mabruk conf adaptive gc

7/30/2019 Mabruk Conf Adaptive GC

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Design of adaptive MIMO system using linear

dispersion code

Mabruk Gheryani, Zhiyuan Wu, and Yousef R. Shayan

Concordia University, Department of Electrical EngineeringMontreal, Quebec, Canada

email: (m gherya, zy wu, yshayan)@ece.concordia.ca

Abstract In this paper, we develop a new design for adap-tation of linear dispersion code. A new adaptive parametercalled space-time symbol rate is applied in our design. We havestudied the statistics of signal-to-interference-noise of a linearMMSE receiver over a Rayleigh fading channel. The averageBER for a given constellation using the MMSE receiver iscalculated numerically. With the statistics as a guideline, twoadaptive techniques using constellation and space-time symbolrate are studied, respectively. If constellation and space-timesymbol rate are considered jointly, more selection modes can

be available. Theoretical analysis demonstrates that the averagetransmission rate of the joint adaptation can be improved in thiscase. Simulation results are provided to show the benefits of ournew design.

I. INTRODUCTION

The demand for bandwidth efficiency in wireless communi-

cations has experienced an unprecedented growth. One signif-

icant advancement to improve radio spectrum efficiency is the

use of multiple-input-multiple-output (MIMO) technology [1]

[2]. Space-time (ST) codes are the most promising technique

for MIMO systems [3] [4]. Due to battery life and device

size, the power available for radio communications is limited.

Under this power constraint, adaptive technique can cooperatewith MIMO technology to further exploit radio spectrum [5]

[6].

In an adaptive system, a feedback channel is utilized to

provide channel state information (CSI) from the receiver

to the transmitter. According to the feedback of CSI, the

transmitter will adjust transmission parameters, such as power

allocation, modulation, coding rate, etc. This is conditioned

by the fact that the channel stays relatively constant before

the transmitter receives the CSI and then transmits next data

block accordingly. That is, the channel is slow. Many

of adaptive MIMO schemes have been proposed, such as

water-filling-based schemes [1] [7] and various beamforming

schemes [6] [8]. The above schemes often need near-perfectCSI feedback for adaptation calculation and consume large

feedback bandwidth. In practice, the channel estimation will

exhibit some inaccuracy depending on the estimation method.

The receiver will need time to process the channel estimate

and the feedback is subject to some transmission delay. The

transmitter needs some time to choose a proper code, and

there are possible errors in the feedback channel. All these

factors make the CSI at the transmitter inaccurate. Addition-

ally, the feedback bandwidth is often limited. In these cases,

adaptive schemes with a set of discrete transmission modes

are often more preferable. We can call them selection-mode

adaptation. At the receiver, the channel is measured and then

one transmission mode with the highest transmission rate is

chosen, which meanwhile meets the BER requirement. The

optimal mode is fed back to the transmitter.

For selection-mode MIMO adaptation, the most convenient

adaptive parameter is constellation size for uncoded systems.

For example, constellation adaptation, such as M-QAM, isapplied to space-time block code (STBC) [9] and to space-

time trellis code (STTC) [10]. The disadvantage of these

schemes, is that they are not flexible for different rates, which

is the key requirement in the future wireless communications.

Additionally, the gap between the available transmission rates

are often very large due to the use of discrete constellations

[8].

In this paper, we propose to apply linear dispersion code

(LDC) [11] [12] for adaptation. This is because it subsumes

many existing block codes as its special cases which allows

suboptimal linear receivers with greatly reduced complexity,

and provides flexible rate-versus-performance tradeoff [11]

[13]. With the application of LDC, a linear MMSE detectoris more attractive due to its simplicity and good performance

[14]. As a guideline, the statistics of SINR for LDCs with a

MMSE receiver is studied and the associated average BER is

calculated numerically. Since the LDC is applied, it makes

ST symbol rate available for adaptation. By adjusting this

new parameter together with constellation size, more available

transmission modes can be provided. Hence, the throughput

under a power constraint can be further improved while the

target bit error rate (BER) is satisfied.

