DMT Optimal Codes Constructions for
Multiple-Access MIMO Channel
Hsiao-feng (Francis) Lu, Member, IEEE, Camilla Johanna Hollanti, Member, IEEE,
Roope Iikanpoika Vehkalahti, Member, IEEE, and Jyrki Lahtonen, Member, IEEE
Abstract—Explicit code constructions for multiple-input mul-tiple-output (MIMO) multiple-access channels (MAC) with users are presented in this paper. The first construction is dedi-cated to the case of symmetric MIMO-MAC where all the users have the same number of transmit antennas and transmit at the same level of per-user multiplexing gain . Furthermore, we assume that the users transmit in an independent fashion and do not cooperate. The construction is systematic for any values of , and . It is proved that this newly proposed construction achieves the optimal MIMO-MAC diversity-multiplexing gain tradeoff (DMT) provided by Tse et al. at high-SNR regime.
In the second part of the paper we take a further step to inves-tigate 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 op-timal MAC-DMT of such channel is explicitly characterized in this paper. 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 paper. Finally, by generalizing the code construc-tion for the symmetric MIMO-MAC, explicit code construcconstruc-tions are provided for the general MIMO-MAC and are proved to be optimal in terms of the general MAC-DMT.
Index Terms—Cyclic division algebras (CDAs), diversity-mul-tiplexing gain tradeoff (DMT), multiple access channel (MAC), multiple-input multiple-output (MIMO) channel, space-time block codes (STBCs).
I. INTRODUCTION
D
URING the last decade extensive research has been car-ried out in the design of point-to-point space-time (ST) codes [1], [2] for multiple-input multiple-output (MIMO) com-munication systems. ST codes based on cyclic division algebrasManuscript received April 24, 2009; revised June 09, 2010; accepted December 01, 2010. Date of current version May 25, 2011. This work was supported in part by the Taiwan National Science Council under Grant NSC 98-2221-E-009-045-MY3, through a grant for H.-F. Lu; in part by the Finnish Academy of Science and Letters and the Academy of Finland (Grant 210280) through grants for C. J. Hollanti; and in part by the Emil Aaltonen Foundation, Finland, through a grant for R. I. Vehkalahti.
The material in this paper was presented in part at the IEEE International Symposium on Information theory, Seoul, Korea, June 2009.
H.-F. Lu is with the Department of Electronical Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan, R.O.C. (e-mail: francis@mail.nctu. edu.tw).
C. J. Hollanti is with the Department of Mathematics, FI-20014 University of Turku, Turku, Finland (e-mail: cajoho@utu.fi).
R. I. Vehkalahti is with the Department of Mathematics, FI-20014 University of Turku, Turku, Finland (e-mail: roiive@utu.fi).
J. Lahtonen is with the Department of Mathematics, FI-20014 University of Turku, Turku, Finland (e-mail: lahtonen@utu.fi).
Communicated by B. S. Rajan, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2011.2136170
(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 perfor-mance 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 and be respectively the numbers of transmit and receive antennas at transmitter and receiver ends.
Let , , denote the multiplexing gain such
that the actual transmission rate equals
SNR (1)
Assuming a MIMO Rayleigh block fading channel, it was shown [4] that at multiplexing gain , the CDA-based ST codes achieve the optimal codeword error probability
SNR (2)
at high-SNR regime, where by we mean the exponential equality defined in [9]. That is, we write SNR SNR if
SNR SNR
The notations of and are similarly defined. The exponent is commonly known as the DMT [9] and is given by a piecewise linear function connecting the points
for . Furthermore,
rep-resents 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 channel 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 Nakagami 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 multiblock CDA code [6] has been shown to be approximately universal as well. In particular, it achieves codeword error probability
SNR (3)
at multiplexing gain , where is the number of independent fading blocks occupied by the code. The exponent
is known as the multi-block DMT [6], [9] when coding is applied over independent fading blocks. Therefore, the multiblock CDA-based ST code is optimal in terms of the multiblock DMT at high-SNR regime. More important, (3) indicates that the code has error probability decreasing to zero as approaches
in-finity whenever . Hence, the multiblock ST code
could potentially achieve the MIMO ergodic channel capacity at high-SNR regime and simultaneously be optimal in terms of the multiblock DMT at every discrete value .
Motivated by the promising outcome in the point-to-point scenario, the aim of this paper is to investigate the code con-struction for the multiple-access channel (MAC) scenario. We will concentrate on the uplink transmission from multiple mo-bile users to a common base station (or access point). Both the mobile users and the base station may be equipped with mul-tiple antennas.
Consider a MIMO-MAC with mobile users. For simplicity, we first focus on the case of symmetric MIMO-MAC [11], where each user is equipped with transmit antennas and communi-cates independently to the base station that has receive an-tennas. 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 the per-user multiplexing gain in
the symmetric MIMO-MAC. Let , be respectively
the ST codes used by the th user, . Each
code , , consists of matrices and
satisfies the following power constraint:
SNR (4)
where by we mean the Frobenius norm of matrix .
Furthermore, we require SNR for all such that
every user transmits at the same multiplexing gain . Let be
the channel matrix of the th user. We assume is
fixed for a block of channel uses. is known completely to the receiver at base station but unknown to all the users. Entries of are modeled as i.i.d. complex Gaussian random variables to model the MIMO Rayleigh block fading channel. Let be the signal matrix transmitted by the th user; then the signal matrix received at base station is given by
(5)
where is the noise matrix with i.i.d.
en-tries. When each user’s information is encoded independently, Tse et al. [11] proved that the tradeoff between the diversity gain and multiplexing gain in a symmetric MIMO-MAC is gov-erned by the following theorem.
Theorem 1 (Symmetric MAC-DMT [11]): In a symmetric
MIMO-MAC with users, each having transmit antennas
Fig. 1. The MAC DMT forK = 3 users with n = 2 and n = 2.
and transmitting independently at multiplexing gain , the max-imal possible diversity gain is given by
(6)
where is the point-to-point DMT for transmit
antennas, receive antennas and multiplexing gain defined as before (or see [9], [11]). Equation (6) is termed optimal sym-metric MAC-DMT. The multiplexing gain for nonnegative di-versity gain is bounded between
(7)
Compared with the point-to-point scenario, the decrease of maximal multiplexing gain by a factor of (see in of (7)) is due to the sharing of receive antennas among
users and the fact that . Equation (6)
also shows that when the level of multiplexing gain is low
such that , each user is able to retain
his single-user performance, i.e., , as
if there were no other users in the channel. On the other hand, when the level of multiplexing gain is high and , the MIMO-MAC system would operate in the antenna pooling region [11], and single-user performance can no longer be maintained. As a consequence, a much lower diversity gain
dominates the error performance in this region.
