[ieee milcom 2005 - 2005 ieee military communications conference - atlantic city, nj, usa (17-20...

6
Optimal Space-Frequency Group Codes for MIMO-OFDM System Yao Chen, Emre Aktas and Ufuk Tureli Stevens Institute of Technology Electrical and Computer Engineering Department Hoboken, New Jersey, 07030 ychen5,[email protected] Abstract— Space-frequency (SF) group codes are designed for multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems. A rather general channel model is assumed, where the channel is frequency-selective Rayleigh fading with arbitrary power-delay prole. It is shown that the SF group code has a symmetric distance structure like the ST group code, if the group consists of diagonal matrices. A scenario where the multiple codewords are loaded onto the subcarriers of the OFDM system in parallel is considered. The optimality condition on the choice of subcarrier allocation is found, and an optimal subcarrier allocation scheme is proposed. A transmit scheme where rotated versions of the same signal are transmitted from different transmit antennas is proposed, and it is shown that it satises the optimality condition. Then matrix groups are designed which guarantee that the resulting SF codes are full rank. Numerical comparisons with recently published techniques in the literature verify our improved performance. I. I NTRODUCTION Space-time (ST) coding is a transmit diversity strategy that has been highly successful in combating at-fading and multiuser interference in radio channels to improve the link reliability of wireless communication systems [1], [2]. How- ever, many practical wireless channels are frequency selective in nature. So space-time codes are subject to inter-symbol interference (ISI) which degrades the performance. Orthogonal Frequency Division Multiplexing (OFDM) can transform the frequency-selective channel into a set of at fading channels, upon which ST techniques developed for at-fading channels can be applied [3] to exploit the transmit diversity. To achieve the frequency diversity due to the frequency-selective channel as well as the transmit diversity, effective space-frequency processing across the OFDM subcarriers and transmit antennas is required [4], [5]. Recently, space-frequency (SF) coding methods garnered much interest [6], [7], [8], where coding is done across transmit antennas and subcarriers of the OFDM system, based on the pairwise error probability (PEP) performance criteria. Similar to a ST code, a good SF code should possess both full diversity and high coding advantage. A unitary space- frequency code was presented in [6] which can exploit full diversity when power-delay prole of multipath channel is assumed to be uniformly distributed. Another example is SF mapping technique which was presented in [7] recently. By repeating the symbols of whatever full diversity ST code L times over adjacent subcarriers, full spatial and frequency diversity can be obtained. A variation of SF code is space- time-frequency (STF) code [8]. STF coding extends the coding to time domain by coding across frequency and transmit antennas corresponding to more than one OFDM blocks, therefore allowing to exploit any time-selectivity present in the channel, as opposed to coding across frequency and transmit antennas for one OFDM block in SF coding. In this paper, we consider the design of optimal group codes for SF coding. The developed SF group codes provide short block codes that capture full transmit and multipath diversity. Such designs can play an important role in MIMO systems when they are used in conjunction with powerful outer codes. The rest of this paper is organized as follows. In Section II, we describe the MIMO-OFDM system model and the design criteria for SF block codes. According to these criteria, optimal SF group codes are developed in Section III. Simulation results and comparisons for the proposed code are in Section IV. Section V concludes this paper. Notation: E denotes the expectation. Tr(X) denotes the trace of the matrix X. |X| stands for the determinant of the matrix X. ||X|| 2 F denotes the Frobenious norm of X, and ||X|| 2 F = Tr(XX ) = Tr(X X). I x is the identity matrix of order x, I x,y is the all one matrix of size x × y, and 0 is the all zero matrix. (·) T , (·) and (·) denote transpose, complex conjugate and complex conjugate transpose, respectively. II. MIMO-OFDM SYSTEM A. Frequency-selective Rayleigh Fading Channel Model We consider a MIMO wireless communication system with T transmit antennas and R receive antennas. When transmitter antennas are separated well enough, the channel between each transmitter-receiver pair is assumed to be quasi-static, wide- sense stationary uncorrelated scattering channel. The channel gain from transmit antenna t to receive antenna r for the nth subcarrier remains constant during the transmission of one OFDM symbol, and can be written as H t,r [n]= L l=1 h l,t,r e j2πnτ l /Tu = L l=1 h l,t,r e j 2π N l , (1)

