Low-complexity fractionalphase estimation for totally blind channelestimation

,2,*

1.Departmentof Electronic Engineering,Fudan University,Shanghai200433,China; 2.Key Laboratory of EMWInformation,Fudan University,Shanghai200433,China

Low-complexity fractionalphase estimation for totally blind channelestimation

Xu Wang1,Tao Yang1,and Bo Hu1,2,*

1.Departmentof Electronic Engineering,Fudan University,Shanghai200433,China; 2.Key Laboratory of EMWInformation,Fudan University,Shanghai200433,China

To remove the scalar ambiguity in conventional blind channel estimation algorithms,totally blind channel estimation (TBCE)is proposed by using multiple constellations.To estimate the unknown scalar,its phase is decomposed into a fractional phase and an integer phase.However,the maximum-likelihood (ML)algorithm for the fractionalphase does not have closed-form solutions and suffers from high computational complexity.By exploring the structures of widely used constellations,this paper proposes a low-complexity fractional phase estimation algorithm which requires no exhaustive search.Analyticalexpressions ofthe asymptotic mean squared error(MSE)are also derived.The theoretical analysis and simulation results indicate that the proposed fractional phase estimation algorithm exhibits almost the same performance as the ML algorithm but with signi fi cantly reduced computationalburden.

orthogonal frequency division multiplexing(OFDM), totally blind channel estimation(TBCE),scalar ambiguity,fractionalphase,low-complexity.

1.Introduction

To accommodate the demand on high speed data communications,coherent detection is preferred in wireless systems,which entails channel state information and thus channel estimation at the receiver.The scheme of pilots or training sequences,though simple,signi ficantly reduces the system throughput.To eliminate the bandwidth-consuming pilotsignals,considerable research efforts have been devoted to the technique ofblind channel estimation[1-6].Forexample,Al-Naffourietal.proposed a blind maximum-likelihood(ML)estimation algorithm for orthogonal frequency division multiplexing(OFDM) systems in[5],where no constraintis imposed on the constellation.Besides,the complexity of the algorithm can be reduced by subcarrier reordering.

Recently,the ambiguity problem in blind channel estimation attracts much attention.For conventional algorithms like Al-Naffouri’s method,there is an unknown scalar which cannot be identi fi ed.If pilots are used to estimate the ambiguous scalar,then the algorithm becomes semi-blind.It was considered as an inherent problem in blind channelestimation for many years and[7]even provided a proof for that.

However,no a prioriinformation of the transmitted signals was assumed in[1-7].Some recent blind channelestimation algorithms fi nd that scalar ambiguity can be removed by taking advantage of information of the constellation,and then the channelis uniquely identi fi ed in a blind fashion,i.e.totally blind channel estimation(TBCE).In space-time block coding(STBC)systems,[8,9]showed thatTBCE can be achieved iftwo phase shiftkeying(PSK) constellations are used and their constellation orders are coprime.In OFDM systems,it was fi rst shown in[10] thatthe quadrature amplitude modulation(QAM)constellation can be used in TBCE.Reference[11]further proposed a multiple-constellation scheme that allows for all kinds of constellations;hence the widely used constellations,such as PSK,pulse amplitude modulation(PAM), amplitude and phase shift keying(APSK)and QAM,are allfeasible for TBCE.

In OFDM systems,to blindly estimate the unknown scalar,its phase is decomposed into a fractionalphase and an integer phase.The integer phase can be solved with closed-form solutions.Besides,its amplitude can be easily obtained by the second-order statistics(SOS).Therefore, the mostdif fi cultproblem is the estimation ofthe fractional phase.The ML method was adopted in[10,11],which requires exhaustive search and suffers from high computationalcomplexity.