In Fig. 1 we demonstrate the above facts of the symmetric
MAC-DMT for the case of users, and . It
and antenna pooling regions is at and
the cut-off point of is at .
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 performance dictated by the MAC-DMT. Thus, a coding scheme is called MAC-DMT
optimal if it achieves the following error performance under
joint decoding
SNR
A. 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 constructions 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., .
1) [13] extended the pairwise-error-probability-based design criteria of point-to-point ST codes to the MAC case for
users and , . An explicit (4 4)
two-user MIMO code,1i.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 (6).
2) In [14] Badr and Belfiore proposed an explicit algebraic
code for and . The idea can be extended
to bigger values of . The determinant of the code matrix is nonzero 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 . Whether this code achieves the optimal MAC-DMT also when 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 . 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 and
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 con-structed 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 obtained by adding a twist ma-trix to the (2 2) Golden ST code [20], [21] such that the
1In this paper, by an(m2n) code we mean a code consisting of (m2n) code
matrices, wherem is the number of transmit antennas required for transmission, andn is the number of channel uses. The number m can be either n or Kn , depending on the discussion. Whenm = n , the code is for each user’s use. Whenm = Kn , we mean the vertical concatenation of all users’ codes as an overall code. Notationn 2 n without parenthesis is used for the channel dimensions.
overall code matrix is nonsingular whenever all the subma-trices associated with each user are nonzero. However, be-cause of this additional matrix, the overall code matrix, though nonsingular, could be ill-conditioned at high-SNR regime, thereby resulting in a vanishing determinant, simi-larly as did their earlier one-antenna code [14] already dis-cussed above.
5) [22] addressed the problem of whether there exists a two-user MIMO-MAC code satisfying the nonvanishing deter-minant (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 matrix, [22] reported another code con-struction and proved its MAC-DMT optimality for and for any values of and . 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 rotation matrix is not neces-sary from the MAC-DMT point of view.
7) In [16], Coronel et al. studied the optimal DMT perfor-mance of a selective fading MIMO-MAC and provided a sufficient criterion for designing MAC-DMT optimal codes for any and . Noting that the Rayleigh block fading channel is a flat fading channel, a simplification of their criterion requires the product concatenation of codes from any subsets of users to satisfy the NVD property such that the error probabilities associated with these sub-codes do not exceed the corresponding outage probability. However, as already pointed out in [22], such codes do not exist for the case of . A further investigation of their criterion can be found in [17].
B. Principal Results
A complete solution to the problem of constructing
MIMO-MAC codes for users that are MAC-DMT
op-timal in Rayleigh MIMO-MAC is presented in this paper. The paper consists of two parts. The first part provides the constructions of MAC-DMT optimal codes for the symmetric MIMO-MAC. The second part is on the code construction for the general MIMO-MAC where the users are allowed to have different numbers of transmit antennas and can transmit at dif-ferent levels of multiplexing gain.
C. Outline
A general result on the nonexistence of NVD MIMO-MAC codes is presented in Section II. 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 Section III, we present a new code construction for the
symmetric MIMO-MAC for any , and . Several nice
properties of the proposed code are presented in this section. We prove that this newly proposed construction is MAC-DMT op-timal and meets the relaxed design criterion given in Section II. For ease of reading, the proof of MAC-DMT optimality is rele-gated to Section V.
In Section IV we investigate the MAC-DMT in a general MIMO-MAC where the users are allowed to have different num-bers of transmit antennas and transmit at different levels of mul-tiplexing gain. The exact general MAC-DMT in such channel will be given in Section IV-B, 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 IV-D 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 Section VI.
II. RELAXED DESIGN CRITERION OFMAC-DMT OPTIMALCODES
In this section, we first present a rigorous, yet negative, re-sult 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 cri-terion will be presented and will be met by all subsequent con-structions of MIMO-MAC codes in this paper.
Consider a symmetric MIMO-MAC with users, each having transmit antennas and communicating independently to the base station at the same level of multiplexing gain . Let , be respectively the space-time codes
used by the th user, , all satisfying the
power constraint (4). If independent Gaussian random code-books were used, i.e., the entries of code matrices are i.i.d. SNR random variables for all , Tse et al. [11] showed that the event of users in error has probability upper bounded by
SNR (8)
where is the event of users in outage. Note that the overall
error event . The union bound on
gives
(9)
Since the right-hand-side of (9) has a negative SNR-exponent equal to defined in (6), (9) 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 prob-ability as the outage events. Thus, for any MIMO-MAC code
, set
where
SNR
To see how is chosen, we offer the following insight. For each
, the code has size SNR so that it is of
multiplexing gain . An explicit construction of was given in [4] where the code is seen as a real algebraic ST lattice code with
dimension . Hence there are SNR PAM
signals selected from each dimension and SNR for all . Thus, the constant is chosen such that the code
satisfies the power constraint (4).
From [4], it is easy to prove the following theorem which in turn gives a sufficient criterion for designing MAC-DMT op-timal codes. We remark that this theorem is an alternative state-ment of the result given in [16] under certain restrictions, and we refer the interested readers to [17] for the connections.
Theorem 2 ([16]): Let be given as above. For
any , let be
the product concatenation of , defined by
.. .
If for all pairs of distinct code matrices ,
, the difference matrix ..
. (10)
satisfies , where by we mean the
Her-mitian transpose of matrix , then the codes are
jointly MAC-DMT optimal.
Proof: Note that the imposed condition implies that the
code satisfies the NVD property for any . Along sim-ilar lines as in [4], it can be shown that the error event associated with code , i.e., the error event of users in in error, has probability upper bounded by
SNR
where is the event of users in in outage. Now taking union bound over all possible as in (9) completes the proof.
Remark 1: The condition of for all is called the full NVD criterion and is actually equivalent to the criterion given by Coronel et al. in [16] with certain re-strictions, 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 condition the exactly
full NVD criterion.
Theorem 3: For any and for any , there do not exist any linear MIMO-MAC codes2that satisfy the exactly full
NVD criterion.
Proof: For ease of reading, the proof is relegated to
Appendix A.
