3566

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 12, DECEMBER 2009

Spatial-Polarizational Correlation-Coefficient Function Between Receiving-Antennas in Radiowave Communications — Geometrically Modeled, Analytically Derived, Simple, Closed-Form, Explicit Formulas Kainam Thomas Wong and Yue Ivan Wu Abstract—This paper analytically derives closed-form expressions of the uplink received-signal’s 𝑝𝑜𝑙𝑎𝑟𝑖𝑧𝑎𝑡𝑖𝑜𝑛-𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑖𝑧𝑒𝑑 spatial-correlation-coefficient functions across the basestation antenna-array’s spatial aperture, based on a geometrical model of idealized spatial relationships among the transmitter, the scatterers, and the receiving antennas. The derived formulas fit well with some empirical data. Index Terms—Communication channels, dispersive channels, fading channels, geometric modeling, multipath channels, scatter channels, spatial correlation.

I. I NTRODUCTION

“G

EOMETRIC modeling" inter-relates the fadingchannel’s various measurable metrics (e.g., the spatial correlation) by abstracting the propagation channel’s spatial geometric relationships among the transmitter, the scatterers, and the receiver – such that this geometric idealization’s very few independent parameters will inter-connectedly affect the various measurable fading metrics to conceptually reveal the channel’s macroscopic statistical structure. The “spatial correlation" here is across the base-station antenna-array’s physical aperture, as opposed to a correlation over the space traversed by a mobile transmitter, over real time, over the propagation delay, or across frequency. The system development of “space diversity" or “directionof-arrival (DOA) diversity" smart-antennas schemes or multiple-input multiple-output (MIMO) schemes can benefit from simple rules-of-thumb that estimate this spatialcorrelation-coefficient function. II. T HE P ROPOSED S TATISTICAL M ODEL

A mobile, located at z𝑀𝑆 = [𝑧𝑀𝑆,𝑥 , 𝑧𝑀𝑆,𝑦 ] on a twodimensional Cartesian plane ℛ2 , emits a fully polarized signal. This emitted signal bounces off each scatterer in parallel, before reaching the base-station at z𝐵𝑆 = (0, 0). Each scatterer here acts as a re-transmitter, producing one multipath towards each receiving-antenna at the base-station. That is, if 𝑆 number of scatterers exist in ℛ2 , then 𝑆 multipaths will travel from the transmitter to each receiving antenna, with each such multipath representing one “bounce" off a different scatterer out of the 𝑆 scatterers. This reflected ray’s polarization and power Paper approved by R. K. Mallik, the Editor for Diversity and Fading Channels of the IEEE Communications Society. Manuscript received July 23, 2008; revised April 6, 2009. The authors are with the Department of Electronic & Information Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong (e-mail: [email protected], [email protected]). This work was supported by the Hong Kong Polytechnic University’s Internal Competitive Research Grant number G.YF52. The authors would like to thank Dr. Petr Tichavský for useful discussions. Digital Object Identifier 10.1109/TCOMM.2009.12.080362

depend on the scatterer’s intrinsic properties (to be defined momentarily) and the polarization of the incident ray. The free-space path-loss is to be overlooked. A. The Scatterer as a Polarization-Sensitive Re-Transmitter From basic electromagnetics, any fully polarized electromagnetic wave may be decomposed as a sum of a vertically polarized component and a horizontally polarized component. When this fully polarized electromagnetic ray reflects (scatterers) off a surface (a scatterer), the reflected (scattered) ray will take on a polarization state that depends on the incident wave’s polarization state, frequency and incident angle, as well as depends on the reflector’s (scatterer’s) electromagnetic properties and surface roughness. More mathematically, let a unitpower incoming ray have vertically polarized power 𝑎2𝑣 and horizontally polarized power 𝑎2ℎ = 1−𝑎2𝑣 . Let the reflected ray have vertically polarized power 𝑏2𝑣 and horizontally polarized power 𝑏2ℎ . The relationship among these is governed by the 2 × 2 scatterer’s scattering matrix S: [ [ ] ][ ] 𝑠𝑣,𝑣 𝑠𝑣,ℎ 𝑏𝑣 𝑎𝑣 = (1) 𝑏ℎ 𝑠ℎ,𝑣 𝑠ℎ,ℎ 𝑎ℎ       =b

