JIA Guo-qing(賈國(guó)慶)，CHEN Shan-ji(陳善繼)，JIXiao-hong(紀(jì)小紅)，CHEN Chao(陳 超)

School of Physics and Electronic Information Engineering，Qinghai University for Nationalities，Xining 810007，China

Training Sequence Design for Coordination on Multiple Point System

JIA Guo-qing(賈國(guó)慶)*，CHEN Shan-ji(陳善繼)，JIXiao-hong(紀(jì)小紅)，CHEN Chao(陳 超)

School of Physics and Electronic Information Engineering，Qinghai University for Nationalities，Xining 810007，China

Although the Cramer-Rao Bound(CRB)can be used as the benchmark of estimation algorithm performance，it's too com p licated for joint training sequence(TS)design for multip le input multiple output(M IMO)orthogonal frequency division multiplexing(OFDM)coordination on multiple point(CoMP) systems.So a m inimum mean square error(MSE)based suboptimal sequence design criterion was proposed，including ideal sequence correlation property and sequence length constraint.The simulation results verify the theory analysis.

coordination on multiple point(CoMP);minimum mean square error(MSE);training sequence(TS)design

Introduction

As one of the most efficient w ireless radio transmission schemes，multiple input multiple output(M IMO)orthogonal frequency divisionmultiplexing(OFDM)system is deployed in many systems，such as the 3rd generation partnership project (3GPP) long term evolution (LTE)， and worldw ide interoperability for m icrowave access(W IMAX).It offers favorable features such as high spectral efficiency，robust performance，and simple receiver architecture in double selective fading channel environment.

In traditional single cell transm ission systems，cell edge user equipment(UE)receives not only the useful signals from the service base station(BS)，but also the interference signals from neighboring BSs.Thus the throughput and the frequency efficiency of cell edge UEs are constrained.To solve this problem，coordination on multiple point(CoMP)transmission technology is proposed as shown in Fig.1.In which，the coordinated BSs transm it useful signals to cell edge UEs simultaneously，thus the interference signals is converted as the useful signals and the performance of cell edge UEs could be enhanced obviously.

In OFDM system，inter-carrier interference is avoided through the inherentorthogonality of sub-carriers.However，the oscillators＇difference between the transmitter and the receiver w ill degrade this orthogonality and introduces significant intercarrier interference，thus the frequency offset should be estimated and compensated accurately.In addition，the channel impulse response(CIR)also should be estimated to demodulate the OFDM system.For the traditional OFDM system，there are many research contributions on carrier frequency offset(CFO) and CIR estimation and joint training sequence design.Maximum likelihood(ML)CFO and CIR estimation along w ith the Cramer-Rao Bound(CRB)for single input single output (SISO)OFDM systems were given in Ref.［1］.Besson and Stoica presented the ML estimation on CFO and CIR forM IMOOFDM system［2］and the joint optimal training sequence(TS) design criterion based on asymptotic CRB［3-4］.The sim ilar contribution was made in Ref.［5］，but it was based on the m inimum mean square error(MSE)criterion.An average CRB for CFO estimation in frequency selective fading channels was derived in Ref.［6］.References［7 -8］gave ML synchronization and channelestimation for SISO OFDMA uplink transmissions and the CRB of CFO，respectively.References［9-10］gave the CFOs CRB analysis in CoMP system.

1 System Model

Although the BS couldn＇t be restricted to single antenna transmission，TS is transmit on only one antenna inmost of the w ireless system［11］，such as LTE.Without loss of generality，we suppose the TS is transmitted on only one antenna of a specific BS，the CoMPsystem includes M coordinated BSs，and the frequency domain TS of antenna m(1≤m≤M)is denoted as Sm，then the time domain TS xmis

where 0≤ n≤ N+Ng-1 is the samples index，εm∈(-0.5 0.5)is the sub-carrier spacing normalized CFO between BS m and UE.ω(n)is the Gauss white noise w ith mean zero，varianceσ2.After removing CP and combining all the received vectors as a total vector at the receiver side，we obtain

where Nris the number of receive antenna，matricesρm，Am， andare given by Eqs.(4)(6).

where1≤ p≤ N，1≤ q≤ Ng，anddenote all zeros matriceswith dimension N1×N2.Symbol［·］p，qdenotes the elementof amatrix in the row p and the column q.|·|Ndenotes themod N operation.

After furthermatrix processing，Eq.(3)can be simplified as Eq.(7)，where

Superscript(·)Hdenotes conjugate， (·)Tdenotes transpose.

2 Optimal TS Design

Although there are some optimal TS design methods based on asymptotic CRB in single cell system［34］，it can＇tbe used in CoMP systems as CRB is directly related to CFOs and CIRs in CoMP systems.Thus the asymptotic CRB based TS design criterion isn＇t applicable in CoMP systems.

2.1 Optimal TS selection criterion

Here we use themethod of minimum MSE to give a suboptimal TS design criterion.The log-likelihood probability function of received vector y can be expressed as

Equation(12)gives the log-likelihood function of received vector y.Defineandare trials ofφandη，thus the optimalandcan be obtained by maximizingΛ(y;，).Here we keepfixed and letvary，so the log-likelihood function Λ(y;，)maximizing when

Substituting Eq.(13)into Eq.(12)and maximizing w ith respect to，we obtain

Then the MSE of the CIR estimation can be calculated as

Thus the TS design criterion considering CIR estimation accuracy is

From Ref.［5］，it＇s found that MSE can approach the m inimum value whenφHφ =P Iwhen transmit power P is restricted，thus

The above equation is equivalent to

That is to say，the optimal TS satisfies

From the above equation，we find the optimal TS based on m inimum MSE criterion not only relates to correlation property of TSs，butalso to CFOs.For a specific system TS design，this is unachievable.Thuswe neglect the factor of CFOs，and give a sub-optimal TS design criterion as follows

This is to say，the sub-optimal TS has the ideal autocorrelation and cross-correlation properties.These kinds of sequence include white sequence，Zadoff-Chu(ZC)sequence，etc.

