a generative mimo channel model: encompassing single satellite and satellite diversity cases

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Space Communications 22 (2009-2013) 133–144DOI 10.3233/SC-130009IOS Press

133

A generative MIMO channel modelencompassing single satellite and satellitediversity cases

G. Carriea,∗, F. Perez-Fontanb, F. Lacostec and J. Lemortona

aONERA, The French Aerospace Lab, Edouard Belin, Toulouse, FrancebUniversity of Vigo, Campus Universitario, Vigo, SpaincCNES Toulouse, Edouard Belin, Toulouse, France

Abstract. This paper addresses the statistical modelling of MIMO-LMS fading channels. In the absence of accurate experimentalresults, a statistical model for the characterization of MIMO-LMS channels is proposed based on consolidation of availableexperimental results for SISO-LMS, SIMO-LMS and MISO-LMS as well as on their extrapolation to the MIMO-LMS andsatellite diversity cases of interest.

Keywords: LMS, MIMO, channel modelling

1. Introduction

The benefits of employing multiple-input multiple-output (MIMO) systems are already state-of-the-artin terrestrial systems. However, land mobile satellite(LMS) broadcasting system exhibit distinct channel,system and geometrical characteristics and are subjectto stringent spatial limitations compared to terrestrialwireless [1]. Therefore, LMS MIMO channel modelsstill need to be investigated in order to drive the devel-opment of future operating systems.

The starting point of the modelling activity is a stateoriented LMS single-input single output (SISO) chan-nel model (such as the ones presented in [17, 18] andmore recently [16] but the approach is not limited tothese models). Within each state (i.e. Good or Bad statedepending on the link budget, see [16–18] for moredetails), the SISO model can encompass large scale and

∗Corresponding author: G. Carrie, ONERA, The FrenchAerospace Lab, 2 Avenue Edouard Belin, 31055 Toulouse, France.E-mails: [email protected]; [email protected] (F. Perez-Fontan); [email protected] (F. Lacoste); [email protected] (J. Lemorton).

small scale effects. The first part of the paper will there-fore focus on the dynamic parameters estimation bothfor the large scale and small scale effects.

To extend a SISO channel model to a MIMO case, itis proposed to introduce a correlation between the chan-nels using a MIMO correlation matrix for each smalland large scale effect. The second part of the paper willtherefore focus on the extension of state oriented SISOchannel models to single satellite dual polarized MIMOchannel models. In this case, the basic assumption is thatall the channels are always in the same state (i.e.: Goodor Bad). Several works have already been done. Espe-cially, typical MIMO correlation matrixes are proposedin [7] where a generative model step by step procedureis also given. A consolidated version of the step by stepprocedure is also proposed in [10] based on the samecorrelation matrix. However, it will be theoreticallyshown in this paper that the MIMO correlation matrixproposed in [7] can be used to validate channel mod-els but they cannot be used as input parameters to stateoriented channels models. Indeed, it will be shown thatthe final MIMO correlation matrix is closely connectedto the States Parameters. Hence, a procedure will beproposed in order to extract MIMO correlation matrix

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134 G. Carrie et al. / A generative MIMO channel model encompassing single satellite

from experimental dataset that is compatible with stateoriented channel models.

The last part of this paper will address the satellitediversity MIMO channel models. The modelling frame-work proposed in this case can encompass both thepolarization diversity and the spatial diversity at thereceiver side, although the experimental results pro-posed here are obtained with a single antenna andsingle polarization at the receiver side. The main dif-ference with the single satellite channels is that inthe satellite diversity case the various channels canbe in different states. As a consequence, the initialSISO model state duration parameters can no more beused. A semi-Markov approach will be proposed in thispaper to model the multi-states duration distributions.Indeed, as for the SISO case, the semi-Markov approachallows more flexibility for the state duration modellingcompared to a classical Markov approach [16]. Fur-thermore, it can be assumed that the different satelliteslinks have different angles of arrival with respect to thereceiver location. Then, the local obstacles will not havethe same effects on all the links and then it is likely thatthe various channels will have different shadowing cor-relation lengths. It will be shown in this paper that, in agenerative model, using different correlation lengths oncorrelated channels will distort the correlation betweenthe channels. However, this distortion can be closedform quantified and then a pre-compensation can beapplied to the MIMO correlation matrix.

