Optimal Diversity Multiplexing Tradeoff of
Constrained Asymmetric MIMO Systems
Hsiao-feng (Francis) Lu
Department of Communications Engineering National Chiao Tung University
Hsinchu, Taiwan Email: francis@cc.nctu.edu.tw
Abstract—In a MIMO downlink channel it is often that the number of transmit antennas is strictly larger than the number of receive, and such channel is termed asymmetric MIMO channel. To employ simple decoding techniques in this channel, such as zero-forcing or sphere decoding, the number of active transmit antennas must be constrained to be no larger than the number of receive, and the resulting system is coined “constrained asymmetric MIMO system.” For the case of two receive antennas and for any number of transmit antennas, an optimal transmission scheme is presented in this paper and is shown to achieve the same performance as the unconstrained ones in terms of the diversity-multiplexing tradeoff. The construction of optimal constrained codes is also provided.
I. INTRODUCTION
Consider a MIMO communication channel that consists of nttransmit and nrreceive antennas. When nt> nr, i.e. when the number of transmit antennas is strictly larger than the number of receive, such (nt×nr) MIMO channel is commonly referred to as the asymmetric MIMO channel, and it can often be found in the MIMO downlink communication.
Assuming that all the nt transmit antennas are active during signal transmission, let x be the length-nt code vector sent from the transmitter to the receiver and let H be the corresponding (nr× nt) channel matrix. The length-nrsignal vector y received at the receiver end is given by
y = Hx + w, (1)
where w is a length-nr vector used to capture the effects of additive white Gaussian noise. Entries of the channel matrix H and the noise vector w are modeled as i.i.d. complex Gaussian random variables with zero mean and unit variances in this paper. Further, the code vector x is assumed to satisfy the following power constraint
TrExx† ≤ SNR, (2)
where by† we mean the Hermitian transpose of a vector. When the channel matrix H is known completely to the receiver but not the transmitter, Telatar [1] first showed that the ergodic channel capacity of such (nt× nr) MIMO chan-nel approximates min{nt, nr} log2SNR at high SNR regime, regardless of the relation between nt and nr. Furthermore, it was shown that such capacity can be achieved by using i.i.d. complex Gaussian random vectors x having covariance
matrix KX = SNRnt Int. On the other hand, assuming that the
transmitter communicates at rate
R = r log2SNR (bits/channel use), (3) where r, 0 ≤ r ≤ min{nt, nr}, is termed the multiplexing
gain, Zheng and Tse proved in their landmark paper [2] that given r, the smallest bit error probability that can be achieved by all possible coding schemes is given by
Pe,min(SNR) = SNR. −d ∗(r) , (4) meaning lim SNR→∞ log Pe,min(SNR) log SNR = −d ∗(r). (5)
The negative exponent d∗(r) is termed the diversity gain and is given by a piecewise linear function connecting the points
{(k, (nt− k)(nr− k)) : k = 0, 1, · · · , min{nt, nr}} . (6)
d∗(r) represents an optimal tradeoff between the multiplexing gain r and the diversity gain, and is thus termed the optimal diversity-multiplexing tradeoff (DMT). It is also proved in [2] that d∗(r) can be achieved by using i.i.d. length-nt com-plex Gaussian random vectors, provided that the asymmetric MIMO channel is quasi-static and the channel matrix H remains fixed for T ≥ nt+ nr− 1 channel uses.
This remarkable result has spurred a considerable amount of research activities of constructing coding schemes [3], [4], [5], [6], [7] to achieve the optimal tradeoff (6). In particular, Elia et al. [6] have provided a sufficient condition for having deterministic DMT optimal codes. Furthermore, for any nt, using a cyclic division algebra (CDA) with degree n2t over the center Q(ı), where ı = √−1, an algebraic construction of (nt× nt) matrix codes meeting this sufficient condition is proposed in [6] for all T ≥ nt.