=S

By definition, 𝑠2𝑣,𝑣 +𝑠2ℎ,𝑣 = 1, 𝑠2𝑣,ℎ +𝑠2ℎ,ℎ = 1, and all elements in S are non-negative real numbers not larger than 1. B. The Scatterers’ Spatial Distribution The scatterers’ spatial density and locations are modeled as a two-dimensional heterogeneous Poisson spatial point process Π(ℬ) indexed on subsets1 of ℛ2 . For any ℬ, the random number Π(ℬ) of points in ℬ is distributed according to a Poisson law with parameter Λ(ℬ) = E[Π(ℬ)], the expected number of scatterers in the set ℬ. The present model specifies only the statistical expectation of the scatterer-field’s spatial density, not the spatial density itself. Such a Poisson model can adapt to the field scatterers’ irregular and random spatial variability from one field location to another. The present work models Π(ℬ) as a sum of two components, Π𝑣 (ℬ) and Πℎ (ℬ). The first component Π𝑣 (ℬ) models a cluster of scatterers, each of which would respond to an incoming ray only if it is vertically polarized. If the incident ray is horizontally polarized, any scatterer in Π𝑣 (ℬ) would produce no reflected ray. If an incoming ray consists of a vertically polarized component 𝑎𝑣 and a horizontally polarized component 𝑎ℎ , the scatterer in Π𝑣 (ℬ) will re-transmit only the vertically polarized component, 𝑎𝑣 . This re-transmitted ray, 1 ℬ belongs to a Borel algebra of sets. That is, ℬ results from a countable number of set-theory operations with rectangle in the plane ℛ2 .

c 2009 IEEE 0090-6778/09$25.00 ⃝

Authorized licensed use limited to: University of Waterloo. Downloaded on January 18, 2010 at 22:44 from IEEE Xplore. Restrictions apply.

WONG and WU: SPATIAL-POLARIZATIONAL CORRELATION-COEFFICIENT FUNCTION BETWEEN RECEIVING-ANTENNAS . . .

however, will have a vertical component 𝑏𝑣,𝑣 and a horizontal component 𝑏ℎ,𝑣 . In matrix notation, the re-transmitted ray will have the polarizational components 𝑏𝑣,𝑣 and 𝑏ℎ,𝑣 , represented in a vector form as: ] ] [ ] [ [ 𝑎𝑣 𝑠𝑣,𝑣 0 𝑏𝑣,𝑣 = (2) 𝑠ℎ,𝑣 0 𝑏ℎ,𝑣 𝑎ℎ          ] [ =a =b𝑣 =S𝑣 = s𝑣 0 Analogously for the second cluster corresponding to Πℎ (ℬ), any scatterer there will reflect only the horizontally polarized component 𝑎ℎ . In vector-form, [ [ ] ] [ ] 𝑏𝑣,ℎ 0 𝑠𝑣,ℎ 𝑎𝑣 = (3) 𝑏ℎ,ℎ 0 𝑠ℎ,ℎ 𝑎ℎ       [ ] =bℎ =Sℎ = 0 sℎ These two clusters will each have a Gaussian Poisson intensity (𝑛𝑜𝑡 a Gaussian 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛). That is, ( ) 1 ∣z − z𝑀𝑆 ∣2 Λ𝑣 (𝑑z) = exp − 𝑑z, (4) 2𝜋Σ𝑣 2Σ2𝑣 ( ) 1 ∣z − z𝑀𝑆 ∣2 exp − Λℎ (𝑑z) = 𝑑z, (5) 2𝜋Σℎ 2Σ2ℎ respectively for the first and the second clusters. These two clusters overlap each other spatially. Both their Poisson intensities are spatially co-centered at the transmitter, but may have different spatial spreads, Σ𝑣 and Σℎ . The two corresponding Poisson fields, Π𝑣 (𝑑z) and Πℎ (𝑑z), are modeled as statistically independent. Hence, the overall scatterers are distributed randomly according to an heterogeneous Poisson law with a Gaussianmixture intensity, ( ) ∣z − z𝑀𝑆 ∣2 1−𝛾 exp − Λ(𝑑z) = 𝑑z 2𝜋Σ𝑣 2Σ2𝑉    =Λ𝑣 (𝑑z)

