Spatial correlation-coefficient across a receiving sensor-array – accounting for propagation loss

w(z) uniformly distributed over (−p, p]. w(z) is statistically independent of the spatial stochastic point process P. w(z1) and w(z2) are statistically independent for two scatterers located at z1 = z2.

V.I. Piterbarg, K.T. Wong and Y.I. Wu

Deriving the spatial correlation coefficient: The transmitted signal bounces off a scatterer at z, producing a multipath towards receivingantenna 1 located at zBS1 ¼ (0, 0), and another multipath towards antenna 2 at zBS2 ¼ (dsp , 0). These two multipaths’ complex-value amplitudes are multiplied by the channel-propagation (complex-valued) coefficients of c1(z) ¼ g1(z)e jw(z), and c2(z) ¼ g2(z)e j(w(z)2Dw(z)), where Dw(z) represents the phase difference between the two retransmitted multipaths at zBS1 ¼ (0, 0) against at zBS2 ¼ (dsp , 0). This Dw(z) depends on the distances traversed by the multipaths reaching the two sensors, and differs from the previously defined w(z). The following derives an expression for this Dw(z) ¼ (2p/l ) [s1(z) 2 s2(z)], where l denotes the wireless carrier wavelength, z ¼ [r cos (f(z)), r sin (f(z))] is in the polar co-ordinates, with the polar axis coinciding with the Cartesian xaxis. Moreover, [si (z)]2 = [a(z)]2 + di2 + 2a(z)di cos(a(z) − gi (z)) and di = ||zMS − zBSi ||, for i ¼ 1, 2. Furthermore, a(z) = ||z − zMS || =  (zx − zMS,x )2 + (zy − zMS,y )2 . Because L ≪ 1/dsp ,

In the open literature on the geometric modelling of wireless radiowave landmobile propagation channels, this reported work is the first to attempt to account for the propagation power loss of the multipaths, in deriving closed-form expressions for an uplink received-signal’s spatial-correlation-coefficient function across the aperture of a basestation antenna-array.

Introduction: The development of ‘smart antenna’ systems would be aided by a simple rule-of-thumb, to estimate the inter-antenna spacing needed for a required level of the spatial correlation coefficient. This Letter offers such a simple rule, in terms of the transmitter’s arrival angle to the receiving antennas. Proposed geometric model: Consider a (mobile) transmitter (located at zMS = [zMS,x , zMS,y ] on the Cartesian x-y plane) emitting omnidirectionally (see Fig. 1). The transmitted signal bounces off any scatterer (located at z ¼ [zx , zy]) in the surrounding environment, producing a multitude of propagation-multipaths towards diverse directions. This model assumes S number of such multipaths to be arriving at each receiving antenna, with each multipath representing one ‘bounce’ off a different scatterer in the channel.

a(z) zMS

z

a(z)

d2

dc

zBS1

s2(z)

=c2 (z)

independent and because w(z) is uniformly distributed, stochastic integration gives E{C1 } = A E{e jw(z) }E{g1 (z)P(dz)} = 0, and E{C2 } =  {e jw(z) }E{g2 (z)e−jDw(z) P(dz)} = 0. The preceding line holds because A Dwz depends statistically only on P. Hence, the spatial covariance and the spatial correlation both equal:

b

zBS1 + zBS2

zBS2

2 dsp

Fig. 1 Geometry relating transmitter, scatterers, and base-station antennas

C(dsp ) = E{C1 C2∗ } = L

Each multipath suffers propagation-path loss, which worsens as the area of the wavefront increases with propagation distance. The propagation-path loss is modelled as: g(z) = |z − zMS |−n ,

n [ {1, 2, 3, · · ·}

y

Vector-sum all multipaths arriving at antenna 1 from all scatterers on  A # R2 , to obtain C1 = A g1 (z)e jw(z) P(dz). Similarly for antenna 2,   =c1 (z)  C2 = A g2 (z)e j(w(z)−Dw(z) ) P(dz). Because w(z) and P are statistically  

s1(z) fMS = f(zMS)

x

(2)

