Space-Time Equalization for AsynchronousMultiuser Bit-Interleaved Coded OFDM

Taiwen Tang, Member, IEEE, and Teng Joon Lim, Senior Member, IEEE

Abstract—For a multiuser bit interleaved coded OFDM (BIC-OFDM) system, we study the methods of time domain equal-ization and per-subcarrier maximum likelihood (ML) frequencydomain equalization when there is timing asynchronism amongthe users. For the time domain equalizer structure, a coded pair-wise error probability (PEP) approximation is derived, whichleads to a training-based space-time design method that takes theerror rate performance into consideration. We compare the timedomain equalization and per-subcarrier ML frequency domainequalization methods, and conclude that at high SNR’s, the timedomain equalization approach is preferable.

I. INTRODUCTION

Multiple antenna technology is being combined with or-thogonal frequency division multiplexing (OFDM) in nextgeneration wireless systems, because of the flexibility sucha combination offers in creating various modes of operationoffering a range of spectral efficiencies and reliabilities. Westudy a MIMO-OFDM uplink, in which multiple single-antenna transmitters send OFDM signals to a receiver withmultiple receive antennas. The system is asynchronous, mean-ing that the received signals from the transmitters are notaligned in time at the receiver due to different propagationdelays.

In conventional cellular systems, such asynchronism may beovercome by adjusting the transmission times at the transmit-ters. This requires a feedback timing adjustment mechanismfrom the receiver to the transmitters. For communicationswithout such a feedback mechanism, we need to study suitabletransceiver architectures. Also for applications where there areasynchronous interferers, we have to consider such asynchro-nism between the transmitters (including interferers) and thereceivers.

Due to the asynchronism among the transmitters, it maynot be possible to find an FFT time window that is freeof inter-OFDM symbol interference (IOSI) and inter-carrierinterference (ICI) for the transmitted signals. There are mainlytwo equalization methods to deal with such ISOI and ICI.The first method is frequency domain equalization. It hasbeen shown in [1] that the per-subcarrier receive beamformingthat only utilizes the signal and channel response on eachsubcarrier cannot completely cancel the ISOI and ICI causedby the asynchronism unless the number of antennas is greater

This material is based upon work supported by Redline CommunicationsInc., the Ontario Centers of Excellence (OCE) and the National Science andEngineering Research Council (NSERC).

The authors are with the Department of Electrical & Computer Engineering,University of Toronto, Toronto, Canada (e-mail: taiwen.tang@utoronto.ca,tj.lim@utoronto.ca).

than the number of time taps in the channel, which is notfeasible in real systems. Another alternative is to place amultiple-antenna finite impulse response (FIR) equalizer ateach subcarrier to deal with the ISOI and ICI [2] [3]. Thedrawbacks of this approach are that the design complexity ishigh since we need to design each multi-antenna FIR equalizerindividually, and it requires a large memory to store theequalizer coefficients [3].

The method of time domain equalization (TDE) was in-troduced in [4]–[9]. The TDE approach uses one space-timeequalizer per user before the FFT to (i) minimize multi-userinterference, in case each subcarrier is occupied by more thanone user in a spatial division multi-access (SDMA) manner,and (ii) shorten the channel response from the desired userto within the cyclic prefix. The equalization delay is adjustedoptimally in the mean square error sense for each user. Thenafter the FFT, in the frequency domain, a per-subcarrier scalarequalization is applied. The design metrics for the space-time equalizer are mainly the signal-to-interference and noisepower ratio or the mean square error between the equalizedsignal and the transmitted signal convolved with the desiredchannel response in these papers. The coded PEP has not beenconsidered as the equalizer design metric.

Our main contribution in this paper is to derive a codedPEP approximation for the multiuser asynchronous OFDMuplink with bit interleaved coded modulation (BICM). Wethen minimize this approximate PEP expression to designthe space-time equalizers. Our approach is different from theminimum bit-error rate design for space time equalization-based multiuser detection in [10]. The latter assumes a Gaus-sian mixture model after space time equalization to derivethe uncoded error probability expression, which does not takeerror control coding into consideration. Also it may not bepractical to OFDM systems due to the optimization complexityscales exponentially with the number of OFDM subcarriers.