Roughly speaking, the proof of Theorem 3 shows that while it is possible to construct DMT-optimal codes for each user, as the existing CDA-based ST codes [4] would do, it
is impossible for the product code to have an
exactly full NVD. Any such product code would have difference
matrices such that 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 sig-nificantly 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 3 is that the exactly full NVD condition can be met only if the users coop-erate 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 con-dition without adversely affecting the DMT performance. To do so, we will partition the error events in a different manner. Given
the set of users , let , , denote the error
event when the users in are in error and the corresponding error matrix (cf. (10)) has rank exactly . Clearly event defined in the proof of Theorem 2 is a disjoint union of
. Now the codes are jointly
MAC-DMT optimal if the following holds.
Theorem 4 (Relaxed Design Criterion): Let be defined as above. Then they are jointly MAC-DMT optimal if the error events have probabilities upper bounded by
SNR (11)
for all and for all
. Furthermore, as for design of MAC-DMT optimal codes we require at least that
SNR (12)
for all and for all .
While (12) might be the most relaxed condition for designing MAC-DMT optimal codes, in this paper we will focus on con-dition (11).
2Here by linear codes we mean codes having linear dispersion forms [8] or
having a lattice structure. Almost all existing ST codes are linear, for example, the Alamouti codes [2], the CDA-based ST codes [3]–[7], [13], [18]–[21], [23], etc.
The rationale behind the above theorem is the observation that
the error probabilities SNR with are
not dominant in the overall DMT performance. Hence we could relax the conditions such that the event
has probability larger than the corresponding outage probability, but no larger than the dominant error probability. That is, we could allow
SNR (13)
but would still require SNR
Relaxation (13) would not affect the overall DMT performance. Compared with the exactly full NVD condition required by The-orems 2, Theorem 4 relaxes greatly the code design criterion in the following ways.
1) We do not require the difference matrix to be non-singular and to satisfy the NVD property when all the
com-ponent matrices are nonzero, which has been
shown to be impossible by Theorem 3.
2) Should the difference matrix happen to be singular, (11) requires the resulting error performance must be no
worse than SNR for some , , in
order to maintain the MAC-DMT optimality.
3) In Theorem 2, events with were required
to have probability absolutely zero. This is too strict and would preclude the existence of MAC-DMT optimal codes.
III. MAC-DMT OPTIMALCODECONSTRUCTION FOR
SYMMETRICMIMO-MAC CHANNELS
For the symmetric MIMO-MAC coded system with users, each having transmit antennas and transmitting at multi-plexing gain , in this section we will propose a systematic code construction that is MAC-DMT optimal for any combi-nations of , , , and . 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.
A. Proposed Construction
Given the number of users , let be the smallest odd integer such that , i.e.,
if even,
if odd. (14)
The construction calls for the following number fields. Let be a number field that is a cyclic Galois extension of
Fig. 2. Field extensions required by the proposed code constructions.
with degree , where . Let be
an-other cyclic Galois extension of with degree . Let and
be the generators of Galois groups and
with degrees and , respectively. The fields and are
chosen3such that . Let be
the compositum of and . See Fig. 2 for the relation among the required number fields. The readers are referred to [4], [22], [24] for the constructions of such number fields.
Let be a cyclic division algebra with
(15) where
(16) (17) for . The element is an indeterminate satisfying
, and is some suitable nonnorm element.
4 By we mean the complex conjugate of and is the
algebraic closure of in [26]–[28]. Notice that and is unimodular. It has been shown [5] that with such unimodular
, is always a cyclic division algebra.
Remark 2: While in the above we have set to be of form such that is unimodular, it might be possible that in some CDAs, the nonnorm element is actually an th root of unity for some integer and is already unimodular. See [29] for such example construction. Should it be the case, we could set , 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 of .
Remark 3: We note that by construction the Galois groups of
the numbers fields are
3A more general condition on and is that the automorphisms and
commute.
4A sufficient criterion for finding a suitable nonnorm element is given in
[25, Theorem 1]. Also, we refer the interested readers to [4, Theorems 10 and 11] for two explicit constructions of .
where in the last line denotes the direct product of the groups generated by and , respectively. It should also be noted that the automorphisms and commute, i.e.,
due to the direct product of two groups.
Given multiplexing gain , let SNR be the base alphabet defined as
SNR SNR SNR
then the corresponding information set is
SNR SNR
(18)
where is an integral basis of . It should
be noted that for SNR
and that SNR . Let
be the left-regular map that maps elements in into square matrices with entries in . Specifically, given with
is given by
..
. ... . .. ... (19)
Note that the field is the center of the division algebra ,
meaning that for any and .
Equiva-lently we have
showing that the matrix-product commutes.
Proposition 5 ([4], [26]): Let and be defined as above. Then
for all , where .
Having defined the above, the encoding of each user’s data stream proceeds as follows. Given the multiplexing gain , the th user first partitions his binary data steam into blocks of SNR bits. Then using the integral basis and set SNR defined above, each block of binary bits is mapped in a one-one fashion to a symbol
SNR . The encoding is performed indepen-dently at each user’s end.
Given SNR , the th user actually sends out the
following signal matrix through his transmit
antenna array in channel uses
(20)
where and where is a normalizing constant such
that
SNR SNR
Hence we have
SNR (21)
Remark 4: The above construction of the MIMO-MAC codes
is reminiscent of the multiblock ST code presented in [6]. Some key differences are highlighted below.
1) In the proposed construction we require the length of the
code to be where must be an odd integer.
2) The number fields and are required such that the automorphisms and commute. This was not needed in [6].
3) The element of the CDA must be unimodular, and we
have set .
We use the following example to illustrate the proposed con-struction.
Example 1: We consider the case of and . By construction is the smallest odd integer such that
. Then it can be shown that with and
the number fields and meet the required
conditions of , and .
Furthermore, we have . The generators
and for the Galois groups and are
given respectively by
The set is an integral basis for .
As the prime ideal of remains inert in and
, following from [4] this gives an appropriate nonnorm
ele-ment . Hence we have . With ,
is a CDA of index 2 which is also a central simple -algebra [26]. Next let
for with SNR . The Galois conjugates of
are for example given by
where and
. With the above, the signal matrix of the first
user is given by , where
SNR and
By vertically concatenating the signal matrices from all users, the overall MIMO-MAC code of the users is
..
. . .. ...
SNR
(22)
For ease of code performance analysis that comes later we set , i.e.,
..
. . .. ...
