DESIGN OF ADAPTIVE MIMO SYSTEM USING LINEAR DISPERSION CODE

Mabruk Gheryani, Zhiyuan Wu, and Yousef R. Shayan

Concordia UniversityDepartment of Electrical Engineering

Montreal, Quebec, Canadaemail: (m gherya, zy wu, yshayan)@ece.concordia.ca

ABSTRACT

In this paper, we develop a new design for adaptation of lin-ear dispersion code. A new adaptive parameter called space-time symbol rate is applied in our design. We have studied thestatistics of signal-to-interference-noise of a linear MMSE re-ceiver over a Rayleigh fading channel. The average BER fora given constellation using the MMSE receiver is calculatednumerically. With the statistics as a guideline, two adaptivetechniques using constellation and space-time symbol rate arestudied, respectively. If constellation and space-time sym-bol rate are considered jointly, more selection modes can beavailable. Theoretical analysis demonstrates that the averagetransmission rate of the joint adaptation can be improved inthis case. Simulation results are provided to show the benefitsof our new design.

Index Terms— MIMO,LDC, FRFD

1. INTRODUCTION

The demand for bandwidth efficiency in wireless communi-cations has experienced an unprecedented growth. One sig-nificant advancement to improve radio spectrum efficiencyis the use of multiple-input-multiple-output (MIMO) technol-ogy [1]. Space-time (ST) codes are the most promising tech-nique for MIMO systems [2]. Due to battery life and devicesize, the power available for radio communications is limited.Under this power constraint, adaptive technique can coop-erate with MIMO technology to further exploit radio spec-trum [3] [4]. In an adaptive system, a feedback channel isutilized to provide channel state information (CSI) from thereceiver to the transmitter. According to the feedback of CSI,the transmitter will adjust transmission parameters, such aspower allocation, modulation, coding rate, etc. This is condi-tioned by the fact that the channel stays relatively constantbefore the transmitter receives the CSI and then transmitsnext data block accordingly. That is, the channel is “slow”.Many of adaptive MIMO schemes have been proposed, suchas water-filling-based schemes [1] [5] and various beamform-ing schemes [4] [6]. The above schemes often need near-perfect CSI feedback for adaptation calculation and consume

large feedback bandwidth.In practice, the channel estimationwill exhibit some inaccuracy depending on the estimationmethod. The receiver will need time to process the channelestimate and the feedback is subject to some transmissiondelay. The transmitter needs some time to choose a propercode, and there are possible errors in the feedback channel.All these factors make the CSI at the transmitter inaccurate.Additionally, the feedback bandwidth is often limited. Inthese cases, adaptive schemes with a set of discrete trans-mission modes are often more preferable. We can call them“selection-mode” adaptation. At the receiver, the channel ismeasured and then one transmission mode with the highesttransmission rate is chosen, which meanwhile meets the BERrequirement. The optimal mode is fed back to the transmitter.

For selection-mode MIMO adaptation, the most conve-nient adaptive parameter is constellation size for uncoded sys-tems. For example, constellation adaptation, such as M-QAM,is applied to space-time block code (STBC) [7] and to space-time trellis code (STTC) [8]. The disadvantage of theseschemes, is that they are not flexible for different rates, whichis the key requirement in the future wireless communications.Additionally, the gap between the available transmission ratesare often very large due to the use of discrete constellations[6].

In this paper, we propose to apply linear dispersion code(LDC) [9] [10] for adaptation. This is because it subsumesmany existing block codes as its special cases which allowssuboptimal linear receivers with greatly reduced complexity,and provides flexible rate-versus-performance tradeoff [9] [11].With the application of LDC, a linear MMSE detector is moreattractive due to its simplicity and good performance [13]. Asa guideline, the statistics of SINR for LDCs with a MMSE re-ceiver is studied and the associated average BER is calculatednumerically. Since the LDC is applied, it makes ST symbolrate available for adaptation. By adjusting this new parametertogether with constellation size, more available transmissionmodes can be provided. Hence, the throughput under a powerconstraint can be further improved while the target bit errorrate (BER) is satisfied.

2. ADAPTIVE SYSTEM MODEL USING LDC

In this study, during one ST modulation block, the channelis assumed to be the same as estimated at the receiver. Fur-thermore, the channel is assumed to be a Rayleigh flat fad-ing channel with Nt transmit and Nr receive antennas. Let’sdenote the complex gain from transmit antenna n to receiverantenna m by hmn and collect them to form an Nr×Nt chan-nel matrix H = [hmn], known perfectly to the receiver. Theentries in H are assumed to be independently identically dis-tributed (i.i.d.) symmetrical complex Gaussian random vari-ables with zero mean and unit variance.

