compact spatial correlation formula for mimo in factory ... · compact spatial correlation formula...

224 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.8, NO.2 August 2010

Compact Spatial Correlation Formula forMIMO in Factory Environment

KY Leng1 , Kei Sakaguchi2 , and Kiyomichi Araki3 , Non-members

ABSTRACT

Multiple-Input Multiple-Output (MIMO) trans-mission scheme is considered as a promised technol-ogy for future wireless communication since it of-fers significant increases in data throughput and linkrange without additional bandwidth or transmittedpower. However, the MIMO channel capacity isseverely degraded by the correlation between indi-vidual sub-channels of its channel matrix. Hence, insystem simulations, the correlation should be takeninto account. Since the conventional method suchas the extended one ring and double ring scatter-ers model require the knowledge of each multipathcomponents, it is obvious that a simpler model, butaccurate enough, should be proposed to reduce thecomplexity and the calculation time of the model. Inthis paper, we propose a compact spatial correlationformula that can be used to calculate the channel ca-pacity of MIMO system in 950 MHz band at factoryenvironment. The validity of the proposed model alsoholds for other environments and frequency bands.With the proposed model, we need to know only threeimportant parameters, such as antenna size, antennaseparation, and Rician K factor to generate the spa-tial correlation of the channel.

Keywords: MIMO Channel Capacity, Spatial Cor-relation, K-factor, Line-of-Sight Component

1. INTRODUCTION

In recent years, Multiple-Input Multiple-Output(MIMO) wireless communication techniques have at-tracted strong attention in research and developmentdue to their potential benefit in spectral efficiency,throughput and quality of service. However, it de-mands a great effort to test the performance of asystem, such as the real behavior of protocols andnetwork performance, using MIMO technology by atest-bed. Hence, a high level protocol simulationis preferred, especially for Wireless Sensor Networks(WSN), with hundreds or thousands of sensor nodes,which use MIMO as a core technology for data trans-mission [1-3].

Manuscript received on May 19, 2009 ; revised on April 4,2010.

1,2,3 The authors are with Tokyo Institute of Technology,Tokyo, 152-8552 Japan. E-mail: [email protected],[email protected] and [email protected]

Since the correlation effect between individual sub-channels of the MIMO channel matrix can degradethe performance of MIMO, the correlation effectshould be taken into account in the system simu-lation. To our knowledge, until now, when MIMOsystem simulation are performed, the spatial correla-tion is included by means of measured data, or basedon a ray-tracing method or by using a one ring ordouble ring scatterers model or Salz-Winters spatialcorrelation model . Some of these approaches are dif-ficult to carry, especially for WSN, since the MIMOchannel statistics are represented by a large numberof parameters due to its multipath components; andothers lack of generality.

Note that, for WSN, the design goals are focusedon energy saving, traffic and environment adaptabil-ity. It is mainly used for data collection and eventcontrol. The maximum transmitted power for suchnarrow band and small coverage is about 5 dBm. Formobile communication, the design goals are mobilityand high speed transmission. It is mainly used forvoice and data communication systems. The max-imum transmitted power for such wider bands andlarger coverage is about 30 dBm to 33 dBm. Themain differences between WSN and cellular commu-nication are the initial motivation keys of this paper.

In this paper, a simplified spatial correlation modelis introduced for WSN using MIMO, then extendedto cover MIMO in the general case. Moreover, thispaper is an extended version of our work in [7] whichfocuses on the development of a pathloss model forWSN using MIMO in a factory environment. We in-tend to remain with the factory environment condi-tion until the formula is derived.

Using test-bed to measure data in a real environ-ment demands a lot of effort and time. In this paper,we use a ray-tracing method to arrive at the raw data.This method is expected to provide data close to thereal measurement.

In [4, 6], the correlation model is a function of theantenna separation and the antenna size only. Inour model, we introduce a Rician K factor, whichis a ratio between the Line of Sight (LOS) compo-nent and diffused multipath components. Unlike [5],which specifies the correlation model and channel ma-trix with the number of scatterers around transmitterand receiver, we arrive at a model free from specifyingmultipath components. Hence, we reduce the compu-tational resource and time.

The rest of this paper is organized as follows.