SNR
(23)
Remark 5: Below we briefly compare the proposed
construc-tion of with another MAC-DMT optimal code constructed for users in [22]. The latter MIMO-MAC code takes the fol-lowing form
SNR
(24)
The construction of requires a number field
with and such that
, and
. Here by “1” of we mean the trivial au-tomorphism. The field and the element are defined as be-fore. The element is taken from the cyclic division algebra for some indeterminate . SNR is the base-in-formation set defined similarly as SNR in (18). Thus, com-pared with the present proposed construction, we see that requires an additional sign change at the second block matrix of the second user’s code. This sign change is essential to en-sure an NVD-like property. It also endows with another nice property that the transmission of code matrices in takes only channel uses, less than that required by . 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 . Everything works perfectly after patching an extra block of transmission when is even. An-other drawback of is the difficulty of generalization to the
cases of .
Let be the channel matrix of the th user. We
(5), given the overall transmitted code matrix , the re-ceived signal matrix at receiver end is
(25) is the noise matrix whose entries are i.i.d. random variables, and is the th block received signal matrix given by
and
B. Properties of the Proposed Construction
To simplify the analysis of code performance, below we de-fine the extended versions of and .
..
. . .. ...
SNR (26) (27)
Given the overall signal matrix , let be any
signal matrix such that the upper submatrix of
equals . Then we can rewrite (25) as
(28) where
if odd, if even.
By we mean the all-zero matrix of proper size. Noting (25) and (28) are equivalent, henceforth we will work only with the extended codes and , rather than and . We next show several nice properties possessed by and .
Property 1: For any , we have
(29)
Proof: We first claim
(30) To see this, notice that
..
. . .. ...
..
. . .. ...
where the last equality follows from the fact that is even, hence the claim (30) is proved. Next, we show
(31) To this end, define
(32) where is the indeterminate defined as in (15). Since from (17)
for all , it is clear that ,
where . Now we have
.. . . .. ... .. . . .. ... . .. .. . . .. ... . ..
where we have used the fact that since
by construction. Thus, as is fixed by both and , we
see that .
Finally, from the definition of (19), the matrix
. ..
has entries in for all and
since SNR
is the ring of algebraic integers in number field . It then follows that
Summarizing the above results, we conclude that
Property 2: Let .. . ... . .. ... (33) and .. . . .. ...
with , SNR , where by we mean the
transpose of vector . Let be the maximal number of rows in that are linearly independent as a left -module; then
(34) where the rank is measured in the complex number field .
Proof: To find out the rank of matrix , we use the el-ementary row operations from Gaussian elimination method. Note that the same row operations can be performed on whose entries are in . Extra care must be taken because multiplica-tion in is noncommutative. Further, we note that elementary row operations on are equivalent to the block elementary row operations on . By this we mean that, say is a
elementary matrix with entries in ; then it is clear
where is the natural extension of to the central
simple matrix algebra over [26], i.e.,
(35)
From hypothesis, assume is the maximal
subset of the rows of that are linearly independent over . Then it follows that there are leading ones in the row-reduced matrix of . Equivalently, the same block elementary opera-tions would reduce matrix into a matrix whose main diagonal consists of identity matrices, each of size , after permuting the columns if necessary. This completes the proof.
Property 2 shows that the overall code matrix might not always have full rank , and the rank of is always a multiple of . This is not too much of a surprise as it is straight-forward to see that in (26) if some ’s are identical, then the overall code matrix cannot be nonsingular.
Compared with the constructions proposed in [18], [19], the matrix of the present construction could be singular even when the component matrices are all distinct and nonzero as shown by Property 2. Nevertheless, we will prove in Section V that in order to achieve the optimal MAC-DMT performance at high-SNR regime, it is unnecessary to construct codes such that is nonsingular whenever all the component matrices are 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 it is impossible for all the code matrices to be non-singular as some component matrices of the th user could be zero. Also, from the pairwise error probability point of view,
for any , can be singular at least when the
information symbols transmitted by some users are the same. The rank of overall code matrices is at best a multiple of . Therefore, intuitively speaking, perhaps it would not hurt to make things a bit worse in the sense that the difference matrix can be singular in other cases. By this we mean that if there are distinct information symbols in the difference
ma-trix , the maximal possible rank of is . We
claim that it would not hurt in the DMT sense if the
construc-tion can provide only rank for some with .
The reason for this actually follows from Theorem 4 that the error events of users in error but getting only rank distance do not dominate the error performance in the final DMT performance. Therefore, we strongly speculate that such difference matrices do not have to achieve the same rank as the Gaussian random code does. The rank can be less, as long as the resulting error performance is not worse than those
of and .
Although we do not need the whole code to 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 be defined as in (33) and assume that is a subset of rows of that are linearly in-dependent as a left -module. Define
..
. (36)
i.e., is the submatrix of consisting of the corresponding linearly independent rows, where is the natural exten-sion of . Then
(37) where by we mean the hermitian transpose of matrix .
Proof: First, it follows from Property 2 that
since has full row rank and by assumption.
To show , we shall first verify that
is fixed under automorphisms and . For , it can be seen from the proof of Property 1 that
and
..
..
. . .. ...
for some column permutation matrix of size ,
where and ,
. Now it follows that
as , and we have proved is fixed by
.
For , again from the proof of Property 1 we see that
and .. . . .. ... . .. . .. where .. . ... . .. ... ... (38)
From (38) it is clear that since by
con-struction. Therefore, we see that
Taking into account that it follows
that . So far, we have proved that
is fixed by both and . This in turn implies that . Finally, the proof is complete after noting that has entries in .
In Property 2 we have shown that the overall code matrix might not have full rank, and when that happens, its rank always equals for some . The number indicates the number of users whose transmitted signal vectors, when regarded as rows of matrix in (33), are linearly independent over . Further, Property 3 shows that even when is singular and fails to have
NVD, i.e., fails to satisfy , the submatrix
formed by the transmitted signal matrices of those users still
satisfies the NVD property. Such result can be further extended to yield the following property on the nonzero eigenvalues of
.
Property 4: Let and be defined as above with
. Let be the
nonzero eigenvalues of . Then
(39)
Proof: Here we take an information theoretic ap-proach to prove the first inequality. To this end, let be a complex Gaussian random vector of length with zero mean and covariance matrix
Without loss of generality we can assume that linearly inde-pendent users are the first users and corresponds to the th
user, . Hence the covariance matrix of the
subvector equals
We have the following inequality for the differential entropies
of and
(40) Notice that the covariance matrix of can be decomposed as
for some unitary matrix . is a diagonal
matrix whose nonzero entries are the ’s. Thus setting we have
Now combining the above results proves the first inequality in (39). The second inequality in (39) follows directly from
Prop-erty 3 and from .