+

( ) 𝛾 ∣z − z𝑀𝑆 ∣2 exp − 𝑑z 2𝜋Σℎ 2Σ2𝐻   

(6)

=Λℎ (𝑑z)

where the model parameter 𝛾 regulates the relative preponderance of the two clusters.2 That which is Gaussian here is the parameter Λ(𝑑z), 𝑛𝑜𝑡 the spatial distribution. The model of (6) takes on these meanings: The 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 number of scatterers decreases farther from the mobile. A scatterer nearer the mobile transmitter is less likely to have an obstacle blocking its line-of-sight path. Hence, those scatterers closer to the mobile transmitter would likely have more impact on the uplink’s overall multipath profile. This is roughly equivalent to having more single-bounce scatterers closer to the mobile transmitter. Rather than modeling each scatterer’s retransmission characteristics to depend on its spatial location, it is mathematically simpler here to model all scatterers as having identical re-transmission property and as more densely populated where closer to the mobile. 2 The “geometric modeling" details here are admittedly not based on electromagnetics, but represent a mathematical construction with sufficient degrees of freedom to fit of measured data.

3567

III. S PATIAL C ORRELATION ACCOUNTING FOR P OLARIZATION OF THE T RANSMITTING / R ECEIVING A NTENNAS A. Preliminary Geometric Analysis Consider a multipath, at whatever polarization, bouncing off a scatterer located at z = (𝑧𝑥 , 𝑧𝑦 ) and arriving at receivingantenna #1 located at z𝐵𝑆1 = (0, 0). This arriving multipath would equal the transmitted signal multiplied by the complexvalued coefficient 𝑐1 (z) = 𝑔1 (z)𝑒𝑗𝜑(z) , where 𝑔1 (z) represents the channel-gain. As a multipath reflects off any specific scatterer located at any particular z, that multipath’s initial phase 𝜑(z) is modeled as uniformly random over (−𝜋, 𝜋] and statistically independent of Π(ℛ2 ). Any two scatterers’ initial phases are modeled as statistically independent. Similarly for a multipath arriving at the receiving-antenna #2 located at z𝐵𝑆2 = (𝑑𝑠𝑝 , 0), 𝑐2 (z)

=

𝑔2 (z)𝑒𝑗(𝜑(z)−Δ𝜑 (z)) ,

with 𝑔2 (z) being the channel gain experienced by this multipath. Referring to Figure 1, 𝑠1 (z)2 𝑠2 (z)2

= =

𝑎(z)2 + 𝑑21 + 2𝑎(z)𝑑1 cos (𝛼(z) − 𝛾1 (z𝑀𝑆 )) 𝑎(z)2 + 𝑑22 + 2𝑎(z)𝑑2 cos (𝛼(z) − 𝛾2 (z𝑀𝑆 )) ,

where 𝑎(z) = ∣z − z𝑀𝑆 ∣ denotes the distance between the scatterer and the transmitter. This gives a multipath’s temporal phase-difference between receiving-antennas #1 and #2 as Δ𝜑 (z)