Compact Spatial Correlation Formula for MIMO in Factory Environment 225

Section 2 presents a review and updated version ofour work in [7], and the evaluation of the proposedpathloss model is presented in Section 3. Section 4proposes a MIMO channel model using the outcomefrom the proposed pathloss and spatial correlationformula, and the validation of this spatial correlationformula is evaluated in Section 5. Finally, the con-clusion is discussed in Section 6.

2. PATHLOSS MODEL AND SPATIAL COR-RELATION FORMULA

The objective of our research in [7] aims to developa pathloss model for WSN using MIMO technology infactory environments. The pathloss model is designedto work at 950 MHz band, whereas the spatial correla-tion is designed for two transmitted (Tx) and two re-ceived (Rx) antennas. These models are designed forthree basic environments, named Roof-to-Roof (RR),Roof-to-Ground (RG) and Ground-to-Ground (GG).Since these correlations are assumed independent [8],and in WSN, all sensor nodes are assumed fixed (notemporal correlation) and the propagation channel isnarrowband (no spectral correlation), we will consideronly the effect of spatial correlation. Unless otherwisestated, the word correlation refers to spatial correla-tion for the rest of this paper.

In this paper, process #5 and #6 in [7] Section3 are repeated. However, we do not fix the antennaspacing D = 1

2λ as before. On the other hand, weregenerate and recalculate our spatial correlation co-efficients (for RR, RG and GG environment at bothTx and Rx side) for D ranging from 0.3λ to 3λ witha step of 0.1λ. For each antenna spacing, we recal-culate all these spatial correlation coefficients in allpossible Tx-Rx distances d.

2.1 Kronecker model validity

In order to verify the validity of the Kroneckermodel as stated in [9, 10], it is necessary to only verifywhether the Direction-of-Arrival (DoA) spectral andthe Direction-of-Departure (DoD) spectral are inde-pendent or not.

Figure. 1 shows the spatial correlation coefficientat Tx and Rx side versus the distance where D = 1

2λand antenna size at both sides N = 2 in RR environ-ment. The reason why we selected RR environmentis because this environment has less numbers of mul-tipaths compared to the other two environments. Soif the Kronecker model is valid in this environment,it must be valid in the other two as well.

For a factory environment as shown in [7], it seemsthat there are so many reflected and diffracted objectsin between Tx and Rx antennas. If this is true, wecan say directly that DoD and DoA spectral are inde-pendent. However, we do not conclude this statementbased on geometry alone. To be transparent, we markall the spatial correlations that are smaller than 0.1

0 100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Tx−Rx Distance [m]

Spa

tial C

orre

latio

n fo

r 2x

2 A

nten

nas

Factory−Environment: Roof−to−Roof (432 points)

Spatial Correlation at RxSpatial Correlation at Tx

Fig.1: Distribution of spatial correlation coefficientvs distance

(blue circle), then in between its mean value ±0.02(black circle), and all the value that are bigger than0.7 (red circle). See also Fig. 1. Finally, we verifythe corresponding position of each spatial correlationpoint with the factory map to see the relationship be-tween these correlation values, and building heights.

As expected, there is no correlation between thesestated relationships because we expect in general thatfor the RR environment the correlation between sub-channel must be close to 1 since there are so littlemultipath components which interfere with the LOScomponents. However, as we can see from Fig. 1 thereare so many points that are smaller than 1 and someof them are even close to 0. When we check the ge-ometric position of those points that are close to 0(smaller than 0.1), we found that, for some Tx-Rxpair, both positions (height of Tx and Rx) are highfrom the ground, and on some, Tx is high but Rx islow (the lowest height for RR environment is 4 m),and vice-versa. We can conclude that the spatial cor-relation for this factory environment is independentof building height.

Moreover, from this figure, we can see that thespatial correlation coefficients at both side seem tobe randomly distributed in all distance values. Thisfigure also shown that the mean value of the spatialcorrelation in each distance block seem to be constantover the propagation distance. Normally, the meanvalue of the spatial correlation changes over distance,however, it does not change here. This is because ofthe propagation distance of our model is small (lessthan 1 km for microcell environment). We can alsosay that, for a given Tx and Tx-Rx distance, the cor-relation at Rx side can have value in between 0 and1 which mean the DoA and DoD spectral are not thesame. In other word, DoA and DoD spectral are inde-pendent for RR environment. With similar reasons,DoA and DoD spectral are independent for RG and


GG environment as well.We can conclude that the Kronecker model is valid

for this factory environment as well as other environ-ments of the same kind.