While all the coding schemes mentioned above, including the Gaussian random codes and the CDA-based codes, are DMT optimal, it should be noted that they require that all the nt transmit antennas are active during each channel use. Such requirement can lead to some unavoidable difficulty in decoding. To see this, note that the channel matrix H is of size (nr×nt) with nt> nr. Therefore H has no left multiplicative matrix inverse, and it is impossible to use zero-forcing (ZF) decoder to decode the code. Similarly, the same requirement
again forbids the possibility of using sphere decoder which relies on the QR decomposition of the matrix H. For the minimum-mean square error (MMSE) detector, due to the number of observations, nr, in each channel use, is strictly less than the number of unknowns, which is ntin this case, the error performance resulting from the use of MMSE decoding technique cannot be good.
In order to use the ZF decoder, sphere decoder, or the MMSE decoder to reduce the decoding complexity, the num-ber of active transmit antennas in each channel use must not be larger than nr. With this additional constraint, the resulting system is termed constrained asymmetric MIMO system in this paper, and coding schemes meeting this requirement are coined constrained asymmetric space-time codes. Similarly, codes without this constraint will be termed unconstrained codes.
In [8], Hollanti and Ranto focused on the case of 4 transmit and 2 receive antennas, i.e. nt= 4 and nr= 2, and proposed a block-diagonal coding method for constructing the constrained asymmetric space-time codes. The construction first partitions the 4 transmit antennas into two groups, say {T1, T2} and {T3, T4}, and then performs a joint-encoding between the two groups by making use of the multi-block space-time codes proposed by the author [9]. Specifically, let X be a (2 × 4) multi-block space-time code where the coding is applied over 2 consecutive (2× 2) independent fading blocks, and let H1 (resp. H2) denote the channel matrix corresponding to the transmit {T1, T2} (resp. {T3, T4}) and the receive antennas. Given the transmitted code matrix X = [X1X2]∈ X , where the submatrix Xi is of size (2× 2), the resulting received signal matrix is
Yi = HiXi+ Wi, i = 1, 2, (7) where Wi is the (2× 2) noise matrix. Clearly we have the original (2× 4) channel matrix H = [H1H2] and given the desired multiplexing gain r, it can be easily shown by using results in [9] that the resulting diversity gain d(r) achieved by X is given by a piecewise linear function connecting the points (k, 2 (2− k) (2 − k)), for k = 0, 1, 2. From Fig. 1 it can be seen that the DMT performance achieved by X is far from being optimal compared to d∗(r) in (6) .
In this paper, we will investigate the optimal DMT of the constrained asymmetric MIMO systems, and in particular, focus on the case of nr= 2, which represents a very common scenario in the MIMO downlink communications. This paper is organized as follows. In Section II, we will present a DMT optimal transmission scheme for any constrained asymmetric MIMO systems with nt> nr= 2 and show that the resulting DMT equals d∗(r), meaning that there is no performance loss with the additional constraint on the number of transmit antennas used in each channel use if the code is properly designed. The proposed DMT optimal transmission scheme is basically a selection pattern of the transmitted antennas used, and the corresponding DMT optimal coding schemes that follow this selection pattern will be briefly discussed in Section III. Finally, in Section IV, we conclude the paper.
0 0.5 1 1.5 2 0 1 2 3 4 5 6 7 8 Multiplexing Gain r Diversity Gain d(r) Unconstrained d*(r) BlockíDiagonal Method
Fig. 1. The DMT performances of unconstrained coding schemes and codes derived from block-diagonal constructions [8].
II. PROPOSEDDMT OPTIMALTRANSMISSIONSCHEME FORCONSTRAINEDASYMMETRICMIMO SYSTEMS In the previous section, we have shown that in order to employ ZF, sphere, or MMSE decoding techniques for decoding the transmitted signal matrix in an asymmetric MIMO channel, the number of transmit antennas used in each channel use cannot be larger than the number of receive antennas nr. In this section, we will focus on the case when
nr = 2. For any nt > 2 we will present a DMT optimal transmission scheme that can achieve the optimal DMT d∗(r) in the (nt× 2) constrained asymmetric MIMO channel. To describe the proposed transmission scheme, we first define the following.