I I . SYSTEM MODEL

In this study, during one ST modulation block, the channelis assumed to be the same as estimated at the receiver.

Furthermore, the channel is assumed to be a Rayleigh flat

fading channel with Nt transmit and Nr receive antennas.Lets denote the complex gain from transmit antenna n toreceiver antenna m by hmn and collect them to form anNr Nt channel matrix H = [hmn], known perfectly to thereceiver. The entries in H are assumed to be independently

identically distributed (i.i.d.) symmetrical complex Gaussian

random variables with zero mean and unit variance.


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In this system, the information bits are first mapped into

symbols. After that, the symbol stream is parsed into blocks of

length L. The symbol vector associated with one modulationblock is denoted by x = [x1, x2, . . . , xL]

T with xi {m|m = 0, 1, . . . , 2Q 1, Q 1}, i.e., a complex constella-tion of size 2Q, such as 2Q-QAM). The average symbol energy

is assumed to be 1, i.e., 12Q2Q1

m=0 |m|2 = 1. Each block of

symbols will be mapped by the ST modulator to a dispersion

matrix of size NtT and then transmitted over the Nt transmitantennas over T channel uses. The following model will beconsidered in this study, i.e.,

X =Li=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 [11]. With a constellation of size 2Q,the data rate of the space-time modulator in bits per channel

use is

Rm = Q L/T (2)Hence, one can adjust ST symbol rate L/T and constellationsize Q according to the feedback from the receiver.

At the receiver, the received signals associated with one

modulation block can be written as

Y =

P

NtHX+ Z =

P

NtH

Li=1

Mixi + Z (3)

where Y is a complex matrix of size N r 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 with

i.i.d. symmetrical complex Gaussian elements of zero meanand variance 2z , and P is the average energy per channeluse at each receive antenna. It is often desirable to write the

matrix input-output relationship in (3) in an equivalent vector

notation. Let vec() be the operator that forms a column vectorby stacking the columns of a matrix and define y = vec(Y),z = vec(Z), and mi = vec(Mi), then (3) can be rewritten as

y =

P

NtHGx+ z =

P

NtHx+ z (4)

where H = IT H with as the Kronecker productoperator and G = [m1,m2, . . . ,mL] will be referred to as

the modulation matrix.

III. THE STATISTICS OF SINR WITH THE MMSE

RECEIVER

Since the LDC is linear, an MMSE detector can be ap-

plied as suboptimal receiver due to its simplicity and good

performance [14]. The main goal of this section is to study

the error-rate probability and the statistics of SINR for LDCs

[11]- [13] using linear MMSE receiver over a Rayleigh fading

channel.

We consider a general system model as shown in Section II.

In our study, Nr Nt is assumed. For simplicity, we chooseT equal to Nt and L equal to NtT.

Equation (4) can also be written as

y =

P

Nthixi +

P

Nt

j=i

hjxj + z (5)

In the sequel, the i-th column of H, denoted as hi, will bereferred to as the signature signal of symbol xi.

Without loss of generality, we consider the estimation of one

symbol, say xi. Collect the rest of the symbols into a columnvector xI and denote HI = [h1,.., hi1, hi+1, ..., hL] as thematrix obtained by removing the i-th column from H.

A linear MMSE receiver is applied and the corresponding

output is given by

xi = wHi y = xi + zi. (6)

where zi is the noise term of zero mean. The correspondingwi can be found as

wi =hihHi +RI1 hi

PNthHi

hih

Hi +RI

1hi

(7)

where RI = HIHHI + Nt2zP I. Note that the scaling factor1

PNthHi (hihHi +RI)

1hi

in the coefficient vector of the MMSE

receiver wi is added to ensure an unbiased detection as

indicated by (6). The variance of the noise term zi can befound from (6) and (7) as

2i = wHi RIwi (8)

With (7) and (8), the SINR of MMSE associated with xi is

i =1

2i= P

Nt

hHi R

1I hi (9)

The average BER over MIMO fading channel for a given

constellation can be found as follows.

