Adaptive MMSE Multiuser Detection for MIMOOFDM over Turbo-Equalization for Single-Carrier

Transmission Wireless ChannelTitus Eneh, Predrag Rapajic, Grace Oletu, Kwashie Anang and Lawal Bello

Mobile and Wireless Communication Dept.,Medway School of Engineering,the University of Greenwich, UK. Email: {T.Eneh, P.Rapajic, og14, K.Anang and bl65}@gre.ac.uk

Abstract—This paper presents an adaptive multiuser receiversscheme for MIMO OFDM over Turbo-Equalization for Single-Carrier Transmission. It involves the joint adaptive minimummean square error multiuser detection and decoding algorithmwith prior information of the channel and interference cance-lation in the spatial domain. A partially filtered gradient LMS(Adaptive) algorithm is also applied to improve the convergencespeed and tracking ability of the adaptive detectors with slightincrease in complexity. The proposed technique is analyzed in aslow and fast Rayleigh fading channels in MIMO OFDM systems.The adaptive multiuser detection for MIMO OFDM (AMUDMIMO OFDM) perform as well as the iterative equalization forsingle-carrier.

Index Terms—MMSE, Single Carrier, MIMO OFDM, Adap-tive filter and Turbo Equalizer.

I. INTRODUCTION

IEEE 802.11 wireless Local Area Network standards em-ploy OFDM, which offers high spectral efficiency and superiortolerance to multi-path fading [1]. In OFDM computationally-efficient Fast Fourier Transform [2] is used to transmit data inparallel over a large number of orthogonal subcarrier whichis maintained even in frequency selective fading [3][4][5].Throughput and capacity can be improved when multipleantennas are applied at the transmitter and receiver side,especially in a rich scattering environment [6][7] as well asfrequency-selective fading channels.

The Conventional approaches implements an equalizer to re-move ISI or use MAP or maximum likelihood (ML) detectionwhile we employ joint detection technique of (AMUD MIMOOFDM) and joint equalization decoding(Turbo equalizer) tototally eliminate ISI. A recent approach that significantlyreduces the complexity of joint equalization and decoding isthe so called ”turbo equalization” algorithm, where MAP/MLdetection and decoding are performed iteratively on the sameset of received data. It has recently been shown that passingsoft information, the use of interleaving, and the controlledfeedback of soft information are essential requirements toachieve performance gains with an iterative system [8][9].

The iterative principle has been extended to encompasssingle carrier equalization techniques while the multicarrierequalization technique employs joint channel estimation asstated in [10]. This allows single carrier systems to combinethe operations of equalization and channel coding to operate

in a wideband channel with performance that could not pre-viously be achieved with traditional equalization and forwarderror correcting (FEC) techniques [11]. Iterative equalizationtechniques have been shown to give excellent error rateperformance for both fixed and fast fading channels [12].

AMUD MIMO OFDM is for demodulation of digitallymodulated signals with multiple access interferences (MAI).Conventionally individual channel estimation as stated by [13]was improved by joint estimation as stated in [10]. Thisscheme was designed for total elimination of MAI in thesystem. In a single user environment, every match filter max-imum likelihood receiver plays the role of Adaptive MMSEmaximum likelihood receiver [14][15]. In the implementationof AMUD MIMO OFDM, provides robustness and mobility ina time variable frequency selective multipath fading channel; itimproves the bit error rate performance and therefore enhanceschannel capacity of a multi-cellular environment. MIMOOFDM mitigates multiple access interference and increasescapacity [16][17].In [15] A MMSE MUD techniques wasused effectively to achieve the performance of a maximumlikelihood estimator but on a linear complexity.

The contribution of this paper is as follows:• Enhanced joint channel estimation and signal detection

makes the new technique effectively mobile and thusone can easily get network wherever he goes becauseof continuous handover (due to training).

• Proposed iterative scheme achieves significant iterationgain

• Bpsk modulation schemes achieves better BER in bothmemory 2 and 4 respectively for the iterative scheme.

• Memory 4 performance at BER 10−5 has approximately1dB gain difference when compared to memory 2 perfor-mance.

• Adaptive iterative receiver provides a significant perfor-mance improvement to SISO linear adaptive receivers.

• The sum rate capacity result in the new technique is veryclose to MIMO theoretical upper bound.

