Semiparametric theory based MIMO model and performance analysis

THE JOURNAL OF CHINA UNIVERSITIES OF POSTS AND TELECOMMUNICATIONSVolume 14, Issue 4, December 2007XU Fang-min, XU Xiao-dong, ZHANG PingSemiparametric theory based MIMO model and perfor-mance analysisCLC number TN929.5Document AArticle ID 105-8885 (2007) 04-0036-05Abstract In this article, a new approach for modeling multi-model will not work well. Contrarily, the linear model willinput multi-output (MIMO) systems with unknown nonlinearcause serious system errors. Such problems are common ininterference is introduced. The semiparametric theory basedMIMO systems, for example, pulse noise, interference causedMIMO model is established, and Kernel estimation is applied toby the correlation between antennas or system coexistence,combat the nonlinear interference. Furthermore, we deriveinherence errors in the systems, etc. Hence, the influenceMIMO capacity for these systems and explore the asymptoticfactor of received signal concludes both linear parts andproperties of the new channel matrix via theoretical analysis.nonlinear parts. Generally, the channel condition is consideredThe simulation results show that the semiparametric theoryto affect the received signal linearly. In the meantime, therebased modeling and kermel estimation are valid to combat thisexist nonlinear parts and random factors (e.g. noise). Sincekind of interference.relationships between the nonlinear parts and the receivedsignal are unknown and there is no reason to treat nonlinearKeywords MIMO, partial linear model, kernel estimation,parts as noise, we treat nonlinear parts as an extra part in theunknown nonlinear interferenceMIMO model.In this article, the semiparametric theory based MIMO1 Introductonmodel is established to describe MIMO systems afected bynonlinear inteference. A kernel estimated method is used toMIMO technology, as an important way to improvecombat unknown nonlinear interference, which has nearly nottransmission reliability and provide high data rates, has beenbeen resolved, and the system capacity is obtained under thean attractive candidate for the physical layer in enhanced 3rdsemiparametric theory based MIMO model.generation(B3G) and 4th generation (4G) wireless systems.The remaining sections of the article are organized asThe pioneering work of Telatar [1] and Foschini [2] has shownfollows. The system model based on semiparametirc theory isthat using multiple antennas offers remarkable spectraldescribed in Sect, 2. Section 3 analyzes the capacity based onefficiency. Since then, research activities to develop optimalthe established model. In Sect. 4, we provide the asymptoticimplementations as well as to extend MIMO systems haveproperties of the new channel matrix H and validate theexploded. Several algorithms for MIMO systems have beenproperties in Sect. 5 via numerical simulation. Anotherstudied, but most are based on the traditional linear modelsimulation result in Sect. 5 shows that Kernel estimation is[3- -5]. However, the practical wireless environment isvalid to combat this kind of interference. In Sect. 6, wecomplex and variable. It will not obey linear rules very wellconclude with a brief summary of results.and can only be treated as linear model approximately. Toimprove the transmission reliability, a precise model to2 Systom model based on the semlparamotlcdescribe MIMO systems is required for 4G wireless systems.theoryTraditionally, the linear model [1-5] is usually used todescribe MIMO systemns to study the relationship between theWe consider a single user, point-to-point communicationreceived signal and its influence factor. It is well known thatchannel interfered by nonlinear interference, with p transmittervhen nonlinear interference exists in the system, the linearantennas and n receiver antennas, and describe the systemmodel as follows:Recceived dace 207-01-11Y=H'X+ e(T)+e1)XU Fang-min (8), Xu Xiao dong, ZHANG PingKey Laboratory of Universal Wireless Communications (Bejing Univenity ofwhe,中國煤化工x):; e=e. ...Posts and Telecommunications) Ministy of Education, Wireless TechnologyInnovation Institute, Bejing 100876, China:MHCN M H ...H.);j T=(T, T.... .E-mail: fangminxu@gmail.comNo. 