Upload: u

Post on 13-Apr-2017

212 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: [IEEE MILCOM 2005 - 2005 IEEE Military Communications Conference - Atlantic City, NJ, USA (17-20 Oct. 2005)] MILCOM 2005 - 2005 IEEE Military Communications Conference - Optimal Space-Frequency

Optimal Space-Frequency Group Codes forMIMO-OFDM System

Yao Chen, Emre Aktas and Ufuk TureliStevens Institute of Technology

Electrical and Computer Engineering DepartmentHoboken, New Jersey, 07030ychen5,[email protected]

Abstract— Space-frequency (SF) group codes are designedfor multiple-input multiple-output orthogonal frequency divisionmultiplexing (MIMO-OFDM) systems. A rather general channelmodel is assumed, where the channel is frequency-selectiveRayleigh fading with arbitrary power-delay profile. It is shownthat the SF group code has a symmetric distance structure likethe ST group code, if the group consists of diagonal matrices.A scenario where the multiple codewords are loaded onto thesubcarriers of the OFDM system in parallel is considered. Theoptimality condition on the choice of subcarrier allocation isfound, and an optimal subcarrier allocation scheme is proposed.A transmit scheme where rotated versions of the same signal aretransmitted from different transmit antennas is proposed, and itis shown that it satisfies the optimality condition. Then matrixgroups are designed which guarantee that the resulting SF codesare full rank. Numerical comparisons with recently publishedtechniques in the literature verify our improved performance.

I. INTRODUCTION

Space-time (ST) coding is a transmit diversity strategythat has been highly successful in combating flat-fading andmultiuser interference in radio channels to improve the linkreliability of wireless communication systems [1], [2]. How-ever, many practical wireless channels are frequency selectivein nature. So space-time codes are subject to inter-symbolinterference (ISI) which degrades the performance. OrthogonalFrequency Division Multiplexing (OFDM) can transform thefrequency-selective channel into a set of flat fading channels,upon which ST techniques developed for flat-fading channelscan be applied [3] to exploit the transmit diversity. To achievethe frequency diversity due to the frequency-selective channelas well as the transmit diversity, effective space-frequencyprocessing across the OFDM subcarriers and transmit antennasis required [4], [5].

Recently, space-frequency (SF) coding methods garneredmuch interest [6], [7], [8], where coding is done acrosstransmit antennas and subcarriers of the OFDM system, basedon the pairwise error probability (PEP) performance criteria.Similar to a ST code, a good SF code should possess bothfull diversity and high coding advantage. A unitary space-frequency code was presented in [6] which can exploit fulldiversity when power-delay profile of multipath channel isassumed to be uniformly distributed. Another example is SFmapping technique which was presented in [7] recently. Byrepeating the symbols of whatever full diversity ST code L

times over adjacent subcarriers, full spatial and frequencydiversity can be obtained. A variation of SF code is space-time-frequency (STF) code [8]. STF coding extends the codingto time domain by coding across frequency and transmitantennas corresponding to more than one OFDM blocks,therefore allowing to exploit any time-selectivity present in thechannel, as opposed to coding across frequency and transmitantennas for one OFDM block in SF coding.

In this paper, we consider the design of optimal group codesfor SF coding. The developed SF group codes provide shortblock codes that capture full transmit and multipath diversity.Such designs can play an important role in MIMO systemswhen they are used in conjunction with powerful outer codes.The rest of this paper is organized as follows. In Section II,we describe the MIMO-OFDM system model and the designcriteria for SF block codes. According to these criteria, optimalSF group codes are developed in Section III. Simulation resultsand comparisons for the proposed code are in Section IV.Section V concludes this paper.

Notation: E denotes the expectation. Tr(X) denotes thetrace of the matrix X. |X| stands for the determinant of thematrix X. ||X||2F denotes the Frobenious norm of X, and||X||2F = Tr(XX′) = Tr(X′X). Ix is the identity matrix oforder x, Ix,y is the all one matrix of size x × y, and 0 is theall zero matrix. (·)T , (·)∗ and (·)† denote transpose, complexconjugate and complex conjugate transpose, respectively.