2. SISO channel model: Dynamic parametersassessment

Channel dynamic parameters are of great interest asthey will mainly affect the design of system parameterssuch as the interleaving duration or the modulation andcoding (modcod) scheme selection.

2.1. Large scale effects

The shadowing component is usually modelled as aGauss Markov process with correlation length Lcorr,and then the theoretical auto-correlation function of theprocess is given by:

RSh(d) = e− |d|Lcorr (1)

Before estimating the correlation length parameter fromexperimental data, a first pre-processing step consistsof extracting the shadowing component from total fad-ing distance series or time series. To this aim, a popular

technique is the so called MoM (Method of Moments,[3]) which enables to extract both the shadowing com-ponent and the fast fading envelope (i.e. the multipathpower). The MoM technique has genuinely been devel-oped for Rice channels (i.e. with constant shadowingcomponent on the observation window) while in realword, the shadowing component is time varying. Then,a trade-off has to be done: on the one hand, the win-dow has to be long enough in order to get sufficientstatistics, but on the other hand, the window has tobe short enough so that the shadowing componentremains quite stable. It has been decided here to givethe priority the sufficient statistics and then the win-dow length is equivalent to a few tens of wavelengths.As a consequence, the estimated shadowing componentis somehow smoothed by the MoM so as to approachRice behaviour. The goal of this section is then toanalyse and compensate for the impact of this smooth-ing effect on the estimation the shadowing correlationlength parameter.

The MoM can either be applied on sliding windowsor on disjoint blocks of data. Let’s first consider theMoM is applied on a sliding window with fixed durationN.Ds where Ds is the sampling distance. It can be seenfrom Fig. 1 that the effect of the MoM on the Shadowingdistance series is the same as that of a simple flat slidingwindow filter.

The following derivations are then based on the the-oretical formulations corresponding to a flat slidingwindow filter h of duration N.Ds, and they also hold for

Fig. 1. Synthetic raw shadowing distance series and filtered versions

using either the MoM on a sliding window or a simple flat sliding

window filter.

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G. Carrie et al. / A generative MIMO channel model encompassing single satellite 135

the MoM processing. The normalized auto-correlationfunction Rh of the filter h is a triangle given by:

h(d) = 1

N.Ds�[0,N.Ds] (d)

Rh(d) = N.Ds × h(d) ∗ h(−d)

Rh(d) =(

1 − |d|N.Ds

)× �[−N.Ds,N.Ds] (d)

(2)

where ‘*’ denotes the convolution operator and � is aflat window function. Then, the auto-correlation func-tion of the filtered Shadowing process is given by thefollowing convolution:

RfSh(d) =∞∫

−∞Rh(y) × RSh(d − y)dy

RfSh(d) =N.Ds∫

−N.Ds

(1 − |y|

N.Ds

)× e− |d−y|

Lcorr dy

(3)

Finally, computing the above integral leads to the fol-lowing expression:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

for : 0 ≤ d ≤ N.Ds

RfSh(d) = e− d+N.DsLcorr + e

d−N.DsLcorr + 2 × N.Ds−d

Lcorr− 2e− d

Lcorr

K0

for : d > N.Ds

RfSh(d) = e− d+N.DsLcorr + e− d−N.Ds

Lcorr − 2e− dLcorr

K0

K0 = 2 ×(

e− N.DsLcorr − 1 + N.Ds

Lcorr

)

(4)

From (4), it can be checked that the filtering step widensthe auto-correlation function of the shadowing process.

Applying the MoM or any other filter on disjointblocks of data would result in undersampling the pro-cess, then in order to restore the genuine data rate,some experimenter may interpolate the data using eitherlinear or spline interpolation techniques. As a conse-quence, the auto-correlation peak would be even morewidened than in the above case and no closed-formexpression can be provided (the short term informationis lost due to the undersampling, and the interpolationbasically consists in trying to guess the behaviour butwithout any guarantee). Then, in the following, we onlyconsider the cases of pre-processing techniques appliedon sliding windows.