The first approximation above is by the Taylor series expansion, √ 1 + x ≈ 1 + (x/2). The second approximation omits all higherthan-first-order terms from an infinitesimal expansion with respect to  (dsp / z2x + z2y ), which has been assumed to be ≪ 1. In other words,  the only geometric requirement now is dsp ≪ r = z2x + z2y .

q

d1

 2p  z2x + z2y − z2x + dsp )2 + z2y l dsp dsp zx  = 2p cos(f(z)) ≃ 2p l z2 + z2 l

Dw (z) =

(1)

where n denotes the path-loss exponent and implicitly regulates the spatial extent within which scatterers are significant. This propagation path loss model in (1) contrasts with customary modelling of g(z) as independent of z (or equivalently, n ¼ 0). The scatterers’ spatial locations z ¼ [zx , zy] are modelled as drawn from an homogeneous Poisson spatial point process P(B) indexed on subsets of a two-dimensional Cartesian plane R2 . For any spatial region B, the random number P(B) of points in B is distributed according to a Poisson law with parameter E[P(B)] = L(B); and L(B) gives the expected number of scatterers in B. For an homogeneous Poisson process, a positive scalar L suffices to denote the point process intensity. This spatial homogeneity would not restrict the spatial extent of the scatterers’ locations. Moreover, only the statistical expectation of the spatial density of the scatterer-field has been specified, not the scatterers’ actual locations. Each scatterer is modelled as an omnidirectional lossless retransmitter of any incoming signal from the transmitter, thereby producing a multipath towards each omnidirectional receiving-antenna. All multipaths, reflecting off a particular scatterer located at z, have an initial phase

=

A

A

E{c1 (z)[c2 (z)]∗ }dz (3)

g1 (z)g2∗ (z)e jDw(z) Ldz

To facilitate subsequent derivation, approximate (1) as g(z) ≈ [|z − zMS |2 + d2 ]−n/2 , for n ¼ 1, 2, 3, . . .. This approximation converges to (1) as d  0. This constant d . 0 is included to preclude any non-integrable pole at zero in the subsequent steps. The limit d  0 will be taken later, such that d will disappear from the final formula. Hence, d is not a model parameter. Along with (2), rewrite (3) as C(dsp ) = L

R2

1 (2pdsp /l) ej cos(f(z))dz (|z − zMS |2 + d2 )n

(4)

where f(z) denotes the angle of the position-vector z ¼ [r cos(f(z)), r sin(f(z))] in the polar co-ordinates. Express zMS as [rMS cos fMS , rMS sin fMS] in the signed-polar co-ordinates, which are inter-related to the Cartesian co-ordinates through = rMS sin fMS , rMS = zMS,x = rMS cos fMS , zMS,y   z2MS,x + z2MS,y sgn(zMS,y ) fMS = arccos (zMS,x / z2MS,x + z2MS,y ), ∀rMS [

1, x ≥ 0 where sgn(x) = (−1, 1), ∀fMS [ [0, p), and −1, x,0 p 1   f (zx , zy )dzx dzy = 0 −1 rf (r cos f, r sin f) dr d f. Hence, rewrite A

ELECTRONICS LETTERS 16th September 2010 Vol. 46 No. 19

where G(.) refers to the gamma function, B(.) symbolises the the beta function, and Pnm(.) denotes the associated Legendre function of the first kind. Furthermore, p df |R(y)| ≤ 2 2 2n−1/2 e(y) [sin (f − f0 ) + y]

(4) as p C(dsp ) = L 0

e j2p(dsp /l) cos f

 1 × p =L 0

 rdr d f, 2 n 2 2 −1 [r − 2rrMS cos(f − fMS ) + rMS + d ]

(2n − 3)!! (e j2p(dsp /l) cos f ) (2n − 2)!!