Remark 6: The above property shows that despite can be singular, the product of the nonzero eigenvalues of is al-ways bounded from below by 1. This can be regarded as a re-laxation of the conventional NVD property. In the design of ST codes, satisfying the NVD criterion is a sufficient condition to achieve the optimal point-to-point DMT performance. To guar-antee NVD in the point-to-point MIMO, we require all the users to cooperate fully as already seen in Theorem 3. However, it is not allowed in MIMO-MAC where users transmit indepen-dently 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 (39).
C. MAC-DMT Optimality of the Proposed Construction
Armed with the properties discussed in the previous section, below we are able to show the proposed code is MAC-DMT optimal.
Theorem 6: Given multiplexing gain , the proposed code
defined as in (22) achieves the following diversity gain (41)
over Rayleigh block fading channel with channel coherence
time channel uses. Thus, is MAC-DMT optimal.
Proof: The proof is relegated to Section V for ease of
reading.
IV. MAC-DMT OPTIMAL CODES FORGENERAL
MIMO-MAC SYSTEMS
In [11], Tse et al. focused on analyzing the DMT in a sym-metric MIMO-MAC system. By symsym-metric we mean that every mobile user in the system has the same number of transmit an-tennas 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 transition stage, migrating from conventional 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 ex-pect the mobile users having different numbers of transmit an-tennas. 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 im-plies a different level of multiplexing gain in the DMT sense. It is then of fundamental importance that we must have a gen-eral 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 multi-plexing gains. In the previous sections we have provided a sys-tematic 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.
A. Decoding in General MIMO-MAC
There can be at least two decoding methods in the gen-eral 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 users, each transmitting using a codebook that consists
of ST code matrices, for . Let
be the signal matrix transmitted by the th user, and let
be the received signal matrix; then the joint ML decoder seeks
the optimal joint ML estimate by
(42)
where . 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 th user, the truly optimal decoder, though having extremely high com-putational complexity, is the individual ML decoder that seeks optimal ML estimate by
(43)
where and
. The difference be-tween the individual and joint ML decoders is analogous to that between the BCJR and Viterbi decoders [30] 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 subsections 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.
B. MAC-DMT for General MIMO-MAC With Joint Decoding
Consider a general MIMO-MAC system with mobile users. Let denote the number of transmit antennas of the
th user, , and let be the corresponding
multiplexing gain. Assuming receive antennas at the base station, the first major result of this section is the following.
Theorem 7 (General Joint MAC-DMT): Let , , and be defined as above. If joint decoding is performed at receiver end, the optimal MAC-DMT of such system is given by
(44) for i.i.d. Rayleigh block fading channel that is fixed for at least
The minimization in (44) is taken over all possible nonempty
subsets , and
(45) is the total number of transmit antennas of users in . The notion of is the conventional point-to-point DMT.
Fig. 3. Joint MAC-DMTd (r ; r ) of general MIMO-MAC with two users.
Fig. 4. Joint MAC-DMTd (0; r ) of general MIMO-MAC with two users.
Prior to proving Theorem 7, we shall give an example illus-trating 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 has transmit antenna and transmits at multiplexing gain ; the second user has transmit antennas and transmits at multiplexing gain . Assume there are receive an-tennas at receiver end. Using (44) the resulting MAC-DMT is shown in Fig. 3. 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 . This effect is shown in Fig. 4. While this is quite unexpected, such phenomenon can be easily explained. Recall that the DMT is an asymptotic result. Strictly speaking, the multiplexing gain is defined as
SNR SNR
and is the actual transmission rate. Therefore, when we say it does not necessarily mean . It simply means that the rate of the first user grows much slower than SNR. For example, an ST code that is fixed and does not vary with
SNR has multiplexing gain 0 since the rate is a constant. But the rate is bounded away from 0.
Having learned the above, in our example given the multi-plexing gain for some positive very close to 0, the DMT performance of joint decoder would be dominated by erroneous decoding of the first user’s signals when is small. It is also easy to confirm this observation from pairwise error probability
(PEP) analysis. Assume , but , i.e., the
codes are fixed and do not vary with SNR. Since the two users do not cooperate, for any distinct pairs of overall code matrices, the maximal possible rank is the minimum of and . Hence the resulting maximal possible diversity gain equals
which equals 2 in this example. Therefore, the PEP analysis confirms that the single-user DMT performance cannot be achieved for small values of as shown in Fig. 4.
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 Section IV-C. The proof of Theorem 7 follows along similar lines of that of symmetric MAC-DMT provided by Tse et al. in [11]. Specifi-cally, let
SNR
denote the actual transmission rate of the th user. Given the subset of users, let denote the following outage event
(46) where
• is the overall channel matrix, is
the channel matrix of size of the th user, • is the total number of transmit antennas defined by
• contains the transmitted signal vectors of users in and is defined as
• is the received signal vector given by
where is the complex Gaussian random noise vector, and • consists of transmitted signals of users not in . Let denote the overall outage event. It is clear that
Following similar arguments as in [11] it is straightfor-ward to see that the error probability of joint decoding
is lower bounded by
SNR
(47) To establish the converse, we take the random codebook ap-proach similar to that used by Tse et al. in [11]. Let be the code-book of the th mobile user, consisting of code matrices that are randomly generated by some complex Gaussian random generator. Further, satisfies the desired multiplexing gain,
SNR
Let denote the event that the signal matrices of users in are erroneously decoded by the joint decoder. Then arguing similarly as in [11], it can be shown that
SNR whenever
Thus, using union bound we have
SNR provided that
This proves Theorem 7.
C. 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 decoding 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 th user, the truly optimal decoder, though having extremely high computational complexity, is the indi-vidual ML decoder that seeks optimal ML estimate by
(48)
where and
. Clearly (48) outperforms (42) in error performance, but at a cost of much higher computational complexity.
Without loss of generality, below we focus on the error per-formance of the individual ML decoding for the th user. To dis-tinguish the DMT performances of decoders (42) and (48), we shall call the DMT of the latter the individual MAC-DMT and
will denote it by .