We also derive a frequency-domain joint ML based equal-ization method that uses only the signal and the channelresponse on each subcarrier, and compare it with the TDEdesigns based on MMSE and PEP minimization. We find thatan error floor caused by the IOSI and ICI due to asynchronismexists for the per-subcarrier ML receiver, and thus the space-time equalization method should be favored in the high SNRregion.

The standard matrix notations are used in this paper, where(·)T denotes transpose and (·)H denotes conjugate transpose.

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TABLE ISUMMARY OF NOTATIONS AND THEIR ATTRIBUTES

Symbol Namewi,j Equalizer Vector on the jth Antenna for the ith UserHj,m Channel Matrixyi(b) Signal Vector After FFT

Rj(b, Δi,j) Time Domain Signal MatrixVm(b) Frequency Domain Diagonal Signal Matrixbi,i Time Domain Shortened Channel for the ith User

Q Partial DFT Basis MatrixTi Subcarrier Selection Matrix for User ifi,i Frequency Domain Channel Response Vector for User iχ QAM Constellation SetΠ Bit Interleaver, Mapping t → (u, p, b)

Λi(u, p, b) Log Likelihood Ratio

II. SYSTEM MODEL

We consider an OFDM system where the receiver has Mr

receive antennas, and receives signals from Mt users each withone transmit antenna. The channel is assumed to be frequencyselective and constant over a block duration of multiple OFDMsymbols and varies independently from block to block. We willdefine the data and training structure within each block later.The channel between the mth user and the jth receive antennaof the base station is denoted by hj,m(l) where l = 0, ..., ν −1 and ν denotes the number of channel taps. The absolutepropagation delay of the link between the mth user and thejth receive antenna is denoted by dj,m. We model the effectsof propagation delays as part of the channel response by settingthe first dj,m taps of the channel response to zero. Assume thatthe number of nonzero taps of the channel response hj,m(l)is νj,m + 1. We have νj,m + dj,m + 1 ≤ ν + 1, where ν =maxj,m(νj,m + dj,m). We make the assumption that ν is lessthan the number of OFDM subcarriers N , but ν may be lessthan, equal to or greater than the cyclic prefix length LCP .The discrete time signal model is

rj(k) =Mt∑

m=1

ν∑l=0

hj,m(l)sm(k − l) + nj(k), (1)

where rj(k) is the received signal on the jth receive antennaat time k. We also assume that the noise term nj(k)’s are i.i.d.circularly symmetric complex Gaussian distributed with zeromean and variance σ2

n.The system diagram for space-time equalization is illus-

trated in Fig. 1. Some notations are summarized in TableI. A bank of Mt space-time equalizers is employed at eachreceive antenna to suppress the various sources of interference.The space-time equalizer at the jth receive antenna for theith user of the equalizer is represented by a vector wi,j =[wi,j(L), ..., wi,j(0)]T of L + 1 taps. Each FIR equalizer wi,j

may have a different equalization delay Δi,j .A packet of B information bits is transmitted via J coded

OFDM symbols per user. In addition, K training OFDM sym-bols are transmitted right before the coded OFDM symbols totrain the space-time equalizer for each user. The channel is as-sumed to be constant over a duration of J+K OFDM symbols.The frequency domain signal of the bth OFDM symbol of the

Decoder.

Freq.DomainScalarEqual.

OFDMDemod

FIR (2,1)

FIR (2,2)

FIR (2,3)

Encoder Interleaver ModulationMapper IFFT

Encoder Interleaver ModulationMapper IFFT

Channel

FIR (1,1)

FIR (1,2)

FIR (1,3)

Add CP

Add CP

RemoveCP +FFT

Freq.DomainScalarEqual.