Definition 1: In an (nt × nr) constrained asymmetric MIMO channel, letT = {T1,· · · , Tnt} be the set of indices
of nttransmit antennas. We say
S := {(T1, n1) ,· · · , (Ts, ns)} (8) is an antenna-selection transmission scheme if the antenna selection patterns Ti are distinct proper subset ofT and have size 1≤ |Ti| ≤ nr < nt for each i. Moreover, each antenna selection patternTi will be used for nitransmissions and it is assumed that the MIMO channel remains fixed for T channel uses with T ≥ s i=1 ns. (9)
For example, the block-diagonal coding method proposed in [8] for the (4× 2) constrained asymmetric MIMO channel can be regarded as an antenna-selection transmission scheme of
SBD = {({T1, T2} , 2) , ({T3, T4} , 2)} . (10) However, it has already been seen in Section I that the above scheme SBD is not DMT optimal in the (2 × 4) constrained asymmetric MIMO channel. On the other hand, for any antenna-selection transmission scheme S =
{(T1, n1) ,· · · , (Ts, ns)} with |Ti| = nr, it is clear that the ergodic channel capacity achieved by S is the same as that achieved by the unconstrained schemes. To see this, let Hi denote the channel matrix associated with the selection pattern Ti and the set of all receive antennas, and let xij, 1≤ i ≤ s
and 1≤ j ≤ nibe i.i.d. zero-mean complex Gaussian random vectors having same covariance matrix K = SNRn
r Inr. Then
following the same approach as in [1] the ergodic channel capacity achieved byS using random xij as transmitted signal vectors is C (SNR) = s1 i=1ns s i=1 Enilog2det Inr+ HiKHi† = E log2det Inr+ SNR nr H1H1† ≈ nrlog2SNR (11)
at high SNR regime and is the same as that achieved by the unconstrained schemes.
To improve the DMT performance, for any nt > nr = 2 below we provide another transmission scheme and we will prove that it can achieve the optimal DMT d∗(r) given in (6). Clearly, in this case, the maximal value of multiplexing gain r is upper bounded by min{nt, nr} = 2, hence 0 ≤ r ≤ 2. The proposed scheme is the following.
Theorem 1 (Main Result): In an (nt×2) constrained asym-metric MIMO system with nt> 2, letT = {T1,· · · , Tnt} be
the set of indices of nt transmit antennas. Given the desired multiplexing gain r,
1) if the multiplexing gain r falls within the range of [1, 2], the following antenna-selection scheme
S1 = {({T1, T2} , 2) , ({T2, T3} , 2) , · · · , ({Tnt−1, Tnt} , 2)} (12) achieves the optimal DMT d∗(r) of (6), and
2) if r∈ [0, 1),
S2 = {({T1, T2} , 4) , ({T2, T3} , 2) , · · · , ({Tnt−2, Tnt−1} , 2) , ({Tnt−1, Tnt} , 4)}
(13) is DMT optimal in terms of d∗(r).
First of all, the only difference between the selection pat-ternsS1andS2 is that when 0≤ r < 1, the sets {T1, T2} and {Tnt−1, Tnt} are used twice more than the others. Secondly,
for the case of (4×2) constrained asymmetric MIMO channel, the scheme in Theorem 1 is given by
S1 = {({T1, T2} , 2) , ({T2, T3} , 2) , ({T3, T4} , 2)} for multiplexing gain r∈ [1, 2] and
S2 = {({T1, T2} , 4) , ({T2, T3} , 2) , ({T3, T4} , 4)} for r∈ [0, 1). Comparing to the block-diagonal method SBD, the proposed scheme requires two more transmissions for r≥ 1 and six more for r < 1. However, the price of using more transmissions is well paid off by having a much better error
performance and achieving the same DMT performance as the unconstrained systems.