To characterize the DMT performance of the individual ML decoder, we only need to consider the outage events (cf. (46)) in which the th user is a member of . Event with is not counted as an outage for the th user for obvious reasons. Thus, along similar lines as in the proof of Theorem 7 we can show the following.
Theorem 8 (General Individual MAC-DMT): Let , , and be defined as before. If individual ML decoding is per-formed at receiver end for the th user, the optimal individual MAC-DMT is given by
(49) where the minimization is taken over all
under the condition and is defined in (45).
Proof: For brevity we only outline the proof. Let denote the outage event of the th user; then following from the above discussion it can be seen that
since if , the th user is not in outage. Now let denote the error probability of the in-dividual decoder for the th user; then it can be shown that
SNR
where the first inequality follows from [9, Lemma 5]. To show the converse, let denote the error event that the signal ma-trices of the users in are erroneously decoded under joint de-coding. Then simply note that the error probability of an indi-vidual ML decoder is upper bounded by that of a joint ML de-coder, i.e.,
where the right-hand-side gives the probability of a joint ML decoder when the signal of the th user is erroneously decoded.
Fig. 5. Comparison between the joint MAC-DMT and the individual MAC-DMT of the second user whenr = 4r = r.
Now using the union bound argument and along similar lines as in the proof of Theorem 7 it can be shown that
SNR
SNR This completes the proof.
With the above result, we now come back to Example 2 to investigate the individual MAC-DMT of the second user.
Example 3 (Continued From Example 2): In Example 2 we
have considered the specific case of , , ,
and . Assuming the second user transmits at mul-tiplexing gain , from Theorem 8 the individual MAC-DMT of the second user is
Hence we see that the single-user performance of the second user is recovered by the use of an individual ML decoder. To illustrate further the difference in MAC-DMT between (42) and (48), in Fig. 5 we compare the MAC-DMT performances of joint
and individual decoders at . It can be clearly
seen that the individual ML decoder outperforms significantly the joint ML decoder at low-multiplexing-gain regime.
Another comparison between the DMT performances of both decoders at is given in Fig. 6. It shows that the
joint ML decoder (given by ) is not optimal for
the second user. The truly optimal individual ML decoder for
the second user has DMT performance .
Further-more, the individual ML decoder for the second user achieves
the single-user DMT performance as long as .
On the other hand, for the first user who has lesser number of
Fig. 6. Comparison between the joint MAC-DMT and the individual MAC-DMT whenr = r = r.
transmit antennas, the DMT performances of the joint and in-dividual decoders are the same and are actually equal to his single-user performance .
Next we could apply Theorem 8 to the case of symmetric MIMO-MAC to see how the error probabilities of joint and individual ML decoders compare. The comparison is given in the following corollary. It shows that in the symmetric MIMO-MAC there is no difference in terms of MAC-DMT performance between the joint and individual ML decoders.
Corollary 9: For symmetric MIMO-MAC with users, each having transmit antennas and transmitting at multiplexing gain , let denote the error probability of the joint ML decoder and denote the error probability of the th individual ML decoder. Then
and in terms of DMT we have
for all .
Proof: It suffices to show only the equality in DMT. First,
from Theorem 8 we have
and the proof is complete after noting that the right-hand-side of the above is the same as the MAC-DMT given in Theorem 1.
Before concluding the subsection we have the following re-marks. First, while the individual ML decoder could achieve a much higher DMT performance as seen in Examples 2 and 3, the computational complexity required by (48) is often extremely high. Thus, the individual ML decoder has widely been consid-ered as being impractical in multiuser detections. The reason for including this receiver is only to clarify the unexpected DMT performance loss of the joint ML decoder in Example 2.
As the individual ML decoder is rarely used, below we will not consider this receiver anymore. We will regard the joint MAC-DMT given in Theorem 7 as the optimal MAC-DMT in
practice, although it is now clear that it is not the best one can actually achieve.
D. General MAC-DMT Optimal Codes
So far we have provided the optimal MAC-DMT (44) for the general MIMO-MAC system with users where the th user has transmit antennas and transmits at multiplexing gain . To have a deterministic code for the general MIMO-MAC, we can extend the code construction given in Section III for sym-metric MIMO-MAC to the present case.
Let , , , and be defined as before. For brevity we only present the construction when is odd. Codes for even can be constructed by simply patching an extra coded block to each user’s code matrices, similar to that described in Section III. Henceforth we will drop the subscript “ ” in , , , ,
, etc. for simplicity.
Given and , we first define
(50)
as the maximal number of transmit antennas among all users. In general, the number can be either preknown to all the users, or explicitly specified among any groups of users. Next, let be the number field that is cyclic Galois over
with degree , and let be another cyclic
Galois extension of with degree . Let be the generator of the Galois group , and similarly let be the generator of . The fields and are required to satisfy
or are required such that and commute. Finally, we set to be the compositum of fields and . Similar to Section III, with some suitable unimodular , we have
(51) as an appropriate central simple division -algebra with for , where is an indeterminate satisfying .
Given the multiplexing gain of the th user, we set the cor-responding base alphabet and information set as follows:
SNR SNR SNR
(52) and
SNR SNR
(53)
where is an integral basis of . Unlike
the construction for symmetrical MIMO-MAC, here the infor-mation set can be different among users as each user has dif-ferent level of multiplexing gain.
Let denote the left-regular map of elements in into
ma-trices of size whose entries are in (similar to
of (19)); then the ST code of the th user is given by
SNR
(54) where
SNR (55)
such that the power constraint (4) is satisfied.
Given , , the overall code is obtained
by vertically concatenating the code matrices from each user,
..
. (56)
The overall code matrix is a square matrix of size . Below we will present some nice properties of which are essential to proving its MAC-DMT optimality.
The first property extends Property 2 of the symmetric MAC code in Section III-B.
Property 5: For any SNR , define ..
. ... . .. ... (57)
and let
..
. . .. ...
be the corresponding overall code matrix where .
Then SNR. Further, let be the maximal number of
rows of that are linearly independent as a left -module. Then (58) where the rank is measured in the complex number field .
Proof: The first claim can be easily verified from the
set-tings of and SNR , and is thus omitted for brevity. For the second, to determine the rank of , it suffices to consider the rank of the unscaled code matrix
..
. . .. ... (59)
Notice that is a code matrix of the code defined in (27) for
the symmetric MIMO-MAC when we set and
The next property generalizes Property 3 in Section III-B where we were interested in the Gram determinant of the un-scaled code matrix. Here, for the purpose of analyzing the general MAC-DMT performance of the proposed code, we will seek directly the Gram determinant of the overall matrix .
