML decoder that seeks optimal ML estimate ˆSi by
ˆ Si = arg max Si∈Si Pr {Si|Y } = arg max Si∈Si X S(i)∈S(i) exp − Y − K−1 X i=0 HiSi 2 F , (4.2)
where S(i) = (S0, S1, · · · , Si−1, Si+1, · · · , SK−1) and S(i) = S0× S1× · · · × Si−1× Si+1× · · · ×
SK−1. The difference between the individual and joint ML decoders is analogous to that between
the BCJR and Viterbi decoders [32] for the decoding of convolutional codes. It is easy to see that the individual ML decoder always outperforms the joint ML decoder.
In the next two sections we will examine the MAC-DMT performances of these two decoders. Obviously we expect there might exist certain performance loss in the joint ML decoder, compared to the individual ML decoder.
4.2
MAC-DMT for General MIMO-MAC with Joint Decoding
Consider a general MIMO-MAC system with K mobile users. Let ni denote the number of
trans-mit antennas of the ith user, i = 0, 1, · · · , K − 1, and let ri be the corresponding multiplexing
gain. Assuming nr receive antennas at the base station, the first major result of this section is the
following.
Theorem 8 (General joint MAC-DMT). Let K, ni,riandnrbe defined as above. If joint decoding
is performed at receiver end, the optimal MAC-DMT of such system is given by
d∗{n0,··· ,n K−1},nr(r0, · · · , rK−1) = minI d ∗ Nt(I),nr X i∈I ri ! (4.3)
for i.i.d. Rayleigh block fading channel that is fixed for at least
T ≥ "K−1 X i=0 ni # + nr− 1 (channel uses).
The minimization in(4.3) is taken over all possible non-empty subsets I ⊆ {0, 1, · · · , K − 1}, and
Nt(I) :=
X
i∈I
ni (4.4)
is the total number of transmit antennas of users in I. The notion of d∗
p,q(r) is the conventional
point-to-point DMT.
Prior to proving Theorem 8, we shall give an example illustrating this theorem and in particular, show some unexpected effects resulting from joint decoding.
Example 2. For simplicity, here we consider a general MIMO-MAC system with two users. The first user hasn0 = 1 transmit antenna and transmits at multiplexing gain r0; the second user has
n1 = 2 transmit antennas and transmits at multiplexing gain r1. Assume there arenr = 2 receive
antennas at receiver end. Using (4.3) the resulting MAC-DMT is shown in Fig. 4.1. First, it is interesting to note that unlike the symmetric MIMO-MAC where all users have same number of transmit antennas and transmit at same level of multiplexing gain, here the second user cannot achieve his single-user DMT performance even when r0 = 0. This effect is shown in Fig. 4.2.
While this is quite unexpected, such phenomenon can be easily explained. Recall that the DMT is an asymptotic result. Strictly speaking, the multiplexing gainriis defined as
ri = lim SNR→∞
Ri
log2SNR,
andRiis the actual transmission rate. Therefore, when we sayr0 = 0 it does not necessarily mean
R0 = 0. It simply means that the rate of the first user grows much slower than log2SNR. For
example, an ST code that is fixed and does not vary with SNR has multiplexing gain 0 since the rateRiis a constant. But the rateRi is bounded away from0.
Having learned the above, in our example given the multiplexing gainr0 = for some positive
very close to0, the DMT performance of joint decoder would be dominated by erroneous decoding of the first user’s signals whenr1is small. It is also easy to confirm this observation from pairwise
error probability (PEP) analysis. Assumer0 = r1 = 0, but R0, R1 > 0, i.e., the codes are fixed and
do not vary withSNR. Since the two users do not cooperate, for any distinct pairs of overall code matrices, the maximal possible rank is the minimum of n0 andn1. Hence the resulting maximal
possible diversity gain equals
dmax = nr· min{n0, n1},
which equals 2 in this example. Therefore, the PEP analysis confirms that the single-user DMT performanced∗2,2(r1) cannot be achieved for small values of r0as shown in Fig. 4.2.