When processing experimental data in order toparameterize a state oriented channel model, it is also

desirable to process separately data corresponding tothe different states. As a consequence, the datasets con-sist of disjoints blocks of continuous data which do notenable to compute easily an average auto-correlationfunction. It is then proposed to use another second ordermoment K2 on the whole datasets as follows:

K2(d) = E{

(y(d0 + d) − y(d0))2}

(5)

where the mathematical expectation E{.} is computedover the travelled distance d0 (the process is assumedergodic). The K2 moment is asymptotically related tothe auto-correlation R function as follows:

K2(d) = 2 × (1 − R(d)) (6)

Hence, using either the auto-correlation function or theK2 moment should not make any difference from theparameter estimation point of view. However, basedon Monte-Carlo numerical simulations, it has beenobserved that the convergence rate of the K2 momentis faster than that of the auto-correlation function (i.e.:less data are required in order to get a good estimation

of the K2 moment). It has also been observed that, usingthe K2 moment, the continuous blocks of data shouldlast about 50 times longer than the estimated correlationlength in order to get an unbiased estimation of the cor-relation length. For shorter data blocks, the correlationlength will systematically be underestimated.

Finally, the following method is proposed in orderto estimate the shadowing correlation length: first thesecond order moment is computed based on (5) andcontinuous blocks of data as long as possible. Then,knowing the sliding window length used to extract toshadowing distance series, and using (6) and (4), the K2curves are fitted in a RMS sense in order to estimate thecorrelation length.

A validation example of this method is illustrated onFig. 2. In this case, synthetic data have been generatedwith a 2 m correlation length. The blue curve represents

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Fig. 2. Validation example of the shadowing correlation length esti-

mator.

the K2 moment of the synthetic raw data. The syntheticdata have then been processed using either the MoM ora flat sliding window with 3 m duration. The resultingred and black curves are fully superimposed. The cyancurve, also superimposed to the red and black curves,represents the result of the data fitting problem. Thecorresponding estimated correlation length is 2.04 mand matches quite well the simulated one. Finally, adirect fit of the red or black curve using (6) and (1) (i.e.:without taking into account the filtering effect) wouldresult in the purple curve and an estimated correlationlength of 6.09 m.

To conclude this first sub-section, it has been shownthat the data pre-processing distorts the shadowingauto-correlation function and can result in a signifi-cant overestimation of the correlation length. However,a closed-form expression of the distortion has beenderived which enables to get unbiased estimations of theshadowing correlation length as far as the continuousblocks of data are long enough.

2.2. Small scale effects

The small scale effects are related to the recom-bination of the received multipaths (MP) and arecharacterized by their Doppler Spectrum. In the terres-trial case, the Doppler Spectrum can be modeled as aJakes Spectrum, while in the LMS case, a Butterworthspectrum may look more reasonable at least for somespecific environments. As a consequence, LMS gener-ative channel models may simulate the MP component

starting from the Butterworth filtering of a white Gaus-sian random process. This approach is valid for a mobilereceiver with constant velocity. However, a new set offilter coefficients is required for each possible mobilevelocity and this approach cannot handle the scenarioof a mobile with velocity varying during the simulation.

In order to remove these limitations, the Sum of Sinu-soids (SoS) approach is a good candidate. Indeed, inthis case a unique set of parameters would enable tomodel any mobile velocity, including the possibility ofchanging the mobile velocity during the simulation. Theproposed “ergodic stochastic” model is built from thetwo models in [14] and [15] so as to take advantage ofboth of them: it is then able to synthesize several uncor-related channels with any possible spectrum from a flatshape to the classical Jakes shape. The analytical formof the model is given in (7):

µ(k)(t) =√

2

N

N∑n=1

[cos(

2πf(k)1,nt + θ

(k)1,n

)

+ j × cos(

2πf(k)2,nt + θ

(k)2,n

) ] (7)

The number N of sinusoids in (7) should be larger than20 [15]. The exponent k indicates the channel number(always one for a SISO channel, but the approach isalso valid for the generation of K multiple uncorrelatedchannels). The discrete sinusoids frequency f and phaseθ parameters are detailed in (8):⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

f(k)i,n =χ×fmax sin

(arcsin (K0)

2N

[(2n − 1) + α

(k)i,0

])

α(k)i,0 = (−1)i−1 k

K + 2

θ(k)i,n

∼= U[−π;π] i.i.d.