In this system, the information bits are first mapped intosymbols. After that, the symbol stream is parsed into blocksof length L. The symbol vector associated with one modu-lation block is denoted by x = [x1, x2, . . . , xL]T with xi ∈Ω ≡ Ωm|m = 0, 1, . . . , 2η − 1, η ≥ 1, i.e., a complexconstellation of size 2η , such as 2η-QAM). The average sym-

bol energy is assumed to be 1, i.e., 12η

2η−1∑m=0

|Ωm|2 = 1. Each

block of symbols will be mapped by the ST modulator to adispersion matrix of size Nt × T and then transmitted overthe Nt transmit antennas over T channel uses. The followingmodel will be considered in this study, i.e.,

X =L∑

i=1

Mixi (1)

where Mi is defined by its L Nt × T dispersion matricesMi = [mi1,mi2, . . . ,miT ]. The so-obtained results can beextended to the model in [9]. With a constellation of size 2η ,the data rate of the space-time modulator in bits per channeluse is

Rm = η · L/T (2)

Hence, one can adjust ST symbol rate L/T and constellationsize η according to the feedback from the receiver.

At the receiver, the received signals associated with onemodulation block can be written as

Y = HX + Z = HL∑

i=1

Mixi + Z (3)

where Y is a complex matrix of size Nr × T whose (m,n)-th entry is the received signal at receive antenna m and timeinstant n, Z is the additive white Gaussian noise matrix withi.i.d. symmetrical complex Gaussian elements of zero meanand variance σ2

z . It is often desirable to write the matrix input-output relationship in (3) in an equivalent vector notation. Letvec() be the operator that forms a column vector by stackingthe columns of a matrix and define y = vec(Y), z = vec(Z),and mi = vec(Mi), then (3) can be rewritten as

y = HGx + z = Hx + z =L∑

i=1

hixi + z (4)

where H = IT⊗H with⊗ as the Kronecker product operator,G = [m1,m2, . . . ,mL] will be referred to as the modulationmatrix and hi is the i-th column of H.

In our design, a full-rate full-diversity (FRFD) LDC is ap-plied. However, ST symbol rate L/T and constellation size ηwill adapt to the channel. To achieve full rate, the followingconditions shall be satisfied.

L = NtT (5)

GGH = I (6)

That is,

trace(MHi Mj) =

1 ∀i = j0 otherwise (7)

Furthermore, to achieve full diversity, we shall maximizethe outage probability Pr(‖hi‖2 ≥ ε), the following lemmais introduced [14].

Lemma 1 Consider vi, ,∀1 ≤ i ≤ n, i.i.d. Gamma randomvariables with density p(x) = e−βxxα−1βα/Γ(α). Let ξ =[ξ1, ξ2, . . . , ξn] with ξi for all i and b =

∑ni=1 ξivi. Pr(b ≥ t)

is Schur-convex in ξ for

t ≥ (nα + 1)(ξ1 + ξ2 + . . . + ξn)β

(8)

Noting that hi = Hmi, we have

‖hi‖2 =Nr∑

m=1

hHmMiMH

i hm (9)

where hm is the m-th column vector of HH . The followingdecomposition is assumed

MiMHi = UDUH (10)

where D = diag[dn],∀n = 1, 2, ..., Nt. By Lemma 5 in [1],since UHhm has the same distribution as hm, we need onlyconsider MiMH

i = D [1]. Then we have

‖hi‖2 =Nt∑

n=1

dn

Nr∑m=1

|hmn|2 (11)

For the power constraint, we haveNt∑

n=1dn = 1. Without loss

of generality, we assume d1 ≥ d2 ≥ . . . ≥ dNt ≥ 0.

By Lemma 1, since the random variableNr∑

m=1|hmn|2 in

(11) is identically chi-square distributed with 2Nr degreesof freedom, the outage probability Pr(‖hi‖2 ≥ ε) is maxi-mized if all the eigenvalues of MiMH

i are equal [11]. Withtr(MiMH

i ) = 1, it implies that di = 1/Nt, ∀1 ≤ i ≤ Nt. Asa result, for each i, ‖hi‖2 has the same value.

