Deterministic Simulations of Spatial Fading Correlation for MIMO Communication Channels Vmrani A. Waheed & Dubey V. K. Nanyang Technological University, Division of Communication Engineering, Radar Laboratory, SI-B4a-03, School of EEE, Block SI Nanyang Avenue, Singapore 639798 [email protected]

Analytical space-timefading correlationfor the multi-input-multi-output (MIMO) channel as funcrion of antenna spacing. angle spread, and Doppler spread 'is derived and is verified with deterministic simulations. We investigate rhe performance of uplink sparial fading correlation of MIMO channel. We consider four uniform circular array WCA) elements ar both mobile sration and base sration. The angle ofarrival and the angle ofdeparture at the base station and the mobile station. are distributed by truncated Gaussian and uniform probabiiiry distribution function(s) respectively. For large values of N, the simulation and analytical results agree. The results show rhat, sparial separation behveen,antenna elements can be minimized ar rhe base starion, by employing hvo or more antennas ar the mobile station. Abstract-

model for spatiotemporally correlated fading generation was proposed, which not only satisfy temporal correlation but also arbitrary spatial correlation. We use this model to verify the analytical results. The analytical space-time fading function derived here can be used for diversity combining analysis as well as for estimating the information-theoretic channel capacity of MIMO systems.

Keywords: Spatial fading correlation: MlMO channels; antenna arrays, deterministic simulafions

1.

INTRODUCTION

Antenna arrays have been widely used in mobile radio communications to improve signal quality, thereby increasing system coverage, capacity, and link quality [I].Ideally antenna elements should be spaced far enough apart so that the randomly fading signals at the antenna elements are uncorrelated. This is termed as sparial diversity. However, zero correlation can be difficult to achieve in practice due to physical limitation. Thus the amount of spatial correlation between antenna elements for practical spacing needs to be understood. To assess the performance of space-time processing algorithms, an understanding of space-time correlation is important. Previous results for spatial correlation for single-input-multi-output (SIMO) channel are reported in [2-4]. In this paper, we derive and extend the results for multi-input-multi-output (MIMO) channel as function of antenna spacing, angle spread, and Doppler spread. Deterministic channel models [5-71 have been proposed to solve the problem encountered in the pseudonoise filtered Gaussian model. In [5], a deterministic channel

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Figure I . 4-element Uniform Circular Array MlMO communication system in mobile radio environment.

2.

MIMO CHANNEL MODEL

We consider the statistical multipath MlMO channel model where the system employs multiple-transmit and multiple-receive antenna array system. We assume a 4element uniform circular array (UCA) transmit and receive

Proceedings IEEE INMIC 2003

arrays in a mobile radio environment, as shown in Figure I . In our analysis, we only consider the case of flat-fading channel. The multipath channel impulse response (normalized) for the plh transmit antenna at the mlh receive antenna element can be expressed as:

where erf(x) is the error function, and a is a parameter of the distribution closely related to angle spread, and 8' is the mean angle-of-arrival (AOA) at the base station. 3.

where the superscript U is for uplink (mobile to base station), L, is the number of scatter, (pl is uniformly distributed random phases over (0, 2x1, F d is the maximum Doppler frequency due to mobility, ":(q)is the array response for mlh receiving antenna and sthscatter path at a base station, ") (8:)is the array response for pIh transmitting antenna and

SPATIAL FADING CORRELATION

The space-time fading correlation consists of the temporal correlation and the spatial correlation. The spacer two time correlation function p,, ( v , m p , m 1 p 3between channels

hl,, and h:,,, for a MlMO channel can be defined

as

*:

sthscatter path at a mobile station, are random angles-ofarrival relative to the motion of the mobile and is determined by et3.In (I), rC;,(t) becomes a complex Gaussian random process when the number of scatters is large. Both

((q)

and (((4) for UCA can be expressed as follows

where Km,(t) is multipath channel impulse response defined in (I), and it is function ofangle spread (spatial domain) and Doppler spread (temporal domain), we can further express as:

c)'

where R, and R,is the circular radius of the antenna array for a base station and a mobile station, respectively. 5 is the elevation angle, and h is the wavelength of carrier frequency. For simplicity, only azimuth angles are considered in the propagation geometry (i.e. 5 = 90.), hut the results can be generalized to three dimensions. (0, is the phase excitation of mlh element in azimuth plane, as shown in Figure I. For the sfhscattering path on the uplink, 0: is the angle-of-departure (AOD) at a mobile access point (MS)and 0,; is the angle-ofarrival (AOA) at base station. Note that y; is the random angle (AOD) relative to the mobile for each scatter and is

denotes the complex conjugate, ip where the superscript is the angle of deparhue (AOD), <,,, is the angle-of-arrival (AOA), and

is the complex Gaussian process for large values of L,. R,(v) is the temporal fading correlation for delay v, and

R,, (mp,m'p')" is the spatial fading correlation. R,(v) is defined in [4,6] as

determined by 0:. Both Os' and 6 : are random variables.