II. MIMO-OFDM SYSTEM

A. Frequency-selective Rayleigh Fading Channel Model

We consider a MIMO wireless communication system withT transmit antennas and R receive antennas. When transmitterantennas are separated well enough, the channel between eachtransmitter-receiver pair is assumed to be quasi-static, wide-sense stationary uncorrelated scattering channel. The channelgain from transmit antenna t to receive antenna r for the nthsubcarrier remains constant during the transmission of oneOFDM symbol, and can be written as

Ht,r[n] =L∑

l=1

hl,t,r e−j2πnτl/Tu =L∑

l=1

hl,t,r e−j 2πN nβl , (1)

Page 2: [IEEE MILCOM 2005 - 2005 IEEE Military Communications Conference - Atlantic City, NJ, USA (17-20 Oct. 2005)] MILCOM 2005 - 2005 IEEE Military Communications Conference - Optimal Space-Frequency

where L is the total number of paths, hl,t,r and τl denotethe lth path complex gain and time delay from transmit-ter t to receiver r, respectively. The path gains are inde-pendent, identically distributed (i.i.d.) zero mean, circularlysymmetric complex Gaussian random variables with variancesE|hl,t,r|2 = α2

l,t,r. The L multipath powers are normalized

such that∑L

l=1 α2l,t,r = 1. βl = τl/Tp, Tu is the FFT interval,

∆f = 1/Tu is the subcarrier spacing, Tp = Tu/N is thesampling period, and N denotes the number of subcarriersof the OFDM block. The model here is the standard MIMO-OFDM model [6] except that we consider an arbitrary powerdelay profile with power vector α2 = (α2

1, α22, ..., α

2L) and

delay vector β = (β1, β2, ..., βL).The discrete-time frequency response of the channel in (1)

can be put into matrix form as follows:

H(n) =L∑

l=1

Hl e−j 2π

N nβl , n = 0, · · · , N − 1, (2)

where H(n) and Hl are both R× T , and the (r, t)th elementof H(n) and Hl are Ht,r[n] and hl,t,r, respectively.

Let C be the space-frequency block code of size M : C =C0,C1, · · · ,CM−1, where each Cm is a Q × T matrix tobe transmitted from Q subcarriers and T transmit antennas,where Q ≤ N . When Q is smaller than N , other codewordsare transmitted in parallel from the remaining subcarriers. The(q, t)th element of Cm is transmitted from subcarrier sq andantenna t, where the the vector of subcarrier indexes, s =(s0, s1, · · · , sQ−1) will be chosen according to the PEP designcriteria derived in the sequel. The rate of the code is definedas

R =1Q

log2 M bits/subcarrier use (3)

For a C ∈ C let C = [c0, c1, · · · , cQ−1]T , where cq isthe length T vector to be transmitted from T antennas fromsubcarrier sq. Given the signal at N subcarriers, the OFDMmodulator applies an N -point IFFT to generate N consecutivesymbols and then inserts a cyclic prefix (CP) (which is a copyof the last L samples of the OFDM symbol) at each transmitter.After FFT demodulation at the receiver, the signal at thesqth subcarrier is subject to flat-fading and additive complexwhite Gaussian noise wq , which is statistically independentamong different receiver antennas and different subcarriers,i.e., E(wqw

†q′ ) = σ2

nIRδ[q − q′]. The R × 1 signal vector yq

received at R receiver antennas at the subcarrier sq can beexpressed as

yq =√

ρ/TH(sq)cq + wq q = 0, 1, · · · , Q − 1, (4)

where ρ is the average SNR at each receive antenna.

B. SF coding design criteria

In the following, we consider the situation when the receiverhas exact knowledge of the channel state information (CSI)while the transmitter has no knowledge of CSI.

Consider the pairwise error probability P (C → E) whenthe receiver erroneously decodes E when C is actually trans-mitted.

The Chernoff upper bound on pairwise error probability is:

P (C → E) ≤R∏

r=1

rank(V)∏i=1

11 + ρ

4λi(V), (5)

where

V = KLα,β,s(C − E) ·KL

α,β,s(C − E)†, (6)

rank(V) is the rank of V , and λi(V) is the ith eigenvalue ofthe matrix V . The Q×TL matrix KL

α,β,s(C) has the followingform:

KLα,β,s(C) =

[α1Aβ1

s C α2Aβ2s C · · · αLAβL

s C],(7)

where As is the Q × Q delay rotation matrix which is afunction of the subcarrier allocation vector s:

As =

⎡⎢⎢⎢⎣e−j 2π

N s0

e−j 2πN s1

. . .

e−j 2πN sQ−1

⎤⎥⎥⎥⎦ . (8)

We term this matrix as the Krylov codeword associated withthe codeword C. The naming stems from the fact that itsstructure is similar to the well-known Krylov matrix. Noticethat we have an equivalent ST code KL

α,β,s(C) which has thesame distance characteristics as the SF code C. The Krylovcodeword associated with the codeword C, and thus the upperbound on the pairwise error probability, is a function of thechannel power profile vector α, channel delay profile vectorβ, and subcarrier allocation vector s.

