International Journal of Automation and Computing 04(1), January 2007, 100-106

DOI: 10.1007/s10453-004-5872-7

Blind Joint Maximum Likelihood Channel Estimation andData Detection for SIMO Systems

Sheng Chen∗ Xiao-Chen Yang Lei Chen Lajos HanzoSchool of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, UK

Abstract: A blind adaptive scheme is proposed for joint maximum likelihood (ML) channel estimation and data detection of single-input multiple-output (SIMO) systems. The joint ML optimisation over channel and data is decomposed into an iterative optimisationloop. An efficient global optimisation algorithm called the repeated weighted boosting search is employed at the upper level to optimallyidentify the unknown SIMO channel model, and the Viterbi algorithm is used at the lower level to produce the maximum likelihoodsequence estimation of the unknown data sequence. A simulation example is used to demonstrate the effectiveness of this joint MLoptimisation scheme for blind adaptive SIMO systems.

Keywords: Blind space-time equalisation, single-input multiple-output (SIMO) systems, maximum likelihood (ML) estimation.

1 Introduction

The single-input multiple-output (SIMO) system, con-sisting of a single-antenna transmitter and a receiverequipped with multiple antennas, has enjoyed popularityowing to its simplicity. A space-time equaliser (STE) basedon this SIMO structure is capable of mitigating the chan-nel impairment arising from hostile multipath propagation.For the sake of improving the achievable system through-put, blind adaptation of the STE is attractive, since thisavoids the reduction of the effective throughput by invokingtraining. Blind space-time equalisation of the SIMO systemcan be performed by directly adjusting the STE’s param-eters using the constant modulus algorithm (CMA) typeadaptive scheme[1−5]. Blind space-time equalisation per-formance can further be improved by aiding the CMA witha soft decision-directed scheme[6]. Research for blind adap-tive SIMO systems has also been focused on blind chan-nel identification[7−9]. Once the SIMO channel impulse re-sponses (CIRs) have been identified, various designs, suchas the minimum mean square error or minimum bit er-ror rate[10], can be invoked for the STE. Alternatively, thedecoupled weighted iterative least squares with projection(DW-ILSP) algorithm[11,12] can be adopted. The DW-ILSPalgorithm is a batch suboptimal EM-type algorithm, whichiteratively performs channel estimation and symbol detec-tion.

This contribution develops a blind adaptive scheme ofjoint maximum likelihood (ML) channel estimation anddata detection for the SIMO system. The proposed al-gorithm decomposes the joint optimisation over channeland data into an iterative optimisation loop by combininga global optimisation method, referred to as the repeatedweighted boosting search (RWBS)[13], for an optimal es-timation of the SIMO channel and the Viterbi algorithm(VA)[14] for the maximum likelihood sequence estimationof the transmitted data sequence. Specifically, at the upperlevel, the RWBS algorithm searches the channel parameterspace to optimise the ML criterion, while at the lower level,

Manuscript received date; revised date*Corresponding author. E-mail address: sqc@ecs.soton.ac.uk

the VA decodes data based on the given channel modeland feeds back the corresponding likelihood metric to theRWBS algorithm. The effectiveness of this joint ML esti-mation scheme for blind equalisation of the SIMO systemis demonstrated by a simulation example. We point outthat a genetic algorithm (GA) can be used in place of theRWBS algorithm to optimise the SIMO channel estimate.In this case the proposed scheme becomes an extension ofthe joint ML channel and data estimation scheme using theGA originally developed for the single-input single-output(SISO) system[15].

2 The proposed blind joint ML estima-tion algorithm

Consider the SIMO system employing a single transmit-ter antenna and L (> 1) receiver antennas. The symbol-ratesampled antennas’ outputs xl(k), 1 ≤ l ≤ L, are given by

xl(k) =

nc−1Xi=0

ci,ls(k − i) + nl(k) (1)

where nl(k) is the complex-valued Gaussian white noise as-sociated with the lth channel and E[|nl(k)|2] = 2σ2

n, {s(k)}is the transmitted symbol sequence and is assumed to takevalues from the quadrature phase shift keying (QPSK) sym-bol set {±1± j}, and ci,l are the CIR taps associated withthe lth receive antenna. For notational simplicity, we haveassumed that each of the L channels has the same lengthof nc. Let

xxx = [x1(1) x1(2) · · ·x1(N) x2(1) · · ·xL(N)]T (2)

sss = [s(−nc + 2) · · · s(0) s(1) · · · s(N)]T (3)

ccc = [c0,1 c1,1 · · · cnc−1,1 c0,2 · · · cnc−1,L]T (4)

be the vector of N × L received signal samples, the corre-sponding transmitted data sequence and the vector of theSIMO CIRs, respectively. The probability density functionof the received data vector xxx conditioned on the SIMO CIR