B ,' is assumed uniformly distributed over [0, 2x1. And the '?r

has truncated Gaussian PDF, which can be expressed as

P,, (v.mp,m'p')' = R , ( v ) R . (mp-m'p'y

(a:-cf .~

j(e:)=cRe

20'

where lo(.)is a Bessel function of the first kind of order zero. Thus, the space-time correlation can be written as

;

-n+8'se:sn+8'

(3)

= J , ( 2 nF,v) R , (mp,m ' p

(9) I)"

where

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K,,, = z n -R[ c o s ( ~ m ) -

whereR,(np,m’p’)”is the spatial correlation between two channels

a

hi,p and h:,p., and is defined as

(16)

K,,,= ~ n R- [ s ~. n ( ~ ~ ) - s i n ( h . ) ]

a

is(WdP’)’=E{ Y ( P )Y(F )4(5% (F)} (Io) I..(P)o,(F)j(p)u~(s“)f(P,F)dFdP

=L

where A&, 6)is the joint probability density function of AOD and AOA. Since & and h. are statistically independent, we can rewrite ( 9 ) as

4.

DETERMINISTIC SIMULATIONS

In order to analyze the performance of antenna array systems, we need a fading channel simulator to generate spatiotemporally correlated fading waveforms. In digital simulation, computing efficiency can he enhanced by not using Doppler filters and interpolators.

R ( ~ ~ # ) ” = ~ ~ ( ~ ) ~ I() ~ channel ) Equation ~ models ~ )(7).as~can ~be ~approximated ( ~ ~ using ~ (deterministic ~ ) ~ ~ ) ~ sum of sinusoids (6,71, Since p ( , ) is a =R(A#)”R(mnl)” where&)

is the PDF of AOD and&,) is the PDF of AOA. ~ , ( m , m ’ ) ’ is the spatial correlation due to

complex Gaussian random process and can be wrillen as p(f) =a,( t ) + j a Q ( fwhere ),

R,(~,~’)’,and

the transmit and receive antennas, and is given by (U),

R,( P. P’)”=

R. (mm’). =

up(5’)

h

U,

d,(r)=~cC,,,cos(2nf,,,r+~,,,) . ‘I

2, ( I ) =

(5’ )f (c’)&’

2

Cp.4

(17)

c o s ( 2 x f p . d+ PP.,)

,.I

(12)

(F )4. ( 5 9 4F)dF

where N designates the number of sinusoids, and paramelem Doppler coefficients such as Doppler frequencies f,,,,f,,.

C,,” ,CO,,, and Doppler phases p,,, ,qQ,, are determined to In [Z], it is shown that the real and imaginary parts of R,(m,m’)’fOr

a UCA with

os’ truncated

Gaussian

distributed can be expressed as

where Zc’is related to the antenna spacing and ar is the relative angle between the p” and p” antenna element defined in [Z]. J.(x) is Bessel function of the first kind, and order n. It can be shown that R,(p,p’)’with

8: uniformly

distributed over [0,24 for a UCA,and can be expressed as R, ( P . P’Y = Jo

(z:)

(14)

satisfy the statistical properties and the given criteria [7]. Depending on how to determine the Doppler frequencies and coefficients, the deterministic models can be chosen h m Jakes’ model [6], an equal gain method, equal distance method, and a Monte Carlo method [7].While the Doppler coefficients of Jakes’ model and the equal distance model vary with k, those of the equal gain method and the Monte Carlo method are constant. In this paper, in order to sample the Doppler phase difference easily, the Doppler frequencies are determined by the equal area model [7]. Recently [5], has proposed a novel deterministic channel model satisfying not only rigorous temporal correlation but also arbitrary spatial correlation by means of Doppler phase difference sampling. We use the technique proposed in [SI,to compute deterministically spatial fading correlation. When a single fading waveform by equal gain method is expanded to M fading waveforms on M arrays, the m” fading waveform is described as [5]

where

2: =

JGZ

(15) to suppress the cross correlation

and

between in-phase and quadratic components of the resultant

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Proceedings IEEE INMlC 2003

waveforms, and f, = F~sin(

kgN)is to meet the temporal

correlation. Then the cross correlation between the mth and nth fading waveforms can be calculated in terms of Doppler phase difference py; - p:;' = Apmnh as

/4

The electrical distance between the array elements. d

determine the spatial correlation between fading waveforms. It is defined as:

where erf(x) is the error function, and LI is a parameter of the distribution closely related to angle spread, and 8' is the mean angle-of-arrival (AOA) at the base station. 5.

where d,,, denotes the distance between the mlh and dharray element, h is the wavelength. Ihe integration in (19) can be approximated into a summation as (20) in such a manner that the Cl, s are sampled more densely in the region where f ( B ; ) increases as follows

where N is the number of the values of Elk s. We need to determine the proper values of dxs and Oks. Prior to selecting the proper ekthe incident angle regions are chosen as

(23) then the distributed powers in the region (d,,d,.,)are equal for k = I , ..., N. By taking the average representative value in the kth region (d, ,d*.,)that is Q, = N Q T f ( B : ) d B ;

(24)

To determine the parameter, :dr) and {El,}, using CDF of truncated Gaussian distribution given by (3). we have from [51,

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RESULTS

For practical uplink applications. we consider an example of a UCA with four antenna elements at the MS and BS as shown in Figure 1. We fix the radius of circle at the mobile station at I .SA and vary the antenna spacing at the base station. To assess the usefulness of simulation model, one million samples are generated for evaluating spatial fading correlation. Analytical results are computed using ( I 1)-( 15).