(8)where “i.i.d.” stands for independent and identicallydistributed and fmax is the spectrum maximum fre-quency component. The K0 parameter allows restrictingthe Jakes spectrum to its lowest frequency compo-nents. Furthermore, with respect to [14] and [15], a newmultiplicative parameter χ of the maximum Dopplerfrequency is introduced in (8). This parameter enablesto expand the spectrum in order to represent almost flatspectra (see bottom part of Fig. 3) or oppositely to com-press the spectrum. It is then mandatory to impose thecondition that the product χ × K0 remains lower than 1to prevent modeling non-physical Doppler components.

Finally, let us assume that the mobile moves at a con-stant speed v during the time interval �t. In this case,

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Fig. 3. Modified restricted Jakes spectrum SoS model.

it can be seen in (9) that the channel model is a func-tion of the wavelength λ and of the travelled distanceonly. Then, in the following we will no more considerthe channel model spectrum, but its space spectrum, asillustrated on Fig. 3.

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

fmax = v

λ

�t = d

v

f(k)i,n × �t = χ sin

(arcsin (K0)

2N

[(2n − 1) + α

(k)i,0

])

× d

λ(9)

The classical Jakes Spectrum is illustrated as a bluecurve on top on Fig. 3. The green circles on Fig. 3represent each discrete sinusoid of the SoS model. Theyall have the same amplitude, and their density increasesin areas where the continuous spectrum to be modeledcontains more energy. On bottom part of Fig. 3, theproposed new model is illustrated with a K0 parameterset to 0.4 in order to restrict the Jakes spectrum to itsalmost flat domain. In this example, the χ parameteris set to 2 and the resulting simulated spectrum is then

flat and covers 80% of the maximum physical Dopplerbandwidth.

It can also be seen from Fig. 3 that the highestspace frequency component of the channel model isthe inverse of the wavelength λ, and then a samplingrate in λ/2 should theoretically satisfy the Shannon cri-terion and enable to get all the channel information withrespect to the MP component. However, this is basedon the capability of the experimental device to measurethe channel complex amplitude. In some experiments,the databank may consist only of signal level mea-surements and then the phase component is lost. TheMP component being modelled as a circular symmetriccomplex Gaussian process with purely real space auto-correlation function Rx(d) and power σ², using [5] it canbe shown that the space auto-correlation function of itssquare norm can be written as:

Rx2 (d) = σ4 + [Rx(d)]2 (10)

As a consequence, the corresponding spectrum is givenby:

Sx2 (fs) = σ4δ(fs) + Sx(fs) ∗ Sx(fs) (11)

where fs denotes the space frequency and δ(.) is theDirac function. A consequence of (11) is that the band-width of the signal is doubled (due to the convolution)

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Fig. 4. Undersampling and averaging effects on second order

statistics.

and then the minimum required distance sampling rateis no more in λ/2 but in λ/4, which is never satis-fied in the available measurements. Furthermore, thechannel measurements do usually not strictly consistof a distance sampling, but of integration (or averag-ing) of the measured power. This process results in alow pass filtering of the data depending on the receivervelocity.

Both the undersampling and averaging effects areillustrated on Fig. 4 with respect to the fade durationand level crossing rate statistics for synthetic C-Banddata. In this case, the integration time is 2.1 ms andthe simulated mobile velocity is 45 km/h. The subscript“ref” in the legend indicates that only the undersam-pling is considered. No subscript indicates that the dataare both under sampled and averaged. The discrepan-cies between the second order statistics curves clearlyshow that both the experimental data sampling distanceand integration time have to be taken into account whileestimating the channel model parameters.