In our design, the dispersion matrices are given by

M(k−1)Nt+i = diag[fk]P−(i−1) (12)

for k = 1, 2, . . . , Nt and i = 1, 2, . . . , Nt, P is the permuta-tion matrix of size Nt and given by

P =(

01×(Nt−1) 1INt−1 0(Nt−1)×1

)(13)

where fk denotes the k-th column vector of F. F = [fmn] isa Fast Fourier Transform (FFT) matrix and fmn is calculatedby

fmn =1√Nt

exp(−2πj(m− 1)(n− 1)/Nt) (14)

3. THE STATISTICS OF SINR WITH THE MMSERECEIVER

Since the LDC is linear, an MMSE detector can be appliedas suboptimal receiver due to its simplicity and good perfor-mance [13]. The main goal of this section is to study the error-rate probability and the statistics of SINR for LDCs [9]- [11]using linear MMSE receiver over a Rayleigh fading channel.

We consider a general system model as shown in Section2. In our study, Nr ≥ Nt is assumed. Without loss of gen-erality, we consider the detection of one symbol xi. Equation(4) can also be written as

y = hixi + HIxI + z (15)

where xI is the rest of the symbols and HI is the matrix ob-tained by removing the i-th column from H.

A linear MMSE receiver is applied and the correspondingnormalized output is given by

xi = wHi y = xi + zi. (16)

where zi is the zero-mean noise term, which is approximatedto be Gaussian [12]. The corresponding wi can be found as

wi =

(hihH

i + RI

)−1

hi

hHi

(hihH

i + RI

)−1

hi

(17)

where RI = HIHHI + σ2

zI. Note that the scaling factor1/[hH

i (hihHi +RI)−1hi] in the coefficient vector of the MMSE

receiver wi is added to ensure an unbiased detection as indi-cated by (16). The variance of the noise term zi can be foundfrom (16) and (17) as

σ2i = wH

i RIwi (18)

Substituting the coefficient vector for the MMSE receiver in(17) into (18), the variance can be written as

σ2i =

1hH

i R−1I hi

(19)

Then, the SINR of MMSE associated with xi is 1/σ2i .

γi =1

σ2i

= hHi R−1

I hi (20)

Closed-form BER for a channel model with Gaussian noiseas (16) can be found in [15]. The average BER over MIMOfading channel for a given constellation can be found as fol-lows.

BERav = Eγi

[1L

∑

i

BER(γi)

](21)

In our LDC design, we have

hHi hj =

Nr∑m=1

hHmMiMH

j hm (22)

Since Mi is symmetric in our LDC design and ‖hi‖2 has thesame value for each i as shown in the previous section, all thesymbols has the same SINR, i.e., γ1 = γ2 = · · · = γL = γ.Equation (21) can be written as

BERav =∫

BER(γ)PΓ(γ)dγ (23)

By using singular value decomposition (SVD), we have

HIHHI =

L−1∑j=1

λjujuHj where u1, . . . ,uL−1 are orthogo-

nal vectors and λi, ∀i = 1, . . . , L − 1 are eigen-valuesof HIHH

I . We assume λ1 ≥ . . . ≥ λr > λr+1 = . . . =λL−1 = 0 and r is the rank of HIHH

I . Equation (20) can bewritten as

γ = hHi

L−1∑

j=1

λjujuHj + σ2

zI

−1

hi

=L−1∑

j=1

|hHi uj |2

λj + σ2z

=r∑

j=1

|hHi uj |2

λj + σ2z

+L−1∑

j=r+1

|hHi uj |2σ2

z

(24)

Since the elements in hi are i.i.d. Gaussian distributed, thecomplex random variables hH

i u1, . . . , hHi uL−1, conditioned

on u1, . . . ,uL−1, are also i.i.d. Gaussian distributed.The closed-form formula for the average BER in (23) de-

pends on the distribution of γ, which is difficult to determine.Here, the above average BER is calculated numerically. Forexample, the average BER for the BERav for 2η-PSK can bewritten as

BERav = Eγ

[2ηQ

(√2η γ sin(

π

2η))]

(25)

0 5 10 15 2010

−6

10−5

10−4

10−3

10−2

10−1

100

SNR

BE

R

3x3

8PSK−Simulation8PSK−Numerical

Fig. 1. Numerical and simulation results for LDC withMMSE receiver

and for rectangular 2η-QAM can be written as

BERav = Eγ

[4ηQ

(√3η γ

2η − 1

)](26)

where Q(·) denotes the Gaussian-Q function. Here, the aboveaverage BER is calculated numerically. In Fig. 1, numericaland simulation results are compared for 8PSK over 3× 3 and4PSK over 4 × 4 fading channels, respectively. As can beseen, the numerical and simulation results match very well.