= =

≃ =

2𝜋 [𝑠1 (z) − 𝑠2 (z)] 𝜆 [ √ 2𝜋 𝑑21 + 2𝑎(z)𝑑1 cos(𝛼(z) − 𝛾1 (z𝑀 𝑆 )) + 𝑎(z)2 𝜆 ] √ − 𝑑22 + 2𝑎(z)𝑑2 cos(𝛼(z) − 𝛾2 (z𝑀 𝑆 )) + 𝑎(z)2 2𝜋 [𝑑1 − 𝑑2 + 𝑎(z) (𝜁𝑥 cos 𝛼(z) − 𝜁𝑦 sin 𝛼(z))] 𝜆 2𝜋 [𝑑1 − 𝑑2 + ⟨𝜻, z − z𝑀 𝑆 ⟩] (7) 𝜆

where 𝜆 symbolizes the wireless signal’s carrier-wavelength, ⟨v1 , v2 ⟩ denotes an inner vector-product between two sizecompatible vectors v1 and v2 , 𝜻 = [𝜁𝑥 , −𝜁𝑦 ], 𝜁𝑥 = 2 2 𝑑1 +𝑑2 [𝑑𝑠𝑝 − (𝑑1 − 𝑑2 ) cos 𝜃 cos 𝛽], and 𝜁𝑦 = 𝑑1 +𝑑2 (𝑑1 − 𝑑2 ) cos 𝜃 sin 𝛽. The above approximation [2] holds for 𝑑𝑐 ≫ max{Σ𝑉 , Σ𝐻 } and 𝑑𝑐 ≫ 𝑑𝑠𝑝 . These two inequalities together require each scatterer to be sufficiently close to the transmitter relative to each scatterer’s distance from the receiver. An example field-scenario would be a base-station receiver elevated on a tower, with no prominent scatterer in the tower’s immediate vicinity. B. Deriving the Closed-Form Explicit Formula The vector-sum of the stochastic fading coefficients of all multipaths arriving at receiving-antenna #1 equals: [ ] ∫ 𝑟1,𝑣 𝑐1 (z)SaΠ(𝑑z) = r1 = 𝑟1,ℎ ℛ2 ∫ = 𝑔1 (z)𝑒𝑗𝜑(z) SaΠ(𝑑z) ℛ2

Authorized licensed use limited to: University of Waterloo. Downloaded on January 18, 2010 at 22:44 from IEEE Xplore. Restrictions apply.

3568

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 12, DECEMBER 2009

1 Empirical data Proposed Model

Spatial Correlation Coefficient, |ρvv(dsp/λ)|

0.9

MS

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

1

2

(a)

2

c

0

0.5

1

1.5 2 2.5 3 Inter−Sensor Separation in Wavelengths, dsp/λ

3.5

4

1 Empirical data Proposed Model

1

BS 1

BS 1

Spatial Correlation Coefficient, |ρhh(dsp/λ)|

0.9

BS 2

BS 2

sp

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

∫

= (1 − 𝛾) ∫ + 𝛾

0.8

0.7

0.6

0.5

0.4

0.3

ℛ2

𝑔1 (z)𝑒

𝑗𝜑(z)

S𝑣 aΠ𝑣 (𝑑z)

0.2

(b)

𝑗𝜑(z)

𝑔1 (z)𝑒 Sℎ aΠℎ (𝑑z) ∫ 𝑔1 (z)𝑒𝑗𝜑(z) Π𝑣 (𝑑z) = (1 − 𝛾)b𝑣 2 ℛ ∫ + 𝛾bℎ 𝑔1 (z)𝑒𝑗𝜑(z) Πℎ (𝑑z), ℛ2

(8)

0

0.5

1

1.5 2 2.5 3 Inter−Sensor Separation in Wavelengths, dsp/λ

3.5

4

  ( ) ( ) 𝑑𝑠𝑝  𝑑𝑠𝑝    Fig. 2. The 𝜌𝑣𝑣 𝜆  formulas in (21) and the 𝜌ℎℎ 𝜆  formula in (22), calibrated together to the empirical data in Figure 3 of [1] with the base𝑣 ℎ station at 30 meters aboveground. Σ = 0.057339, Σ = 0.75787, 𝑎𝑣 = 𝑑𝑐 𝑑𝑐 ∘ 0.4631, 𝑠𝑣,𝑣 = 0.39693, 𝑠𝑣,ℎ = 0.5742, 𝛽 = 39.996 , 𝛾 = 0.36398.