3. Extension to mimo cases

The models discussed in [16–18] and in previoussection correspond to the SISO case. To produce cor-related series the corresponding RMIMO matrix willbe forced into the uncorrelated outputs of the SISOchannel simulator. To achieve this objective, the guide-lines provided in [7] will be partly followed. In thereference the cross-correlation properties between thevarious transmit-receive antenna pairs are separated

into two different matrices: RSH and RMP, describ-ing the cross-correlations of shadowing and multipatheffects respectively. The MIMO correlation matrix,RMIMO, contains elements ρkl

ij which are defined asfollows:

ρklij = E

[(hij − µij

)(hkl − µkl)∗

]σijσkl

(12)

where ij indicates one of the transmit-receive antennapairs and kl the other pair, and hij , are their correspond-ing complex envelope values. In the specific case wheredirect signal and diffuse multipath are characterizedseparately, similar definitions are used.

Thus, if in several, Gaussian, uncorrelated channels,represented in matrix form by X, we want to forcegiven cross-correlation properties described by a cross-correlation matrix, R, we have to perform the Choleskyfactorization of matrix R (i.e. C = Ch(R), such that forpurely real and positive definite matrix: R = C × CT)then we need to convert matrix X into a vector form,i.e. vect(X), and perform the multiplication of the twomatrices to get a set of partially correlated channels,Y:

Y = vect−1(vect(X) × Ch(C)) (13)

Note that X is an n × p × m matrix, where n repre-sents the number of transmit antennas, p the numberof receive antennas, m represents the time dimension,i.e., m time samples, and the cross-correlation matrixR is a np×np matrix. This procedure is also detailedin [6] and applied to the whole channel coefficientswhile in this paper, we make the distinction between theslow fading (shadowing) and the fast fading (multipath)components due to the different physical mechanismsinvolved for each component.

The global MIMO generative channel model schemeis illustrated on Fig. 5, where:

• G(0,1) denotes a Gaussian process synthesizerwith zero mean and unit variance. The synthesizedindependent processes are n’i = 1, ... ,4.

• C(RSH) stands the Cholesky factorization ofthe shadowing cross-correlation matrix (resp.C(RMP) for the multipath cross-correlationmatrix). Applying C(RSH) to the independentprocesses n’I = 1, ... ,4 results in the correlated pro-cesses ni = 1, ... ,4. These processes have still zeromean and unit variance.

• HLcorr(Z) is a first order low pass digital filterdefined in (26).

• Ai = 1, ... ,4 denote the shadowing time series in dB.

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Table 1

Parameters of the MIMO channel model

Domain Parameter

1st order, discrete distance 2-state (m, σ)G,B Mean and std of log-normal (duration of states)

semi-Markov chain

Loo distribution (M, �) G,B Normal distribution parameters for <A> (dB), the average value of the

direct signal in the states

std(A) = g G,B (<A>) Expression linking the <A> in one state to the standard deviation of A.

<K> = h(<A>) Expression linking state average values of A and K (the carrier-to-multipath ratio)

Lcorr Correlation length for both GOOD and BAD states

Sp = SP(�A) (dB/m) Slope in the transitions between states

K0, χ Multipath Doppler Spectrum parameters for the Sum of Sinusoids synthesizer

(Modified Restricted Jakes Spectrum)

MIMO RSH Overall Correlation matrix of slow variations

Note. G stands for the GOOD state and B stands for the BAD state.

• Finally, hi = 1, ... ,4 denote the channel complexenvelope time series in natural values obtained asthe sum of the slow variations ai = 1, ... ,4 and thefast variations υi = 1, ... ,4.

The definition of the model parameters is given inTable 1.

The order of the different stages in Fig. 5 is of primaryimportance and must not be changed. Especially, whenconsidering the shadowing component, the first stagemust necessarily consist in correlating the n × p centredreduced Gaussian processes. And then, the processescan be filtered depending on the channels correlationlengths, and the processes can be normalized by the Looparameters. Reversing the stages order would result inmixing all the channel parameters (either the Loo or thecorrelation length parameters).