In this paper,a low-complexity fractionalphase estimation algorithm is proposed.By exploiting the structures of widely used constellations,such as PSK,PAM,APSK andQAM,no exhaustive search is needed when estimating the fractional phase.To improve the estimation accuracy,the weighted average is exploited and the weights are carefully designed.We also derive the analytical expression of the asymptotic mean squared error(MSE).Numerical results con fi rm thatthe proposed fractionalphase estimation algorithm exhibits almostthe same performance as the ML algorithm,butwith signi fi cantly reduced computationalburden.

2.Fractionalphase in TBCE

2.1 System model

Let us consider an OFDM system.The input signal s(n):=[s0(n),s1(n),...,sN?1(n)]Ttakes N-point inverse fastFouriertransform(IFFT).Afteradding the cyclic pre fi x(CP),the resulting time domain signalis transmitted over the multipath channel.At the receiver,the signal on the k th subcarrier is given by

There are many algorithms in the literature which can estimate the channel within a scalar ambiguity[12-14]. Here we assume that the channel H has been identi fi ed up to an unknown scalar c by one of these algorithms,i.e. ?H=H/c.Then(1)can be rewritten as

where wk

2.2 TBCE in OFDMsystems

Here,the main ideas of TBCE from[11]are brie fl y introduced.To remove the scalar ambiguity,it is proposed to use multiple constellations in differentsubcarriers[11].At fi rst,the precise meaning of the constellation is given as follows.

Definition 1[11]A constellation S is a fi nite set of complex numbers with two or more elements.

Note that the trivial case that S contains only a single element is excluded in De fi nition 1.Next,the concept of the symmetric number is introduced.Denote cS:={cs: s∈S}.

Definition 2[11]If a phaseα∈[0,2π)satis fi es

then it is a symmetric phase of constellation S.All the symmetric phases constitute the symmetry set,denoted by AS.The cardinality of ASis called the symmetric number, denoted by QS.

For PAM,PSK and QAM constellations,the symmetry set and the symmetric number are[11]:PAM,AS=M;square QAM,AS={0,π/2,π,3π/2},QS=4. For these constellations,ASand QSsatisfy the following relationship

In fact,itcan be proved that(4)holds foran arbitrary constellation[11].

The main idea of TBCE relies on the exploitation of multiple constellations.More speci fi cally,to remove the scalar ambiguity,differentconstellations are used in differentsubcarriers.These constellations are carefully selected so that the scalar c can be uniquely determined.The following theorem gives the criterion for the constellation selection.

Theorem 1[11]Assume that each subcarrier sk(n) uses constellation Sk(k=0,1,...,N-1),respectively. If the input sequence{s(n)}nis suf fi ciently long in the time index n such that sk(n)contains all the possible values of Sk,then the unknown scalar c can be blindly and uniquely identi fi ed if and only if

where Qkis the symmetric number of Sk,and gcd(·) stands for the greatcommon divisor.

For example,we can use 16-QAMin a few subcarriers, and 3-PSK in other subcarriers,then gcd(4,3)=1,and the scalar c can be uniquely determined.In Theorem 1,the constellations used in different subcarriers S0,...,SN?1are notnecessarily distinct.

Remark 1The adaptive modulation(AM)technique also exploits multiple constellations[15,16],where the relative positions of the constellations are determined by the channelcondition.For example,16-QAM should be used in subcarriers with a large channelgain,while 3-PSK can tolerate subcarriers with the poorer channelcondition.On the other hand,for TBCE,the constellations are selected according to condition(5),and their relative positions are irrelevant because gcd(x,y)=gcd(y,x).Therefore,we can use AM and TBCE together.Note that the signal-tonoise ratio(SNR)ofeach subcarrieris determined by|Hk|, which is independentof the phase ambiguity.Besides,the amplitude of the scalar c can be easily solved with SOS. Thus the channelstate information needed by AM can be readily obtained by conventionalblind channelestimation algorithms.Then this information is fed back to the transmitter and the constellations are selected according to thecriteria of TBCE and AM.For example,we can use 16-QAM in subcarriers with the good channelcondition,and 3-PSK with the smaller channelgain.

