Channelmodeling and in fl uenced factoranalysis ofthe broadband dual-orthogonalpolarized MIMO land mobile satellite channel

1.College of Astronautics,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China;

2.College of Hydrology and Water Resource,Hohai University,Nanjing 210089,China

Channelmodeling and in fl uenced factoranalysis ofthe broadband dual-orthogonalpolarized MIMO land mobile satellite channel

Qingfeng Jing1,*,Jiajia Wu1,Yuping Lu1,Xin Liu1,and Xiaoju Yan2

1.College of Astronautics,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China;

2.College of Hydrology and Water Resource,Hohai University,Nanjing 210089,China

An accurate,complete and realistic channelmodelis required to accurately analyze the system performance ofa multiple input multiple output(MIMO)broadband satellite mobile communication system with dual-orthogonal polarized antennas(DPAs). In mostcurrent studies,the channelcharacteristic matrix(CCM)is always formed by an independentidenticaldistribution(i.i.d)model of Rayleigh or Rice distribution and nevertheless incomplete and inaccurate to describe a broadband dual-orthogonal polarized MIMO land mobile satellite(BDM-LMS)channel.This paper focuses on establishing the BDM-LMS channel statistical model, which combines the 4-state broadband LMS channel model,the time selective fading features,the channelcovariance information (CCI)channel model and polarization correlations between antennas.The modeling steps of the channelmodelare introduced. The main emphasis is placed on the effects ofthe factors,such as antenna numbers,temporal correlations,terminal environments, elevation angles and polarization correlations between the DPAs, on the channelcapacity in the BDM-LMS system.Many simulation results are provided to illustrate the effects ofthese factors through comparisons of the transmit rate,ergodic capacity and outage capacity with different factor values.Besides,the MIMO outage capacity advantages,which indicate the bene fi ts of MIMO compared with a single input single output(SISO)system under the same channelcondition,are also studied under i.i.d or BDM-LMS channel.

land mobile satellite(LMS),multiple input multiple output(MIMO),dual-orthogonalpolarized antenna(DPA),channel covariance information(CCI)channel,ergodic capacity(EC).

1.Introduction

Substantially increased information capacity and high speed transmissions[1]are required in broadband satellite communication systems.Both the satellite and mobile terminals may sufferserious coupling among the antennas, due to limited volume and smaller antenna array spacing. Perfectchannel independence among the antennas cannot be guaranteed[2].In order to solve this problem,two adjacentantennas separately using the orthogonalpolarization component of the electromagnetic wave can be placed on satellites,or on mobile terminals,to form dual-orthogonal polarized antennas(DPAs)structure.The multiple input multiple output(MIMO)system structure with DPAs is shown in Fig.1.Even though there exists almostno multipath and,hence,only a dominating line of sight(LOS) signal,the DPAs can achieve good independence between antennas with relative small space in transmitand receive antenna arrays[3].Therefore,the MIMOsystem with multiple polarization antennas can also obtain the diversity gain and channelcapacity same as the MIMO system with enough space among a single polarization antennas array, even withoutenough multipath signals[4-6].Some studies prove thata space of three wavelengths(e.g.18 cm of 5 GHz carrier)between DPAs can achieve optimal channel capacity,which greatly reduces the antennas space to obtain a better spatial diversity gain,and provide a strong technical and theoretical support for realizing the handheld,vehicle and other small or micro MIMO device with large capacity[7,8].

Moreover,in order to accurately analyze the system performance of a broadband dual-orthogonal polarized

Fig.1 Structure ofa broadband satellite mobile communication system with DPAs

MIMO land mobile satellite(BDM-LMS)system,to obtain the transmission performance boundary(capacity)in the actual environment,and to evaluate and compare differenttransmit methods,an accurate,complete and realistic channelprobability statisticalmodelis required[9,10].

Besides the DPAs structure,a MIMO LMS system can be formed by multiple satellites,but the synchronization between the satellites is a complicated process.Horvath et al.considered that a single satellite with multi-polarized antennas could form the MIMO structure instead of multiple satellites constellation[9].From the pointview of the satellite payload,multi-polarized antennas can be applied in a single beam and complex multi-beam satellite system [10].Using orthogonal coding signals by multi-polarized antennas,the space diversity effects of a MIMO system can be further improved[11].However,the in fl uencing factors of the system increase when using DPAs in satellite communication system,such as the polarization mode, cross-polarization discrimination(XPD),and polarization coupling(due to re fl ection,and atmospheric effects).Their in fl uences on channel correlation must be analyzed.As a result,the effects of DPAs on the signal transmission performance(channeltransmission characteristics)can be obtained.

Atpresent,studies on the dualpolarization MIMOLMS channelare concentrated in signalmeasurementand probability statistics.The basic distribution models are always the simple single state narrow-band model and not suitable to be applied in the broadband satellite communications system.Oestges et al.[6]obtained related LMS channelpolarization parametervalues,and use simple Rice and Rayleigh distribution to establish a simple small scale fading model.Sellathurai et al.[11]and King et al. [12]assumed that the amplitude of a large scale fading and a small scale fading conforms to the Nakagami distribution and the Rayleigh distribution,respectively.The model is simple in form,and cannot entirely describe the LMS channelcharacteristics.Cheffena etal.[13]only examined the trees shadowing and scattering effectofthe LMS channel,which is a specialenvironmentalcondition.From the studies at present,a more complete and multiple state broadband channel model is required to accurately describe the double polarization MIMO LMS channel.The Fontan model[14]based on Loo model[15]is suitable for statistical modeling the broadband satellite mobile communication channel,which is selected as the fundamental channelmodel.

2.Properties of a polarized MIMO satellite mobile communication channel

The capacity gain obtained from multiple antennas heavily depends on the available channelinformation ateither the receiverorthe transmitter,the channelsignal-to-noise ratio (SNR),and the correlation between the channels on each antenna element[11].Among these three factors,SNR is not determined by the channel characteristics,but by the performance of the transmitter and receiver.Therefore,another two factors should be described in detailbefore analyzing the channelcapacity.