Below we provide a proof to Theorem 1
Proof: As the difference between the schemes S1 and S2 lies only in the number of times used for each antenna selection pattern Ti, here we consider the following general form:
S := {({T1, T2}, n1) ,· · · ({Tnt−1, Tnt}, nnt−1)} . (14)
For the ith selection, let xibe the length-nr, zero-mean, com-plex Gaussian random vector with covariance Ki = SNR2 Inr
as the random code vector; then the corresponding received signal vector is given by
y
i = Hixi+ wi (15)
where Hi :=
hi hi+1 and hi is length-2 vector consisting of the fading coefficients between the ith transmit antenna Ti and the receive antennas. wi is the zero-mean complex Gaussian random vector of length 2 used to model the effect of additive white Gaussian noise. Thus, given the channel matrix Hi, the mutual information between the transmit and receive signal vectors is I xi; yi|Hi ≈ log2det I2+ SNRHiHi† , (16)
where we have neglected the 2 appearing in the denominator of SNR2 as here we are only interested in the high SNR regime for the sake of DMT performance analysis. Define
N :=
nt−1
j=1
nj. (17)
Given the desired multiplexing gain r, the channel outage probability ofS is Pout(r) := n t−1 i=1 nilog det I2+ SNRHiHi† ≤ N r log SNR . = SNR−d(r). (18)
In particular, the mutual information associated with{T1, T2} can be rewritten as log det I2+ SNRH1H1† = log det I2+ SNRh1h†1+ SNRh2h†2 = log det (I2+ SNRD1) + log 1 + SNRh†2U1(I2+ SNRD1)−1U1†h2 = log 1 + SNR|h1|2F + log 1 + SNRg†2(I2+ SNRD1)−1g2 ,
where U1D1U1† is the eigen-decomposition of the rank-1 matrix h1h†1 and where g2 := U1†h2 has the same joint probability density function as that of h1. By|h1|F we mean the Frobenius norm of vector h1. Hence, without affect the
calculation of (18), we can set the channel matrix associated with the second selection pattern {T2, T3} as
H2 = g 2 h3 (19) and the corresponding mutual information changes to
I x2; y 2|H2 = log 1 + SNRg 2 2 F + log 1 + SNRg†3(I2+ SNRD2)−1g3 ,
where U2D2U2† is the eigen-decomposition of the rank-1 matrix g2g†2 and g3 = U2†h3. Continuing in this fashion, we can rewrite the overall mutual information associated with schemeS as nt−1 i=1 nilog det I2+ SNRHiHi† = nt−1 i=1 ni log 1 + SNRgi2 F + log ⎛ ⎜ ⎝1 + SNR|gi+1,1|2 1 + SNRg i 2 F + SNR|gi+1,2|2 ⎞ ⎟ ⎠ ⎤ ⎥ ⎦ ,(20) where we have set g1 = h1, and for i = 2,· · · , nt, gi =
Ui−1† hi= [gi,1 gi,2]t. UiDiUi† is the eigen-decomposition of
g ig † i. Now define |gi,j|2 = SNR. −αi,j (21) and we can rewrite (20) as
1 log SNR nt−1 i=1 nilog det I2+ SNRHiHi† ≈ nt−1 i=1 ni ⎡ ⎢ ⎢ ⎢ ⎣ ⎛ ⎜ ⎜ ⎜ ⎝maxj (1− αi,j)+ !" # :=(1−βi)+ ⎞ ⎟ ⎟ ⎟ ⎠+ max $ 1− αi+1,1− (1 − βi)+ + , (1− αi+1,2)+ % , where (x)+:= max{0, x}. Thus, the diversity gain achieved by the general schemeS is