II. SYSTEM STRUCTURE

A. AMUD MIMO OFDM

Fig. 1, shows the system model for an AMUD MIMOOFDM, with Nt and Nr transmit and receive antennas with

978-1-4577-9538-2/11/$26.00 ©2011 IEEE 1408

Fig. 1. System Model for AMUD MIMO OFDM

k subcarriers in one OFDM block. At time t, a data blockb′[n, k] : k = 0, 1, ..., n transformed into different signalsx1[n, k] : k = 0, ..., k − 1 and i = 1, 2, ..., n, and i arenumbers of sub channels of OFDM system. Signals transmittedare modulated by x1[n, k]. The FFT received at each receiveantenna is the superposition of the transmitted signals. Thereceive signal at jth receive antenna is

yj [n, k] =Nt∑i=1

Hi,j [n, k]xi[n, k] + nj [n, k] (1)

Where Hi,j [n, k] is the frequency response between antennasi and j, nj [n, k] is the additive Gaussian noise with zero meanand unit variance σ2

n .

B. Principal of Turbo Equalization

Fig. 2. Iterative structure

The iterative equalization receiver structure used in thisstudy shows that both the equalizer and the decoder employsthe optimal symbol by symbol Maximum A-Posteriori (MAP)soft input soft output (SISO) algorithm [18]. Soft input sym-bols are fed into the decoder from a sampled receive filterstream r(t) and bit-wise hard decisions are produced as thefinal output. It is possible to equalize and decode in an iterativemanner that is similar to turbo decoding.

The equalizer provides soft outputs, i.e., reliable informationon the coded bits for the channel decoder. The soft informationon the bit ck is usually given as a log-likelihood ratio (LLR)or L-value as:

λE(ck) = logP (ck) = +1|r)P (ck) = −1|r)

(2)

which is the ratio between the conditional bit probabilities inthe logarithmic domain. These L-values are deinterleaved andgiven to the channel decoder, which uses them to recover theinformation bits u’.

At the first iteration there is no feedback information fromthe channel decoder available, so the equalizer calculatesthe L-values λE(c′) as given by (4) which are based onthe received samples r from the channel. The L-values aredeinterleaved to break consecutive bits far apart and thusgiving the channel decoder independent input values. Theinterleaving is an essential part in the iterative receiver scheme,since the extra information on an individual data bit is due tothe different neighboring bits in the detection and decodingprocesses feedback branch to the equalizer. Therefore we needto use the more complex SISO decoder instead of the conven-tional hard output decoder. The equalizer is able to producethe L-values λE(c′) based on the received samples from thechannel, so that information is not repeated in the feedback.Hence, the feedback only contains the extra information thatis obtained from the surrounding bits in the channel decoding.The input L-values and the obtained extra information arecalled intrinsic and extrinsic information, respectively. Theextrinsic information from the channel decoder is given as[8][9]

λDa ck = λD(c)− λE

a ck (3)

where λEa ck denotes the extrinsic information from the equal-

izer. The turbo equalization technique is based on the utiliza-tion of this extrinsic information at the next iteration round[8]. So it is passed through the interleaver to the equalizer asa priori information on the bit reliiabilities. By exploiting thisside information in the detection, more reliable decisions areachieved. Also in the equalizer output the extrinsic informationλE

a ck is extracted from the output as follows:

λEa ck = λE(c)− λD

a ck (4)

This equalizer information is again used in the SISO decoderto produce new soft outputs and furthermore, the new extrinsicinformation according to (3). As soon as this feedback infor-mation becomes available, the new iteration can be started.The number of iterations may depend on the processing poweravailable or the achieved performance improvement. At thefinal stage, there is no need for the SISO decoder, since onlyhard decisions u on the information bits are needed. The Turboequalization receiver is able to improve the performance, butat the cost of higher complexity.

III. ADAPTIVE MMSE RECEIVER FOR MIMO OFDM

In adaptive filter the parameters are continuously changingdue to the received training sequence from the transmitterwhich informs the receiver to adjust the parameter of the filtersto match the desired signal. In the single user environmentevery match filter Maximum Likelihood (ML) plays the roleof an Adaptive MMSE ML receiver [14][15].

An adaptive turbo receiver structure used in this paper showsthat the a bank of adaptive linear MMSE filters trains the filtercoefficients and retrieves the signatures. Training is employedusing least mean square (LMS) approach to adaptively updatethe linear coefficient, an MMSE convergence of the filtercoefficients provides estimates of the received signatures.

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An adaptive MMSE filter minimize the error by an adaptivealgorithm, the steepest decent algorithm is used to minimizethe mean square error (MSE). For simplicity, fractionallyspaced adaptive linear transversal filter for Adaptive MMSEdetection is used, which is insensitive to the time differencesin the signal arrival times of different users, thus the receivertiming recovery is extremely simplified [14][19]. Consider thereceived signals symbols y1[n, k], y2[n, k] and y3[n, k] and lettheir general form for any node and any path in the network isyN [n, k] = rn[m]. Where n is the specific number assigned tothe signals at nodes. y1[n, k], y2[n, k] and y3[n, k] receiveddigital output symbol block from adaptive filters is b′[n, k].