2.1

Channelstate information(CSI)of satellite channel

The degree of CSI varies from no CSI up to perfect CSI depending on whether only statistical distribution information of the channel(CDI)is available or exact channel gain values can be obtained at the transmitter(CSIT)/receiver(CSIR)for every channel realization.When CSIR and CSIT are perfectly known,the MIMO channel may be decomposed into rank parallel single input single output(SISO)channels obtained through singular value decomposition(SVD).A relatively higher transmit rate can be reached through the optimalenergy allocation applying the Water fi lling algorithm at transmitter[12].Anyway,a perfect CSIR is usually assumed in MIMO systems since the channelgains can be estimated fairly easily through pilotsequences.In this case,the strategy thatmaximizes capacity is to allocate equal power to each transmit antenna [13,14].

In broadband satellite communication systems,a perfect CSIR is possible to acquire,butthe CSIfeedback from the receiver to the transmitter has to suffer a long time delay, which may lead to incorrect CSIT,especially in fast varied channels.However,in highly mobile channels,the assumption of perfect CSIR can also be unrealistic.Even for a rapidly fl uctuating channel where reliable channel estimation is notpossible,itmightbe possible forthe receiver to track the short-term distribution of the channelfades as the CDI,since the channeldistribution changesmuch more slowly than the channelgains[14].

Therefore,the satellite communication channel can be regarded as with CSIR or channeldistribution information of the receiver(CDIR)and no CSIT.The salient features ofthe modelare as follows.The channeldistribution is defi ned as Rayleigh or other channelstatistical model.Conditioned on these models,the channel characteristic matrix(CCM)H at different time instants is independently and identically distributed(i.i.d.)as Rayleigh distribution or under an environmentstate(LMS model)[11]within a data frame.

2.2 Correlated fading of polarized MIMO satellite channel

In MIMO communication systems,both the transmitter and the receiver have multiple antennas,therefore thecorrelation between the antennas seriously in fl uence the performance of the MIMO system,and thus cannot be avoided.Moreover,in a BDM-LMS system,the received signals stillundergo the polarization correlation fading due to the following two aspects.First,the relative smallspace between the DPAs may lead to the leakage of cross polarization signals,which can deduce the cross polarization discrimination(XPD)of the system,and weaken the channelindependence.Second,the depolarization effect,due to multipath re fl ection and scatters,may change the polarization mode and bring aboutthe cross polarization coupling (XPC).For instance,in marine or aero communication at a low elevation angle situation,the re fl ections are always strong and XPC is quite severe.Targeted at the parameters of a MIMOsystem,the polarization correlation fading has greatin fl uence on the correlation coef fi cients between the antennas.Therefore,the channel should be characterized as a channelcovariance information(CCI)model.The mostcommonly used CCImodelis the Kroneckerproduct form,and the CCM H may be written as

The total channel correlation matrix can be written in the Kroneckerproduct(?)form.

where Hwis an NR×NT(NTand NRdenote the number of transmitted and received antennas,respectively)matrix of i.i.d.zero-mean,unit variance complex circularly symmetric Gaussian random variables(CSIR)or random process induced by a channel statistical model(CDIR).Rrand Rtare referred as the received and transmitted antenna correlation matrices,respectively,which are assumed to be known when CSIR or CDIR is available.Under the CCI model,the channel is assumed to be varying too rapidly to track its mean,so the mean is setto zero and the information regarding the relative geometry of the propagation paths is captured by a non-white covariance matrix[11]. Although not completely general,this simple correlation modelhas been validated through fi eld measurements as a suf fi ciently accurate representation of the fade correlations seen in actualcellular systems[15].

3.Broadband LMS modeling

3.1 4-state Fontan model

Differentcombinations ofsimple mathematicalprobability statistical distribution,i.e.lognormal,Rayleigh,Rice distribution,form the current LMS models,such as the Loo model,Corazza model,and Lutz model.However,these models are only suitable to describe the narrowband channel,without considering the broadband factors,such as the excess delay due to multipath propagation.If the excess delay is signi fi cant compared to the inverse of the symbol rate,the channel is no longer fl at but frequencyselective,therefore the channelcan be regarded as a broadband channel.Moreover,more and more low Earth orbit (LEO)satellites are used to provide communication signals,then the Doppler frequency shift cannot be ignored further more.The temporalcorrelation caused by Doppler frequency shift should also be considered in the channel mode[14].

A simple distribution model is not enough to describe the statistical LMS channel when the link between the moving satellite and terminalexperiencesdifferentcircumstances,such as in urban,suburban,open area,underdense or sparse tree shadow area.This explains why LMS models try to classify propagation events according to the degree of shadowing or obstacles(environments),and quantify these events independently with 3-states as LOS,intermediate shadow and deep shadow.The state probability W and state transition probability P among the 3-states are also different according to different elevation angles between satellite and terminals.

Furthermore,the transition between the narrow and the broadband channels cannotbe ignored,because of the increasing data speed and the frequency spectrum utilizing in the currentsatellite communication system.By combining the narrow and the broadband channels,the Fontan model is based on a 4-state Markov chain as shown in Fig.2[14]. The Markov chain describes the transition between the narrowband state(NS)and the broadband state(BS),according to whether the multipath signals are with enough average power delay pro fi les(APDPs)under different environments or elevation angles.The modeldescribes the BSs by assuming thatthe excess delays of individualmultipath ray are exponentially distributed.In Fig.2,S4 as a broad channel state cannot transit to S2 or S3,but only transit with S1,which can be detailexplained in Section 3.4.

Fig.2 A 4-state broadband Fontan model

3.2 Generation and statistics of the random process with Loo distribution

In each NS of the Fontan model,the directsignalmay undergo shadowing/blockage effects and the multipath(specularand diffuse),which can be described as the Loo model. In the Loo model,suppose the amplitude of the LOS signal sLOSis 1,the received signal r(t)is the sum of theLOS phasor signal z(t)with lognormal distribution amplitude and the multipath echoes phasor signals w(t)with Rayleigh distribution amplitude as shown in(3).The phase of z(t)and w(t)are uniformly distributed between-πand π[15].