Training-based optimization

D/A & RF

D/A & RF

RF& A/D

RF& A/D

RF& A/D

User 1

User 2

Decoder

Fig. 1. Illustration of the transceiver block diagram. Two users send signals toa receiver with three receive antennas. For each user, there is an FIR equalizerbank at the receiver.

mth user is denoted by vm(b) = [vm(0, b), ...., vm(N−1, b)]T ,where vm(p, b) is the symbol on the pth subcarrier.

The transmitted information bit vector for each user is ofa length B. For the coded OFDM symbols, the transmittedbits for each user are encoded with binary linear code C witha rate rC , then these bits are interleaved by a bit interleaverΠ. The coded bits are denoted by the sequence cm for themth user. The total number of coded bits is denoted by B =B/rC . Note that, for simplicity, we assume that B, C andΠ are the same for all users. After interleaving, the bits aremapped to frequency domain symbols by a Gray mapping ruleμ using a QAM constellation χ of size 2mQ with the minimumdistance dmin. Note that, in this paper, we do not assume theknowledge of channel information at the transmitter, therefore,we do not adapt the modulation order and coding rate forthe packets as in prior work, e.g., [11]. Effectively, the bitinterleaver specifies a one-to-one correspondence between thebit location t and the uth bit of the pth subcarrier of the bth

OFDM symbol, i.e., Π : t → (u, p, b).The time domain signal can be obtained by sm(b) =

QHvm(b), where Q denotes the DFT basis matrix andsm(b) = [sm(0, b), ...., sm(N − 1, b)]T . For the samplesm(p, b), the relation between b, p and k is that k = b(N +LCP ) + LCP + p and k denotes the universal time index inthe system. We also make the assumption that only a portionof the subcarriers are used for data transmission because inpractical OFDM systems, there are unused bands/guard bandsfor spectrum shaping. The set of used subcarriers is denotedby Nm. Assume that the cardinalities of the set Nm are thesame for all users and |Nm| = Ns. We denote the subcarrierselection matrix by a diagonal matrix Tm for the mth user,which consists of only 0 and 1’s.

We can construct a Toeplitz channel matrix Hj,m of a size(L + 1) × (L + ν + 1) for the mth transmit antenna and thejth receive antenna constructed from hj,m(l). The transmittedsignal of the mth user is stacked into a vector sm(k−L− ν :k) = [sm(k−L−ν), ..., sm(k)]T , where p : q denotes the pth

to the qth sample of the signal in MATLAB notation. The timedomain signal vector at the jth receive antenna is denoted by

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rj(k − L : k) and we have

rj(k−L : k) =Mt∑

m=1

Hj,msm(k−L− ν : k) +nj(k−L : k),

(2)where nj(k − L : k) = [nj(k − L), ..., nj(k)]T is the noisevector at the jth receive antenna.

The equalizer bank performs the interference suppressionand the output from the equalizer at time k is denoted byxi(k) where

xi(k) =Mr∑j=1

wTi,jrj(k + Δi,j − L : k + Δi,j). (3)

We define the post-equalization equivalent channel responsefor the ith user (i.e., between si(k) and xi(k)) as bi,i =[bi,i(LCP ), ..., bi,i(0)]T of a length LCP +1. The partial DFTbasis is denoted by Q, formed from the first LCP +1 columnsof Q, which transforms the time domain channel response bi,i

into the frequency domain response fi,i = [fi,i(0), ..., fi,i(N−1)]T . The frequency domain equivalent model after the space-time equalization and FFT transformation is thus

yi(p, b) = fi,i(p)vi(p, b) + ei(p, b) (4)

on the pth subcarrier, where ei(p, b) denotes the noise andinterference on the pth subcarrier of the bth OFDM symbol.

Let vi(u, p, b) be the uth bit of the symbol at the pth subcar-rier of bth OFDM symbol of the ith user after the interleaving.The log likelihood ratio for this bit can be approximated asthe following by using the max-log approximation [12]

Λi(u, p, b) = minvi(p,b)∈χu

1 ,vi(u,p,b)=1

|yi(p, b) − fi,i(p)vi(p, b)|22σ2

i (p)

− minvi(p,b)∈χu

0 ,vi(u,p,b)=0

|yi(p, b) − fi,i(p)vi(p, b)|22σ2

i (p), (5)

where σ2i (p) denotes the noise plus interference power on the

pth subcarrier. The terms χu1 and χu

0 denote the complex QAMsignal set in χ whose uth bit is 1 and 0, respectively. Let usdefine a matrix Wi,j of size N × (N + L) as

Wi,j =

⎡⎢⎣ wi,j(L) ... wi,j(0) ... 0

......