BERav = Ei

1

L

i

BE R(i( H, ))

(10)

where = PNt2z

.

In our LDC design, all the symbols has the same SINR, i.e.,

1 = 2 = = L = . Equation (10) can be written as

BE Rav = BE R(, )P()d (11)By using singular value decomposition (SVD), (9) can be

written as

=

P

Nt

hHi U

1UHhi (12)

where UH is an N2t 1 N2t 1 unitary matrix and thematrix is (N2t 1) (N2t 1) with nonnegative numberson the diagonal and zeros off the diagonal. Lets define

h = UHhi


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0 5 10 15 2010

6

105

104

103

102

101

100

SNR

BER

3x3

8PSKSimulation

8PSKNumerical

Fig. 1. Numerical and simulation results for LDC with MMSE receiver

which is the transformed propagation vector with components

hl, l = 1, ...........NrNt.The vector h has the same statistics as the original vector

hi [1]. We can replace |hl|2 by |hil|2. Now, we can write (12)as

=

ri=1

|hil|

2

(l + 1)+

NtNrr+1

|hil|2 (13)

where r is the rank of HIHHI . The closed-form formula forthe average BER in (11) is difficult to find. For example, the

BE Rav for 2-PSK can be written as

BE Rav =2

Q

2 sin(

2)

P() d (14)

and for rectangular 2-QAM can be written as

BE Rav =4

Q

3

2 1

P() d (15)

where Q() denotes the Gaussian-Q function. Here, the aboveaverage BER is calculated numerically. In Fig. 1, numerical

and simulation results are compared for 8PSK over 3 3 and4PSK over 44 fading channels, respectively. As can be seen,the numerical and simulation results match very well.

IV. DESIGN OF SELECTION-M OD E ADAPTATION

The general idea of selection-mode adaptation is to maxi-

mize the average transmission rate by choosing a proper trans-

mission mode from a set of available modes. Based on some

certain strategy, the transmitter is informed by the receiver to

increase or decrease the transmission rate depending on the

channel condition, i.e., CSI. For selection-mode adaptation,the signal-to-noise ratio (SNR) will be considered as a proper

metric. The corresponding adaptive algorithm is proposed as

follows.

1) Find the SNR, saying o, at the receiver;2) Find the BERs of each mode at the obtained SNR o

from BER curves by experiment;

3) Select a proper transmission mode with the maximum

rate while satisfying the target BER;

4) Feed back the selected mode to the transmitter.

We can formulate the selection of transmission modes as

follows.

opt = arg max{n,n=1,2,...,N}

Rn (16)

subject to

BE Rn(o) BE Rtarget (17)

where {n, n = 1, 2, . . . , N } is the set of transmissionmodes, Rn is the rate of transmission mode n, BERn(o)is the BER of transmission mode n at SNR o andBERtarget is the target BER. Without loss of generality, weassume R1 < R2 < . . . < RN. opt is the optimaltransmission mode at SNR o.

Below, we consider the average transmission rate using the

proposed adaptive algorithm. Let n denote the minimumSNR satisfying the following condition.

n = arg min

[BERn() BE Rtarget)] (18)

That is, for the SNR region n n+1, the trans-mission rate Rn (i.e., the transmission mode n) should beselected while the target BER is satisfied.

Then, the average transmission rate is

R =Nn=1

Rn

n+1n

p()d (19)

where p() is the probability density function (PDF) ofthe SNR and N+1 = . Maximization of the averagetransmission rate R can be solved using Lagrange multi-pliers. However, due to the structure of both the objective

function and the inequality constraint, an analytical solution

is extremely difficult to find. Therefore, we will find the

SNR region corresponding to each transmission mode bymeasurement.