If G is a standard normally distribution random process with average value 0 and variance 1,G?is a normally distribution random process with average valueμand variance d.Then Z can be obtained by(4),which is a lognormal distribution random process with average value eμ+d/2and variance e2μ+d·ed?1.

And W is a Rayleigh distribution random process with average power b of the multipath echos.If X and Y are two uncorrelated standard normally distribution random processes with average value 0 and variance 1,W can be obtained by

By deduction in[15],the cumulative distribution functions (CDFs)of r(amplitude of r(t))can be described as

where I0(·)is the modi fi ed Bessel function of the zeroth order.Rewriting the parameters ofμ,d and b,we can obtain the variancesα,Φand MP in dB unit:

The values of the modelparameters above can be found in many references.In fact,the extraction of these values is made by plotting the CDFs ofindividualmathematicalcalculated series(corresponding to speci fi c environments,elevations,and states)and fi nding the CDF thatmore closely fi ts the measured CDF.After this process,the Loo distribution modelparametervalues are available for each environment,elevation,and state[14].

Fig.3(a)and Fig.3(b)show the comparison between the probability calculation results by Loo equations as(6)and the statisticalresults by randomdistribution processes simulation as(3)-(5).In the fi gures,the amplitude ofthe LOS signalis 1,and the y axis represents the relative value of r to LOS signalamplitude in dB form.Fig.3 shows thatthe calculation and statisticalcurves almostcoincide with each other,which indicates thatthe complex random process of r(t)is suitable to describe the Loo model,and the random process of r(t)is capable to produce the time-series forthe LMS channel,which willbe detailed in Section 3.4.Therefore,based on the Loo model,the 4-state Fontan modelis suitable for describing an LMS channel and is capable to help the performance evaluation of a satellite mobile communication system.

Fig.3 Comparison between the probability calculation results by Loo equations and the statistical results of random distribution processes

Fig.4 shows the effects of the mean valueα,the standard varianceΦof the LOS signal,and the average power MP on CDF curves.The parameters in this fi gure are listed in Table 1.The curves in Groups A,B and C representthe effectofΦ,αand MP,respectively.From Fig.4, the following conclusions can be obtained:

(i)From the curves of Groups A and C,it is clear that the ordinate of the intersection point(4 dB or-4 dB)is the αvalue,which is the mean value of the LOS signals.And the x axis value of the intersection pointof Group A is 0.5.

Table 1 Parameters of Fig.4 dB

Fig.4 Effects ofα,Φ,and MP values on CDF curves

(ii)Whenαis increasing,the curves move up,which indicates that the probability bigger than a certain y axis value is increasing.In other words,the average amplitude increase of the LOS signal is bene fi cial for the received signalenhancement.

(iii)However,whenΦand MP are increasing,the curves present different variation tendency separated by the interaction points,which illustrates that the signal power are expanded to a wide amplitude range,i.e.expanded from[-6-2]dB to[-10 0]dB in Group C. In other words,the multipath echoes have made the signal energy dispersed,and partially bene fi t for the signal in a wide amplitude area.

3.3 Generation of the 4-state Markov chain

The terminal may experience differentenvironments such as in urban,suburban,open area,underdense tree orsparse tree shadowing during the communication maintaining period.And the elevation angles between the satellite and terminalmay undergo substantialvariation when the satellite is a non-geostationary Earth orbit(GEO)one.The changes above can be considered as different states in Markov chain.

As an example,a 4-state Markov chain describes the state transition during the relative motion between the satellite and terminalreceiver.Two matrices determine the state transition in Markov chain:the state transition probability square matrix P and state probability row vector W. Each elementin P,pijrepresents the probability when the transition from the state i to the state j happens(if i=j, no state transition happens).And each element in W,wirepresents the probability when the channelcondition is in the state i.

Three properties of Markov chains are highlighted here [14]:

(i)the sum of allelements in every P row mustbe equal to 1,

(ii)the sum of N elements in row vector W must be equalto 1,i.e.

(iii)the asymptotic behavior(convergence property)of the Markov chain is de fi ned by

where I is a unit square matrix.From(8),W is a nonzero vector,thus square matrix[P?I]is an invertible matrix(singular matrix),which indicates the row vectors in[P?I]is correlated.Therefore,if P is given,W can be solved by(7)and(8).

In 4-state Fontan model,S1,S2,S3,and S4(i,j= 1,2,3,4)represent the LOS existing state,the intermediate shadow state,the deep shadow state,and the BS,respectively.Typical P and W values under differentenvironments and elevation angles at S,L-band were given in [14].In the NS and BS,P and W under the same environment and elevation angles are listed respectively,such as P and W in an urban area atelevation 60°in NS are

Without S4(BS),the state probability and transition probability among S1,S2,and S3 are described as the matrices above.As illustrated in Fig.2,S4 as a broad channel state cannot transit to S2 or S3,but only transit with S1. Therefore,with the help of the probability of S1(without multipath)and S4(with multipath)in[14],the transition probability among the 4-state can be obtained.The probabilities of S1 and S4 in urban area at elevation 60°are 78.33%and 21.67%,respectively.Then the 4-state probability matrix in urban area atelevation 60°can be obtained:

Fig.5 shows the probability comparison of three narrowband states in differentenvironments and elevation angles.From Fig.5,the following conclusions can be obtained:

(i)The LOS signal state(S1)probabilities are almost constant in different environments and different elevation angles,which indicates the obstacles can slightly in fl uence the LOS signalin the satellite communication system,due to the signi fi cantheightof the satellite.

(ii)The probabilities of LOS state are higher than those ofintermediate ordeep shadow state,which proves thatthe satellite communication relies on LOS signals.

(iii)Deep shadow is increasing with the elevation angle in urban area,suburban area and open area,which may be in contrast with the regular logic.This phenomenon may be caused by the effectof the increasing multipath echoes when increasing the elevation angle in these three areas. This conclusion can also be provided in Fig.6.