2 International Journal of Automation and Computing 04(1), January 2007

ccc and the symbol vector sss is

p(xxx|ccc,sss) =1

(2πσ2n)NL

×

e− 1

2σ2n

PNk=1

PLl=1|xl(k)−Pnc−1

i=0 ci,ls(k−i)|2. (5)

The joint ML estimate of ccc and sss is obtained by max-imising p(xxx|ccc,sss) over ccc and sss jointly. Equivalently, the jointML estimate is the minimum of the cost function

JML(ccc, sss) =1

N

NX

k=1

LX

l=1

˛˛˛xl(k)−

nc−1Xi=0

ci,ls(k − i)

˛˛˛

2

(6)

namely

(ccc∗, sss∗) = arg

»minccc,sss

JML(ccc, sss)

–. (7)

The joint minimisation process (7) can also be solved usingan iterative loop first over the data sequences sss and thenover all the possible channels ccc:

(ccc∗, sss∗) = arg

»min

ccc

„min

sssJML(ccc, sss)

«–. (8)

The inner or lower-level optimisation can readily be carriedout using the standard VA. In order to guarantee a joint MLestimate, the search algorithm used in the outer or upper-level optimisation should be capable of finding a global-optimal or near optimal channel estimate efficiently. Weemploy the RWBS guided random search algorithm[13] toperform the outer optimisation task. The detailed RWBSalgorithm is given in Appendix. The proposed blind jointML optimisation scheme can now be summarised.Outer level optimisation. The RWBS algorithmsearches the SIMO channel parameter space to find a globaloptimal estimate ccc∗ by minimising the mean square error(MSE)

JMSE(ccc) = JML(ccc, sss∗) (9)

Inner level optimisation. Given the channel estimateccc, the VA provides the ML decoded data sequence sss∗, andfeeds back the corresponding value of the likelihood metricJML(ccc, sss∗) to the upper level.

Let CVA be the complexity of the VA required to de-code a data sequence of N × L samples, and denote NVA

the total number of VA calls required for the RWBS algo-rithm to converge. The complexity of the proposed schemeis obviously NVA × CVA. The RWBS algorithm is a sim-ple yet efficient global search algorithm. In several globaloptimisation applications investigated in [13], including theblind joint ML channel estimation and data detection forthe SISO system, the RWBS algorithm achieved a similarconvergence speed as the GA and was seen to be more ac-curate than the GA. The RWBS algorithm has additionaladvantages of requiring minimum programming effort andhaving fewer algorithmic parameters that require to set.

3 Simulation example

In the simulation, the number of receiver antennas wasL = 4, the transmitted data symbols were QPSK, and theSIMO CIRs, listed in Table 1, were simulated. The length

of data samples was N = 50. In practice, the value ofthe likelihood metric JMSE(ccc) is all that the upper leveloptimiser can see, and the convergence of the algorithmcan only be observed through the MSE (9). In simulation,the performance of the algorithm can also be assessed bythe mean tap error defined as

MTE = ‖ccc− a · ccc‖2 (10)

where

a =

8>>><>>>:

+1, if c → +c

−1, if c → −c

−j, if c → +jc

+j, if c → −jc

(11)

Note that since (ccc∗, sss∗), (−ccc∗,−sss∗), (−jccc∗, +jsss∗) and(+jccc∗,−jsss∗) are all the solutions of the joint ML estima-tion problem (7), the channel estimate ccc can converges toccc, −ccc, jccc or −jccc.

Fig. 1 MSE against number of VA evaluations averaged over

50 runs using the RWBS for the SIMO channel listed in

Table 1. The length of data samples N = 50.

Figs. 1 and 2 show the evolutions of the MSE and MTEaveraged over 50 runs and for different signal to noise ratios(SNR), respectively, obtained by the proposed blind jointML optimisation scheme using the RWBS. From Fig. 1,it can be seen that the MSE converges to the noise floor.Phase ambiguity of 90◦, 180◦ or 270◦ associated with theblind ML estimate for sss cannot be resolved by the blindadaptive scheme itself. In practice, this ambiguity is re-solved either by adopting differential encoding or by em-ploying a few pilot training symbols. We adopted the com-plete blind adaptive scheme of using differential encoding.Fig. 3 depicts the bit error rate (BER) of the blind jointML optimisation scheme with differential encoding, in com-parison with the BERs of the optimal maximum likelihoodsequence estimation in the known channel case with andwithout differential encoding. It is seen that the proposedblind scheme only induces half dB degradation in SNR com-pared with the optimal solution with differential encoding.