In our simulations, we assume Nt = Nr = 4 and use thefollowing dispersion matrices for our design

M(k1)Nt+i = diag[fk]P(i1) (20)

for k = 1, 2, . . . , N t and i = 1, 2, . . . , N t,P is the permutationmatrix of size Nt and given by

P =

01(Nt1) 1INt1 0(Nt1)1

(21)

where fk denotes the k-th column vector ofF. F = [fmn] is

a Fast Fourier Transform (FFT) matrix and fmn is calculatedby

fmn =1Nt

exp(2j(m 1)(n 1)/Nt) (22)

First, we perform constellation adaptation alone with a fixed

ST symbol rate. Secondly, we perform the ST symbol rate

adaptation alone with a fixed constellation. Finally, we will

consider these two parameter jointly to maximize the average

transmission rate meanwhile maintaining the target BER,

which is equal to 103 in our design examples.


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4 2 0 2 4 6 8 10 1 2 1 4 1 6

104

103

102

101

100

SNR(dB)

BER

8PSK

QPSK

BPSK

16QAM

(a) L/T = 1

2 0 2 4 6 8 10 1 2 1 4 16 1 8 2 010

4

103

102

101

100

SNR(dB)

BER

BPSK

QPSK

8PSK

16QAM

(b) L/T = 2

5 0 5 10 15 20 25 3010

4

103

102

101

100

SNR(dB)

BER

BPSK

QPSK

8PSK

16QAM

(c) L/T = 3

0 5 10 15 20 25 30 35

103

102

101

100

SNR(dB)

BER

BPSK

QPSK

8PSK

16QAM

(d) L/T = 4

Fig. 2. Adaptive Constellation.

A. Adaptation Using Variable Constellations

Although the system design for continuous-rate scenario

provide intuitive and useful guidelines [8], the associated

constellation mapper requires high implementation complex-

ity. 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 to

maximize the transmission rate meanwhile keeping the target

BER satisfied. The proposed adaptive algorithm is applied to

the case. Here, we only consider BPSK (Q = 1), QPSK (Q =2), 8PSK (Q = 3) and 16QAM (Q = 4) as examples. Thatis, n {BPSK,QPSK, 8PSK, 16QAM} with a fixed STsymbol rate. The optimal transmission mode is selected by the

proposed adaptive algorithm, i.e., by equation (16) and (17).

Simulation results are shown in Fig. 2, where each subfigure

has its own ST symbol rate.

B. Adaptation Using Variable ST Symbol RateIn other existing schemes, only the orthogonal designs,

such as Alamouti scheme, are applied as the ST modulation.

In this case, the most convenient adaptive parameter is the

constellation size. For our adaptive scheme, the application

of LDC makes another adaptive parameter available, i.e., ST

symbol rate. In this subsection, we fix the constellation size

but adjust the ST symbol rate for adaptation. Additionally,

one advantage of using ST symbol rate is that it is easier to

change ST symbol rate than constellation size for adaptation.

2 0 2 4 6 8 10

104

103

102

101

100

SNR(dB)

BER

3 Layer4 layer1 layer2 layer

(a) BPSK (Q = 1)

2 0 2 4 6 8 10 1 2 1 4 1 6 18 2 010

6

105

104

103

102

101

100

SNR(dB)

BER

2 layer3 layer4 layer1 layer

(b) QPSK (Q = 2)

0 5 10 15 20 2510

4

103

102

101

100

SNR(dB)

BER

1 layer2 layer3 layer4 layer

(c) 8PSK (Q = 3)

0 5 10 15 20 25 30 3510

4

103

102

101

100

SNR(dB)

BER

3 layer

1 layer

2 layer

4 layer

(d) 16QAM (Q = 4)

Fig. 3. Adaptive ST symbol rate.