Fig.5 State probability comparison of three narrowband states in different environments and elevation angles

Fig.6 Variation ofthe LOS mean valueαand the average power ofthe multipath echoes MP in different environments and elevation angles

Fig.6 shows the variation of the LOS mean valueαand the average power of the multipath echoes MP in different environments and elevation angles.From the fi gures, the following conclusions can be obtained:

(i)The LOS signalamplitude is decreasing when the environmentgets poorer(from open to urban area);

(ii)The multipath echoes power is increasing when the environment gets poorer(from open to dense tree cover area),butthe multipath signalpower weakens in the urban area,which may due to the fact that the dense obstacles block the multipath signals;

(iii)The MP values in large elevation angles(70°or 80°)are almostlarger(notalways)than those in small elevation angles(40°or 60°),which proves the conclusion (iii)in state probability analysis above.

3.4 Particularity of the broadband channelmodelA channelcan be regarded as a broadband one,only when the following two requirements are satis fi ed.Firstly,the multipath echoes should be received in the dynamic range of the receiver and distinguished from the noise background.In shadowed states(S1,S2,and S3),the multipath echoes can hardly reach the receiving threshold because the multipath echoes as same as the LOS signal must experience the shadow and suffer from the power attenuation.S4 cannot happen in these states.Nevertheless,the LOS signal can be received in S1,and the energy of the multipath echoes in this state has a certain probability to reach the threshold of the receiver dynamic range.Thus, the broadband state as labeled with S4 can only happen when LOS exists,and S4 can only transit with S1.Secondly,the excess delay is signi fi cant compared to the inverse of the symbol rate,in other words,the signal bandwidth is biggerthan the inverse of the excess delay.If both two requirements above are satis fi ed in a channel,the channel is no longer fl at but frequency-selective and no longer a narrowband channel but a broadband one.Moreover,if only the fi rstrequirementis satis fi ed,mostreceived multipath echoes cannotbe regarded as usefulsignals when the receive fi lter is bandwidth-limited.

Generally,the transfer functions of different multipath echoes within coherent time are similar,so the echoes received within the coherence time may have a strong correlation.Then the echoes arrived atthe receiver can be regarded as useful signal.However,the sum of the echoes can only bene fi t for level enhancement when the channel can be estimated and equalized wellenough.On the other hand,if the echoes intervalare greater than the coherence time,they could not affect the sample decision process, since they are considered as a kind of noise.In this situation,as in S1,the average power of the multipath echoes MP is much smaller than that in S4,a broadband state. The typical MP value comparison between S1 and S4 is listed in Tables 2-3.

Table 2 L-band MP comparison between S1 and S4 dB

Table 3 S-band MP comparison between S1 and S4 dB

3.5 Generation of the time-series of the channel based on the 4-state broadband LMS model

For current LMS engineering applications,models solely providing CDFs are not suf fi cient.New statistical models must be suitable for interfacing system studies,including link-level and network-level simulations.Fig.3 indicates that the complex random process of r(t)is suitable to describe the Loo model,so the random process of r(t)is capable to produce the random process(time-series)for the LMS channel.

Based on the Loo model,the 4-state Fontan model is suitable for describing an LMS channel and is capable to help the performance simulation of a satellite mobile communication system.Therefore,the state transition and the random process in each state mustbe involved in the timeseries generation.

When a relative motion between the satellite and the terminalhappens,the channelmay experience differentstates as a Markov chain describes.As two examples,Fig.7 shows the generation of the time-series according to state transition in differentenvironments,and comparison of 3-state modelwith 4-state model.From Fig.7,when the two requirements(detailed in Section 3.4)ofa broadband channelare satis fi ed,S4 willbe appeared based on S1,and cannot transit to S2 or S3.Moreover,the circle notations in the 4th sub-fi gure in Fig.7(a)show that the highest average power is S4,followed by S1,S2,and S3 in sequence. Obviously,the signal powers of S2 and S3 in open areas are much higherthan those in urban areas.

Fig.7 Generation of the time-series according to the states transition in different environment

Based on the analysis above,the received signals between satellites and terminals experience a typicalMarkov chain channel,due to the relative motion of the transmitter or the receiver.Then,the discrete received signals can be obtained by multiplying the discrete transmit signal with the random process(time-series)number under different Markov states of the channelas shown in Fig.7.As a result,the amplitudes of the received signals are given certain properties of an LMS channel,which are suitable for simulation and analysis on computer.

3.6 Multipath echoes in broadband channelmodelIn fact,a broadband communication terminal can receive hundreds of multipath components within coherent time. The tapped delay line(TDL)modeldivides the huge number of multipath components into a few(e.g.,5-11)multipath echoes with excess time delay,thus transform a frequency selective fading channel into several fl at slow fading channels.Therefore,each multipath echo in the TDL modelcan be described by a fl atslow fading channel model,and the power proportion of the multipath echoes is determined by a power delay pro fi le(PDP)distribution.

The mostcommonly used broadband TDL modelis the L-tap Rayleigh fading model[16].L is the number of multipath echoes of fl at Rayleigh fading.The impulse response of the system with the LOS signal and multipath echoes in TDL modelcan be described as

In the simple simulation process,the inverse of the excess delay 1/τkis the integertimes ofthe signalbandwidth W,for example,W is the symbolwidth in an orthogonal frequency division multiplexing(OFDM)modulation system.Supposeτk=k/W(k=1,2,...,L),the L-tap TDL modelis shown in Fig.8. In Fig.8,the received signal r(t)can be described as

Fig.8 Structure ofthe broadband L-tap TDL model

where ck=αkej?(k)(k=1,2,...,L),αkis the amplitude of each multipath echo,and also represents the amplitude of the multipath components with Rayleigh distribution in each echo(for example,Rayleigh distribution pa-?(k)is the uniformly distribution phase with[?π,π],and z(t)is the phasor additive white Gaussian noise(AWGN).

whereαkis determined by the PDF model,such as exponentialdistribution model,which is shown in Fig.9.whereτkis the excess delay of the k th echoes(τ1=0), τavis the exponentialdistribution factor.In order to simplify the simulation process,the excess delayΔτkbetween each echo are supposed to be constant,which is n-times of the sample time Ts(the excess delay is n-times of the sample period).De fi ned A=τav/nTsin(12),sogeometric progression whose fi rstterm isand common ratio is e?2A,as shown in(13).If the sum power of the multipath echoes is obtained,α1can be easily calculated by(13).Both MP andτavmeasured in typical environments at L or S band can be found in[14].