Design Criteria For Frequency-Selective Rayleigh FadingChannel

The Rank Criterion: In order to achieve the maximumtransmit diversity TL, the Krylov codeword KL

α,β,s(C − E)has to be the full-rank (rank TL) for any codeword pair C andE, since rank(V) = rank

(KL

α,β,s(C− E)). Here we use the

property of rank(XX†) = rank(X). Therefore, the number ofsubcarriers Q used for a SF code should be equal to or largerthan the desired level of transmit and multipath diversity. Ifwe desire full diversity, then Q ≥ TL is necessary. Howeverthe number of SF codewords transmitted in parallel, N/Q,decreases with Q. Thus the choice of Q allows a trade-offbetween rate and diversity. We define the full-rank associatedwith the Krylov code as Krylov-full-rank.

The Product Distance Criterion: Suppose that coding gainis our primary goal, then with respect to any Krylov codewordpair, the minimum product of the nonzero eigenvalues of V ,which is called the minimum Krylov product distance ΛK, hasto be maximized to get coding advantage as large as possibleunder the transmission energy constraint. The constraint isgiven as:

Energy Constraint Ω1:

1TQ

Tr(CC†) = 1, for every C ∈ C (9)

Page 3: [IEEE MILCOM 2005 - 2005 IEEE Military Communications Conference - Atlantic City, NJ, USA (17-20 Oct. 2005)] MILCOM 2005 - 2005 IEEE Military Communications Conference - Optimal Space-Frequency

and the minimum Krylov product distance is defined as

ΛK = minC, E∈C

ΛK(C,E), (10)

where

ΛK(C,E) =∣∣KL

α,β,s(C− E)† ·KLα,β,s(C − E)

∣∣1/TL. (11)

An SF code is said to be optimal if it is Krylov-full-rank forall pair of codes and has the largest Krylov product distanceunder transmit energy constraint Ω1.

III. SF GROUP CODES FOR MIMO-OFDM

A. Background of Group Codes

Group codes were initially introduced as Slepian-type groupcode for one-antenna use [10], then were extended to multi-antennas as ST group codes for flat-fading channels [2], [1].In this paper we extend them to be employed MIMO-OFDMsystems in frequency-selective fading channels.

The codewords C ∈ C in a multiantenna Q×T SF (or ST)group code C have the following structure:

C = G D (12)

where G ∈ G, and G is a set of Q × Q unitary generatingmatrices which satisfy GG† = G†G = IQ. The Q×T matrixD is a fixed starting matrix. In particular, if the group codeC is unitary, then D is unitary as well.

B. Conditions for Symmetric Distance Structure and Optimal-ity for SF Group Codes

In this paper, we wish to construct optimal SF group codesfor frequency-selective MIMO channel in coherent modu-lation. Our goal is to extend the favorable properties andresults for ST group codes to SF group codes. This is not astraightforward task because, as shown in the previous section,the product distance is dependent on not only the codewordpairs but also the power and delay profiles of the channel, andthe subcarrier allocation vector. We first give the constrainton the SF code to have a symmetric distance structure inTheorem 1. In Theorem 2, we show that the channel powerprofile decouples from the rest of the variables in the designcriteria, so the code and subcarrier allocation design is notaffected by the channel power profile. Then, in Theorem 3, wepresent the constraints on the starting matrix and the subcarrierallocation vector, and propose a starting matrix along with achoice of subcarrier allocation and transmission scheme whichsatisfy the constraint. In Theorem 4 we present the conditionon the matrix group G in order to achieve full rank. Finally,we perform numerical searches for codes within the codeframework developed.