2.2 Spatial correlation distribution

In the previous Subsection, we found that the spa-tial correlation values are randomly distributed for agiven Tx-Rx distance d and seem to be the same fordifferent value of d. It is better to represent thesespatial correlation values by a statistical means.

Unlike [7], this time, we evaluate the performanceof the fitted distribution by means of Maximum Like-lihood Estimation (MLE); and the candidate distri-bution function to be chosen for this time are Gaus-sian, Rayleigh, and Rician since the Rician K factoris about 0 (mean Rician become Rayleigh) in [7].

From Fig. 2(a), we can see that the Maximum Like-lihood Value (MLV) using Rician distribution is al-ways higher or equal to the MLV using Gaussian andRayleigh. In some cases, the MLV using Rayleigh ishigher than the MLV using Gaussian. However, wecan say in general that the MLV using Gaussian arevery close to that using Rician.

From Fig. 2(b), we can see that the standard de-viation in Gaussian case is more stable than that inRician, moreover, from Fig. 2(c), the variation of themean value in Gaussian case can be represented by amodified zero-order Bessel function of the first kind,which is more convenient to represent than the vari-ation of V value in Rician case. See also Fig. 3 wherethe variation of the mean (or median for Gaussiandistribution) value of both correlation at Tx and Rxside are plotted with different Rician K values. Wewill discuss more about this figure in the Section 4.

Based on these reasons, Gaussian distribution ischosen with the condition that the spatial correla-tions generated using Gaussian are in between thevalues 0 and 1. The work in [7] also confirms thisfinal selection.

Since the antenna separation D = 12λ is far enough

to ignore the mutual coupling effect between thenearby antenna, we can use Fig. 2(c) to get the spa-tial correlation for the other antennas as well. Forexample, in a 4 × 4 MIMO system, the correlationmatrix < is formed by it component <ij where i, j =1, 2, 3, and 4. By symmetry and the propriety ofthe correlation matrix, we need to know only <12

which corresponds to D12 = 12λ, <13 which corre-

sponds to D13 = 22λ = λ, and <14 which corresponds

to D14 = 32λ.

Based on this consideration, and that it is morethan enough at the present time for WSN to use atmaximum 4 antennas at each node, we will show onlythe spatial correlation for the 4 × 4 MIMO system.Moreover, as the Kronecker model is selected as aproper correlation model to be used, we will showdirectly the square root of the corresponding spatial

0 0.5 1 1.5 2 2.5 3−1

0

1

2

3

4x 10

4

Antenna separation distance over Lamda

Max

Log

Lik

elih

ood

Roof−to−Ground: Rx side

GaussianRayleighRician

0 0.5 1 1.5 2 2.5 3−1

0

1

2

3

4x 10

4


Max

Log

Lik

elih

ood

Roof−to−Ground: Tx side


(a)Maximum Loglikelihood

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8


Sta

ndar

d D

evia

tion



0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8


Sta

ndar

d D

evia

tion



(b)Standard deviation

0 0.5 1 1.5 2 2.5 30.2

0.4

0.6

0.8

1

Antenna saparation distance over Lamda

Mu

& V


Mu−GaussianV−Rician

0 0.5 1 1.5 2 2.5 30.2

0.4

0.6

0.8

1

Antenna saparation distance over Lamda

Mu

& V


Mu−GaussianV−Rician

(c)Mean value (Gaussian) vs V (Rician or Rayleigh) value

Fig.2: Fitting results from Roof-to-Ground environ-ment Data

correlation at Tx and Rx in RR, RG, and GG envi-ronment as listed below.


0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Spa

tial C

orre

latio

n

Roof−to−Ground: Spatial Correlation vs Rician K−factor

MedianCorr RxMedianCorr TxK=0.5K=0.8K=1K=1.2K=1.5

Fig.3: Spatial correlation coefficient vs distance

<12TxRR

=

0.9430 0.1412 0.2611 0.15020.1412 0.9442 0.1426 0.26110.2611 0.1426 0.9442 0.14120.1502 0.2611 0.1412 0.9430

(1)

<12RxRR

=

0.9371 0.1460 0.2798 0.14910.1460 0.9375 0.1465 0.27980.2798 0.1465 0.9375 0.14600.1491 0.2798 0.1460 0.9371