d(r) = inf A(r) nt i=1 2 j=1 αi,j, (22) where A(r) = (α1,1,· · · , αnt,2) : nt−1 i=1 ni (1− βi)++ max$1− αi+1,1− (1 − βi)+ + , (1− αi+1,2)+ % ≤ Nr, and αi,j≥ 0 & . (23)
While the optimization of d(r) subject to the constraint (23) appears to be a non-linear optimization problem, below we will convert it to a problem of linear programming. First note that for each αi,j, the probability of αi,j< 0 is zero. Secondly, to minimize the diversity gain d(r), we do not need αi,jto be larger than 1 as (1− αi,j)+ = 0 for αi,j ≥ 1 and setting
αi,j = 1 minimizes the cost of d(r). Thus, we have the following sets of linear constraints:
0≤ αi,j≤ 1 for all i = 1, · · · , nt− 1, j = 1, 2. (24) Next, for i = 1, 2,· · · , nt− 1, setting
(1− βi)+= max{(1 − αi,1)+, (1− αi,2)+} := ri,1 (25) yields the following linear constraints:
αi,1 ≥ 1 − ri,1 (26)
αi,2 ≥ 1 − ri,1 (27)
1 ≥ ri,1≥ 0. (28)
Again, for i = 1, 2,· · · , nt− 1, setting max $ 1− αi+1,1− (1 − βi)+ + , (1− αi+1,2)+ = ri,2 (29) gives the following linear constraints:
αi+1,1 ≥ 1 − ri,1− ri,2 (30)
αi+1,2 ≥ 1 − ri,2 (31)
1 ≥ ri+1,2≥ 0. (32)
To achieve the desired multiplexing gain r, the linear con-straint on the ri,j is given by
nt−1
i=1
ni(ri,1+ ri,2) ≤ Nr.. (33) Using standard linear programming techniques to minimize d(r) of (22) subject to the constraints of (24), (26), (27), (28), (30), (31), (32), and (33), it can be shown that
1) for the scheme S1, i.e. ni = 2 for all i, we have N = 2(nt− 1) and
d(r)≥ (nt− 1)(2 − r) and d(r) ≥ 2nt− 2(nt− 1)r. (34) Hence for the region of 1≤ r ≤ 2, the DMT achieved byS1is given by
d(r) ≥ max {(nt− 1)(2 − r), 2nt− 2(nt− 1)r} = (nt− 1)(2 − r), for 1 ≤ r ≤ 2. (35) 2) for the schemeS2, i.e. the case when n1= 4, nnt−1= 4
and the remaining ni = 2, we have N = 2nt+ 2, and
d(r)≥ 2nt− (nt+ 1)r and 2d(r)≥ (nt+ 1) (2− r) . (36) Thus for the region of 0 ≤ r ≤ 1, the DMT achieved by scheme S2 is given by d(r) ≥ max $ 2nt− (nt+ 1)r, nt+ 1 2 (2− r)
= 2nt− (nt+ 1)r, for 0≤ r ≤ 1. (37) The proof is now complete after noting that the DMTs (36) and (37) achieved respectively by schemes S1 andS2 in the region of r∈ [1, 2] and r ∈ [0, 1) match exactly the optimal DMT d∗(r) given in (6).
In Fig. 2 we have provided the exact DMT performances of the transmission schemes S1 and S2 proposed in Theorem 1 for the (4× 2) constrained asymmetric MIMO system. It can be easily seen that the schemes are DMT optimal and achieves the optimal DMT d∗(r) of (6) within the designated regions.
0 0.5 1 1.5 2 0 1 2 3 4 5 6 7 8 Multiplexing Gain r Diversity Gain d(r) Unconstrained d*(r) Proposed S1 (a) 0 0.5 1 1.5 2 0 1 2 3 4 5 6 7 8 Multiplexing Gain r Diversity Gain d(r) Unconstrained d*(r) Proposed S2 (b)
Fig. 2. DMT performances of (a) the proposed scheme S1 and (b) the proposed schemeS2for the(4 × 2) constrained asymmetric MIMO system.