Theorem 1—The code C has symmetric Krylov productdistance if and only if every G is diagonal:

The Krylov product distance of the SF code C = GD, issymmetric, i.e., ΛK(GD,G′D) = ΛK(D,G†G′D), if andonly if G ∈ G is diagonal. (See proof in [14])

Theorem 2—The channel power profile decouples from therest of the variables in the Krylov product distance:

ΛK = (α12α2

2 · · ·αL2)1/L

· minG∈G,G =I

∣∣∣KLβ,s(D)

†(I − G)†(I − G)KL

β,s(D)∣∣∣1/TL

= (α12α2

2 · · ·αL2)1/L

· minG∈G,G =I

ΛP(KL

β,s(D),GKLβ,s(D)

)︸ ︷︷ ︸

Λ∗K

(13)

(See proof in [14])It is observed that the Krylov product distance is the product

of two elements: one is geometric mean of multipath powers,the other is the effective Krylov product distance Λ∗

K, which isthe product distance of KL

β,s(D). We term the matrix KLβ,s(D)

as the effective Krylov codeword corresponding to D. Notethat the effective Krylov codeword, and therefore the effectiveKrylov product distance, is not a function of the channel powerprofile. Thus we are able to separate the effect of channelpower profile from the code construction.

We can express the transmit power constraint Ω1 in termsof the effective Krylov codeword corresponding to D:

Energy Constraint Ω2:

1TLQ

Tr(KL

β,s(D)KLβ,s(D)†

)= 1 (14)

Thus the goal is ensure full rank and maximize the effectiveKrylov distance Λ∗

K under the constraint Ω2.Theorem 3—For optimal SF group codes KL

β,s(D) is uni-tary. (See proof in [14])

Theorem 3 provides us a guideline to choose optimal D ands. The goal is to construct starting signal D and subcarrier al-location vector s such that the corresponding effective Krylovcodeword KL

β,s(D) is unitary.1) Optimal D and s for single transmit antenna: To sim-

plify the problem, we first consider single transmit antennacase, where the starting signal D becomes a vector d, andthe corresponding effective Krylov codeword KL

β,s(d) has theform:

KLβ,s(d) =

[Aβ1

s d Aβ2s d · · · AβL

s d]

(15)

To construct KLβ,s(d) such that KL

β,s(d)† ·KLβ,s(d) = Q ·I,

two requirements should be satisfied:

1) d†d = Q,2) d†A−βl

s Aβl′s d = 0 for 1 ≤ l = l′ ≤ L.

We considerd = IQ,1. (16)

That choice easily satisfies the first requirement, provides auniform distribution of energy over the Q subcarriers, andsimplifies the second requirement. For d = IQ,1, the secondrequirement becomes:

Tr(Aβl′−βl

s

)= 0 for 1 ≤ l = l′ ≤ L. (17)

Now the task is to find a subcarrier allocation vector s =(s0, s1, . . . , sQ−1) such that (17) is satisfied. Note that we

Page 4: [IEEE MILCOM 2005 - 2005 IEEE Military Communications Conference - Atlantic City, NJ, USA (17-20 Oct. 2005)] MILCOM 2005 - 2005 IEEE Military Communications Conference - Optimal Space-Frequency

transmitting N/Q SF codewords in parallel, each of which isloaded onto its own set of subcarriers. Therefore we need tofind N/Q different s vectors each satisfying (17). We propose:

sq = s0 + q · N/Q (18)

where s0 can take N/Q different values, one for each of theN/Q codewords transmitted in parallel: s0 = 0, 1, . . . , N/Q−1. It is straightforward to show that using the subcarrierallocation scheme in (18) we have Tr

(Aβl′−βl

s

)= 0 if the

following condition holds:

βl − βl′ = kQ for 1 ≤ l = l′ ≤ L, (19)

in other words, τl − τl′ = kQTu/N , where k is an integer.It is easy to choose the codeword length Q such that (19) issatisfied; for example by letting Q = max

l,l′βl − βl′ + 1 we

satisfy (19).