ℛ2

with

∫

𝐸{r1 }

2𝜋(𝑑1 −𝑑2 )

= (1 − 𝛾)b𝑣 𝐸{𝑒𝑗𝜑(z) }𝐸{𝑔1 (z)Π𝑣 (𝑑z)} ℛ2    =0 ∫ + 𝛾bℎ 𝐸{𝑒𝑗𝜑(z) }𝐸{𝑔1 (z)Πℎ (𝑑z)} ℛ2    =0

= 0.

(9)

Analogously at base-station-antenna #2, ∫ 𝐸{𝑒𝑗𝜑(z) }𝐸{𝑔2 (z)𝑒−𝑗Δ𝜑(z) SaΠ(𝑑z)} 𝐸{r2 } = ℛ2

= 0,

(10)

because Δ𝜑z depends statistically only on Π. If 𝑑𝑐 ≫ max{Σ𝑣 , Σℎ }, (7) gives [2]: [ ] 𝑟2,𝑣 r2 = 𝑟2,ℎ ∫ 2𝜋(𝑑1 −𝑑2 ) 2𝜋 𝜆 = 𝑒−𝑗 𝑔2 (z)𝑒𝑗𝜑(z)−𝑗 𝜆 ⟨z−z𝑀 𝑆 ,𝜻⟩ SaΠ(𝑑z) ℛ2

−𝑗

2𝜋(𝑑1 −𝑑2 ) 𝜆

b𝑉 = (1 − 𝛾)𝑒 ∫ 2𝜋 ⋅ 𝑔2 (z)𝑒𝑗𝜑(z)−𝑗 𝜆 ⟨z−z𝑀 𝑆 ,𝜻⟩ Π𝑣 (𝑑z) ℛ2

−𝑗 𝜆 + bℎ ∫ 𝛾𝑒 2𝜋 ⋅ 𝑔2 (z)𝑒𝑗𝜑(z)−𝑗 𝜆 ⟨z−z𝑀 𝑆 ,𝜻⟩ Πℎ (𝑑z)

(11)

ℛ2

The 2 × 2 spatio-polarizational cross-correlation matrix thus equals: ( ) 𝑑𝑠𝑝 R = 𝐸{r1 r2 } 𝜆 ⎡ ( ) ( ) ⎤ 𝑑 𝑑 𝑅𝑣𝑣 𝜆𝑠𝑝 𝑅𝑣ℎ 𝜆𝑠𝑝 ⎣ ( ) ( ) ⎦ = 𝑑 𝑑 𝑅ℎ𝑣 𝜆𝑠𝑝 𝑅ℎℎ 𝜆𝑠𝑝 ∫ 𝑔1 (z)𝑔2 (z)𝑒𝑗Δ𝜑(z) Λ𝑣 (𝑑z) = (1 − 𝛾)2 b𝑣 b𝐻 𝑣 2 ℛ ∫ + 𝛾 2 bℎ b𝐻 𝑔1 (z)𝑔2 (z)𝑒𝑗Δ𝜑(z) Λℎ (𝑑z) ℎ 2

ℛ2 2𝜋(𝑑1 −𝑑2 ) 𝐻 𝑗 𝜆 b𝑣 b𝑣 𝑒

= (1 − 𝛾) ∫ 2𝜋 ⋅ 𝑔1 (z)𝑔2 (z)𝑒𝑗 𝜆 ⟨𝜻,z−z𝑀 𝑆 ⟩ Λ𝑣 (𝑑z) ℛ2

2𝜋(𝑑1 −𝑑2 )

𝑗 𝜆 + 𝛾 2 bℎ b𝐻 ℎ 𝑒 ∫ 2𝜋 ⋅ 𝑔1 (z)𝑔2 (z)𝑒𝑗 𝜆 ⟨𝜻,z−z𝑀 𝑆 ⟩ Λℎ (𝑑z) ℛ2

Authorized licensed use limited to: University of Waterloo. Downloaded on January 18, 2010 at 22:44 from IEEE Xplore. Restrictions apply.