III. DMT OPTIMALCODES FORCONSTRAINED ASYMMETRICMIMO SYSTEMS
In Section II, we have identified the DMT optimal transmis-sion schemes for the constrained asymmetric MIMO systems with nt> nr= 2. To achieve this optimal DMT performance, it turns out that the multi-block space-time codes [9] that are originally designed to achieve the optimal DMT in multi-block fading channel can be modified to cater to the constrained channel. Due to limited space, below we provide without proof the construction of DMT optimal constrained codes for the case when nt> nr= 2.
Theorem 2: Given nt> nr= 2 and the desired multiplex-ing gain r, let S be the DMT optimal transmission scheme specified in Theorem 1. LetC be a (2×2L) multi-block space-time code given in [9] with number of coded fading blocks L = nt− 1 if r ∈ [1, 2] and L = nt+ 1 if r∈ [0, 1). For any code matrix C ∈ C, transmit C according to the transmission scheme S. Then the codeword error probability achieved by C is
Pcwe(SNR) = SNR. −d
∗(r)
, (38)
meaning that the constrained codeC is DMT optimal. IV. CONCLUSION
When the number of transmit antennas nt is strictly larger than the number of receive nr, all the currently available DMT optimal codes require that all the transmit antennas are active during transmission, hence forbid the possibility of having a ZF, sphere, or MMSE decoder. To remedy this, the number of active transmit antennas must be constrained to be less than or equal to nr. When nr = 2 and for all nt > 2, an optimal transmission scheme satisfying the above constraint was presented in this paper and was shown to achieve the same DMT performance as the unconstrained. A systematic construction of DMT optimal constrained codes was also provided.
ACKNOWLEDGMENT
This research was supported by Taiwan National Science Council under Grants NSC 97-2219-E-009-014 and NSC 96-2628-E-009-172-MY3.
REFERENCES
[1] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Europ. Trans.
Telecomm., vol. 10, no. 6, pp. 585–595, Nov.-Dec. 1999.
[2] L. Zheng and D. Tse, “Diversity and multiplexing: a fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1073–1096, May 2003.
[3] J.-C. Belfiore, G. Rekaya, and E.Viterbo, “The Golden code: a2 × 2 full-rate space-time code with non-vanishing determinants,” IEEE Trans.
Inf. Theory, vol. 51, no. 4, pp. 1432–1436, Apr. 2005.
[4] H. El Gamal, G. Caire, and M. Damen, “Lattice coding and decoding achieve the optimal diversity-multilpexing tradeoff of MIMO channels,”
IEEE Trans. Inf. Theory, vol. 50, no. 6, pp. 968–985, Jun. 2004.
[5] T. Kiran and B. S. Rajan, “STBC-schemes with non-vanishing determi-nant for certain number of transmit antennas,” IEEE Trans. Inf. Theory, vol. 51, no. 8, pp. 2984–2992, Aug. 2005.
[6] P. Elia, K. R. Kumar, S. A. Pawar, P. V. Kumar, and H.-F. Lu, “Explicit construction of space-time block codes achieving the diversity-multiplexing gain tradeoff,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 3869–3884, Sep. 2006.
[7] S. Tavildar and P. Viswanath, “Approximately universal codes over slow fading channels,” IEEE Trans. Inf. Theory, vol. 52, no. 7, pp. 3233– 3258, Jul. 2006.
[8] C. Hollanti and K. Ranto, “Asymmetric space-time block codes for MIMO systems,” in Proc. 2007 IEEE Inform. Theory Workshop, Sol-strand, Norway, Jul. 2007, pp. 101–105.
[9] H. F. Lu, “Explicit constructions of multi-block space-time codes that achieve the diversity-multiplexing tradeoff,” IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3790–3796, Aug. 2008.
![Fig. 1. The DMT performances of unconstrained coding schemes and codes derived from block-diagonal constructions [8].](https://thumb-ap.123doks.com/thumbv2/9libinfo/7712343.146305/2.918.495.798.83.329/performances-unconstrained-coding-schemes-codes-derived-diagonal-constructions.webp)