2.3 Fractionalphase

To estimate the scalar c,its phase is decomposed into a fractionalphase and an integer phase.For the sake of clarity,letus focus on a single subcarrier fi rst.

Definition 3[11]By(4),the phase of the scalar c can be decomposed as

whereθ∈[0,2π/QS)is called the fractional phase,and K∈{0,1,...,QS-1}is called the integer phase.

The integer phase does not change the shape of a constellation,i.e.ej2π/QSS=S.Thus it cannot be identifi ed.On the other hand,if a constellation is rotated by a fractional phase,it no longer looks the same.Thus the fractional phase can be uniquely determined.Let us take 16-QAM as an example.Assume that arg(c)=3π/4= π/4+π/2,thenθ=π/4,K=1.First,the fractional phaseπ/4 can be identi fi ed,because the shape of the constellation is changed when it is rotated byπ/4.However, 16-QAMis invariantunder a rotation of phaseπ/2,so the integer phase K may be estimated as 0,1,2 or 3.Thus the value of K cannot be uniquely determined based on the received signalonly.

If multiple constellations are used and they satisfy condition(5),then the integer phase can be uniquely determined.For example,16-QAMis invariantundera rotation of phases{0,π/2,π,3π/2},and 3-PSK is invariantunder phases{0,2π/3,4π/3}.Hence,if both 16-QAM and 3-PSK are used,then the integer phase mustbe“0”.The interested reader can refer to[11]for more discussions.

Now let us consider the estimation of the scalar c.The amplitude can be directly obtained via the SOS[10].The integerphase K can also be solved in the closed-form,provided thatthe fractionalphases ofallsubcarriers have been estimated[11].Thus the computationalcomplexities ofthe amplitude and the integerphase are very low.However,the estimation ofthe fractionalphaseθis much harder.On the k th subcarrier,letthe phase of c be written in the form of (6)with respectto Sk,then(2)can be rewritten as

where sk(n)∈Sk,and Qkis the symmetric number of Sk.In[10,11],the ordinary ML estimation method is adopted.The estimate ofθis obtained by maximizing the log-likelihood function of rk(n).Unfortunately,the fractional phaseθcannot be solved in closed-form due to the highly nonlinearrelationship between rk(n)andθ,and exhaustive search is needed to determineθ[10,11].

3.Low-complexity fractionalphase estimation

The conventional ML method suffers from the high computationalcomplexity since the estimate ofθcannotbe obtained in the closed-form.To overcome this problem,we propose a novel low-complexity fractional phase estimation algorithm.By exploiting the structures of widely used constellations such as PSK,PAM,APSK and QAM,no exhaustive search is required when estimatingθ,and the computational complexity is greatly reduced.Besides,to improve the estimation accuracy,weighted average is exploited and the weights are carefully designed.

3.1 PSK constellations

For M-PSK constellations,the transmitted information is conveyed by the phase of the inputsignal sk(n),where the constellation points are uniformly spaced on the unitcircle,Noting that the symmetric number Qk=M,(7)can be rewritten as

The fractionalphaseθis estimated as

or equivalently,

where≡denotes congruence[17],andis the greatest integer notexceeding x.

3.2 PAMconstellations

For M-PAM constellations,the transmitted information is conveyed by the amplitude of the input signals sk(n), where all the constellation points lie on the real line,and.Noting that the symmetric number Qk=2,(7)can be rewritten as

Thenθis estimated as

If multiple observations are available,we can take the average to reduce the noise effect.However,unlike PSK, the amplitude of PAMis no longer a constant,so the SNRs of different data symbols are not the same.To get better performance,the weighted average is preferred,and the weights should be carefully designed.According to(31)(see Section 4.2),for a given transmitted signal sk(n),the variance of estimator(12)satis fi es the following relationship,

where Quan{}is the quantization operation to the PAM constellation,andβis a normalizing constant such that β?βn=1.