In multi-cellular environment transmitter transmits informa-tion independently. Therefore, non orthogonally transmittedsignal from independent users arrives asynchronously at thereceiver [20][21] and the delay cannot be neglected. Due tonon orthogonality of the spreading code the correlation existbetween the spreading code at the receiver. The co-efficientof correlation is given by : E [rM [m]r∗N [m]]=%(M,N) =R(M,N) =

∫sMsNdt and R correlation matrix. The digital

output of the nth filter for the mth symbol period on relay ordestination is given by

rn[m] = RHxn[m] + vn[m] (5)

where H is matrix of respective channels and vn[m] is noise.The error between the reference signal and the output ofadaptive filter is

ε = (xn[m]− xn[m])

ε = ([xn[m]− aHn [m]rn[m]) (6)

aHn [m] is M dimensional complex valued weight vector atmth symbol time when the variable filter estimates the desiredsignal by convolving the input signal with impulse response.an[m] = [a1, a2......., aM ] M are tap of filter and rn[m] =[r1, r2........., rM ]. During the adaptation mode the weightparameters are adjusted such that mean square error Jan

isminimized in mth symbol time. For simplicity of descriptionm is with every term but we are not mentioning it here

Jan = E [εnε∗n] (7)

from equation (5)

Jcn= E [(xn − aH

n rn)][(xn − aHn rn)∗] (8)

Jcn= E [xnx∗n]+aH

n E [rnrHn ]an−aH

n E [rnx∗n]−E [xnrHn ]an (9)

The first term in equation E [xnx∗n], represent the variance ofthe desired signal. The expectation E [rnrH

n ] denotes M by Ncorrelation matrix of the received signal given by

R =

%(1, 1) %(1, 2) ... %(1, N)%(2, 1) %(2, 2) ... %(2, N)... ... ... ...

%(M, 1) %(M, 2) ... %(M,N)

(10)

The matrix R is Hermitian and can be uniquely defined byspecifying the values of the correlation coefficients %(M,N),where %(M,N) = %∗(N,M)

Z = E [rnx∗n] (11)

The expectation is M by N cross-correlation matrix vec-tor between the received components and the referencesequence, and the expectation E [x(n)rH

(n)] = ZH . whereZH = [z∗1 , z

∗2 , .......z

∗M ]. The co-efficient zi are given by

zi = E [r(n)x∗(n)]. For stationary input and reference signalsthe surface obtained by plotting the mean square error J(an)

versus the weight co-efficient has a fixed shape and curvatureith a unique minimum point; the adaptive process seeksthat minimum point at which the weight vector is optimal.Differentiating the mean squared error function J(an) withrespect to each coefficient of the weight vector an yields thegradient ∇n.

∇n =∂J(an)∂an

=

∂J(an)

∂a1∂J(an)

∂a2

.....∂J(an)∂aM

(12)

The optimal weight vector aopt can be determined by settingthe gradient ∇n equal to zero,

∇n = −2Z + 2Ran = 0 (13)

where 0 is an M by 1 null vector at the minimum point ofthe error surface, the adaptive MUD is optimum in the meansquared error sense, and equation can be simplified in the formRaopt = Z Which is Weinner-Hopf equation or the normalequation, where the vector representing the estimation error isnormal to the vector representing the output of the combiner.One possible solution of this equation is matrix inversion,

Z = R−1aopt (14)

Another simple solution that does not require matrix inver-sion or explicit calculations of the correlation coefficients isthe steepest decent method. The Steepest Decent Method isrecursive procedure that can be used to calculate the optimalweight vector aopt. Let an and ∇n denote the values of theweight vector and the gradient vector at time m, respectively.Then succeeding values of the weight vector are obtained bythe recursive relation. After each symbol period m the weightof the filter updated till optimum coefficient get best crosscorrelation value. And then filter can go to decision directedmode where coefficient continuously change with the variationof channel.

an = an − µ∇n (15)

Where µ is step size constant that controls stability and therate of adaptation. If we express ∇n in terms of instantaneousestimates Z = rnx

∗n and R = rnr

Hn Then the equation can be

simplified as

an[m+ 1] = an[m] + 2µrn[m][x∗n[m]− rHn [m]an[m] (16)

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which can be expressed in term of ε∗n[m] as,

an[m+ 1] = an[m] + 2µrn[m]ε∗n[m] (17)