(2)

<12TxRG

=

0.9412 0.1401 0.2530 0.17450.1401 0.9462 0.1450 0.25300.2530 0.1450 0.9462 0.14010.1745 0.2530 0.1401 0.9412

(3)

<12RxRG

=

0.9426 0.1402 0.2473 0.17500.1402 0.9477 0.1450 0.24730.2473 0.1450 0.9477 0.14020.1750 0.2473 0.1402 0.9426

(4)

<12TxGG

=

0.9348 0.1463 0.2855 0.15270.1463 0.9356 0.1474 0.28550.2855 0.1474 0.9356 0.14630.1527 0.2855 0.1463 0.9348

(5)

<12RxGG

=

0.9376 0.1460 0.2760 0.15280.1460 0.9386 0.1471 0.27600.2760 0.1471 0.9386 0.14600.1528 0.2760 0.1460 0.9376

(6)

3. VALIDATION OF THE PATHLOSS MODEL

In our previous work [7], the proposed pathlossmodel was verified with the free space and d−4 model.In this paper, we compared our model (RG environ-ment for a fair comparison with other models) with

101

102

103

40

60

80

100

120

140

160

180

200

Distance [m]

Pat

hLos

s [d

B]

LossdB = 10*log10

(3.7*10−5*d3.96 + d1.82) + 6.5 + Const

RapLab Field−SumRapLab Power−SumProposed Model [MSE = 10.9783]FreeSpace [MSE = 13.1235]d−n law (n=3.96) [MSE = 12.5805]Hata−Urban (d > 1km) [MSE = 20.5448]MicroCell NLOS [MSE = 23.2088]MicroCell LOS [MSE = 11.2746]

Fig.4: Roof-to-Ground pathloss in factory environ-ment

the Hata model for urban areas and the COST 231Walfish-Ikegami in Line of Sight (LOS) and NoneLine of Sight (NLOS) microcell. Note that ourpathloss model aims to improve the accuracy of thehigh level protocol simulation of WSN as mentionedin [2] whereas Hata model and COST 231 model aremainly used in cellular communication systems [11,12].

In order to see more clearly the difference betweenthese models, we decide to use the Mean Square Error(MSE) value to judge which one is the best represen-tation of the real pathloss in factory environments asshown in Fig. 4.

Note that Hata model for Urban area (Hata-Urban) is valid for distance between 1 km till 20km. However, we want to know only how far the per-formance of this model is from the proposed model.From this figure, we see that the best fit to the rawdata is our proposed model, the next good one isCOST 231 Walfish-Ikegami for microcell LOS (Mi-crocell LOS), and the third good one is d−n law withpathloss exponent n = 3.96.

From these observations, we can say that this RGfactory environment can be considered as microcellLOS environment and is desirable to represent thispathloss by d−n law since this law considers thepathloss in two different situations called pathloss be-fore and after break point distance db. It is more de-sirable to represent this pathloss by our model sinceour model considers a smooth transition between thetwo environment before and after db. See also [7].

The formulas and parameters used in this paperare taken from [11, 12]. With Base Station (BS) an-tenna height hB = 12.5 m, Mobile Station (MS) an-tenna height hM = 1.5 m, and other necessary param-eter as listed in [12], the Hata-Urban, the microcellNLOS and LOS pathloss are respectively simplified


to:

LdBNLOS= −58.5 + 37.7log10d

+26.1log10fc (7)

LdBNLOS= −55.9 + 38log10d

+(24.5 +1.5925

fc)log10fc (8)

LdBLOS = −35.4 + 26log10d + 20log10fc (9)

where fc is the operating frequency in MHz, and dis the distance between Tx and Rx in m. In thoseformulae, d is at least 20 m.

With the parameters stated above, the break pointdistance db, measured in [m], can be calculated usingthe following equation.

dB =4hMhBfc

300(10)

This implies db = 237.5 m. If n = 4 was chosen,the resulting MSE of d−n law is 13.0411. In Fig. 4,db = 1√

C= 164.4 m was chosen, where C = 3.7×10−5

is the coefficients in the proposed model in [7]. Withthis setting and n = 3.96 was chosen, the MSE of d−n

law is now reduced to 12.58, which means that thepropose model is flexible and reflects the behaviors ofthe environment than any existing models.