Rew rite Eq.(7)as follows

where

Then the optimal estimation ofρ-is

where RΩ=ηηHis the channel correlation matrix，thus the optimal TS design can be got by m inimum the follow ing equation

For system design，the channel correlation property can＇t be known.Here we suppose the channel is ideal，that is RΩbeing an identity matrix，so

This criterion is equivalent to Eq.(21).Thus the optimal sequence design for CFOs and CIRs estimation is the same，which is ideal correlation property.That is to say，Gauss white is the optimal TS.

2.2 TS length design criterion

Equation(13)mustmeet the condition thatmatrixφHφis full rank，

For the above block diagonalmatrix，the full rank property is equivalent to sub-block matrix QHQ full rank.That is equivalent to Q full column rank，i.e.，rank{Q}=MLh.Thus the condition MLh≤N must be fulfilled.

3 Simulation Results

The simulation parameters are set as shown in Table 1.

In addition，we suppose the CIRs between different BSs are independent and irrelevant to each other.For each path of a specific CIR，it＇s a Gauss random variable w ith zerosmean and variance as follows

Figure 2 gives the CRB for CFOs and CIRs estimation for ZC and Gauss white sequence，from which we find that the Gausswhite sequence outperforms ZC in every scenario.So we conclude that Gausswhite sequence is the optimal sequence for CoMP system.

In addition，the performance of MLE compared w ith CRB under white sequence is given in Figs.3-4，and the conclusions are:

1)for single cell system，the ML estimation performance approaches CRB very close，that is to say the ML is the m inimum variance unbiased(MVU)estimation;

2)for CoMP system，the MLE estimation performance curve is a bit far from CRB curve because of the orthogonality loss by multiple CFOs and CIRs.

4 Conclusions

Although the CRB can be used as the benchmark ofestimation algorithm performance，it＇s too complicated for joint TS design in CoMP system.So a m inimum MSE based suboptimal sequence design criterion is proposed in this paper.We find that the Gauss sequence gives a minimum CRB in this design criterion.ML estimation of the CFOs and CIRs are performed in numerical simulation to verify the analysis result.

［1］Morelli M， Mengali U.Carrier-Frequency Estimation for Transm issions over Selective Channel［J］.IEEE Transaction on Communications，2000，48(9):1580-1589.

［2］Besson O，Stoica P.On Parameter Estimation of M IMO Flat Fading Channels with Frequency Offsets［J］.IEEE Transaction on Signal Processin g，2003，51(3):602-613.

［3］Stoica P，Besson O.Training Sequence Design for Frequency Offset and Frequency-Selective Channel Estimation［J］.IEEE Transaction on Communications，2003，51(11):1910-1917.

［4］Besson O，Stoica P.Training Sequence Selection for Frequency Offset Estimation in Frequency Selective Channels［J］.Digital Signal Processing，2003，13(1):106-127.

［5］Cui T，Chintha T.Joint Channel and Frequency Offset Estimation and Training Sequence Design for M IMO Systems over Frequency Selective Channels［C］.IEEE Global Telecommunications Conference，Dallas，TX，USA，2004:2344-2348.

［6］Li Y H，Hlaing M，Zeng JQ.An Average Cramer-Rao Bound for Frequency Offset Estimation in Frequency-Selective Fading Channels［J］.IEEE Transactions on Wireless Communications，2010，9(3):871-875.

［7］Kim K J，Pun M O，Iltis R A.Joint Carrier Frequency Offsetand Channel Estimation for Uplink M IMO-OFDMA Systems Using Parallel Schm idt Rao-Blackwellized Particle Filters［J］.IEEE Transaction on Communications，2010，58(9):2697-2708.

［8］ Pun M O，Morelli M，Kuo C C.Maximum-Likelihood Synchronization and Channel Estimation for OFDMA Uplink Transm issions［J］.IEEE Transaction on Communications，2006，54(4):726-736.

［9］Zarikoff W B，Cavers J K.Coordinated Multi-cell Systems: Carrier Frequency Offset Estimation and Correction［J］.IEEE on Selective Areas in Communications，2010，28(9):1490-1501.

［10］Parker A P，Patrick M，Bliss D W，et al.On Bounds and Algorithms for Frequency Synchronization for Collaborative Communication Systems［J］.IEEE Transactions on Signal Processing，2008，56(8):3742-3752.

［11］ Kay S M.Fundamentals of Statistical Signal Processing: Estimation Theory［M］.Englewood Cliffs:Prentice-Hall，1993: 27-82.

TN919 Document code:A

1672-5220(2015)02-0337-04

date:2014-10-10

s:International Science＆Technology Cooperation Projects of Qinghai，China(Nos.2013-H-811，2014-HZ-821);Chunhui Plan Projects，China(Nos.Z2014013，Z2014014)

* Correspondence should be addressed to JIA Guo-qing，E-mail:13709759761@139.com