Moreover,as in a frequency-selective fading channel, the excess delayΔτ=nTsis signi fi cantcompared to the inverse of the symbol rate 1/Rs,i.e.,nTs＞1/Rs.The sample theory illustrates thatthe sample frequency should be larger than twice of the symbol rate,i.e.Ts＜1/2Rs, therefore,n should be larger than 2,or else the signal is under-sampled.

Fig.9 Exponential PDP modelin different environments

3.7 Temporalcorrelation in broadband channel

As the relative motion between the mobile terminal and satellite happens,the transmit signals may be in fl uenced by the serious Doppler frequency shift.Consequently,the spectrum will be spread in the frequency domain,and the signal in the time domain will suffer fast fading phenomenon,i.e.,time-selective fading.In a time-selective fading channel,the inverse of the symbolrate 1/Rsis signi fi cantto the coherence time of the channel,in which the coherence time is de fi ned as the inverse of the maximum Dopplerfrequency shift Tc=1/fm,i.e.1/Rs＞1/fm.In such a time-selective fading channel,the large-scale fading components undergo temporal correlation.To model the speci fi c coherence time Tc,the large-scale fading components should experience a low-pass fi ltering process[17]. In this regard,for each state,a Gaussian random sequence of samples with zero mean and unitvariance is fi rstgenerated.One sample per transmission block of the duration T is assumed.These Gaussian samples are then processed in order to introduce the temporal correlation,using the following one-coef fi cientlow-pass in fi nite impulse response (IIR)fi lter form(IIR parameters b=1,a=e(?vT/rc)) [1]:

where v is the relative motion speed,and rc=v×Tcis the coherentdistance,thus

In(14),the currentsample ynis coherentwith the sample yn?1.The processed samples are then scaled by(1?A2) in order to restore the statistics of the samples[18].

For example,if the satellite communication happens in S-band,the required parameters are listed in Table 4.

Table 4 Required parameters in S-band satellite communication example

After calculating the coef fi cients of the low pass fi lter by(14)-(15),the amplitude response of the IIR fi lter is shown in Fig.10(a).

Fig.10 Amplitude and phase response ofone-order IIR filter

4.Effects of DPAs on channelmodeling

The bene fi ts of using DPAs in the MIMO communication system,especially in the long distance communication system,i.e.,MIMO satellite mobile communication system, are detailed in Section 1,such as decreasing the nearby antennas spaces,enhancing the channel independence,and increasing the channel capacity.However,DPAs in LMS channels willalso in fl uence the channelproperties,such as the polarized property and the channelcorrelation among the antennas.Therefore,the effects of DPAs on modeling of the MIMO broadband satellite mobile channel mustbe studied in order to properly simulate and evaluate a dualpolarized MIMO communication system.

A typical 2×2 dual-orthogonalpolarized MIMO channelmodelis shown in Fig.11.In the fi gure,the inputdata series as a1a2...are coded by space-time code,such as space time block code(STBC),and transmitted by two DPAs T1and T2.Suppose T1and R1are right-handed circular polarization or horizontally polarized antennas,then T2and R2are circular polarization or vertically polarized antennas,and vice versa,which form the DPAs structure.

Fig.11 Typical2×2 dual-polarized MIMO LMS channelmodel

When modeling the dual-polarized MIMO broadband satellite mobile channel,some premises mustbe followed:

(i)The spaces between the transmitantennas T1and T2or between receiving antennas R1and R2can be ignored;

(ii)The transmit signal power between T1and T2are equally allocated,and the receive abilities(antenna gains) of R1and R2are almostthe same;

(iii)Generally,it is obvious that the properties of the LMS sub-channels are symmetrical,as h11=h22and h12=h21,in which hijrepresents the time domain subchannelresponse between the i th received antenna and the j th transmitantenna,according to premises(i)and(ii).

Besides,three aspects should be considered when modeling the dual-polarized MIMO broadband satellite mobile channel:

(i)Sub-channel response hijundergoes a large-scale fading and a small-scale fading,which can be described by the sum ofa log normalrandom process zijand a Rayleigh random process wijas the Loo model presents in Section 3.1.And hijshould experience differentstates as the 4-state Fontan model which is described in Sections 3.3 and 3.5,when the relative motion between the satellite and mobile terminals happens.

(ii)The received power of each antenna should be seriously in fl uenced by the XPD and XPC.

(iii)Although the antennas are orthogonalpolarized,the channels between nearby antennas are notcompletely mutually independent.Parametersρrxandρtxdescribe the polarized correlation coef fi cients(channel correlation coef fi cients)between the received and transmitantennas,respectively.

The consideration(i)above has been detailed in Section 3,and the considerations(ii)and(iii)will be studied as follows.

4.1 Effects of XPD and XPC

Parameter XPD proves that a polarized antenna cannot only obtain the co-polarization signals,butalso receive the cross-polarization signals,in other words,the XPD factor in fl uences the power allocation of the received signals, including the large-scale and the small-scale fading components,which can be described as(17),where factorβ denotes the power allocation ratio of the large scale fading signals[11].

where PCOand PCRrepresent the power of the copolarization signals and the cross-polarization signals,respectively,which can be described as

Equation(18)also indicates that the total power of a receive antenna is composed of two parts:the signal from the co-polarization antenna and the signal from the crosspolarization antennas.Therefore,the total power of the large-scale components of one receive antenna PLis the average power of a log normal random process as in(19), assisted by(4).

Parameter XPC proves that a part of polarized signals, transform to orthogonal polarized ones when the signals are re fl ected by obstacles or pass through the ionosphere. Then,the small-scale fading components may affected by both XPC and XPD.In(20),the factorγdenotes the power allocation ratio of XPC transforming[11].PCOand PCRrepresent the power of the co-polarization signals remaining and the cross-polarization signals transformed by XPC, separately.