4. DESIGN OF SELECTION-MODE ADAPTATION

The general idea of selection-mode adaptation is to maximizethe average transmission rate by choosing a proper transmis-sion mode from a set of available modes. Based on somecertain strategy, the transmitter is informed by the receiver toincrease or decrease the transmission rate depending on thechannel condition, i.e., CSI. For selection-mode adaptation,the signal-to-noise ratio (SNR) will be considered as a propermetric. The corresponding adaptive algorithm is proposed asfollows.

1. Find the SNR, say γo, at the receiver;

2. Find the BERs of each mode at the obtained SNR γo

from BER curves by experiment;

3. Select a proper transmission mode with the maximumrate while satisfying the target BER; and

4. Feed back the selected mode to the transmitter.

We can formulate the selection of transmission modes asfollows.

Θopt = arg maxΘn,∀n=1,2,...,N

RΘn (27)

subject toBERΘn(γo) ≤ BERtarget (28)

where Θn,∀n = 1, 2, . . . , N is the set of transmissionmodes, RΘn

is the rate of transmission mode Θn,BERΘn(γo) is the BER of transmission mode Θn at SNR γo

andBERtarget is the target BER. Without loss of generality, weassumeRΘ1 < RΘ2 < . . . < RΘN

. Θopt is the optimal transmissionmode at SNR γo.

Below, we consider the average transmission rate usingthe proposed adaptive algorithm. Let γΘn denote the mini-mum SNR satisfying the following condition.

γΘn= arg min

γ[BERΘn

(γ) ≤ BERtarget)] (29)

That is, for the SNR region γΘn≤ γ ≤ γΘn+1 , the trans-

mission rate RΘn(i.e., the transmission mode Θn) should be

selected while the target BER is satisfied.Then, the average transmission rate is

R =N∑

n=1

RΘn

∫ γΘn+1

γΘn

pΓ(γ)dγ (30)

where pΓ(γ) is the probability density function (PDF) of theSNR γ and γΘN+1 = ∞. Maximization of the average trans-mission rate R can be solved using Lagrange multipliers. How-ever, due to the structure of both the objective function andthe inequality constraint, an analytical solution is extremelydifficult to find. Therefore, we will find the SNR region cor-responding to each transmission mode by measurement.

In our simulations, we assume Nt = Nr = 4 and use thedispersion matrices defined in section 2. First, we performconstellation adaptation alone with a fixed ST symbol rate.Secondly, we perform the ST symbol rate adaptation alonewith a fixed constellation. Finally, we will consider thesetwo parameter jointly to maximize the average transmissionrate meanwhile maintaining the target BER, which is equal to10−3 in our design examples.

4.1. Adaptation Using Variable Constellations

Although the system design for continuous-rate scenario pro-vide intuitive and useful guidelines [6], the associated con-stellation mapper requires high implementation complexity.In practice, discrete constellations are preferable. That is,Q only takes integer number, such as Q = 1, 2, 3, ..... Fora given adaptive system, we can adjust the constellation tomaximize the transmission rate meanwhile keeping the targetBER satisfied. The proposed adaptive algorithm is appliedto the case. Here, we only consider BPSK (Q = 1), QPSK(Q = 2), 8PSK (Q = 3) and 16QAM (Q = 4) as examples.That is, Θn ∈ BPSK,QPSK, 8PSK, 16QAM with afixed ST symbol rate. The optimal transmission mode is se-lected by the proposed adaptive algorithm, i.e., by equation(27) and (28). Simulation results are shown in Fig. 2, whereeach subfigure has its own ST symbol rate.

(a) L/T = 1 (b) L/T = 2

(c) L/T = 3 (d) L/T = 4

Fig. 2. Adaptive Constellation.

4.2. Adaptation Using Variable ST Symbol Rate

In other existing schemes, only the orthogonal designs, suchas Alamouti scheme, are applied as the ST modulation. In thiscase, the most convenient adaptive parameter is the constel-lation size. For our adaptive scheme, the application of LDCmakes another adaptive parameter available, i.e., ST symbolrate. In this subsection, we fix the constellation size but adjustthe ST symbol rate for adaptation. Additionally, one advan-tage of using ST symbol rate is that it is easier to change STsymbol rate than constellation size for adaptation. The pro-posed adaptive algorithm described by (27) and (28) can beapplied to ST symbol rate adaptation.