3.3 APSK and QAMconstellations

Now we turn to the more complicated APSK and QAM constellations,in which both the amplitude and the phase are modulated with information.The APSK constellation consists of several circles and is adopted in digital video broadcasting(DVB)systems[19].We take 32-APSK as an example to discuss the fractional phase estimation of APSK constellations.

The 32-APSK constellation is constituted of three circles,which are 4-PSK with radius R1,12-PSKwith radius R2,and 16-PSK with radius R3,respectively(see Fig.1). Thus 32-APSKcan be treated as a combination of PSKand PAM constellations.Its symmetric number is Qk=4.To estimate the fractional phaseθ,let us consider each circle fi rst.

Fig.1 Scatter diagram of the 32-APSK constellation

If|rk(n)/c|＜(R1+R2)/2,then sk(n)belongs to the inner circle which is equivalent to a 4-PSK constellation,so we can get an estimate?θinby(10).If|rk(n)/c|＞ (R2+R3)/2,then sk(n)belongs to the outer circle,

Considering the phases ofboth sides,we have

Note thatfor any integer n,

Thus the value of m can be restricted to m∈{0,1,2,3}. Furthermore,m can be uniquely determined by lettingin(17).Otherwise,suppose thatthere are two solutionsboth satisfying(17),thenis an integer multiple ofπ/2,which is impossible becauseBy substituting the value of m back into(17),we getan estimateFor the middle circle,an estimatecan be obtained in a similar way.

Next,letus combine the three estimates.However,they have different estimation accuracy since the radii of the three circles are not the same.Hence,the weighted average should be adopted.Similar to the PAM case,the best linear combiner[18]which minimizes the estimation variance is given by

The QAM constellations can be estimated similar to APSK.Taking 16-QAMforexample,if only the inner and outer circles are exploited(see Fig.2),thenθcan be estimated as

Fig.2 Fractionalphase estimation for 16-QAM

4.Performance analysis

In this section,in orderto assess the estimation accuracy of the proposed algorithm,analyticalexpressions of the MSE are derived.It is proved that for most constellations,the proposed algorithm achieves the same asymptotic MSE as the ML method.

4.1 Constrained Cramer-Rao bound

First,let us investigate the Cramer-Rao bound(CRB)for the fractional phase.In(7),there are three unknown parametersθ,K and sk(n).To obtain the CRB,we need to calculate the Fisher information matrix with respect to all the three unknown parameters[18].However,both K and sk(n)are not continuous,and their values are confi ned to discrete sets,i.e.there are constraints on the unknown parameters.Therefore,the constrained CRB,which was introduced in[20,21],should be used.Let L be the number of received OFDM blocks,and denoteγk:=the SNR of the k th subcarrier.

Theorem 2The constrained CRB for fractionalphase estimation is

ProofWe introduce the following result from[22].If an unknown parameter is restricted to a fi nite set in the complex plane,then the constrained CRB can be found by deleting the signalsamples from the corresponding Fisher information matrix,as if this parameter was known.

From this result,since both K and sk(n)are con fi ned to fi nite sets,we treat them as known parameters in the derivation of CRB.Note that they are not needed by the algorithms developed in the previous section.

Nowθis the only unknown parameter in(7).Let r= [rk(1),...,rk(L)]Tbe the received signalvector.Its loglikelihood function is

4.2 Asymptotic MSE

To gain more insight into the proposed algorithm,we derive the asymptotic MSE(AMSE)in the high SNR region.

Theorem 3The AMSE of the proposed algorithm is

ProofDenoteθthe true value ofθ,andσ2sthe transmitted signal power.Then,we will discuss the four kinds of constellations,respectively.

PSK:Because the asymptotic property is investigated, the noise poweris assumed to be small.Therefore,

and(24)follows immediately.

PAM:If sk(n)=d,thenβn=|d|2because the noise power is small.Therefore,

Similar to(29),it can be shown that for a given signal sk(n),we have

Therefore,when sk(n)is drawn from the entire constellation,

APSK:Denotethe estimate from the i th circle.Since the noise power is small,the estimate of m is correct. Hence,the variance o fis the same as PSK,i.e.

where Riis the radius of the i th circle,andμiis the number of points belonging to the i th circle.Note that

QAM:Equation(27)can be proved in a similar way to the APSK case.