Before concluding this example we remark that the loss in DMT for the second user can in fact be recovered if an individual ML decoder is used. We will come back to this in Chapter 4.3.
0 1 2 0 0.5 1 0 1 2 Multiplexing Gain r1 Multiplexing Gain r0 Diversity gain d
Figure 4.1: Joint MAC-DMT d∗{1,2},2(r0, r1) of general MIMO-MAC with two users.
The proof of Theorem 8 follows along similar lines of that of symmetric MAC-DMT provided by Tse et al. in [11]. Specifically, let
0 0.5 1 1.5 2 0 1 2 3 4 Multiplexing Gain r1 Diversity gain d d{1,2},2* (0,r1) d2,2* (r1)
Figure 4.2: Joint MAC-DMT d∗{1,2},2(0, r1) of general MIMO-MAC with two users.
denote the actual transmission rate of the ith user. Given the subset I of users, let O(I) denote the following outage event
O(I) := ( H ∈ Cnr×N : I S I; y|SIc, H ≤ X i∈I Ri ) , (4.5) where
• H = [H0 · · · HK−1] is the overall channel matrix, Hiis the channel matrix of size (nr× ni)
of the ith user,
• N is the total number of transmit antennas defined by
N :=
K−1
X
i=0
ni,
• SI contains the transmitted signal vectors of users in I and is defined as
SI := {si : i ∈ I} ,
• y is the received signal vector given by
y =
K−1
X
i=0
Hisi+ w,
where w is the complex Gaussian random noise vector, and • SIc consists of transmitted signals of users not in I.
Let O denote the overall outage event. It is clear that
O = [
I
Following similar arguments as in [11] it is straightforward to see that the error probability of joint decoding Pe(r0, · · · , rK−1) is lower bounded by
Pe(r0, · · · , rK−1) ≥ Pr {O} ≥ max
I Pr {O(I)}
.
= SNR− minId∗Nt(I),nr(Pi∈Iri).
(4.6) To establish the converse, we take the random codebook approach similar to that used by Tse et al. in [11]. Let Si be the codebook of the ith mobile user, consisting of (ni× T ) code matrices
that are randomly generated by some complex Gaussian random generator. Further, Si satisfies the
desired multiplexing gain,
1
T log2|Si| = Ri = rilog2SNR.
Let E (I) denote the event that the signal matrices of users in I are erroneously decoded by the joint decoder. Then arguing similarly as in [11], it can be shown that
Pr {E (I)} ˙≤ SNR−d∗Nt(I),nr(
P
i∈Iri)
whenever
T ≥ Nt(I) + nr− 1.
Thus, using union bound we have
Pe(r0, · · · , rK−1) ≤
X
I
Pr {E (I)} .
= SNR− minId∗Nt(I),nr(Pi∈Iri),
provided that
T ≥ max
I Nt(I) + nr− 1 = N + nr− 1.
This proves Theorem 8.
4.3
MAC-DMT for General MIMO-MAC with Individual ML
Decoding
In the previous section we investigated the MAC-DMT for a general MIMO-MAC with joint de-coding at the receiver end. We also observed in Example 2 that certain DMT performance loss could result from the use of joint decoder. However, such loss can be safely avoided by the use of individual ML decoder.
Recall that for the ith user, the truly optimal decoder, though having extremely high computa-tional complexity, is the individual ML decoder that seeks optimal ML estimate ˆSiby
ˆ Si = arg max Si∈Si Pr {Si|Y } = arg max Si∈Si X S(i)∈S(i) exp − Y − K−1 X i=0 HiSi 2 F , (4.7) where S(i) = (S
0, S1, · · · , Si−1, Si+1, · · · , SK−1) and S(i) = S0 × S1 × · · · × Si−1 × Si+1×
· · · × SK−1. Clearly (4.7) outperforms (4.1) in error performance, but at a cost of much higher