Furthermore, on the contrary to what can be seenin some publications (such as in [7] and [10]), itwill be shown in the following that an experimentalMIMO correlation compatible with a state orientedchannel model has to be estimated from normalizedtime series (each state episode is normalized by itsmean and standard deviation in order to provide a cen-tred reduced time series). In order to prove the abovestatement, it will be shown that applying a MIMO cor-relation matrix to reduced centred Gaussian processes,and then applying the Markov (or Semi-Markov) stateparameters to the correlated processes would stronglyaffect the whole time-series correlation matrix. To thisaim, let us consider a very simplified SIMO (respec-tively MISO) model with two states. We only considerthe shadowing component, then the channel time-

series belong to the real space (i.e.: not complex). Weassume that both SIMO channels are always in thesame Markov state (which is basically the case fordual polarized channels or for short baseline antennadiversity) with the same mean and standard deviationparameters.

In the GOOD State (resp. in the BAD State withsubscript B), the channel model is then given by:

hG(t) =(

h1 G(t)

h2 G(t)=(

µG + σGn1(t)

µG + σGn2(t)(14)

where n1 and n2 are uncorrelated real reduced centredGaussian processes:⎧⎪⎪⎪⎨

⎪⎪⎪⎩E (n1) = E (n2) = 0

E(n2

1

)= E

(n2

2

)= 1

E (n1n2) = 0

(15)

We have then assumed that the SIMO channels are fullyuncorrelated within each state:

RSIMO G = RSIMO B

=(

1 0

0 1

)=(

E(n2

1

)E (n1n2)

E (n1n2) E(n2

2

))

(16)

In the following, we assume that the channel is in theGOOD state with the probability pG, it is then in theBAD state with the probability 1 − pG. We now wantto compute the channels cross-correlation coefficientaccording to the definition in (12). Let us start with thecovariance coefficient ρ:

ρ = E ([h1 − E [h1]] × [h2 − E [h2]]) (17)

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Fig. 5. Block diagram of the MIMO channel model.

Considering the Markov states for the channels expec-tation and the notations in (14), one gets:⎧⎪⎪⎨

⎪⎪⎩E [h2] = E [h1] = µ0

µ0 = pGE [h1 G] + (1 − pG) E [h1 B]

µ0 = pG.µG + (1 − pG) µB

(18)

Injecting (18) in (17) and considering now the Markovstates for the centred channels expectation one gets:

ρ = pG.E ([h1 G − µ0] × [h2 G − µ0]) +(1 − pG) E ([h1 B − µ0] × [h2 B − µ0])

(19)

Computing the expectations in (19) based on (14) and(15) leads to:

ρ = pG. (µG − µ0)2 + (1 − pG) (µB − µ0)2 (20)

Finally, by injecting the expression (18) of �0 in (20)and rearranging the expression, one gets:

ρ = pG. (1 − pG) (µG − µB)2 (21)

Then, the covariance coefficient (and then the cross-correlation coefficient) is null only if one stateprobability is equal to one, or if both states have thesame mean shadowing value.

So, in the general case, starting from uncorrelatedSIMO channels within each state, one gets correlatedSIMO time-series. This conclusion (but not the closedform expression) can be generalised to any MIMOconfiguration and to more complex Markov or semi-Markov chain models: if a MIMO correlation matrixis used to generate correlated time-series within eachstate, the whole time-series will be even more corre-

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lated if all the channels are always in the same Markovchain. If the MIMO channels can be in different Markovstates, then the whole time-series correlation will still bedifferent than the input correlation but not necessarilyhigher.

As a consequence, a correlation matrix estimatedfrom global experimental time-series regardless of theMarkov state of the time series is not compatible withstate oriented generative channel models (i.e.: it can notbe used as input parameter of such models). In the spe-cial case of dual-polarized channels, using such a matrixwould result in over-correlating the channels, and then,in under-estimating the capacity gain of MIMO LMSchannels with respect to SISO LMS channels. How-ever this matrix can be used as validation element ofsuch models: whatever its input parameters, any MIMOchannel model should be able to reproduce the globalchannel correlation matrix.