Note that, this system with 4 transmit antennas can have16 choices of ST symbol rates, i.e., ( 1

4 ← · · · → 164 ). For

convenience and less complexity, we use 4 choices, i.e., LT =

1, 2, 3, 4. That is, Θn ∈ LT = 1, L

T = 2, LT = 3, L

T = 4with a fixed constellation. In the following context, the in-teger of L

T is referred as “layer”. The simulation results areshown in Fig. 3, where each subfigure has its own constella-tion.

5. JOINT ADAPTIVE TECHNIQUE

As shown in the previous two subsections, either constella-tion adaptation or ST symbol rate adaptation can increase theaverage transmission rate while the given BER is satisfied ascompared to non-adaptive schemes. However, we can fur-ther improve the average transmission rate by applying a joint

−2 0 2 4 6 8 10

10−4

10−3

10−2

10−1

100

SNR(dB)

BE

R

3 Layer 4 layer1 layer2 layer

(a) BPSK (Q = 1)

−2 0 2 4 6 8 10 12 14 16 18 2010

−6

10−5

10−4

10−3

10−2

10−1

100

SNR(dB)

BE

R

2 layer3 layer4 layer1 layer

(b) QPSK (Q = 2)

0 5 10 15 20 2510

−4

10−3

10−2

10−1

100

SNR(dB)

BE

R

1 layer2 layer3 layer4 layer

(c) 8PSK (Q = 3)

0 5 10 15 20 25 30 3510

−4

10−3

10−2

10−1

100

SNR(dB)

BE

R

3 layer1 layer2 layer4 layer

(d) 16QAM (Q = 4)

Fig. 3. Adaptive ST symbol rate.

adaptation. The joint adaptation is performed by choosing thebest pair of constellation size and ST symbol rate. The avail-able transmission modes are increased. That is,

Θn ∈ (BPSK, LT = 1), . . . , (BPSK, L

T = 4),(QPSK, L

T = 1), . . . , (QPSK, LT = 4),

(8PSK, LT = 1), . . . , (8PSK, L

T = 4),(16QAM, L

T = 1), . . . , (16QAM, LT = 4)

We can reduce the gap between the selection modes furtherby adding more choices of the transmission rates. For thetarget BER, a scheme with the joint adaptation can improvethe average transmission rate significantly as compared to thetwo techniques in the previous subsections.

We conclude the result in Table 1. In the following con-

text, γLT

Q denotes the SNR associated with the transmissionmode with 2Q constellation and L

T ST symbol rate.From the simulation results, we have the following obser-

vations:

• If the ST symbol rate is reduced, the slope of the as-sociated BER curve becomes steeper, which suggests alarger diversity;

• If the constellation size is reduced, the BER curve willshift to left with the similar slope, which suggests thediversity keeps the same but the coding gain is improved.

There exists a tradeoff between diversity gain and multiplex-ing gain [16]. However, this tradeoff can not provide insightfor the adaptive system with discrete constellations. From the

Table 1. Joint Adaptation of ST symbol rate and constellationsize

MODE γQ L/T Rm γLT

Q

0 - - - γ < −0.63091 BPSK 1 1 −0.63 ≤ γ1

1 < −0.192 QPSK 1 2 −0.19 ≤ γ2

1 < 1.43 QPSK 2 4 1.4 ≤ γ2

2 < 4.54 QPSK 3 6 4.5 ≤ γ2

3 < 9.05 8PSK 3 9 9.0 ≤ γ3

3 < 24.36 8PSK 4 12 24.3 ≤ γ3

4 < 30.87 16QAM 4 16 γ4

4 ≥ 30.8

above observations, we find that we can improve data rateby using the two adaptive parameters jointly. Specifically, insome cases, we can adjust constellation size to improve rateand performance; which in the other cases, we will adjust STsymbol rate, i.e., multiplexing gain, for adaptation.

6. CONCLUSIONS

In this paper, we proposed a new adaptive design with LDC.We studied the statistics of SINR of LDC with MMSE re-ceiver as a guideline. Since the LDC is applied, it makesspace-time symbol rate available for adaptation. Two adap-tation techniques using constellation and space-time symbolrate are studied, respectively. With joint adaptation of space-time symbol rate and constellation size, more transmissionmodes can be provided to reduce rate gap among transmis-sion modes and thus improve the average throughput. Addi-tionally, with space-time symbol rate of the linear dispersioncode, the adaptive design can be simplified and various levelsof diversity and multiplexing gain can be provided. Simula-tion results were provided to demonstrate merits of the jointadaptation of constellation and space-time symbol rate.

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