. . .. . .

...0 ... wi,j(L) ... wi,j(0)

⎤⎥⎦ . (6)

The noise covariance matrix after the space-time equalizationand DFT operation can be obtained as

Gi = σ2n

Mr∑j=1

QWi,jWHi,jQ

H . (7)

For simplicity, we assume that the interference power afterthe space-time equalization is negligible, so that the noise andinterference power on subcarrier p for the ith user can berepresented as σ2

i (p) ≈ Gi(p, p), i.e., the pth diagonal elementof the matrix Gi. Another reason is that for simplicity again,we do not estimate the residual interference power after thespace-time equalization using the training sequences, hence

the residual interference power is unknown to our algorithm.Overall, this leads to a slight loss in error rate performance. Wedenote the vector of approximate noise/interference powers onall subcarriers for the ith user as Σi = [σ2

i (0), ..., σ2i (N−1)]T .

We use the Viterbi decoding algorithm to decode the data bits.The ML codeword decision is

ci = argmaxc∈C

B∑t=1

Λi(Π(t)). (8)

III. EQUALIZATION METHODS

In this section, we first describe the mean square error basedequalizer training technique. Then we describe the trainingsequence-based equalizer design technique that brings the biterror rate performance into consideration.

A. Mean Square Error Based Equalizer Design

We stack the received signal vector rj(k + Δi,j − L : k +Δi,j) (k = 0, 1, ..., N − 1) into a matrix Rj(b,Δi,j) of sizeN×(L+1). We also stack the time-domain transmitted signals(k − LCP : k) into a matrix Si(b) of size N × (LCP + 1).We stack the frequency domain signal for the ith user into amatrix Vi(b) = diag{vi(0, b), ...., vi(N −1, b)}. We transformthe received signal after the space-time equalization to thefrequency domain and obtain

yi(b) = QMr∑j=1

Rj(b,Δi,j)wi,j

= Vi(b)Qbi,i + ei(b)= Vi(b)fi,i + ei(b) (9)

where yi(b) is a N×1 vector, which has the samples of the bth

OFDM symbol of the ith user, and fi,i is the channel responsein the frequency domain for the ith user.

The design objective is to minimize the mean square errorbetween the received signal and the transmitted signal con-volved with the desired post equalization channel response.Define the cost function g1(Δi,j ,wi,j ,bi,i) as the following

g1 =K−1∑b=0

‖TiQMr∑j=1

Rj(b,Δi,j)wi,j − TiVi(b)Qbi,i‖2.

(10)Define wi = [wi,1, ...,wi,Mr

]. To lower the search com-plexity we assume that Δi,j = Δi,∀j, the optimizationproblem is formulated as [9]

minΔi,wi,bi,ig1(Δi,wi,bi,i),

s.t., ‖bi,i‖2 = 1. (11)

The solution to this problem is given in [9].

B. PEP Based Space-time Equalizer Design

We first introduce the tight PEP upper bound for the bit-interleaved coded OFDM in [11]. We then obtain an approx-imation to the PEP from this result. Finally, we will discussthe training based equalizer design that considers the derivedPEP result. The main result is given in equation (18).

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Define the sequence of coded bits as ci =[c1,i, ..., ct,i, ..., cB,i] before interleaving. We considerthat the decoder is in favor of ci = [c1,i, ..., ct,i, ...cB,i],where ci and ci differ by d bits. Without loss of generality,ci and ci differ in the first d consecutive positions.