Then, in order to extract MIMO correlation matrixfrom experimental dataset that is compatible with stateoriented channel models, one must follow the steps ofgenerative channel model in strictly reverse order. Thatis, based on the scheme and notations of the block dia-gram on Fig. 5, a MIMO correlation matrix compatiblewith this state oriented channel model has to be esti-mated from normalized shadowing time-series wherethe normalization is performed on a state episode basis.For each state episode k, the shadowing Ak

i on channeli is normalised into Ak

i as:

Aki = Ak

i − < Aki >

std(Ak

i

) (22)

where < Aki > and std

(Ak

i

)are the local average and

standard deviation of the shadowing for the consid-ered state episode. Finally, the coefficient of line i andcolumn j of the shadowing MIMO correlation matrixRSH, is computed as:

ρi,j =E[(

Ai − E[Ai

])× (Aj − E[Aj

])∗]std(Ai) × std(Aj)

(23)

4. Extension to satellite diversity cases

4.1. State transitions

The last part of this paper addresses the satellitediversity MIMO channel models. The main differencewith the single satellite channels is that in this casethe various channels can be in different states. As

a consequence, the initial SISO model state durationparameters can no more be used.

Past activities related to this kind of channels weremainly aimed at reducing the number of parameters ofa Markov chain state transition matrix (STM). This isespecially the case of the Lutz model in [11], whichhas been used and validated in some peculiar cases in[12] by Meenan. It is basically a Markov chain modelbut instead of requiring 12 parameters (in the case of atwo-satellites link, with two possible channel states perlink), it only needs 5 parameters: the two parameters ofthe STM of each satellite link and a correlation coef-ficient between both satellite links state series. Then,a methodology detailed in [11] enables to generate thesatellite diversity STM. The advantage compared to aclassical Markov model obviously lies in the reducednumber of parameters. But, as any Markov approach,this model is very unlikely to reproduce the experi-mental state duration distributions due to its intrinsicinability to reproduce long events [16].

More recently, satellite diversity channels perfor-mances were evaluated based on measurements in theMILADY project [2, 19] and a Master-Slave approachhad initially been proposed. In this case, the basic SISOmodel was still based on a 3-states Markov chain.

However, following the logic of previous sections ofthis paper, 2-states are still envisioned for each satel-lite link, resulting in a 4-states semi-Markov model forthe satellite diversity link. Which is also the approachfinally chosen in the MILADY CCN [4] for the firsttwo satellites (and the Master-slave approach is usedfor more than two satellites). The basic assumption ofthe 4-states semi-Markov model is that at the end of astate, the satellite diversity link configuration (e.g.: GG– GB – BG – BB) will necessarily change. Then, themodel is based on a STM structure such as in (24) andstate duration distributions for each of the 4 possiblestates. The total number of required parameters is 16 : 8parameters for the STM and 8 parameters for the stateduration distributions.

STM =

⎛⎜⎜⎜⎜⎝

0 p01 p02 1 − p01 − p02

p10 0 p12 1 − p10 − p12

p20 p21 0 1 − p20 − p21

p30 p31 1 − p30 − p31 0

⎞⎟⎟⎟⎟⎠

(24)The state duration distributions of each of the 4 possiblestates have been found to be log-normal (see the vali-dation examples on Figs. 6 and 7). In order to validatethe model, state distance series have been synthesized

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Fig. 6. Satellite diversity states duration distribution example for a

wooded environment at C-Band.

on a distance equivalent to 10 times the distance of theexperimental data. The satellite diversity state durationdistributions have then been computed and comparedto the experimental ones. The results are illustratedon Fig. 6 for an example of wooded environment atC-Band where the azimuth separation between bothsatellite links is 20◦ [8, 9]. It has also been checkedif the model was also able to reproduce the state dura-tion distribution of each single satellite link (modelledas log-normal distributions in the single satellite case,see Fig. 5). Theoretically the synthetic distance seriesdistribution is a weighted sum of 2 log-normal distribu-tions among the 4 possible states duration distributionsand then the resulting distribution should not be log-normal. However, it can be checked on Fig. 7 that thesynthetic distance series for each single satellite linkfairly well matches the experimental distributions.