We also used the GA to perform the upper-level opti-misation, and the results obtained by this GA-based blind

S. Chen et al./ Blind Joint Maximum Likelihood Channel Estimation and Data Detection for SIMO Systems 3

Table 1 The simulated SIMO system

Channel Channel impulse response

1 0.365 2−0.273 9j 0.730 4+0.182 5j -0.440 2+0.176 1j

2 0.278 3+0.237 6j −0.636 2+0.103 9j 0.667 1−0.074 1j

3 −0.639 3+0.249 4j −0.516 9−0.308 4j 0.365 1+0.182 6j

4 −0.153 9+0.692 8j −0.538 9−0.077 0j 0.268 3−0.357 8j

Fig. 2 MTE against number of VA evaluations averaged over

50 runs using the RWBS for the SIMO channel listed in

Table 1. The length of data samples N = 50.

joint ML estimation scheme are presented in Figs. 4 and5. Comparing Fig. 1 with Fig. 4, it can be seen that boththe RWBS and GA based schemes have similar convergencespeed in terms of the total number of required VA evalua-tions. It can also be seen that the true estimation accuracyof the RWBS-based scheme is more accurate than that ofthe GA-based one, as confirmed by comparing Fig. 2 withFig. 5.

All the above results were obtained under the assump-tion that the correct SIMO CIR length nc = 3 was known.In reality, the CIR length nc is unknown and has to be esti-mated. Fig. 6 shows the MSE of the blind joint ML schemeusing the RWBS optimiser, as a function of the estimatedchannel length. It can be seen that the true channel lengthnc = 3 was correctly identified by the blind joint ML opti-misation scheme.

4 Conclusions

A batch scheme using the global optimisation method,called the RWBS, has been developed for blind space-timeequalisation of the SIMO system based on the joint MLchannel estimation and data detection. The proposed al-gorithm provides the best performance over other types ofblind adaptive schemes for SIMO systems at the expense ofcomputational complexity. Our simulation study has shownthat this blind joint ML optimisation scheme requires veryfew received data samples to achieve a near optimal so-lution of the maximum likelihood sequence estimation fordata detection.

Fig. 3 Comparison of bit error rate performance using the ML

sequence detection for the SIMO channel listed in Table 1. The

length of data samples for the blind scheme is N = 50.

Appendix. Repeated weighted boostingsearch

Solve the generic optimisation problem

minuuu∈U

J(uuu) (12)

where U defines the feasible set of uuu, with the RWBSalgorithm[13]. The algorithm is detailed in the following.

Specify the algorithmic parameters: population size PS ,number of generations in the repeated search NG, and num-ber of iterations in the weighted boosting search NB .Outer loop: generations For (g = 1; g ≤ NG; g = g+1){Generation initialisation: Initialise the population by set-ting uuu

(g)1 = uuu

(g−1)best and randomly generating rest of the pop-

ulation members uuu(g)i , 2 ≤ i ≤ PS , where uuu

(g−1)best denotes the

solution found in the previous generation. If g = 1, uuu(g)1 is

also randomly chosenWeighted boosting search initialisation: Assign the initialdistribution weightings δi(0) = 1

PS, 1 ≤ i ≤ PS , for the

population, and calculate the cost function value of eachpoint

Ji = J(uuu(g)i ), 1 ≤ i ≤ PS

Inner loop: weighted boosting search For (t = 1;t ≤ NB ; t = t + 1) {Step 1. Boosting1) Find

ibest = arg min1≤i≤PS

Ji and iworst = arg max1≤i≤PS

Ji

4 International Journal of Automation and Computing 04(1), January 2007

Fig. 4 MSE against number of VA evaluations averaged over

50 runs using the GA for the SIMO channel listed in Table 1.

The length of data samples N = 50.

Denote uuu(g)best = uuu

(g)ibest

and uuu(g)worst = uuu

(g)iworst

2) Normalise the cost function values

Ji =JiPPS

m=1 Jm

, 1 ≤ i ≤ PS

3) Compute a weighting factor βt according to

ηt =

PSXi=1

δi(t− 1)Ji, βt =ηt

1− ηt

4) Update the distribution weightings for 1 ≤ i ≤ PS

δi(t) =

(δi(t− 1)βJi

t , for βt ≤ 1

δi(t− 1)β1−Jit , for βt > 1

and normalise them

δi(t) =δi(t)PPS

m=1 δm(t), 1 ≤ i ≤ PS

Step 2. Parameter updating1) Construct the (PS + 1)th point using the formula

uuuPS+1 =

PSXi=1

δi(t)uuu(g)i

2) Construct the (PS + 2)th point using the formula

uuuPS+2 = uuu(g)best +

“uuu

(g)best − uuuPS+1

”

3) Compute the cost function values J(uuuPS+1) andJ(uuuPS+2) for these two points and find

i∗ = arg mini=PS+1,PS+2

J(uuui)

4) The pair (uuui∗ , J(uuui∗)) then replaces (uuu(g)worst, Jiworst) in

the population

Fig. 5 MTE against number of VA evaluations averaged over

50 runs using the GA for the SIMO channel listed in Table 1.