(5)

prMS cos(f − fMS ) × 2 df [rMS sin2 (f − fMS ) + d2 ]2n−1/2 where the preceding equality comes from [4] (equation (5) in section 3.252). the variance (i.e. the power), Set dsp = 0 in Dw (z) to obtain  E{|C1 |2 } = E{|C2 |2 } = s2 = L A g1 (z)g2∗ (z)dz. Using the polar coordinates, 1 rdr 2pL 2−2n = s2 = 2pL d (6) 2 2 n−1 0 (r + d )n Divide (5) by (6), to give the spatial correlation coefficient,    C(dsp ) n − 1 (2n − 3)!! r(dsp ) = = 2 (2n − 2)!! s2  r 2−2n d 2 sp d MS × A , 2 , fMS d l rMS

with x ¼ (dsp/l ), y ¼ (d/rMS)2. Set (d/rMS ) ≪ 1 as d d 0, limit fMS [ [0, p], and define a small √ e ( y) such that (e(y)/ y) ≪ 1, nonetheless. These give

MS

× ≃

 e j2px cos f

cos(f − fMS )d f [sin2 (f − fMS ) + y]2n−1/2

fMS −e(y)

e j2px cosMS

fMS −e(y)



+

MS |≥e(y)

df [(f − fMS )2 + y](2n−1/2)

e j2px cos f |f−fMS |≥e(y)

+e(y)

cos(f − fMS )d f [sin (f − fMS ) + y](2n−1/2) 2

df + R(y) 2n−1/2 −e(y) (f + y)  √  1 y dt + R(y) ≃ e j2px cos fMS (2n−1/2) 2n−1/2 2 y −1 (t + 1) = e j2px cos fMS

2

= cn (e j2px cos fMS )y1−n + R(y) From [1] (equation (10) in section 3.252 and equation (1) in section 8.756), 1 dt = 2n−1 G(n) cn = 2 + 1)(2n−1/2) (t −1 1−n (0) = × B(1, 2n − 2)P1−n

2p 4p ≤ 2n−1 2n−1/2 1 (y) [sin 1(y) + y] 2

which needs be much smaller than the first summand. Hence, 1(y) needs to be chosen such that 1(y) ≫ yn−1/2n−1 . This requirement is already sat√ isfied by the earlier stipulation that e(y)/ y ≫ 1. Hence, choosing 1(y) such that it tends to zero slowly,   2n (n − 2)! j2px cos fMS d 2(1−n) e (8) A(x, y, fMS ) ≃ (2n − 3)!! rMS With (2n 2 2)!! ¼ 2n21(n 2 1)!, and substituting (8) in (7), r 2−2n (n − 1)(2n − 3)!! MS r(dsp ) ≃ d 2(2n − 2)!!   n 2 (n − 2)! j2px cos fMS d 2(1−n) × e (2n − 3)!! rMS

(9)

2n−1 (n − 1)! j2p(dsp /l) cos(fMS ) e = (2n − 2)!!

(7)

where (2m)!! ¼ 2 . 4 . 6 . . . . . (2m), and (2m 2 1)!! ¼ 1 . 3 . 5 . . . . . (2m 2 1), for any natural number m; p cos(f − fMS ) A(x, y, fMS ) = e j2px cos f df 2 [sin ( f − fMS ) + y]2n−1/2 0

A(x, y, fMS )   f +e(y) = fMS −e(y) + |f−f

≤

= ej

(2pdsp /l)

cos fMS

The above is independent of L. From (9), the magnitude of the spatial correlation coefficient function equals   dsp cos(f(zMS )) (10) Re[r(dsp )] ≃ cos 2p l

Conclusion: This derived formula is independent of the power-loss exponent n, which implicitly regulates the spatial extent within which a scatterer must be located, for the scatterer to be a notable retransmitter. This n assigns increasing influence to the retransmitted propagation-path of a scatterer, as that scatterer lies closer to the transmitter. As n increases, this partiality for close-by scatterers becomes more pronounced. # The Institution of Engineering and Technology 2010 19 July 2010 doi: 10.1049/el.2010.1875 V.I. Piterbarg (Faculty of Mechanics & Mathematics, M. V. Lomonosov Moscow State University, Moscow, Russia) K.T. Wong (Department of Electronic & Information Engineering, Hong Kong Polytechnic University, Hong Kong, China) E-mail: [email protected] Y.I. Wu (School of Communication & Information Engineering, University of Electronic Science & Technology of China, Chengdu, Sichuan, China) Reference 1 Gradshteyn, I.S., and Ryzhik, I.M.: ‘Table of integrals, series, and products’ (Academic press, San Diego, CA, 2000, 6th edn)

2n (n − 2)! (2n − 3)!!

ELECTRONICS LETTERS 16th September 2010 Vol. 46 No. 19