Property 6: Let be defined as in (57) and assume that is a subset of rows of that are linearly in-dependent as a left -module. Let
..
. . .. ... (60)
be the submatrix of consisting of the corresponding rows. Then
SNR (61)
Proof: Arguing similarly to the proof of Property 5, set
..
. . .. ... (62)
Then we have
. .. and
by Property 3 since is a submatrix of the code matrix (cf. (59)) of the code (cf. (27)) for the symmetric MIMO-MAC
when setting and . The result now
follows from
and from the definition of in (55).
The two properties above are exactly what we need to prove the MAC-DMT optimality of the proposed general MIMO-MAC code in (56). Hence, with these properties we can prove the following theorem.
Theorem 10: Given and , with odd, the proposed code defined in (56) achieves the general joint MAC-DMT
(63)
over a Rayleigh block fading channel that remains static for at
least channel uses. is MAC-DMT optimal.
Proof: The proof is similar to that of Theorem 6 and is
relegated to Section VI for ease of reading.
The proof to Theorem 10 can in fact be further extended to show that the proposed code (56) achieves the optimal indi-vidual MAC-DMT(49), provided that an indiindi-vidual ML decoder for each user is used at the receiver end. This result along with the proof will be presented in Corollary 12 of Section VI.
V. PROOF OFTHEOREM6
Here we only prove the case of odd. The case of even
and can be proved using similar arguments, and
will therefore be briefly handled in a remark following the proof.
A. Proof Overview
In this subsection we provide an overview of the proof for the case of odd, along with a few insights to the proof. Given the overall channel matrix
(64) we will provide an upper bound on the codeword error proba-bility of the joint decoder at receiver end. Let
be a subset of users, and let denote the event that 1) the signal of the th user is erroneously decoded if and only
if , and further that
2) the rank distance between overall transmitted code matrix and the erroneously decoded overall signal matrix is only
for some .
Specifically, let SNR denote the information symbol transmitted by the th user, and let be the corresponding de-coding output at receiver; then the event can be formu-lated as follows:
(65)
where from the proposed construction (cf. (27)) we have ..
. . .. ...
and
..
. . .. ...
Note that the difference matrix has exactly nonzero rows, and by Property 2 we see the rank distance
. Hence it makes sense
for the second requirement of error event that
Thus, it can be seen from the union bound argument that the codeword error probability is upper bounded by
(66)
The event is a further partition of the event considered by Tse et al. in [11]. We discuss this in more detail in the fol-lowing remark.
Remark 7: With regard to the Gaussian random codebook
considered by Tse et al. [11], it is straightforward to see
is empty with probability one if , since the compo-nent matrices associated with each user are complex Gaussian
random matrices of size for some . In other
words, if for all and otherwise, then
the error matrix would have rank with probability one. Therefore, one can rewrite (66) as
(67)
and recover the same union bound used in [11].
Unlike [11] where the authors analyzed each summand of (67) by a union bound argument with a Gaussian random codebook, here we will focus on the error probability of a deterministic codebook (cf. (27)), and
at-tempt to upper bound the probability by using a
joint ML decoder. To this end, in Section V-B we will examine the minimum Euclidean distance among the noise-free received code matrices contained in . It should be noted that here by minimum Euclidean distance, we mean the minimum Euclidean distance among only the pairs of code matrices in , not the whole code . Thus, the minimum Euclidean distance will be a function of , , and .
Once we obtain the minimum Euclidean distance, we will analyze the error performance of a bounded distance decoder, which will be used as an upper bound on that of the ML de-coder. The bounded distance decoder results in an error only when the noise matrix has norm larger than half of the min-imum Euclidean distance. More precisely, let be the overall channel matrix defined in (64) and be known to the decoder; let be the overall MIMO-MAC code, where is the codebook of the th user. The minimum Euclidean dis-tance among all code matrices in is defined as
which is dependent upon . Given the received signal matrix , the bounded distance decoder outputs
if , and declares a decoding failure
oth-erwise. Thus, only the received signal matrices that are within distance from the original transmitted overall code ma-trix can be correctly decoded in the bounded distance decoder. Other received signal matrices would result in either a decoding
error (i.e., decoding into an erroneous code matrix) or a de-coding failure (i.e., cannot find a code matrix within distance ). Though this decoder is suboptimal compared to the ML decoder, its error performance can be mathematically ana-lyzed.
The error performance analysis following this outline will be given in Section V-C. Finally, in Section V-F we briefly discuss the proof for the case of even .
B. Lower Bounds on the Minimum Distance Among Noise-Free Received Signal Matrices
For any with
..
. . .. ...
and
..
. . .. ...
let , , and let , where
is the channel matrix associated with the th user. Given the channel matrix , below we provide a lower bound on the squared Euclidean distance between and , i.e.,
(68)
We distinguish the following two cases which correspond to the
error events and with , respectively.
1) For event we have for
, otherwise, and
. In this case, let and be defined as in (36) and let
be the equivalent channel matrix; then we have
Let be the set of ordered nonzero
eigenvalues of where , and
let be the ordered nonzero
eigenvalues of . Then we have
(69) Note that
(70) where the first inequality follows from Property 3, and the second exponential equality is because is fixed and is independent of SNR.
By repeatedly using the arithmetic mean-geometric mean
inequality and (70) as in [4], [6] for , we
have
(71)
(72)
SNR SNR
SNR (73)
where (72) follows from (70) and where in (73) we have set
SNR
Hence the SNR exponent of is lower bounded
by
(74)
2) The second case corresponds to event which
means for ,
otherwise, and . In other
words, the nonzero rows
are not linearly independent over . From Property 2 we can assume without loss of generality that
are linearly independent for some .
Let and let and be defined as in (36)
with respect to the set . Set
and . Property 2 in turn implies that
. .. ..
. ... ...
(75)
for some square matrices , where ..
. . .. ... (76)
Similar to the previous case, let
be the equivalent channel matrix; then the difference of the noise-free received signal matrices can be rewritten as
(77) where
(78) is an alternative channel equivalent matrix and
(79)
for .
Let be the set of ordered nonzero
eigenvalues of , where , and
let be the ordered nonzero
eigen-values of . Notice that
from Property 3. Arguing similarly as in the first case shows that SNR (80) for , where SNR (81) (82) and (83)
Remark 8: We remark that (83) shows the last term is , instead of being as in (74). For readers who may wonder why these two terms are different given both events concern the case of users in error, the major reason is due to the distance bounding techniques, i.e., the repeated arithmetic mean-geo-metric mean inequalities, we have used in the above.