Finally, the ability of this model to reproduce the cor-relation between the state series of both satellite linkshas been tested. The result is inserted as text in bottomon the right on the example of Fig. 7. Whatever theenvironment and the azimuth separation, it has beenchecked that, while not being directly an input param-eter of the model, the synthetic correlation coefficientdoes not exactly match the experimental one but is inthe good order of magnitude. The proposed approach isthen valid for the satellite diversity LMS channel. Thus,the scheme in Fig. 5 can be adapted to generate satel-lite diversity distance series just by randomly drawingnew states based on the STM (24) and by using theappropriate Loo parameters for each satellite link.

Fig. 7. Satellite diversity state model validation example for a wooded

environment at C-Band.

4.2. Channels correlation coefficient

Basically, for the satellite diversity LMS channels,both satellites links have different angles of arrival withrespect to the receiver location. It is then likely thatthe various channels will have different shadowing cor-relation lengths as they are not affected by the sameobstacles. In order to analyse the impact of it on thecorrelation coefficient between the channels of bothsatellite links, let us consider two white centred Gaus-sian processes x and y such that:{

E(x) = E(y) = 0

E(xy) = σ(25)

The processes are then filtered respectively with Ha andHb filters, where Ha (resp. the Hb) is a simplified nota-tion for the first order low pass digital filter HLcorr aalready introduced in Fig. 5 and defined as:

Ha(Z) =√

1 − a2

1 − aZ−1 with a = exp

(− Ds

Lcorr a

)(26)

or equivalently, using the reduced angular frequency θ:

Ha(θ) =√

1 − a2

1 − a exp(−jθ)(27)

Let us note xa and yb the filter outputs, Sxayb(θ) their

inter-spectral power density, and Sxy(θ) the x and y inter-spectral power density. We can write the relation:

AU

THO

R C

OP

Y


⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

E(xy) = σ =π∫

−π

Sxy(θ)dθ

2π

E(xayb) =

π∫−π

Sxayb(θ)

dθ

2π

(28)

Thanks to the Wiener-Lee relations [13], we can alsowrite:

E(xayb) =

π∫−π

Sxayb(θ)

dθ

2π=

π∫−π

Ha(θ)H∗b (θ)Sxy(θ)

dθ

2π

(29)with:

Ha(θ)H∗b (θ)

=√

1 − a2√

1 − b2

1 + ab − (a + b) cos(θ) + j(a − b) sin(θ)(30)

where the denominator is real and even and the imag-inary part of the numerator is odd, then its integral isnull. So finally, when integrating the real part of (30)and injecting the result in (29), one gets:

E(xayb) = σ

√1 − a2 × √

1 − b2

1 − ab(31)

Then, depending on the first order low pass filterparameters; the generative scheme of Fig. 5 will distortthe correlation between channels having different cor-relation lengths. However, this distortion can be closedform quantified and then a pre-compensation can beapplied to the channels correlation matrix by multiply-ing each coefficient with the correction factor Ccorrect

defined as:

Ccorrect = 1 − ab√1 − a2 × √

1 − b2(32)

In this case, not correcting the correlation matrix wouldresult in under-correlating the channels, and then, inover-estimating the capacity gain of satellite diversityLMS channels with respect to SISO LMS channels.

5. Conclusions

This paper has addressed the statistical modelling ofLMS fading channels. A versatile generative approachhas been proposed. Just by adapting the dimension ofthe input parameters (scalar or vectorized), the modelcan encompass single satellite SISO to MIMO chan-nels as well as satellite diversity channels. For the

SISO channel modelling, special care has been given tothe channel dynamic parameters as they mainly affectthe design of system parameters. In case of multiplechannels, special care has been given to the channelscross-correlation coefficients as they mainly affect thecapacity gain with respect to SISO channels. Finally,the proposed approach is statistically reliable; it enablesto reproduce the channel behaviour without biasing itscritical parameters.

Acknowledgments

This activity has been carried out under CNES fund-ing and in the framework of the European action COSTIC0802: “Propagation tools and data for integratedTelecommunication, Navigation and Earth Observationsystems”. It is based on the S- and C-Band data collectedby CNES in 2008 in the town of Auch in the south ofFrance [8, 9].

References

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http://telecom.esa.int/telecom/www/object/index.cfm?fobjectid=29020

a generative mimo channel model: encompassing single satellite and satellite diversity cases

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