2) Optimal D and s for T transmit antennas: We next wishto generalize the result to T transmit antennas. We exploitthe structure of the vectors in KL

β,s(d) in (15) correspondingto each path with delay βl, and the fact that we guaranteedorthogonality among these vectors for single transmit antennaby choosing (16) and (18). We can choose the vectors inD by borrowing the structure of the vectors in KL

β,s(d)corresponding to the multiple paths. The orthogonality ofevery vector in KL

β,s(D) will thus be guaranteed. We propose

D =[d AZ1

s d AZ2s d · · · AZT−1

s d], (20)

where d is the vector in (16). Similar to the single transmit an-tenna case, we can show that Tr

(Aβl′−βl

s

)= 0, which ensures

that KLβ,s(D) is unitary. The condition in (19) becomes:

βl+Zt−βl′−Zt′ = kQ for l = l′ or t = t′ or both. (21)

The transmit delay vector Z = (Z1, . . . , ZT−1) should bechosen according to the delay profile β of the channel. Thetransmit delay vector should assure that (21) holds for k = 0. Ifthe channel delay profile is uniform (i.e. β = (0, 1, . . . , L−1)), then a straightforward choice for the transmit delay vectoris Z = (L, L+1, . . . , L+T − 2). If the channel delay profileis sparse and includes zero channel taps, those zero taps canbe utilized in the transmit delay vector. After Z is chosen, theminimum value that Q should take is found as

Q ≥ maxl,l′,t,t′

βl + Zt − βl′ − Zt′ + 1. (22)

We have shown conditions on the starting matrix D andthe subcarrier allocation vector s for optimality of the SFgroup code, and proposed a choice of D and s which satisfythe conditions. However, the question of how a construct thegroup G still remains. The first concern is to ensure Krylov-full-rank, the second is to maximize the minimum Krylovdistance. From Theorem 1, we shall restrict our attention todiagonal G in order to achieve symmetric distance. Thus weconsider diagonal cyclic codes [1]. In Theorem 4 we show that

a restricted class of diagonal cyclic codes can achieve Krylov-full-rank. Then, we conduct numerical searches to maximizethe minimum distance within that class.

Theorem 4—Structure of Generating Signal G that EnsureKrylov Full Diversity:

The cyclic code (M ; ϑq, q = 0, 1, · · · , Q − 1) defined asG = I,G0,G2

0, · · · ,GM−10 , has Krylov-full-rank, if at least

TL number of ϑq’s are integers in (1, 2, · · · , M − 1) that donot factorize M , where L is the number of channel taps, T isthe number of transmit antennas, and

G0 =

⎡⎢⎢⎢⎣

e−j 2πM ϑ0 0 · · · 00 e−j 2π

M ϑ1 · · · 0...

.... . .

...0 0 · · · e−j 2π

M ϑQ−1

⎤⎥⎥⎥⎦ (23)

(See proof in [14])

IV. SIMULATIONS

In the following, we give some simulation results. Wealso have comparisons with other SF coding techniques inthe literature for both uncoded (no FEC code) and codedMIMO-OFDM systems. Consider a MIMO-OFDM systembased on IEEE 802.11a standard with transmission bandwidthof 20MHz. The frequency bandwidth is divided into 64 sub-carriers, yielding a subchannel spacing ∆f of 312.5kHz. Twotransmit antennas and one receiver antenna are placed suchthat their channel transfer functions can be considered asuncorrelated. Two wideband channel models for 5GHz WLANare employed. The first one is a 10-tap NLOS Nokia rooftopchannel model in a suburban environment [15]. Its mean rootmean-squared (rms) delay spread is about 49ns. The secondone is a 18-tap NLOS ETSI BRAN E model in an outdoorenvironment [15]. The mean rms delay spread is about 250ns.Since in practice, worthwhile improvements diminish aboveapproximately fourth-order diversity, we consider the numberof subcarriers in each subcarrier subblock is four, implyingthat the maximum diversity which can be exploited is fouras well. Consequently, there are sixteen SF block signalstransmitted simultaneously within an OFDM symbol. A 3/4rate convolutional FEC code (octal polynomials 133 and 171)with constraint length seven is adopted in the simulations.

Four different SF codes are employed in the simulations,including 1) 2×2 SF Alamouti code which is exactly the sameas ST Alamouti code, yet the signals originally transmittedin time domain are now transmitted over adjacent OFDMsubcarriers [13]. Modulation is BPSK. Note that Alamouti

code is a full rate code and can be expressed as

[a b

−b∗ a∗

];

2) 4 × 2 repeated SF Alamouti code which is obtained byrepeating SF Alamouti code twice over adjacent subcarriers,resulting in a SF code with coding rate of 1/2. To make thecomparison fair in terms of bit rate per OFDM symbol, QPSKmodulation is used for repeated SF Alamouti code; 3) 4 × 2SF mapping code obtained by repeating a 2 × 2 unitary STgroup code twice over adjacent subcarriers; 4) 4 × 2 optimalSF group code. Note that the coding rates of both optimal SF