Let Ot,i = (ut,i, pt,i, bt,i) denote the utht,i bit of the pth

t,i

subcarrier of the btht,i OFDM symbol. Let Oi denote the

Cartesian product of d such tuples for [c1,i, ..., cd,i], thus

Oi = (u1,i, p1,i, b1,i) × · · · × (ud,i, pd,i, bd,i). (12)

Given Oi, let Fi = [fi,i(p1,i), · · · , fi,i(pd,i)]T be the d dimen-sional vector whose elements are the complex channel gainsselected by Oi and define Σi = [σ2

i (p1,i), · · · , σ2i (pd,i)]T ,

which represents the d dimensional vector whose elementsare the noise plus interference power on a particular subcarrierselected by Oi. We also define

χOici

= χu1,ic1,i

× · · · × χud,icd,i , (13)

χOi

ci= χ

u1,i

c1,i× · · · × χ

ud,i

cd,i. (14)

We assume that the noise/interference components in allsubcarriers are independent to simplify the analysis. (Afterequalization the noise will become correlated, but due to theinterleaving, the noise and interference terms may be con-sidered to be uncorrelated. This is a standard approximationfor analyzing the BICM systems.) Adapting the derivation ofthe pair-wise error probability (PEP) in [11] to the system ofinterest, we have the PEP result (after the expurgation [11])as

P (ci → ci|Oi, fi,i,Σi)

≤ 2−d(mQ−1)∑

vi∈χOici

P (vi → zi|Oi,Fi, Σi)

= 2−d(mQ−1)

×∑

vi∈χOici

Q

⎛⎝√√√√ d∑

t=1

|fi,i(pt,i)|2ηi|vt,i − zt,i|22σ2

i (pt,i)

⎞⎠

≤ 2−d(mQ−1)∑

vi∈χOici

Q

⎛⎝√√√√ d∑

t=1

|fi,i(pt,i)|2ηid2min

2σ2i (pt,i)

⎞⎠

= Q

⎛⎝√√√√ d∑

t=1

|fi,i(pt,i)|2ηid2min

2σ2i (pt,i)

⎞⎠ , (15)

where vi is a sequence of transmitted QAM symbols forc1, · · · cd and zi is the nearest neighbor in χOi

cifor vi (under

the assumption of Gray mapping) [13] [11]. The quantity ηi

denote the transmit power scaling factor. We further work onthe conditional PEP

P (c → c|fi,i,Σi)

= (JNsmQ)−d∑Oi

P (ci → ci|Oi, fi,i,Σi)

≤ (JNsmQ)−d∑Oi

Q

⎛⎝√√√√ d∑

t=1

|fi,i(pt,i)|2ηid2min

2σ2i (pt,i)

⎞⎠. (16)

We can optimize the equalizer parameters with respect to thecost function in (16), however, this gives high computationalcomplexity. We want to derive a bound for the PEP so thatthe cost function has lower computational complexity. Sincethe arithmetic mean is greater than or equal to the harmonicmean, we have the following inequality

1d

d∑t=1

|fi,i(pt,i)|2ηid2min

2σ2i (pt,i)

≥(

1d

d∑t=1

2σ2i (pt,i)

|fi,i(pt,i)|2ηid2min

)−1

.

(17)Therefore, we arrive at

P (c → c|fi,i,Σi) ≤ (JNsmQ)−d

×∑Oi

Q

⎛⎜⎝√√√√d

(1d

d∑t=1

2σ2i (pt,i)

|fi,i(pt,i)|2ηid2min

)−1⎞⎟⎠

≈ Q

⎛⎜⎜⎝√√√√√d

⎛⎝1

d

∑p∈Ni

2σ2i (p)

|fi,i(p)|2ηid2min

⎞⎠−1

⎞⎟⎟⎠ . (18)

Since the subcarriers with large interference and noise to signalpower ratio dominates the PEP performance, the approxima-tion in the last step follows. The PEP approximation impliesthat the sum of the inverse of the per subcarrier SINR is asuitable criterion for the space-time equalizer design. Givena training sequence, the design objective is to minimize thesummation of the inverse of the per-subcarrier SINR. Thismethod has been used to design the space-time equalizer foruncoded MIMO-OFDM systems [14]. We elaborate on thisdesign method below.