Theorem 3 shows that the proposed algorithm attains the CRB for PSK,PAMand APSK constellations ata high SNR.Thus the weights designed in the previous section are optimal.On the other hand,the ML estimator is known to achieve the CRB asymptotically[18].Therefore,for these constellations,the proposed algorithm and the ML method have the same AMSE performance,which implies thatthe computationalcomplexity involved in the estimation ofthe fractionalphase can be reduced by the proposed algorithm withoutperformance loss.

Unfortunately,the AMSE for QAMconstellations loses this desired property because only a partof the data is exploited.If the betterperformance is required,then the proposed algorithm can provide a coarse estimate for the ML algorithm to reduce the search range ofθ.

5.Simulation results

In this section,computer simulations are carried out to evaluate the performance of the proposed fractionalphase estimation algorithm.All the simulation results are averaged over 3 000 Monte Carlo trials.

5.1 MSE performance

To validate the theoreticalanalysis in the previous section, here we investigate the MSE performance of the proposed algorithm with differentkinds of constellations.The number of received OFDM blocks is chosen as L=60.

Fig.3 shows the performance for PSK constellations. At the low SNR,4-PSK performs better than 8-PSK,and BPSK has the bestperformance.When the SNR is higher than 20 dB,allof them achieve the AMSE as predicted by Theorem 3.Fig.4 shows the simulation results for PAM constellations.Unlike PSK,the performance of PAM only degrades slightly as the constellation size increases,and all the three constellations approach the AMSE closely in the whole range of SNR.

Fig.3 MSE of PSK constellations

Fig.4 MSE of PAMconstellations

Fig.5 depicts the MSE for APSK constellations. In accordance with DVB systems[19],for 16-APSK,R2/R1=2.57;for 32-APSK,R2/R1=2.53, R3/R1=4.3.We observe that 16-APSK has better performance than 32-APSK when the noise power is strong. In the high SNR region,both constellations achieve the AMSE.Fig.6 shows the performance of 16-QAM.Similar as before,the proposed algorithm approaches the AMSE as the SNR grows.

Fig.5 MSE of APSK constellations

Fig.6 MSE of QAMconstellations

5.2 Overallsystem performance

Now we compare the performance of the proposed algorithm with thatof the ML method.Since the ultimate goal ofdigitalcommunicationsis to recoverthe transmitted signals,the biterror rate(BER)is chosen as the performance measure.The OFDMsystem undertesthas N=64 subcarriers,where four subcarriers are virtual subcarriers which are distributed evenly atthe spectrum edges.The other 60 subcarriers are used for data transmission.The channelorder is set to 16.The number of received OFDM blocks is L=300.To totally blindly estimate the channel,the subspace method in[24]is used fi rst to estimate the multipath channel up to a scalar.Then the amplitude and the integer phase of the unknown scalar are estimated by the algorithms in[11],while the fractionalphase is estimated by the proposed algorithm and the ML algorithm.

In the simulations,for odd order PSK constellations, e.g.3-PSK,a point“0”is added to the constellation(see Fig.7),so thateach symbolrepresents an integermultiples ofbits,and the symmetric numberremains unchanged.The modi fi ed constellation is called generalized PSK(GPSK) hereafter.The algorithm and performance analysis developed for PSK constellations in previous sections can be extended to GPSK constellations straightforwardly.

Fig.7 Scatter diagram of the 3-GPSK constellation

Fig.8 shows the BER performance when 16-APSK and 3-GPSK constellations are used,whose symmetric numbers are four and three,respectively.Note that gcd(4,3)= 1,so condition(5)for TBCE is satis fi ed.The 16-APSK constellation is used over54 subcarriers in the centerofthespectrum,while 3-GPSK is used over six subcarriers located at the edges.It can be seen that the proposed algorithm has the same BER performance as the ML method. The reason is thatfor APSK and GPSK constellations,the AMSE of the two algorithms are the same.