The length of data samples N = 50.

Fig. 6 MSE as a function of the estimated SIMO channel

length using the RWBS for the SIMO channel listed in Table 1.

The length of data samples N = 50.

} End of inner loop The solution found in the gth

generation is uuu = uuu(g)best

} End of outer loop This yields the solution uuu = uuu(NG)best

The motivations and analysis of the RWBS algorithm asa global optimiser were detailed in [13]. To guarantee aglobal optimal solution as well as to achieve a fast conver-gence, the algorithmic parameters, PS , NG and NB , needto be set carefully. The appropriate values for these algo-rithmic parameters depend on the dimension of uuu and howhard the objective function is optimised. Generally, thesealgorithmic parameters have to be found empirically, justas in any global optimisation algorithm. The elitist initial-isation is very useful, as it keeps the information obtainedby the previous search generation, which otherwise wouldbe lost due to the randomly sampling initialisation. In theinner loop optimisation, there is no need for every memberof the population to converge to a (local) minimum, and it

S. Chen et al./ Blind Joint Maximum Likelihood Channel Estimation and Data Detection for SIMO Systems 5

is sufficient to locate where the minimum lies. Thus, thenumber of weighted boosting iterations, NB , can be set toa relatively small integer. This makes the search efficient,achieving convergence with a small number of the cost func-tion evaluations. The population size PS and the number ofgenerations NG should be chosen sufficiently large so thatthe parameter space is sampled sufficiently to guarantee aglobal-optimal solution.

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Sheng Chen received his B. Eng degreein control engineering from the East ChinaPetroleum Institute, Dongying, China, in1982, and the Ph.D. degree in control en-gineering from the City University, London,in 1986. In 2005, he was awarded the Doc-tor of Sciences (DSc) degree by the Univer-sity Southampton, UK. Since 1999 he hasbeen with the School of Electronics and Com-puter Science, the University of Southamp-ton, UK. He previously held research and aca-

demic appointments at the Universities of Sheffield, Edinburgh andPortsmouth, all in UK.

He has published over 260 research papers. His recent research in-terests include adaptive signal processing, wireless communications,modelling and identification of nonlinear systems, neural networkand machine learning, finite-precision digital controller design, evo-lutionary computation methods, and optimization.

Dr. Chen is a senior member of IEEE. In the database of theworld’s most highly cited researchers in various disciplines, compiledby Institute for Scientific Information (ISI) of the USA, Dr Chen ison the list of the highly cited researchers in the engineering category.

Xiao-Chen Yang received his B. Sc. degreein electrical engineering from Tsinghua Uni-versity, China, in 2003, and the M. Sc. degreein wireless communications from the Univer-sity of Southampton, UK, in 2004. He is cur-rently studying in Civil Engineering Depart-ment at the University of Washington, Seattle,USA.

Lei Chen received his B. Sc. degree fromZhejiang University, China, in 2004, and theM. Sc. degree in wireless communicationsfrom the University of Southampton, UK, in2005. Since 2006, he has been pursuing thePh.D. degree at Institute of Digital Commu-nications, School of Engineering and Electron-ics, the University of Edinburgh, UK.

His current research interests include signalprocessing and wireless communications, withemphasis on identification of time-varying sys-

tem, channel estimation and equalisation, and their applications inwireless communications.

Lajos Hanzo received his M. Sc. degree inelectronics in 1976 and the Ph.D. degree in1983. In 2004 he was awarded the Doctorof Sciences (DSc) degree by the UniversitySouthampton, UK. During his 28-year careerin telecommunications he has held various re-search and academic posts in Hungary, Ger-many and UK. Since 1986 he has been with theSchool of Electronics and Computer Science,the University of Southampton, UK, where heholds the chair in telecommunications.

He has co-authored 11 John Wiley/IEEE Press books totallingabout 9000 pages on mobile radio communications, published in ex-cess of 550 research papers, organised and chaired conference ses-sions, presented overview lectures and has been awarded a numberof distinctions. He is an enthusiastic supporter of industrial-academicliaison. He also offers a range of industrial research overview courses.

Prof. Hanzo is a Fellow of the Royal Academy of Engineering(FREng), UK. He is an IEEE Distinguished Lecturer of both theCommunications Society and the Vehicular Technology Society aswell as a Fellow of both the IEEE and IEE. He is a non-executivedirector of the Virtual Centre of Excellence (VCE) in mobile commu-nications, UK, a governor of the IEEE VT society and an executiveboard member of the Pan-European Network of Excellence known asNEWCOM.