The proposed adaptive algorithm described by (16) and (17)

can be applied to ST symbol rate adaptation.

Note that, this system with 4 transmit antennas can have16 choices of ST symbol rates, i.e., (14 164 ). Forconvenience and less complexity, we use 4 choices, i.e., L

T=

1, 2, 3, 4. That is, n {

L

T

= 1, LT

= 2, LT

= 3, LT

= 4}

with

a fixed constellation. In the following context, the integer ofLT

is referred as layer. The simulation results are shown in

Fig. 3, where each subfigure has its own constellation.

V. JOINT ADAPTIVE TECHNIQUE

As shown in the previous two subsections, either constella-

tion adaptation or ST symbol rate adaptation can increase the

average transmission rate while the given BER is satisfied as

compared to non-adaptive schemes. However, we can further

improve the average transmission rate by applying a joint

adaptation. The joint adaptation is performed by choosing

the best pair of constellation size and ST symbol rate. The

available transmission modes are increased. That is,

n {(BPSK, LT = 1), . . . , (BPSK, LT = 4),(QPSK, L

T= 1), . . . , (QPSK, L

T= 4),

(8PSK, LT

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

= 4),(16QAM, L

T= 1), . . . , (16QAM, L

T= 4)}

We can reduce the gap between the selection modes further

by adding more choices of the transmission rates. For the

target BER, a scheme with the joint adaptation can improve

the average transmission rate significantly as compared to the

two techniques in the previous subsections.


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TABLE I

JOINT ADAPTATION OF ST SYMBOL RATE AND CONSTELLATION SIZE

MODE Constellation L/T Rm LTQ

0 - - - < 0.63091 BPSK 1 1 0.6309 1

1< 0.1893

2 QPSK 1 2 0.1893 21

< 1.40583 QPSK 2 4 1.4058 2

2< 4.4833

4 QPSK 3 6 4.4833 23

< 8.9696

5 8PSK 3 9 8.9696 3

3 < 24.25336 8PSK 4 12 24.2533 3

4< 30.8208

7 16QAM 4 16 44 30.8208

We conclude the result in Table I. In the following context,

LT

Q denotes the SNR associated with the transmission mode

with 2Q constellation and LT

ST symbol rate.

From the simulation results, we have the following obser-

vations:

If the ST symbol rate is reduced, the slope of the

associated BER curve becomes steeper, which suggests

a larger diversity;

If the constellation size is reduced, the BER curve willshift to left with the similar slope, which suggests the

diversity keeps the same but the coding gain is improved.

There exists a tradeoff between diversity gain and multiplexing

gain [15]. However, this tradeoff can not provide insight for

the adaptive system with discrete constellations. From the

above observations, we find that we can improve data rate

by using the two adaptive parameters jointly. Specifically, in

some cases, we can adjust constellation size to improve rate

and performance; which in the other cases, we will adjust ST

symbol rate, i.e., multiplexing gain, for adaptation. To proceed,

we have the following proposition.

Proposition 1: The average transmission rate in the adap-

tive selection-mode system can be improved by adding more

possible transmission modes providing higher data rate than

the corresponding original mode at the same SNR region.

Proof: Let us define the SNR regions of our adaptive system

using one set of selection modes as follows.

i i < < i+1 associated with RiIf we add more possible selection modes, the SNR regions

will be changed as follows.

i i < <

i associated with Rii

i < < i+1 associated with R

i

We assume R

i > Ri for any i. The total average rate fororiginal scheme can be written as

R =i

Ri

i+1i

p()d

The total average rate when for the scheme with more

transmission modes can be written as

A =i

(Ri

ii

p()d+ R

i

i+1

i

p()d)

It is obvious that

A > R

VI . CONCLUSIONS

In this paper, we proposed a new adaptive design with LDC.