By combination of the effects of XPC and XPD,the small-scale fading signals(re fl ectofscatter paths)received by a vertical-polarized antenna(as an example)can be separated by four parts:

(i)the components transmitted by co-polarized antenna(vertical-polarized antennas),without transforming by XPC phenomenon and received by co-polarized antenna within the XPD limit;

(ii)the components transmitted by co-polarized antenna (vertical-polarized antennas),transformed to horizontalpolarized signals by XPC phenomenon and received by the co-polarized antenna exceed the XPD limit;

(iii)the components transmitted by cross-polarized antenna(horizontal-polarized antennas),without transforming by XPC phenomenon and received by co-polarized antenna exceed the XPD limit;

(iv)the components transmitted by cross-polarized antenna(horizontal-polarized antennas),transformed to horizontal-polarized signals by XPC phenomenon and received by the co-polarized antenna within the XPD limit;

As the premises detailed at the beginning of Section 4, the components power partition between part(i)and part (ii)(transmitted by co-polarized antennas)is(1?β)(1?γ)/βγ,and the partition between part(iii)and part(iv) (transmitted by cross-polarized antennas)isβ(1?γ)/(1? β)γ.De fi ne the factorηasη=β(1?γ)+(1?β)γ, 1?η=(1?β)(1?γ)+βγ,then the total power partition between the small-scale fading signals transmitted from the co-polarization antenna and the signal from the cross-polarization antennas can be described as

And the totalpower of the small-scale fading components ofa receive antenna PSis the average powerof a Rayleigh random process as in(22),assisted by(5).

The typical parameter values ofβandηin different environments are listed in Table 5[11].From the table,the effect parameters of DPAs in different environments are almostthe same,which indicates thatthe effects of the environments are limited.Moreover,because the XPD and XPC effects are limited to different frequency band signals,the parameters in L-band and S-band are also similar.

Table 5 Typical parameter value in different environments

4.2 Polarization correlation property of the channelwith 2×2 DPAs

By means of the time-series generation of 4-state broadband LMS channel model as detailed in Section 3.5,the sub-channelresponse hijas random processes is easily to be obtained,then the CCM H with 2×2 DPAs can be described as

However,due to relative smallspaces between the DPAs and the polarization characteristics ofthe antennas,the polarization correlation between the antennas cannot be ignored,when modeling the BDM-LMS channel.The channel response HC,considering the polarization correlation property can be described as

where the operator T represents the transpose of a matrix, and

where Cov(x,y)represents the covariance between x and y,ρrxrepresents the polarized correlation coef fi cients (channelcorrelation coef fi cients)between the received antennas,then(25)can be rewritten as

By the same calculation steps,the results ofcan be obtained:

The polarization correlation coef fi cientsρrxandρtxhave an inconsistent de fi nition in the statistics fi eld,but usually explain the relation between the values.For example,if the correlation valuesρrxandρtxare between 0.3 and 0.5,the antennas are real-correlation with each other,ifthe values are less than 0.3,the antennas are weakcorrelated,if the values are higher than 0.5,the antennas are obvious-correlated.The measured values ofρrxandρtxunder different environments are listed in Table 5,which indicate thatthe DPAs are real-correlation with each other.

5.Analysis ofthe influenced factors on channelcapacity of BDM-LMS channel

5.1 Channelmodels used in simulation

In this paper,the CCM H used in our simulation can be obtained by an i.i.d model,a narrowband LMS modelwithout temporaland polarization correlation or the statistical BDM-LMS channelmodelbuiltin Section 3.Azero-mean, unitvariance complex circularly symmetric Gaussian random process(normalized Rayleigh distribution)is adopted to describe an i.i.d channel.Moreover,the parameters of the statistical BDM-LMS channel model of different environments in Section 3 are shown in Tables 6-11.The parameters of the narrowband LMS model(S1-S3)are shown in Tables 7-11.

Table6 Typical simulation parameters

Table7 parameters of urban area(elevation angle=40°)at S-band

Table8 parameters of urban area(elevation angle=40°)at S-band

Table9 parameters of urban area(elevation angle=40°)at S-band

Table10 parameters of urban area(elevation angle=40°)at S-band

?

5.2 Ergodic and outage capacity

Shannon(ergodic)capacity is de fi ned as the maximum mutual information averaged over all channel states[19]. In the satellite communication system,the CSI is limited to CSIR or CDIR and no CSIT as detailed in Section 2.Therefore,the transmit power of each antenna should be equally allocated,so the ergodic capacity(EC)of the MIMO satellite channelcan be described as[20]

where H is the CCM which is determined by the CSIR or CDIR.Ri,Hiis the communication rate(bps/Hz)as the instantaneous CCM is Hi.ρdenotes the signal-to-noise ratio(SNR).Ntand Nrdenote the number of transmitted and received antennas,respectively.N=min(Nt,Nr). EH{x}indicates that the EC is the mean value of x according to its probability pi.Generally,the CDF of transmission rate can be statistically acquired by calculation of Ri,as shown in Fig.12.Besides,with the help of the random series number of H,the channel EC can be obtained from the mean value of the CDF curves.The simulations are carried outwith Monte Carlo method.

We de fi ne the p%-outage capacity to be the transmission rate that can be supported(100-p%)of the time.The outage capacity,de fi ned as the probability that the transmission rate cannot be supported and hence the transmitted data are received in error,is p/100[19].Therefore,the p%-outage capacity(x axis)can be found by searching the p%position(y axis)in the CDF fi gure of the transmission rate.

Fig.13 shows the comparison of EC and 10%-outage capacity with differentantenna numbers.In this fi gure,the CCM H is de fi ned as i.i.d or normalized Rayleigh distribution,therefore the random numbers hijin H are zeromean,unit variance complex circularly symmetric Gaussian random series.If x and y are two uncorrelated standard normally distribution random numbers with average value 0 and deviation 1,hijcan be obtained by

Fig.12 CDF of transmission rate with different antenna numbers in i.i.d channel

Fig.13 Comparison of EC and outage capacity with different antenna numbers in i.i.d channel

From Fig.13,the values associated with outage capacity are typically smallerthan those ofthe EC due to additional constraintassociated with their de fi nitions.The same conclusion can also be drawn from the calculation methods of EC and outage capacity.In the CDF fi gure of rate,outage capacity is described by the x axis value when the y axis value is p%(i.e.p?10),and EC is approximately atthe 50%position.