Define the interference and noise signal vector post-FFTand after the equalization as the following

ei(Δi, b) = TiQMr∑j=1

Rj(b,Δi)wi,j − TiVi(b)Qbi,i. (19)

We define a diagonal matrix Λi as the following

Λi = diag{|fi,i(0)|−2, ..., |fi,i(N − 1)|−2} (20)

where fi,i = Qbi,i. The cost function which represents thesummation of the inverse of the per subcarrier SINR can bewritten as

g2(Δi,wi,bi,i) =K−1∑b=0

ei(Δi, b)HΛiei(Δi, b). (21)

Therefore, the design optimization problem can be written asthe following

minΔi,wi,bi,ig2(Δi,wi,bi,i),

s.t., ‖bi,i‖2 �= 0. (22)

By fixing the first tap of bi,i to be one (i.e., bi,i(LCP ) = 1),the problem can be solved by the steepest descent algorithmwith the armijo step sizing after applying the technique ofseparation of variables [9].

To optimize the cost function over all delays is very costly.We hence first perform the MSE based method in III-A

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and find the best delay and the post equalization channelresponse. Then use this channel response as the initial inputto the optimizer that assumes the PEP based design method.The rationale here is that we want to maximize the postequalization channel response energy and then use this channelresponse for the optimization.

C. Joint Maximum Likelihood (ML) Frequency DomainEqualization

In this subsection, we derive the ML-based frequency do-main equalization method that only utilizes the signal andchannel response on each subcarrier for all users. It is differentfrom the per-subcarrier equalization used in [3].

Define the cross product of the QAM constellation χu,i1 =

χ× · · · χu1︸︷︷︸

ith

× · · ·χ where the uth bit of the ith user is 1 and

χu,i0 = χ × · · · χu

0︸︷︷︸ith

× · · ·χ where the uth bit of the ith user

is 0. From (1), we remove the cyclic prefix and transform thesignal rj(k) into the frequency domain signal yj(p, b) of thepth subcarrier of the bth OFDM symbol. We can transform thetime domain channel hj,m into the frequency domain channelresponse fj,m(p) on the pth subcarrier. We stack the receivedsignal yj(p, b) into y(p, b) and stack the channel responsesfj,m(p) into a matrix F(p) whose (j,m)th element is fj,m(p).We also stack the transmitted signal vm(p, b) into a vectorv(p, b) of size Mt × 1. With perfect channel information, wecan calculate the log likelihood ratio for each user based onthe maximum likelihood criterion as the following

Λi(u, p, b) = minv(p,b)∈χu,i

1 ,vi(u,p,b)=1

‖y(p, b) − F(p)v(p, b)‖2

2σ2n

− minv(p,b)∈χu,i

0 ,vi(u,p,b)=0

‖y(p, b) − F(p)v(p, b)‖2

2σ2n

, (23)

where the effect of ICI caused by the asynchronism is notmodelled into the denominator of the noise and interferencecomponent for simplicity. This is the max-log approximationto log likelihood ratio. When the effective channel response ofall users lie within the cyclic prefix, the equation (23) accu-rately represents the ML based equalization method. The MLbased approach offers very desirable error rate performance,but it has a high complexity because searching over the productspace of the constellation is required.

IV. SIMULATION RESULTS

In our simulation, we use either QPSK or 16-QAM modu-lation. The number of subcarriers N is fixed to be 128 and thecyclic prefix length is LCP = 16. A block Rayleigh fadingchannel model is assumed in the simulation. The number oftraining OFDM symbols is one (K = 1) and the numberof data symbols is 18 (J = 18) throughout the simulation.The channel is quasi-static fading, constant over blocks of 19OFDM symbols but independent between blocks. For eachchannel realization, we use QPSK signals for the training.The training samples are generated randomly from the QPSK

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

Mag

nitu

de

Tap Number

Channel Response After Equalizer Bank 1 with MSE−based Design

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

1.2

1.4

Mag

nitu

de

Tap Number

Channel Response After Equalizer Bank 2 with MSE−based Design

post equalization channel for the first userpost equalization channel for the second user

post equalization channel for the first userpost equalization channel for the second user

Fig. 2. The post-equalization channel responses of the MSE-based design.