Page 5: [IEEE MILCOM 2005 - 2005 IEEE Military Communications Conference - Atlantic City, NJ, USA (17-20 Oct. 2005)] MILCOM 2005 - 2005 IEEE Military Communications Conference - Optimal Space-Frequency

4 6 8 10 12 14 16 18 2010

−6

10−5

10−4

10−3

10−2

10−1

100

SNR [dB]

Bit

Err

or R

ate

SF Alamouti code, BPSK

repeated SF Alamouti code, QPSK

SF Mapping code, 16PSK

SF optimal group code, 16PSK

Fig. 1. Performance comparison of space-frequency codes in uncodedMIMO-OFDM system for Nokia rooftop channel model

group code and SF mapping code are 1/4, therefore in orderto have fair comparison, 16PSK modulation is used in bothcases.

Fig. 1 depicts the BER performances of the aforementionedspace-frequency codes in Nokia rooftop channel, when FECcode is not used. It is not surprising that SF mapping codeperforms worst since it has a small coding advantage ofonly 0.09 at 16PSK [14]. SF Alamouti code and repeatedSF Alamouti code have moderate performances since Nokiarooftop channel has a very low delay spread of 49ns, hencethe channel can be treated as frequency-flat and the adjacentsubcarriers are thus correlated. Compared with SF Alamouticode, optimal SF group code exhibits 5dB performance gainat BER around 10−4.

Since SF Alamouti code is a full rate (on the subcarrierbasis) code while SF mapping code and optimal SF groupcode have smaller rates, SF Alamouti code has the potentialfor improving performance by prefixing an outer FEC codeto exploit frequency diversity. For instance, when using 3/4convolutional code (133, 171)8, a resulting performance gainof 4dB at BER of 10−4 can be achieved, as shown in Fig. 2,although there still exists a performance gap of about 3dB atBER of 10−5, compared with uncoded optimal SF group codeusing 8PSK (based on same bit rate per OFDM symbol).

Fig. 3 depicts the BER performances of space-frequencycodes in ETSI BRAN E Channel, when FEC code is notused. Compared with Nokia rooftop channel, the performanceof SF mapping code in ETSI BRAN E channel improvessignificantly since its coding advantage is increased from0.09 to 2.37 at 16PSK [14]. On the contrary, the perfor-mances of SF Alamouti code and repeated SF Alamouti codedegrade considerably because ETSI BRAN E channel is afrequency-selective fading channel with long delay spread of250ns, resulting in nearly uncorrelated adjacent subcarriers.This degradation can only be limitedly alleviated by using a3/4 convolutional FEC code (133, 171)8 to exploit frequencydiversity, as shown in Fig. 4. The performance gaps betweenuncoded SF mapping code and uncoded optimal SF groupcode are 4dB at BER of 10−4 and 3.5dB at BER of 10−5, for16PSK and 8PSK respectively.

V. CONCLUSION

In this paper we have developed optimal group codes forMIMO-OFDM systems. We have considered a scenario where

4 5 6 7 8 9 10 11 12 13 1410

−7

10−6

10−5

10−4

10−3

10−2

10−1

SNR [dB]

Bit

Err

or R

ate

SF Alamouti code, uncoded, BPSK

SF Alamouti code, 3/4 coded, BPSK

SF Mapping code, uncoded, 8PSK

SF optimal group code, uncoded, 8PSK

Fig. 2. Performance comparison of space-frequency codes in coded MIMO-OFDM system for Nokia rooftop channel model

4 5 6 7 8 9 10 11 1210

−6

10−5

10−4

10−3

10−2

10−1

SNR [dB]

Bit

Err

or R

ate

SF Alamouti code, BPSK

repeated SF Alamouti code, QPSK

SF Mapping code, 16PSK

SF optimal group code, 16PSK

Fig. 3. Performance comparison of space-frequency codes in uncodedMIMO-OFDM system for ETSI BRAN E channel model

multiple block codes are transmitted in parallel from thesubcarriers of the OFDM system. We presented an optimalallocation of subcarriers for the multiple block codes. Wehave derived optimality conditions on the group codes anddesigned codes accordingly. The optimal signaling structureof the code is build up based on optimal interleaving ofsubcarriers and an exploitation of the transmitters and themultipaths, where multiple transmitters in effect uncorrelatedadditional multipaths. A condition for the generator matrices inthe group to result in full rank is given, and optimal generatormatrices in the group have been found through exhaustivecomputer search. Although we have focused exclusively on thecoherent space-frequency coding and modulation, our codeswith group structure clearly could extend to noncoherent ordifferential cases.