5.3 Number selection of transmitted and received signalin the narrowband scene

In contrast to the results for the spatially white fading model where adding more transmit antennas beyond the coherence intervallength does notincrease capacity,additional transmit antennas always increase capacity as long as theirchannelfading coef fi cients are spatially correlated. However,the EC is also closely related to the CCM and the correlation between the antennas[5].The EC performances of the MIMO communication system with different antennas numbers under i.i.d channeland narrowband LMS channelare shown in Fig.14.

Fig.14 EC of MIMO communication system with different antennas numbers under i.i.d and LMS channels(SNR=5 dB,urban area, elevation angle=40°)

As illustrated in Fig.12 and Fig.14,a 2×2 MIMO communication system is optimal after balancing the system complexity and channel capacity among all groups of antennas below 4×4[21].Besides,a pair of vertical and horizontal polarization antennas(2×2)or a pair of right-hand circular polarization and left-hand circular polarization(2×2)are polarized-independent with each other,and suitable for DPAs structures.Therefore,2×2 antennas are selected in the MIMO satellite communication system to increase the channel capacity,and are adopted in the following simulations.Fig.15 shows the CDF of transmission rate performance of the MIMO communication system with different antennas numbers under LMS channel,which appears obvious decrease of the transmission rate due to the blockage effectin the urban area compared with Fig.12.

Fig.15 CDF of transmission rate performance of MIMO communication system with different antennas numbers under LMS channel with Loo model(SNR=5 dB,urban area,elevation angle=40°)

5.4 Effect of different communication environmentin narrowband scene

According to LMS model parameters list in Tables 6-11 and the analysis in Section 3,differentenvironments,as urban area,suburban area,intermediate shadow area,heavy shadow area and open area,may have great in fl uence on the boundary of transmission rate(EC)or the outage capacity in MIMO satellite communication systems.Figs. 16,17 and 18 show the CDF of transmission rate,the EC,and the 10%-outage capacity of a 2×2 MIMO satellite channel in a different environmentwith elevation angle 40°,respectively.From these three fi gures,it is obvious thatthe CDF of transmission rate,the ECversus SNR, and the 10%-outage capacity versus SNR are in fl uenced seriously by the shadow,scatter or re fl ection level of differentenvironments.As a result,the in fl uences are weaker in the sequence of the heavy shadow area,the urban area, the intermediate area,the suburban area and the open area.

Fig.16 CDF oftransmission rate in different environments(SNR= 5 dB,elevation angle=40°)

Fig.17 EC versus SNR in different environments(elevation angle= 40°)

Fig.18 10%-outage capacity versus SNR in different environments (elevation angle=40°)

5.5 Effect of different elevation angles in narrowband scene

The differentelevation angle can lead to differentstrength of the LOS signal and the multipath signals and in fl uence the CCM H.However,the effects are differentaccording to the environments.For example,the received signals are mainly LOS signals under high elevation angles in urban areas.Nevertheless,under low elevation angles,the multipath signals due to re fl ection and scatter signals dominate the received signals in the same environment.Therefore, the effect of the elevation angles on the CCM H in the urban area is signi fi cant.However,in an open area,the received signals are mainly the LOS signals,whateverunder high or low elevation angles.The LOS signals may only suffer differentpath loss due to differentelevation angles. Therefore,the elevation angles have few effects on channel EC in open area.The effects of different elevation angles on the CDF of the transmission rate in urban and open areas are shown in Fig.19,which can prove the conclusions above.

Fig.19 Effects ofdifferent elevation angles on the CDF oftransmission rate(SNR=5 dB)

5.6 Effectoftemporalcorrelation in broadband scene

Generally speaking,temporalcorrelations willincrease the capacity when no CSIR is available.However,capacity with temporally uncorrelated channels cannot be smaller than the capacity with temporally correlated channels,because of interleaving codewords,it is often possible to transform the correlated channelinto an uncorrelated one. Conditioned on the channelinformation available atthe receiver,the channel randomness is mainly due to the additive noise which is memoryless from one symbol to the next.Therefore,temporal correlations do not affect the EC.Strong temporalcorrelations signify a slowly varying channelthatwould require longercodewords to realize the EC.Many low complexity coding schemes(such as the orthogonalspace-time codes)rely on the channelremaining constantover several symbols,and therefore may perform better for slowly varying(high temporalcorrelation) channels[19].

Fig.20 shows the CDF of transmission rate with temporal correlation,as described in Section 3.7,in a different environment with elevation angle 40°.It is obvious that each curve in Fig.20 is almostthe same as thatin Fig.16. A more intuitive observation of temporal correlation effects is shown in Fig.21,which shows the CDF of transmission rate comparison with and without temporal correlation in the urban area with elevation angle 40°.The fi gures prove the conclusion that temporal correlation has few effects on the EC or transmission rate.

Fig.20 CDF of transmission rate with temporal correlation in different environments(SNR=5 dB,elevation angle=40°)

Fig.21 CDF of transmission rate comparison with or without temporalcorrelation(SNR=5 dB,urban area,elevation angle=40°)

In Section 4.2,the fourcovariance matriceshave certain effects on CCM H,if the channel is polarization correlation,especially in the MIMO communication system with the DPAs.These four covariance matrices have relations with the following three kinds of factors:the Loo modeling parameters such asμ,d and b,which are determined by the terminal environment;the power allocation ratio of the large scale fading signalsβ and the power allocation ratio of XPC transformingγ, which are almost same under different environment as listed in Table 12;and the polarization correlation coeffi cientsρrxandρtx.

The polarization correlation coef fi cientsρrxandρtxhave inconsistent de fi nitions in the statistics fi eld,but usually explain the correlation level among the antennas groups.For example,if the correlation valuesρrxandρtxare between 0.3 and 0.5,the antennas are real-correlation with each other,if the values are less than 0.3,the antennas are weak-correlation,if the values are higher than 0.5, the antennas are obvious-correlated.The measured values ofρrxandρtxunder different environments are listed in Table 12,which indicate thatthe DPAs are real-correlation with each other.