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1Channel Response After Equalizer Bank 1 with PEP−based Design

Tap Number

Mag

nitu

de

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

1.2

1.4

Tap Number

Mag

nitu

de

Channel Response After Equalizer Bank 2 with PEP−based Design

post−equalization channel for the second userpost−equalization channel for the first user

post−equalization channel for the second userpost−equalization channel for the first user

Fig. 3. The post-equalization channel responses of the PEP-based design

signal set. Convolutional coding with generator polynomial(133,171) and the IEEE 802.11a interleaving scheme areused in our simulations. We use the Viterbi decoder for theconvolutional decoding. We assume that each user uses allsubcarriers, thus, the selection matrix Ti = I. We also assumethat the same power scaling is used for the training and thedata transmission.

We first illustrate the effect of space-time equalization forone channel realization of a two user scenario with two receiveantennas. The post-equalization channel responses are plottedin Fig. 2 and Fig. 3. We observe that for both the MSE and PEPbased design, the equalizers suppress the other user’s channeland preserve the desired users’s channel.

We illustrate the bit error rate (BER) comparison of theMSE and PEP based space-time equalizer design for thesynchronous case in Fig. 4, i.e., dj,m = 0. We assume thatthere are two synchronous users in the system. The number ofchannel taps are 6 and 8 respectively for the first and secondusers. The number of receive antennas is fixed to be three.From the BER comparison, we observe 5-6 dB performancegain for the BER based equalizer design versus the MSEbased equalizer design at BER = 10−4. We observe that theML based equalization method significantly outperforms bothMSE and BER based space-time equalization methods. Theperformance gain is about 11 dB at BER = 10−4, however, itrequires higher implementation complexity.

For asynchronous cases, in Fig. 5, we assume that the delayof the first user is zero and the delay for the second user

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0 5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

BE

R

16QAM PEP−based16QAM MSE−basedQPSK PEP−basedQPSK MSE−basedQPSK joint ML (perfect channel)16QAM joint ML (perfect channel)

Fig. 4. The BER comparison for the 2x3 case with 16QAM and QPSKmodulation. The delays dj,m are set to be 0. The channels of the two usersare of 6 and 8 taps.

is 15. The number of receive antennas is 3. The 16-QAMmodulation is adopted. The performance comparison showsthat the PEP based approach outperforms the MSE basedmethod by 4 dB at BER = 10−3. In high SNR region, the PEPbased space-time equalization method performs better than thejoint ML equalization method. Due to the ICI caused by theasynchronism, the BER curve of the joint ML equalizationmethod has an error floor. In Fig. 6, we compare the case oftwo receive antennas and QPSK modulation. The joint MLbased approach has very good performance in the low tomedium SNR region, but due to the error floor, its performanceis less desirable in the high SNR region.

V. CONCLUSION

In this paper, we studied the MSE and PEP based space-time equalizer design methods for the multiuser asynchronousbit interleaved coded OFDM system. The PEP based designsignificantly outperforms the MSE based design in termsof BER performance. We also compared the time domainequalization approach with the joint ML frequency domainequalization for the asynchronous case. From the simulations,we observe that in the high SNR region, time domain equal-ization should be favored since it suppresses the inter-OFDM-symbol interference and the inter-carrier interference.

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10 12 14 16 18 20 22 24 26 2810

−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

BE

R

MSE−basedPEP−basedJoint ML (perfect channel)

Fig. 5. The BER comparison for the 2x3 case with 16QAM modulation.The delay dj,m is set to be 0 and 15. The channels of the two users are of6 and 8 taps.

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Fig. 6. The BER comparison for the 2x2 case with QPSK modulation. Thedelay dj,m is set to be 0 and 15. The channels of the two users are of 6 and8 taps.

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.978-1-4244-4148-8/09/$25.00 ©2009