REFERENCES

[1] B. L. Hughes, “Optimal Space-Time Constellations From Groups,” IEEETrans. Inform. Theory, vol. 49, pp. 401–410, Feb. 2003.

[2] B. L. Hughes, “Differential Space-Time Modulation,” IEEE Trans.Inform. Theory, vol. 46, pp. 2567–2578, Nov. 2000.

4 5 6 7 8 9 10 11 1210

−7

10−6

10−5

10−4

10−3

10−2

10−1

SNR [dB]

Bit

Err

or R

ate

SF Alamouti code, uncoded, BPSK

SF Alamouti code, 3/4 coded, BPSK

SF Mapping code, uncoded, 8PSK

SF optimal group code, uncoded, 8PSK

Fig. 4. Performance comparison of space-frequency codes in coded MIMO-OFDM system for ETSI BRAN E channel model

Page 6: [IEEE MILCOM 2005 - 2005 IEEE Military Communications Conference - Atlantic City, NJ, USA (17-20 Oct. 2005)] MILCOM 2005 - 2005 IEEE Military Communications Conference - Optimal Space-Frequency

[3] D. Agrawal, V. Tarokh, A. Naguib, and N. Seshadri, “Space-timecoded OFDM for high data-rate wireless communication over widebandchannels,” in Proceedings IEEE Vehicular Technology Conference, May1998, pp. 2232–2236.

[4] S. Kaiser, “Spatial transmit diversity techniques for broadband OFDMsystems,” in IEEE GLOBECOM 2000, San Francisco, CA, pp. 1824–1828, Nov. 2000.

[5] Y. Li, J. C. Chuang and N. R. Sollenberger, “Transmitter Diversity forOFDM Systems and Its Impact on High-Rate Data Wireless Networks,”IEEE J. Select. Areas Comm., vol. 17, pp. 1233–1243, Jul. 1999.

[6] H. Bolcskei, A. J. Paulraj, “Space-Frequency Coded MIMO-OFDMwith Variable Multiplexing-Diversity Tradeoff,” in IEEE InternationalConference on Communications, ICC ’03, 2003, pp. 2837–2841.

[7] W. F. Su, Z. Safar, M. Olfat and K. J. R. Liu, “Obtaining Full-DiversitySpace-Frequency Codes from Space-Time Codes via Mapping,” IEEETrans. Signal Processing, vol. 11, pp. 2905–2916, Nov. 2003.

[8] Z. Q. Liu, Y. Xin and G. B. Giannakis, “Space-Time-Frequency CodedOFDM over Frequency-Selective Fading Channels,” IEEE Trans. SignalProcessing, vol. 50, pp. 2465–2476, Oct. 2002.

[9] H. E. Gamal and A. R. Hammons, “On the design of algebraic space-time codes for MIMO block-fading channels,” IEEE Trans. Inform.Theory, vol. 49, pp. 151–163, Jan. 2003.

[10] D. Slepian, “Group codes for the Gaussian channel,” Bell Syst. Tech. J.,vol. 47, pp. 575–602, Apr. 1968.

[11] H. Bolcskei, A. J. Paulraj, “Space-frequency coded broadband OFDMsystems,” in IEEE Wireless Communications and Networking Confer-ence, WCNC’00, 2000, pp. 1-6.

[12] H. Bolcskei, A. J. Paulraj, “Space-Frequency Codes for BroadbandFading Channels,” in ISIT 2001, Washington, DC, pp. 219, Jun. 2001.

[13] H. Bolcskei, M. Borgmann and A. J. Paulraj, “Impact of the PropagationEnvironment on the Performance of Space-Frequency Coded MIMO-OFDM,” IEEE J. Select. Areas Comm., vol. 21, pp. 427–439, Apr. 2003.

[14] Y. Chen, E. Aktas and U. Tureli, “Optimal Space-Frequency GroupCodes for MIMO-OFDM System,” IEEE Trans. Communications, inrevision.

[15] J. Unkeri, “5GHz Radio Channel Modeling for WLANs,” S-72.333Postgraduate Course in Radio Communications, 2004.