Table 12 Typicalparameter value in different environments

Fig.22 shows the EC versus SNR comparison with and without polarization correlation in urban areas,when elevation angle is 40°.The lines of wide ad narrow LMSchannels are nearly the same,and the lines of correlated narrow and wide LMS channels are nearly the same,so it seems that only three lines existing in this fi gures.From the fi gures,it can be known that the effects of temporal correlation on the EC are limited;however,polarization correlation has great in fl uences on the channel EC. Fig.23 demonstrates the EC versus SNR curves under different polarization correlation coef fi cientsρrxandρtxin urban areas,when elevation angle is 40°.From this fi gure,the effect of the polarization correlation coef fi cients ρrxandρtxon the channel EC is intuitive that the EC is decreased when the polarization correlation are increased (increasing the polarization correlation coef fi cients).

Fig.22 EC versus SNR with and without polarization correlation (urban area,elevation angle=40°)

Fig.23 EC versus SNR curves under different polarization correlation coefficientsρrxandρtx(urban area,elevation angle=40°)

5.8 MIMO outage capacity advantage

To facilitate a convenientillustration of the MIMO versus the SISOcapacity gain,the MIMOoutage capacity advantage is de fi ned as the ratio Om(p%)/Os(p%),assuming the same outage probability Pout=p%of the same SNR levels,where Omand Osare obtained through the calculation(detailed Section in 4.2)by the CDF of transmission rate Rmand Rsas(34)describes[22],and assuming perfect CSIR

Fig.24 depicts the 10%MIMO outage capacity advantage for the MIMOsatellite channelunder BDM-LMS and i.i.d.channels,which reveals the differences of MIMO outage capacity advantage between an i.i.d channel and a BDM-LMS channel.From the fi gures,the following conclusions can be obtained:

Fig.24 10%MIMOoutage capacity advantage for the MIMO satellite channel(elevation angle=40°)

(i)the MIMO outage capacity advantage of BDM-LMS channelis generally much lowerthan thatforthe i.i.d channel.

(ii)the MIMO outage capacity of the BDM-LMS channel is insensitive to SNR and,thus,achieves a constant outage capacity advantage of almost1.5 exceptin an open area,which may due to the factthatthe LOS signalis dominated and SNRbetween the LOS signaland additive noise plays a more importantrole in open areas than in otherenvironments.The same conclusions have been almostillustrated in[10].

6.Conclusions

The papermainly studies on the in fl uenced factors and the built-up steps ofa dual-orthogonalpolarized MIMObroadband satellite mobile channel modeling.The particularity of this channelmodeling is its broadband characterand the DPAs structure.The modeling steps are based on a 4-state Markov chain and the classicalLoo model.The broadband properties require thatthe multipath effects and the temporal correlation should be considered seriously in channel modeling steps.The DPAs structure brings severe in fl uences on the modeling,including XPD and XPC.Besides, this study analyzes the effects of some factors,including antenna numbers,temporal correlation,terminal environments,elevation angles and polarization correlation between the DPAs on i.i.d and BDM-LMS channelcapacity. Based on the analysis in this paper,the following typical conclusions can be obtained:

(i)The effectparameters of DPAs on channelmodeling in different environments are almost the same,which indicates that the effects of the environments on DPAs are limited.Moreover,because XPD and XPC effects are limited to different frequency band signals,the parameters in L-band and S-band are also similar.

(ii)2×2 DPAs is optimal in BDM-LMS system after balancing the polarization methods,the system complexity and channelcapacity among all groups of antennas below 4×4.

(iii)The CDF of transmission rate is in fl uenced seriously by the shadow,scatter or re fl ection level of differentenvironments.As a result,the in fl uences are weaker in the order of the heavy shadow area,the urban area,the intermediate area,the suburban area and the open area.The effects of the elevation angles on channelcapacity are differentin differentenvironments

(iv)Conditioned on the CSIR,the channel randomness is mainly due to the additive noise which is memoryless from one symbolto the next.Therefore,temporalcorrelations do notseriously affectthe EC.

(v)The MIMO outage capacity advantage of BDMLMS channel is generally much lower than that of the i.i.d channel.And the MIMOoutage capacity ofthe BDMLMS channelis insensitive to SNR.

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Biographies

QingfengJingwas born in 1981.He received his M.S.and Ph.D.degrees in the School of Electronics and Information Engineering,Harbin Institute of Technology in 2005 and 2009,respectively.He has been working as an associate professor since 2009 in the College of Astronautics,Nanjing University of Aeronautics and Astronautics.He is now in charge of several funded projects.His research interests are digitalsignal processing,satellite communication,cooperative communication and broadband multi-carrier communication.

E-mail:jing nuaa@163.com

JiajiaWuwas born in 1990.She is a graduate studentin the College of Astronautics,Nanjing University of Aeronautics and Astronautics.Her research interests are satellite communication and broadband multi-carrier communication.

E-mail:jjw090803@126.com

YupingLuwas born in 1957.He has received his M.S.degree in 1985.He has been the doctoralsupervisor since 2000.He is now the professor ofthe College of Astronautics,Nanjing University of Aeronautics and Astronautics.He is also the expertofthe National High Technology Research and Development Program.His research interests are navigation guidance and control,controltheory and controlengineering and satellite communication. E-mail:yplac@nuaa.edu.cn

XinLiuwas born in 1984.He received his M.S.degree in Harbin Institute of Technology.And he received his Ph.D.degree in the School of Electronics and Information Engineering,Harbin Institute of Technology in 2012.He has been working as a lecturersince 2013 in the College of Astronautics,Nanjing University of Aeronautics and Astronautics.His research interests are satellite communication,space optical communication and cognitive radio. E-mail:liuxinstar1984@nuaa.edu.cn

XiaojuYanwas born in 1981.She received his M.S. and Ph.D.degrees in the School of Municipal and Environment Engineering,Harbin Institute of Technology in 2006 and 2009,respectively.She has been working as an associate professorsince 2009 in College of Hydrology and Water Resources,HohaiUniversity.Her research interests are water treatment and satellite remote technology.

E-mail:wshyxj@126.com

10.1109/JSEE.2015.00073

Manuscriptreceived February 07,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61301105)and the China Postdoctoral Science Foundation Funded Project(2013M531351).