We studied the statistics of SINR of LDC with MMSE receiver

as a guideline. Since the LDC is applied, it makes space-

time symbol rate available for adaptation. Two adaptation

techniques using constellation and space-time symbol rate are

studied, respectively. With joint adaptation of space-time sym-

bol rate and constellation size, more transmission modes can

be provided to reduce rate gap among transmission modes and

thus improve the average throughput. Additionally, with space-

time symbol rate of the linear dispersion code, the adaptive

design can be simplified and various levels of diversity and

multiplexing gain can be provided. Simulation results were

provided to demonstrate merits of the joint adaptation of

constellation and space-time symbol rate.

REFERENCES

[1] I. E. Telatar, Capacity of multi-antenna Gaussian channels, Eur. Trans.

Telecom., vol 10, pp. 585-595, Nov. 1999.[2] G. J. Foschini, M. J. Gans, On limits of wireless communications in

a fading environment when using multiple antennas, Wireless PersonalCommunications , vol. 6, no. 3, pp. 311-335, 1998.

[3] S. Alamouti, A simple transmitter diversity scheme for wireless commu-nications, IEEE J. Select. Areas Commun., vol. 16, pp. 1451-1458, Oct.1998.

[4] V. Tarokh, N. Seshadri, and A. Calderbank, Space-time codes forhigh data rate wireless communications: Performance criterion and codeconstruction, IEEE Trans. Inform. Theory, vol. 44, pp. 744-765, Mar.1998.

[5] D. Gesbert, R. W. Heath,and S. Catreux,V. Erceg Adaptive modulationand MIMO coding for broadband wireless data networks, in 2002 IEEECommunications Magazine vol. 40, pp. 108-115 , June 2002.

[6] S. Zhou and G. B. Giannakis, Optimal transmitter eigen-beamformingand space-time block coding based on channel mean feedback IEEETransactions on Signal Processing, vol. 50, no. 10, October 2002.

[7] X. Zhang and B. Ottersten, Power allocation and bit loading for spatialmultiplexing in MIMO systems, IEEE Int. Conf.on Acoustics, Speech,and Signal Processing, 2003. Proceedings (ICASSP 03) vol.5 pp. 54-56,Apr. 2003.

[8] P. Xia and G. B. Giannakis, Multiantenna adaptive modulation withbeamforming based on bandwidth-constrained feedback IEEE Transac-tions on Communications, vol. 53, no.3, March 2005.

[9] Youngwook KO and Cihan Tepedelenlioglu, Space-time block codedrate-adaptive modulation with uncertain SNR feedback IEEE Signals,Systems and Computers, vol.1, pp 1032- 1036, Nov. 2003.

[10] T. H. Liew, B. L. Yeap, C. H. Wong, and L. Hanzo, Turbo-codedadaptive modulation versus spacetime trellis codes for transmission overdispersive channels, IEEE Trans. om Wireless Comm., vol.3, no. 6, pp2019-2029, Nov. 2004.

[11] B. Hassibi and B. Hochwald, High-rate codes that are linear in spaceand time, IEEE Trans. Inform. Theory, vol. 48, pp. 1804-1824, July2002.

[12] R. W. Heath and A. Paulraj, Linear dispersion codes for MIMO systemsbased on frame theory, IEEE Trans. on Signal Processing, vol. 50, No.10, pp. 2429-2441, October 2002.

[13] Z. Wu and X. F. Wang, Design of coded space-time modulation,IEEE International Conference on Wireless Networks, Communicationsand Mobile Computing, vol. 2, pp. 1059-1064, Jun. 13-16, 2005.

[14] R. Lupas and S. Verdu, Linear multiuser detectors for synchronouscode-division multiple-access channels, IEEE Trans. inform. Theory, vol.35, pp. 123-136, Jan. 1989.

[15] L. Zheng and D. Tse, Diversity and multiplexing: A fundamentaltradeoff in multiple antenna channels IEEE Trans. Inform. Theory, vol.49, pp. 1073-96, May 2003.

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