Here 2µrn[m]ε∗n[m] is correction factor, where m =0, 1, 2........ till aopt achieved. The equation explains that theupdated weight vector is computed from the current weightvector by adding the input vector scaled by the complexconjugate value of the error error and by µ which controls thesize of correction. The iteration of the equation produces thevalue of the Mean Square Error at which the vector tends to itsoptimal value aopt. Minimum error value cannot be reached bya finite number of iterations though approachable. To achieveproper adaptation, the weight vector must be updated at arate fast enough to track the channel variations. The methodof steepest descent can be viewed as feedback model whichmay become unstable. The stability of the steepest descentalgorithm depends on the step size parameter µ and the autocorrelation matrix R. The eigenvalues of R are all real andpositive, the condition for convergence and stability of thesteepest descent algorithm depends on the step size parameterµ.

IV. CAPACITY EQUATION

System capacity will be significantly improved by MIMOchannels [13][22]. In OFDM [2][3][23], the entire channel isdivided into many narrow parallel sub channels. The capacityformula for the proposed scheme as stated in [15], basedon the assumption that the channel matrix which consists ofindependent and identically distributed iid Rayleigh fadingcoefficients and Fig.6 is the sum rate capacity figure of thedeveloped scheme which is very close to MIMO capacity. Thecapacity formulae for adaptive multiuser detection by Predragis as stated in [14][15]:

C = log

(1σ2

e

)(18)

where σ2e is the noise variance of the signal at the receiver.

V. PERFORMANCE COMPARISON AND SIMULATIONS

The achievable bit error rate of the following three caseswere compared by Monte Carlo simulation: OFDM SISO,OFDM MIMO and AMUD OFDM MIMO systems. Perfectchannel state information was assumed, BPSK Modulation andflat fading channel model were employed in the simulation.The configurations considered for OFDM system with 64subcarriers, 16 symbol time periods and 4 symbol period forantenna configuration Nt = Nr = 2 and 8.

Fig 3 and 4 Shows the SNR in dB versus the BER ofthe SISO OFDM, 2 X 2 MIMO and AMUD OFDM. Thesimulation results shows that at average Bit Error Rate of 105,AMUD OFDM MIMO performs better than other schemesand provides a 2dB SNR gain to the conventional MIMOOFDM scheme. Fig.5, is the sum rate capacity comparativeresults of the proposed technique with the other schemes inbits/seccond/Hz which is very close to MIMO capacity upperbound.

For channel coding we used a rate recursive systematicconvolutional code with memory mc = 2, constraint length L= 4, generator polynomial G = [7 5], block length = 4096. Thesimulation result in Fig. 6 shows that the principles of iterativedecision linear equalization have been extended to interleavedspace time code over AMUD MIMO OFDM broadband wire-less channel. ISI resolution relies on jointly MMSE criterionand operates on multidimensional modulation symbols, whoseindividual components can be detected in accordance withanother criterion, when the optimum MAP criterion is chosen,substantial performances gains over conventional space timeturbo transmission scenarios was achieved.

VI. CONCLUSION

AMUD MIMO OFDM achieves joint detection by implicitassumption of the use of an optimum MUD in comparisonto conventional individual parameter estimation by Telatar.The developed technique has good performance in terms ofbit error rate, SNR and capacity in Raleigh channel, and itscomplexity is linear. The schemes 8 x 8 antenna configurationat BER of 105 provides a 2dB SNR gain compared to theconventional MIMO OFDM. The sum rate capacity of theAMUD MIMO OFDM is very close to MIMO theoreticalupper bound (21.5 bits/s/Hz at signal to noise ratio of 20dB).

The turbo equalization algorithm is effective in achievingreasonably iteration gain when signal power of unknowninterference is as large as those of the known users. TheAMUD MIMO OFDM performs as well MAP equalizer suchthat the memory4 has 1dB gain to memory2 signifying thehigher the memory the better the performance in SNR, AMUDMIMO OFDM equally shows the higher the number of theantennas the better the SNR gain as shown by our 8 X 8AMUD MIMO OFDM. Future study will be on performancecomparisons of the proposed system in terms of Capacity anddifferent modulation schemes.

Fig. 3. The schemes 2× 2 Bit error rate comparison

1) OFDM SISO2) OFDM MIMO3) AMUD OFDM MIMO

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Fig. 4. The schemes 8× 8 Bit error rate comparison

1) OFDM SISO2) OFDM MIMO3) AMUD OFDM MIMO

Fig. 5. Capacity comparisons (in bits/sec/Hz) (4× 4)

1) OFDM SISO - Sum rate capacity2) OFDM MIMO - Sum rate capacity3) AMUD OFDM MIMO - Sum rate capacity

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