(12)

WONG and WU: SPATIAL-POLARIZATIONAL CORRELATION-COEFFICIENT FUNCTION BETWEEN RECEIVING-ANTENNAS . . .

The third equality above is due to the statistical independence between Π𝑣 (𝑑z) and Πℎ (𝑑z). There exists no cross-term above because of (9) and (10). With 𝑔1 (z) = 𝑔2 (z) = 1, the Fourier transform of (6) produces the spatial correlation coefficient function across the receiving antenna-array’s aperture: ( ) 𝑑𝑠𝑝 R 𝜆 ∫ 2𝜋(𝑑1 −𝑑2 ) 2𝜋 2 𝜆 s𝑣 s𝐻 𝑎 𝑒𝑗 𝜆 ⟨𝜻,z−z𝑀 𝑆 ⟩ Λ𝑣 (𝑑z) = (1 − 𝛾)2 𝑒𝑗 𝑣 𝑣 2 ∫ ℛ 2𝜋(𝑑1 −𝑑2 ) 2𝜋 2 𝑗 𝐻 2 𝜆 + 𝛾 𝑒 s ℎ s ℎ 𝑎ℎ 𝑒𝑗 𝜆 ⟨𝜻,z−z𝑀 𝑆 ⟩ Λℎ (𝑑z) =

𝑒𝑗 (1 − 𝛾)2 + 𝛾2

=

𝑒

2𝜋Σ𝑣

2𝜋(𝑑1 −𝑑2 ) 𝜆

2𝜋Σ𝑣

(1 − 𝛾)2 𝑒𝑗2𝜋 + 𝛾2𝑒

=

𝑗

2𝜋(𝑑1 −𝑑2 ) 𝜆

𝑗2𝜋

∫

2 s𝑣 s𝐻 𝑣 𝑎𝑣

2 sℎ s𝐻 ℎ 𝑎ℎ

∫

ℛ2

ℛ2

𝑒

𝑒

𝑗

𝑗 2𝜋 𝜆 ⟨𝜻,z⟩

∣z∣2 2Σ2 𝑉

∣z∣2 2𝜋 𝜆 ⟨𝜻,z⟩ 2Σ2 ℎ

𝑑z

𝑑z

( )2 Σ 𝜋 𝜆ℎ ∥𝜻∥

(𝑑1 −𝑑2 ) −2 𝜆

2 sℎ s𝐻 ℎ 𝑎ℎ 2 (𝑑1 −𝑑2 ) Σ𝑣 2 (1 − 𝛾)2 𝑒−2(𝜋 𝜆 ∥𝜻∥) s𝑣 s𝐻 𝑒𝑗2𝜋 𝜆 𝑣 𝑎𝑣 } ( )2 Σ −2 𝜋 𝜆ℎ ∥𝜻∥ 2 + 𝛾2𝑒 sℎ s𝐻 ℎ 𝑎ℎ

{

∥𝜻∥

2

)2 𝑑1 + 𝑑2 𝑑𝑐 𝑑1 − 𝑑2 𝑑𝑐

=

= =

cos2 𝜃

=

𝑄1

=

𝑄2

=

Hence,  ( )   R 𝑑𝑠𝑝   𝜆 

(13)

(14)

)2 [( )2 𝑑1 − 𝑑2 𝑑𝑐 cos2 𝜃 𝑑1 + 𝑑2 𝑑𝑐 ] ( )2 𝑑𝑠𝑝 𝑑1 − 𝑑2 + − 2𝑄1 cos 𝜃 𝑑𝑐 𝑑𝑐 √ 2𝑄2 + 2 𝑄22 − 𝑄21 ( 2