4XU Fang- min, et al: Semiparametric theory based MIMO model and performnance analysis37T)"; Y and X describe the received signal vector and theY=Hx+e; 1≤i≤n(6transmitted vector, respectively; T describes the nonlinearTherefore, Eq. (6) is equal to the case that the channelinterference factor vector; H describes a pXn random matrixcondition of transmitted signal X is H. In this case, thecorresponding to the channel condition; {(H,T), 1≤i≤n}received signal is Y, and no nonlinear interference is left inand {ep 1≤i≤n} are independent; 8(T) is an unknownfunction defined in close interval (without loss of generality,the system.As the channel condition is known well at the receiver butassume that the close interval is [0, 1]) and describes unknownnonlinear interference; er, ....n.. are independent identicallynot known at the transnitter, the system capacity of Eq. (6)can be expressed bydistributed (i.i.d.) random samples; en describes additive whiteGaussian noise (AWGN) with mean 0 and variance o2 forC=E| logdet|1,+-+ pT(7)po21≤i≤n. Equation (1) can be described as follows:received signal-transmitted signal x channel condition+where P, is the transmitter power.unknown nonlinear interference + noise.Applying singular-value decomposition to H，we canIn the next section, we will try to combat the unknownrewrite Eq. (7) as:nonlinear interference by kernel estimation and derive thecapacity of these systems.C=E(8)(藝"(+器))3 Capacty analysoswhere h is the eigenvalueof畝.In fact, Bq. (1) is a kind of partial linear model in Statistics.4 Asymptote proportesFurthermore, this kind of partial linear model [6, 7], alsoalleld the semiparametric regression model, concludes notWe will provide some asymptotic propertiesof F in thisonly the parametric components, such as HTx in Eq. (1), butsection.also nonparametric components such as 8(T) in Eq. (1). Hence,Firstly, the following assumptions, easy to be satisfied, areit has stronger explanations in application than single-required in Theorem 1.character regression models (e.g. linear models) and can take1) Ti has a continuous density function舊with tE[0,1]ull advantages of data information. As a result, theand 0< inf r(t)≤sup r(t)<∞;05161semiparametric theory has been applied widely [8,9].2) For any tE[0,1], let 8()=E(HyJT=), for 1≤j≤p. g()In the following, a commonly used weighted function in theand 8y(t) satisfy the Lipschitz condition of order apartial linear model is used to estimate the interference, and(1/2znsζ'q'“工2粼國中(9Z)<(*8/1*12183*9>12()"m.學(xué)篇pue(8I)甜0←127*9>1xeu=(2)"Mxeul= 'qq ponoupques H xuneu y0 ]u3u2p-([) ?41J0OLd(H)x'I WJO9LJO jooud oqp ap!AoJd 9M 'MONsouS(tz)se ()o="2(L)"mxeu>2憶aAu咖'∞> p= 2的01 8upoov u1s8u!enqo 2M (z) '咀pue *(61) bg“(8I)婦qJM poqpo8oI習J0 2-'=) pue (n*>1'b)Y= 2 2nu>a J0ol(EZ)(9I)sB "()0='(l)"mZixeu0AEU oM 'Poyspes am b pue‘E 'I suopdunse JI r BwwIPoonpap S!(I) b的'的ou2H018uppJoav°∞←18ou2Yru hq(z)B '0l2xpomonogy (+y)o= (-o4(r1(u8o[8o1_u)opury Dupo aq u0AB叫2M "(Iz)婦pue (LI)咀8uuquoo(SI,4D+ ro(r(u8oq0_[_1)o >|"P(ym-1)(川)x{ 1dis o4.dW+()d-()"J1drspoIDO> |(un- )dP(m)x[ Idns rouydW‘0<9 Kue 10} °Eq anON(02)(∞←)0←Kr+(n-)d -(m )"*)(n)x{ Iispy4,. W<()"IPInsy°●1-川(=)xdns w= J1-21(號)水a(chǎn)ABq‘I 'I JO uonnq!ns!P peundua仰sB "u 8ugouapdxo"g |au了rq pure 'I J0 uopouny uonnquns!p觀(guān)se a 8unouap hg(t]) SB "《oy)0=()$-(D{)號)r學(xué)dieIanoud 01 paau Auo明“(I)婦aAoud o1 'I uopdunsse pue (ZI)咀8umoO0qI(EI)SB °0← +()4-()2| dns{[.<'[oI] 'FP8 " EIT WaJOaul 04 8u!po00z 3Aeq 3M y a8rel Apuapyyns J0J 'htpenbau! u!21SW2q8u!sn : (-(1)5>*9)1="θ 2nou2a ' "9nu>l"a|xeu pueDoqM(=)2-0》"xous(112了時(shí)為陽(yáng)0Aeq om 10qo qpoa(ZI)(伴- (-0-(0)學(xué)喘jo uapuadapup ame w..4 oous“{u>!>I "L}=%v()4_uUaA!D 4>8>I“《“29- *2)(L)*"M= " anousp ">!> 10H+0)F-(D(-)3- ()5 -CD(0"m了L0OZJdNHD Jo peunor PUL88No.4xU Fang-min, e al: Semiparameric theory based MIMO model and performance aysis39lage numbers, we get, for I≤i,j≤p,estimation is valid to combat this kind of interference.-切共之BB→EB.A,=E.. as(27)1.2r.0-Value of true function gwhere E =cov(Hu- E(Hu IT),H,- E(H,IT) denotes thex Estimation value ofg.8-(ij)-element of matrix，.Here,we deal with9 ,o ,and 9.0.6It is shown straightforwardly that.4--2B1→EIB.Km眩IKi