Fig.8 BER performance of 16-APSK and 3-GPSK

Fig.9 shows the BER performance when 16-QAMand 7-GPSK constellations are used,whose symmetric numbers are fourand seven,respectively.Note thatgcd(4,7)= 1,so condition(5)is also satis fi ed.From the fi gure,we fi nd that the proposed algorithm closely approaches the ML method when the SNR is higher than 25 dB.At the lower SNR,slight performance degradation is observed. This is because for the 16-QAM constellation,only parts of the constellation points are exploited by the proposed algorithm.

Fig.9 BER performance of 16-QAMand 7-GPSK

Finally,the computationalcomplexities of the proposed algorithm and the ML method are compared.Table 1 shows the runtime consumed in the experiment.The simulation is carried outin a personalcomputer(PC)with an Intel Core 2 Duo E8400 processor at3 GHz.The program is written in C++,and the IT++library[25]is invoked.From the table,it can be seen that the proposed algorithm requiresmuch less computationaleffortthan the ML method. For example,if 16-QAM and 7-GPSK are used,the time consumed by the proposed algorithm is only 1/20 of that consumed by the ML method;for 8-PSK and 3-GPSK,the ratio is only 1/30.

Table 1 Computationaltime min

6.Conclusions

There is a scalar ambiguity in conventional blind channel estimation algorithms.Recently,TBCE is proposed to solve this ambiguity problem by using multiple constellations.The main disadvantage of TBCE is the high computational complexity involved in the estimation of the unknown scalar,which willlimitits implementation in practice.For example,itcannotbe employed in low-cost,lowpower scenarios.Besides,due to the long processing time, itis notsuitable for delay-sensitive applications.

For OFDMsystems,the computationalburden of TBCE is dominated by the estimation of the fractionalphase.In this paper,by exploiting the structures of widely used constellations such as PSK,PAM,APSK and QAM,we propose a low-complexity algorithm which does not need exhaustive search.To improve the estimation accuracy,the weighted average is exploited and the weights are carefully designed.We also derive the constrained CRB and AMSE.Simulation results indicate thatthe proposed fractionalphase estimation algorithm exhibits almostthe same performance as the ML algorithm butwith signi fi cantly reduced computational burden.Thus,with the help of the proposed algorithm,TBCE can be used in a much wider range of applications.

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Biographies

Xu Wang was born in 1986.He is currently a Ph.D.candidate at Fudan University,Shanghai, China.He is a studentmember of the IEEE Communications Society and Signal Processing Society.His research interests include blind channelestimation and synchronization in OFDMsystems.

E-mail:xwang.fdu@gmail.com

Tao Yang was born in 1972.He received his B.S.degree from Shaanxi Institute of Technology in 1994 and M.S.degree from Shandong University in 2000,both in automation.He received his Ph.D.degree in control theory and application from Shanghai Jiaotong University in 2004. In 2007,He joined Fudan University,where he is currently an associate professor.His research interests include signal processing for wireless communications,intelligent signalprocessing,network information sensing and fusion.

E-mail:taoyang@fudan.edu.cn

Bo Hu was born in 1968.He received his B.Sc.and Ph.D.degrees in electronic engineering from Fudan University,Shanghai,China in 1990 and 1996,respectively.He is currently a professor with the Department of Electronic Engineering,Fudan University and serves as the vice dean of Information Science and Technology School.He is a memberofthe IEEE Communications Society and Signal Processing Society.His research interests include blind signalprocessing,digital image processing,wireless communications,and digital system design.

E-mail:bohu@fudan.edu.cn

10.1109/JSEE.2015.00028

Manuscriptreceived January 22,2014.

*Corresponding author.

This work was supported by the National Science and Technology Major Project of China(2013ZX03003006-003).