Using the method in [7], Section 4, it is very easyto verify that the above formulae are special casesof our proposed model. We can conclude that theproposed pathloss model is valid for any environmentand frequency band with so little modification of itscoefficients.

4. MIMO CHANNEL MODEL AND FINALPATHLOSS MODEL RESULT

For a fixed linear channel matrix H with zero-meanadditive white Gaussian noise, and when the trans-mitted signal vector is composed of statically inde-pendent equal power components each with a Gaus-sian distribution, the MIMO channel capacity can bedenoted by [5, 13] as:

CMIMO

= log2

[det

[IN +

ρ

MHHH

]](11)

where det[.] denotes the determinant operator, [.]H

denotes Hermitian transpose of a matrix, M is thenumber of transmitting antennas, N is the numberof receiving antennas, H is N ×M normalized chan-nel matrix between the transmitter (Tx) and receiver(Rx), ρ is the Signal-to-Noise Ratio (SNR) per Rxantenna, and IN is N ×N identity matrix.

For a correlated channel, let <Tx and <Rx bethe correlation matrix at Transmitter side (Tx) andReceiver side (Rx) respectively. So, the normalizedchannel matrix can be generated by [9] as:

H = <12RxU<

12Tx (12)

Table 1: Rician K factor in different environmentEnvironment Rician K factorRoof-to-Roof (RR) 0.85Roof-to-Ground (RG) 0.80Ground-to-Ground (GG) 0.90Factory (RR, RG, GG) 0.85

where U is a stochastic matrix with i.i.d. complexGaussian zero-mean unit variance elements. Simi-larly, for an uncorrelated channel, the channel matrixH can be defined as:

H = U (13)

In this paper, since the result from ray-tracingshows that the spatial correlation at the Tx side arevery close to that at the Rx side, we will assume that< = <Tx = <Rx. The validity of this assumption willbe checked in Section 5. The correlation matrix atTx or Rx side, can be denoted as <, is defined (wheni 6= j), based on [5, 7], as:

<ij =J0(

2πDij

λ ) + K

(1 + K)(14)

where J0 is zeroth-order Bessel function of the firstkind, K is the Rician factor of the working environ-ment, Dij is the distance between antenna i and an-tenna j at each side (Tx or Rx side), and λ is thewavelength which correspond to the operating fre-quency. In order to get the correlation matrix, weneed only the basic information such as antenna sizeat Tx or Rx side, antenna spacing, and Rician K fac-tor of the environment. The knowledge of multipathcomponents is not required.

From the result in Fig. 3, the K value correspond-ing to RG environment is in between 0.5 and 1. How-ever, on average, K takes the value of 0.8. This re-sult also means that this environment can be con-sidered as a LOS microcell which conform with theresult in Section 3. Since the variation of K valueis small around its mean value, we decided to repre-sent the corresponding environment by only one Kvalue. Note that in [5], K is initially used to char-acterize only the environment between each antennapair. Later on, they admit that the Rician factor foreach antenna pair is the same, so it is obvious to useone K value for one environment. Table 1 shows theRician K factor in RR, RG, and GG environment aswell as in the whole factory environment.

If the upper bound of MIMO channel capacity isneeded, Jensen inequality is preferred and this up-per bound capacity is defined, based on [13] and theinequality & concavity of logdet function [14], as:

〈CMIMO 〉 ≤ log2

[det

[IN +

ρ

M〈HHH〉

]]

= log2

[det

[IN +

ρ

M<RxNIN

]](15)


Table 2: Final proposed pathloss model for factoryenvironment at 950 MHz band

LdB = 10log10(Cdn2 + Bdn1) + A + Const

+(Xσ − σ√

π/2)

σ = bd + a

< = <Tx = <Rx = N(µ0, σ20)

µ0 = J0(2πD

λ )+K

(1+K)

Parameters RR RG GGn2 3.73 3.96 6.15n1 1.77 1.82 2.45C 7× 10−5 3.7× 10−5 6× 10−7

B 1 1 1A 5 6.5 -7b 0.00497 0.0232 0.0520a 7.104 7.440 19.912σ0 0.2 0.2 0.2K 0.85 0.85 0.85

Const 32 32 32

where 〈.〉 is the expectation over the channel matrix.For the reason of simplicity, we will use the upper

bound of the MIMO ergodic channel capacity in thedesign and analysis of the system. The simplified ver-sion of the MIMO channel capacity can be rewrittenwith a very simple expression as:

〈CMIMO

〉 ∼= log2

[det

[IN +

ρ

M<RxNIN

]](16)

Finally, the summary of the proposed pathlossmodel is presented in Table 2 for future use.