√ √ 𝑄2 + 𝑄1 − 𝑄2 − 𝑄1 )2 ( 𝑑 cos2 𝛽 + 𝑄1 1 + 12 𝑑𝑠𝑝 𝑐 𝑄1 + 𝑄2 𝑑𝑠𝑝 cos 𝛽 𝑑𝑐 ( )2 1 𝑑𝑠𝑝 1+ 2 𝑑𝑐 ( )2 Σ𝑉 2 −2 𝜋 𝜆 ∥𝜻∥

= (1 − 𝛾) 𝑒 + 𝛾2𝑒

( )2 Σ −2 𝜋 𝜆ℎ ∥𝜻∥

2 s𝑣 s𝐻 𝑣 𝑎𝑣

2 sℎ s𝐻 ℎ 𝑎ℎ

(15)

where the absolute-magnitude operator above is applied element-by-element to the 2 × 2 matrix. 2 2 For 𝑑𝑠𝑝 ≪ 𝑑𝑐 : cos 𝜃 ≈ 1, 𝑑1𝑑+𝑑 ≈ 2, and 𝑑1𝑑−𝑑 ≈ 𝑐 𝑐 𝑑𝑠𝑝 𝑑𝑐 cos 𝛽. Therefore, (15) becomes:  ( ) ( 𝑑 )2 𝑠𝑝 Σ𝑣   2 −2 𝜋 𝑑𝑐 𝜆 sin 𝛽 R 𝑑𝑠𝑝  ≈ (1 − 𝛾)2 s𝑣 s𝐻 𝑎 𝑒 𝑣 𝑣  𝜆  2 + 𝛾 2 sℎ s𝐻 ℎ 𝑎ℎ 𝑒

( 𝑑 −2 𝜋 𝑑𝑠𝑝 𝑐

Σℎ 𝜆

sin 𝛽

The entries in the 2 × 2 matrix of (16) are:  ( ) ( 𝑑 𝑠𝑝 Σ𝑣   𝑅𝑣𝑣 𝑑𝑠𝑝  ≈ (1 − 𝛾)2 (𝑠𝑣,𝑣 𝑎𝑣 )2 𝑒−2 𝜋 𝜆 𝑑𝑐   𝜆 + 𝛾 2 (𝑠𝑣,ℎ 𝑎ℎ )2 𝑒

( 𝑑 𝑠𝑝 −2 𝜋 𝜆

)2

(16)

Σℎ 𝑑𝑐

sin 𝛽

sin 𝛽

+ 𝛾 2 (𝑠ℎ,ℎ 𝑎ℎ )2 𝑒  ( ( )  )     𝑅ℎ𝑣 𝑑𝑠𝑝  = 𝑅𝑣ℎ 𝑑𝑠𝑝     𝜆 𝜆 

( 𝑑 𝑠𝑝 −2 𝜋 𝜆

Σℎ 𝑑𝑐

)2

)2

 ( ) ( 𝑑 𝑠𝑝 Σ𝑣   𝑅ℎℎ 𝑑𝑠𝑝  ≈ (1 − 𝛾)2 (𝑠ℎ,𝑣 𝑎𝑣 )2 𝑒−2 𝜋 𝜆 𝑑𝑐   𝜆

sin 𝛽

sin 𝛽

)2

(17)

)2

(18) (19)