We define IR,(ab. cd)" the uplink spatial fading correlation between the link from transmit antenna b to receive antenna a and the link from transmit antenna d and receive antenna c. For example, IR,(21, 12)"l is the spatial correlation between the link from transmit antenna I to receive antenna 2, and the link from transmit antenna 2 to receive antenna 1. We assume that both the links are independent of each other. Figure 2 present the analytical and simulation results of spatial fading correlation for IR,(21. 12)"j, when the mean AOA 8' = 90°. LI = 5'. The simulation results are computed for two values ofN (i.e. N = 8, and N = 36). As the value of N increases, the result matches more closely to the analytical results (N = 36). This is due to the fact that, larger points in the summation of equation (17) yield better simulation results and exhibit good agreement with analytical results. Figure 3, shows the result for lRs(31. 41)"j when the mean AOA 0' = 90°. LI = 5". The result in Figure 3. degenerate to the case when using antennas at the base station only (i.e. single-input-multi-output (SIMO) systems). This can be taken as a reference to compare the results with MlMO systems. By comparing the two results, it is suggested that. spatial separation between antenna elements can be minimized at the base station. by employing M O or more antennas at the mobile station.

Figure 2. Uplink spatial correlation o f MIMO channel,

Ik(21.

w.

Figure 4. Uplink spatial correlation of MlMO channel, IR,(I I. 33)"l.

-.. ,.......................... ... ........ 1

1

00

.

.

.

.

os

I

1.5

2

2.5

.,

.

.

1

35

.

.

.

.

.

' 5

c.

Figure 3. Uplink spatial correlation of MlMO channel, i~(3~41)i.

Figure 5. Uplink spatial correlation o f MlMO channel, lR,(3 I , 41)I.

Figure 4, shows the analytical result for Q I ( 1 1 , 33)"I e'= (30'. 4 9 , 60'7,and a = 5 9 The uplink spatial fading correlation in this case corresponds to the case o f (2,2) MlMO channel with U L A (at both BS and MS), having a spacing of 2R between the antenna elements. We can see from the Figure 4, that a very low correlation is observed (approximately 0.1 at 2'1 SA. spacing at the BS, for e'= 30° and (r = 5").

Figure 5-6 shows the analytical results for spatial fading correlation for IR,(31, 41)"/ for both the ULA (at BS instead of UCA) and the UCA for various values o f angle spread U. It is shown that the spatial correlation decreases as a increases and the ULA experiences high correlation for large values of 0'. while the UCA experiences the lowest correlation at 0' = 4 5 O and the highest at Or= Oo and e'= 90°, This is due to the fact that the correlation along lines perpendicular to the central AOA is lower than along lines parallel to the AOA.

when the mean AOA

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Proceedings IEEE INMIC 2003

minimized at the base station. by employing two or more antennas at mobile station. For large values of N, the simulation and analytical results agree. It is also shown that the spatial correlation decreases as angle spread increases.

REFERENCES [I] J. C. Libeni and T. S. Rappaport. Smarr Antenna for Wireless Communicorions. Prentice-Hall 1999.

[2] J:An Tsai and B. D. Woemer, “The fading correlation function of circular antenna array in mobile radio environment”. Proceedings of IEEE GLOBECOM 2001, pages 3232-3236, USA 2001

Figure 6. Uplink spatial correlation of MlMO channel, lR3(3I,41)l.

[3] J. Salz and J. H. Winters, “Effect of fading correlation on adaptive arrays in digital mobile radio,” IEEE Trans. Veh. Tech., VT-43(4), pp. 1049-1057, November 1994. [4] A. F. Naguib. “Adaptive antennas for CDMA wireless networks”. PhD Thesis, Stanford, USA 1996.

6. CONCLUSlON

[SI J-Ku Han, 1. Yook, Han-kue Park, “A deterministic In this paper, we derive an analytical space-time fading correlation for the MlMO channel as function of antenna spacing, angle spread, and Doppler spread. Using the derived results, we investigate the performance of uplink spatial fading correlation of statistical MlMO channel. A 4Element UCA MIMO configuration is considered. The analytical results are compared with deterministic simulations with temporal correlation. The results show that, spatial separation between antenna elements can be

Proceedings IEEE INMIC 2003

channel simulation for spatially correlated Rayleigh fading”. IEEE Communication Letters, vol. 6, no. 2, pp. 58-60, February 2002. [6] W. C. Jakes. Microwave Mobile Communications. Wiley I“edition, 1974.

[7] M. Patzold. Mobile Fading Channels. John Wiley, 2002.

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