In general, when the equivalent channel matrix of (78), and similarly when the channel matrix with , has
rank , the rank of the product matrix would be
since is of full rank . Thus our lower bound on
the norm would only capture the smaller
eigen-values of , which are all nonzero. Furthermore, one reason for introducing the equivalent channel matrix , rather than working with is that, algebraically speaking, the norm could be zero as is singular and all the rows of could lie in the left-null space of . However, since is random, this occurs with probability zero. In other words, if we apply the series of arithmetic mean-geometric mean inequalities to the matrix product , we could end up with the trivial algebraic inequality
even the right-hand-side has probability 0. Whether the above could happen depends on the relations among , , , and . While there is nothing wrong with the algebraic inequality itself, this bound can actually be further tightened by introducing the equivalent channel so that we can focus on error events that have probability larger than zero.
Remark 9: Another heuristic way to see why the last term of
equals follows from the base-alphabet SNR de-fined in Section III-A. Recall that in the construction of
CDA-based ST code for point-to-point channel [4], [6], to achieve the DMT optimality therein we would set the base-al-phabet as
SNR SNR SNR
such that the resulting exponent equals
then along the same lines as in [4], [6] one can prove such code is approximately universal and achieves diversity gain . However, it is because we set the base-alphabet as SNR , which has size
SNR SNR
meaning an -fold increase in the multiplexing gain, we expect the error probability associated with event has diversity
gain .
C. Upper Bounds on Codeword Error Probability
Having obtained the squared minimum Euclidean distances among the signal matrices associated with error
event , and among the signal matrices
associated with error event , below we proceed to an-alyze the error performance of the proposed construction. The analysis resembles the sphere bounding technique used in [4], [6] which is essentially a bounded-distance decoding technique. That is, the bounded-distance decoder declares an error only when the noise has norm larger than half of the minimum Eu-clidean distance. Clearly, the error performance of a bounded-distance decoder serves as an upper bound on that of a joint ML decoder.
First, since the lower bounds on the Euclidean distance hold for all , we define
Then, using the bounded distance decoder discussed in Section V-A, the probability of error event given channel matrix can be upper bounded by
(84) where the inequality follows from the property of a bounded
distance decoder. Hence we see that if
. On the other hand, we may replace the above
upper bound of with the trivial upper bound
when . Thus, it implies
Since the above bounds do not depend on the specific choices of , from (66) the union bound on the codeword error
prob-ability gives
(85)
Remark 10: One can regard the probability
as a further upper bound on the union bound
in (66), and the second type of probability
(86)
as an upper bound on . Furthermore,
the event of users in error has probability upper bounded by
(87) It should be noted that in (86) we have over-estimated the number of choices of -linearly independent rows out of nonzero rows in the difference matrix
that can happen in the event . Even with this over-estimate, noting
for all within the range of interest, we can rewrite (85) as
(88) Below we investigate the diversity orders of each term in (88).
D. Diversity Gain of the First Case
For each , , we have
(89)
SNR SNR
SNR (90)
where , , and
where for . Equation (89) follows
from [31]–[33], and (90) is given in [9] since is a matrix of
size having entries that are i.i.d. complex
Gaussian random variables. The quantity
repre-sents the point-to-point DMT of an MIMO Rayleigh fading channel at multiplexing gain .
E. Diversity Gain of the Second Case
Similarly, for the second set of maximizations in (88) we have
for each that
SNR SNR
(91) where is defined in (78) and is of size . Noting that the entries of are correlated complex Gaussian random variables, we invoke the following result which was shown inde-pendently in [34, Corollary 1] and [35, Theorem 3] to simplify the analysis.
Theorem 11 ( [34], [35]): The diversity order of outage
prob-ability for Rayleigh fading channels with arbitrary full rank cor-relations is unchanged from the case of i.i.d. Rayleigh fading. Moreover, if the channel matrix can be decoupled as
where has independent and regular entries, then the optimal DMT for channel is the same as that for .
Armed with Theorem 11, the analysis of the diversity gain of the second case is now easy. A direct application of the above theorem gives
SNR SNR
Summarizing results of (88), (90) and (92) gives SNR
and
This completes the proof.
Remark 11: The above proof shows that
SNR
for error events , , and
SNR
for . This is exactly what is shown in Theorem 2. Furthermore, in the event of users in error, the proposed code has error probability
SNR SNR
F. Proof Outline for Even
The proof of Theorem 6 can be adapted to cater to the case when the number of users is even. Here we discuss only briefly what the changes are. Firstly, with
in mind, i.e., , the result (68) of the squared Euclidean distance between and remains to hold. Similarly, the fur-ther lower bounds on in (74) and (83) stay without changes except that one should keep the following in mind.
1) The parameter of the first case, where
, and out of ’s are distinct, has value from 1
up to . This is because , and we
can always assume without affecting the
value of . Thus the diversity gain resulting from the first case is
Compared with the case of odd , (90) has up to . 2) The parameters and in the second case can be argued
similarly as the above, and we have
. Hence the diversity gain of this case is
Overall, it shows the MAC-DMT optimality of the proposed construction remains to hold.
VI. PROOF OFTHEOREM10
The proof of Theorem 10 is similar to that of Theorem 6. Therefore, we will skip the most of the details and highlight only the key differences.
First, for any subset
of users, again let denote the event that the decoder has made an error in decoding users signals, but only the rows formed by the difference signal matrices of some users, say user , are linearly independent over . In other words, let be a pair of distinct information symbols of
the th user for . Set the submatrices and as in
(62) with and . Then the event
corresponds to the case when .
Let denote the channel matrix of the th user
that is known completely to receiver. Since the code matrices
of the th users are of size , here we
will assume without loss of generality that only the first rows of are used for transmission via the transmit antennas of the th user, and the remaining rows are discarded during either encoding or transmission. On the other hand, we could extend the channel matrix to an equivalent channel matrix of size by adding on the right an all-zero matrix with appropriate size. That is, we set
and the received signal matrix can be written as
where is the noise matrix of size .
For the event , let
be the overall equivalent channel matrix, and let
be the ordered nonzero eigenvalues of with
Similarly, let be the ordered
nonzero eigenvalues of . Then the
min-imum squared Euclidean distance between and is
bounded by
SNR
(93)