The parameters using in this Table are defined asfollows: A, B, and C are the coefficients that definethe characteristic of the environment, n1 and n2 de-fine the pathloss exponent in the zone before andafter break point distance db respectively, d is thedistance between Tx and Rx, and D is the separa-tion between two antennas. Const is the system lossconstant which is defined as 20log10( 4πf

c ) where f isthe operating frequency in Hz, and c is the speedof light in vacuum (3 × 108 m/s). For the operat-ing frequency of 950 MHz, Const = 32 dB. In somecase, n2, n1, A, B, and C can be functions of f aswell. Xσ characterizes the loss of the received sur-rounding environment (shadowing) [7, 15, 16]. It is aRayleigh random variable with standard deviation σ(σ = bd+a, where a, and b are the coefficients definedthe characteristic of the environment).

Note that, this model is valid for other frequencyband and environment, but the coefficients used inthis Table are valid for the frequency band of 950MHz and factory environment where the averagebuilding height is 8.4 m (or the average antennaheight on the roof is 13.4 m) with the standard devi-ation of 6.38 m.

1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

35

40

Number of Antenna pairs

Erg

odic

cha

nnel

cap

acity

[bps

/Hz]

RG environment with K=0.8

Uncorrelated Channel−Monte CarloCorrelated−Ray Tracing−Monte CarloCorrelated−Bessel−Monte CarloCorrelated−Ray tracing−JensenCorrelated−Bessel−Jensen

SNR=30dB

SNR=20dB

SNR=10dB

(a)Roof-to-Ground environment with K=0.80

1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

35

40

Number of Antenna pairs

Erg

odic

cha

nnel

cap

acity

[bps

/Hz]

RG environment with K=0.85

Uncorrelated Channel−Monte CarloCorrelated−Ray Tracing−Monte CarloCorrelated−Bessel−Monte CarloCorrelated−Ray tracing−JensenCorrelated−Bessel−Jensen

SNR=10dB

SNR=20dB

SNR=30dB

(b)Roof-to-Ground environment with K=0.85

Fig.5: Channel capacity vs antenna size

5. EVALUATION OF THE SPATIAL COR-RELATION FORMULA

In order to evaluate the validity of the proposedspatial correlation formula in Section 4, we decide toplot the ergodic channel capacity of a MIMO systemwith different antenna pairs and SNR. Since Rician Kfactor for RR, RG, and GG environment are different,we decided to plot K = 0.80 and K = 0.85 with RGenvironment data in Fig. 5. Note that K = 0.80 isthe optimum value for RG environment only, whereasK = 0.85 is the average value among RR, RG, andGG. This value defines the characteristic of factoryenvironment.

In this figure, the graph named ‘UncorrelatedChannel-Monte Carlo’ is generated using Equ. 11and Equ. 13; whereas the graph named ‘Correlated-Ray Tracing-Monte Carlo’ is generated using Equ. 11,Equ. 12, Equ. 3, and Equ. 4; and the graph named‘Correlated-Bessel-Monte Carlo’ is generated usingEqu. 11, Equ. 12, and Equ. 14. Similarly, graph


named ‘Correlated-Ray Tracing-Jensen’ is generatedusing Equ. 16, and Equ. 4; and the graph named‘Correlated-Bessel-Jensen’ is generated using Equ. 16,and Equ. 14.

From this figure, we see that the graph‘Correlated-Bessel-Monte Carlo’ gives a result closeto ‘Correlated-Ray Tracing-Monte Carlo’ in anySNR, antenna pair and K value. Similarly, thegraph ‘Correlated-Bessel-Jensen’ gives a result closeto ‘Correlated-Ray Tracing-Jensen’. Moreover, thegraphs using Bessel function are closer to the graphsusing Ray-tracing data when K = 0.80 than K =0.85. However, the differences are little.

From this figure, we can also see the effect ofspatial correlation on the MIMO channel capacity.Normally, the channel capacity of an uncorrelatedchannel increases linearly with the antenna size, butby adding the correlation effect to the channel, theMIMO capacity decreases and the difference betweenthese two become larger and larger. If we increasethe antenna size, the result for the MIMO correlatedchannel capacity does not increase linearly with theantenna size. For 2× 2 MIMO, this difference can benegligible.

We can conclude that instead of using data fromRay-tracing (Equ. 1 to Equ. 6) which demands a lotof efforts, Equ. 14 can be used to generate the cor-relation matrix < as well as correlated channel ma-trix H with a very simple calculation complexity, andyet good accuracy. Since an environment can alwaysbe modeled by a Rician K factor, we also can gen-erate the correlation matrix without complexity andloss of precision using this compact spatial correlationformula (Equ. 14). This statement holds for other en-vironments and frequencies as well.

6. CONCLUSIONS

In this paper, a complete pathloss model for RR,RG, and GG that include the shadowing and spa-tial correlation for a specific factory environment waspresented.

The proposed pathloss formula can be used tomodel any existing pathloss model up to this time.In this paper, Hata model for urban area and COST231 Walfish-Ikegami in LOS and NLOS microcell areused in comparison with our model. The results showthat our model fits better to the raw data (data fromray-tracing which is considered as the most accurateand reliable in prediction of the pathloss) than anymodels, including Free space and d−n law.

With the propose spatial correlation formula, it isvery easy to generate any correlated channel matrixcompared to other methods because this formula re-quires only the basic proprieties such as the antennasize, antenna separation, and Rician K factor. Wealso note that this formula can be used in any en-vironment and frequency. Only one parameter thatcharacterizes the environment is K.

For a chosen factory environment, the Rician Kfactor is found to be 0.85, which means that this fac-tory environment can be considered as a LOS micro-cell. This conclusion is also confirmed with the resultin Section 2 (distribution of the spatial correlationcan be modeled as Rician or Gaussian, which meansthe channel must have the LOS components) and Sec-tion 3 (the second good result from the fitting methodis COST 231 Walfish-Ikegami in LOS microcell).

With these formulae, we can perform a high levelsimulation for any system using MIMO technology,such as future WSN with accuracy and speed.

References

[1] D. S. Baum, J. Hansen, J. Salo, and P. Kyosti,“An Interim Channel Model for Beyond-3G Sys-tems: Extending the 3GPP Spatial ChannelModel (SCM),” Vehicular Technology Confer-ence, VTC-2005, IEEE 61st, Vol. 5, pp. 3132-3136, May-Jun. 2005.

[2] J. M. Molina-Garcia-Pardo, A. Martinez-Sala,M. V. Bueno-Delgado, E. Egea-Lopez, L. Juan-Llacer, and J. Garcıa-Haro, “Channel Modelat 868 MHz for Wireless Sensor Networks inOutdoor Scenarios,” International Workshop onWireless Ad-hoc Networks, IWWAN2005, May2005.

[3] K. Mizutani, K. Sakaguchi, Jun-ichi Takada, andK. Araki, “Development of MIMO-SDR Plat-form and its Application to Real-Time ChannelMeasurements,” IEICE Trans. Commun., Vol.E89-B, pp. 3197-3207, Dec. 2006.

[4] A. Giorgetti, M. Chiani, M. Shafi, and P. J.Smith, “Characterizing MIMO Capacity Un-der the Influence of Spatial/Temporal Correla-tion,” Australian Communication Theory Work-shop Proceeding, AusCTW03, 2003.

[5] Li-Chun Wang, and Yun-Huai Cheng, “Mod-elling and Capacity Analysis of MIMO Ri-cian Fading Channels for Mobile-to-Mobile Com-munications,” Vehicular Technology Conference,VTC-2005, IEEE 62nd, Vol. 2, pp. 1279-1283,Sept. 2005.

[6] S. Loyka, and G. Tsoulos, “Estimating MIMOSystem Performance Using the Correlation Ma-trix Approach,” Communications Letters, IEEE,Vol. 6, pp. 19-21, Jan. 2002.

[7] K. Leng, K. Sakaguchi, and K. Araki, “Pathlossmodel for factory environment,” ISMAC Confer-ence, Bangkok, Thailand, Jan. 2009.

[8] T. D. Abhayapala, R. A. Kenney, and J. T. Y.Ho, “On capacity of multi-antenna wireless chan-nels: Effects of antenna separation and spatialcorrelation,” Proc. 3rd AusCTW, 2002.

[9] C. Oestges, “Validity of the Kronecker Model forMIMO Correlated Channels,” Vehicular Tech-nology Conference, VTC 2006, IEEE 63rd, Vol.6, pp. 2818-2822, May 2006.


[10] Min-Goo Choi, Sung ho Chae, Byung-Tae Yunand S. O. Park, “Validity of Kronecker modelat 3.805GHz Indoor measurement 4 by 4 MIMOsystem,” IEEE Antenna and Propagation Sym-posium 2008, San Diego, America, July 5-12,2008.

[11] http://en.wikipedia.org/wiki/Hata_Model_for_Urban_Area

[12] “Spatial channel model for Multiple Input Mul-tiple Output (MIMO) simulations (Release 6),”http://www.3gpp.org

[13] S. Loyka, and A. Kouki, “On the use of Jensen’sinequality for MIMO channel capacity estima-tion,” Canadian Conf. on Electrical and Com-puter Engineering (CCECE 2001), pp. 475-480,Toronto, Canada, May 13-16, 2001.

[14] T. M. Cover, and J. A. Thomas, “Elements ofInformation Theory,” John Wiley & Sons, NewYork, 1991.

[15] S. G. C. Fraiha, J. C. Rodrigues, R. N. S. Bar-bosa, H. S. Gomes, and G. P. S. Cavalcante,“An Empirical Model for Propagation-Loss Pre-diction in Indoor Mobile Communications UsingThe Pade Approximant,” Microwave and Opto-electronics, 2005 SBMO/IEEE MTT-S Interna-tional Conference, pp. 625-628, Jul. 2005.

[16] X. Gao, J. Zhang, G. Liu, D. Xu, P. Zhang, Y.Lu, and W. Dong, “Large-scale Characteristicsof 5.25 GHz Based on Wideband MIMO ChannelMeasurements,” Antennas and Wireless Propa-gation Letters, IEEE, vol. 6, pp. 263-266, Dec.2007.

KY Leng was born in Battambang,Cambodia, on May 06, 1979. He re-ceived a B.E. degree in electrical engi-neering from the Institute of Technologyof Cambodia, in 2001, and M.E. degreein electrical engineering from the Chu-lalongkorn University, Bangkok, Thai-land in 2005. In 2001-2003 and 2005-2006, he was a lecturer at the Institute ofTechnology of Cambodia. From 2006 tillnow, he is a Ph.D student at the Tokyo

Institute of Technology, Tokyo, Japan. His research interestsare communication system, electronic and computer program-ming.

Kei Sakaguchi was born in Osaka,Japan, on November 27, 1973. He re-ceived a B.E. degree in electrical andcomputer engineering from the NagoyaInstitute of Technology, Japan, in 1996,and a M.E. degree in information pro-cessing from the Tokyo Institute of Tech-nology, in 1998. In 2000-2007, he was anAssistant Professor at the Tokyo Insti-tute of Technology. He is currently anAssociate Professor at the Tokyo Insti-

tute of Technology. He received the Young Engineer Awardsboth from IEICE Japan and IEEE AP-S Japan chapter in2001 and 2002 respectively, the Outstanding Paper Award from

SDR Forum in 2004 and 2005 respectively, and the ExcellentPaper Award from IEICE Japan in 2005. His current researchinterests are in MIMO propagation measurement, MIMO com-munication system, and software defined radio. He is a memberof IEEE.

Kiyomichi Araki was born in Na-gasaki on January 7, 1949. He receiveda B.E. degree in electrical engineeringfrom Saitama University, in 1971, andM.E. and Ph.D. degrees in physical elec-tronics both from the Tokyo Institute ofTechnology, in 1973 and 1978, respec-tively. In 1973-1975, and 1978-1985, hewas a Research Associate at the TokyoInstitute of Technology, and in 1985-1995 he was an Associate Professor at

Saitama University. In 1979-1980 and 1993-1994 he was a vis-iting research scholar at the University of Texa, Austin andUniversity of Illinois, Urbana, respectively. He is currently aProfessor at the Tokyo Institute of Technology. His research in-terests are information security, coding theory, communicationtheory, circuit theory, electromagnetic theory, and microwavecircuits, etc. He is a member of IEEE and Information Pro-cessing Society of Japan.

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