[

≈ (1 − 𝛾)2 𝑠𝑣,𝑣 𝑠ℎ,𝑣 𝑎2𝑉 𝑒 + 𝛾 2 𝑠ℎ,ℎ 𝑠𝑣,ℎ 𝑎2ℎ 𝑒

(𝜋 Σ𝜆𝑣 ∥𝜻∥)2 s s𝐻 𝑎2 𝑣 𝑣 𝑣

(𝑑1 −𝑑2 ) −2 𝜆

Referring to Figure 1,

(

ℛ2

3569

( 𝑑 𝑠𝑝 −2 𝜋 𝜆

[ ( 𝑑𝑠𝑝 −2 𝜋 𝜆

Σℎ 𝑑𝑐

Σ𝑣 𝑑𝑐

sin 𝛽

sin 𝛽

)2 ]

)2 ]

(20)

The corresponding spatial correlation coefficient functions may be found by normalizing the diagonal elements of ∣R (0)∣ in (16) to unity magnitude. That is, ⎡ ( ) ( ) ⎤  ( ) 𝑑 𝑑 𝜌𝑣𝑣 𝜆𝑠𝑝 𝜌𝑣ℎ 𝜆𝑠𝑝   𝑑 𝑠𝑝 𝝆  = ⎣ ( ) ( ) ⎦  𝑑 𝑑 𝜆  𝜌ℎ𝑣 𝜆𝑠𝑝 𝜌ℎℎ 𝜆𝑠𝑝 [ ]−1/2  ( )   ∣𝑅𝑣𝑣 (0)∣ 0 R 𝑑𝑠𝑝  =  0 ∣𝑅ℎℎ (0)∣ 𝜆  ⎡  ( ) ⎤−1/2   𝑑 0 𝑅𝑣𝑣 𝜆𝑠𝑝   ( ) ⎦ ⋅⎣   𝑑 0 𝑅ℎℎ 𝜆𝑠𝑝   ( )  𝑑  The above 2 × 2 matrix, 𝝆 𝜆𝑠𝑝 , has the elements shown in equation (21) to (24). ( ) 𝑑 In the above, 𝜌𝑣𝑣 𝜆𝑠𝑝 refers to the spatial correlation coefficient between two vertically polarized antennas (apart 𝑑 by a( horizontal distance of 𝜆𝑠𝑝 ) at the receiver. Similarly, ) 𝑑 𝜌ℎℎ 𝜆𝑠𝑝 refers to the spatial correlation coefficient between two horizontally polarized antennas (apart ( by)a horizontal ( dis) 𝑑𝑠𝑝 𝑑𝑠𝑝 𝑑 tance of 𝜆 ) at the receiver. Lastly, 𝜌𝑣ℎ 𝜆 and 𝜌ℎ𝑣 𝜆𝑠𝑝 each refers to the spatial correlation coefficient between one vertically polarized antenna and one horizontally polarized 𝑑 antenna (apart by a horizontal distance of 𝜆𝑠𝑝 ) at the receiver. This model’s independent parameters are (i) Σ𝜆𝑣 and Σ𝜆ℎ , which refer to the channel’s wavelengthnormalized scattering environment; (ii) 𝛽, which geometrically relates the mobile station and the base-station; and 𝑑 , which represents a normalized design parameter for (iii) 𝑑𝑠𝑝 𝑐 the base-station receiving antenna-array’s aperture. The closed-form formulas in (21) and (24), explicitly expressed in terms of the geometric-model’s independent parameters, can thus serve as a first-step design-formulas in the development of MIMO or smart-antennas systems. ( )   𝑑 Figures 3 and 2 show the above-derived 𝜌𝑣𝑣 𝜆𝑠𝑝  of (21)  ( )   𝑑 and 𝜌ℎℎ 𝜆𝑠𝑝  of (22) , calibrated (by least-squares fit) to the empirical data shown in Figure 3 of [1]. These empirical measurements were for signals transmitted at street level and

Authorized licensed use limited to: University of Waterloo. Downloaded on January 18, 2010 at 22:44 from IEEE Xplore. Restrictions apply.

3570

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 12, DECEMBER 2009

 ( )   𝜌𝑣𝑣 𝑑𝑠𝑝   𝜆   ( )   𝜌ℎℎ 𝑑𝑠𝑝   𝜆   ) (   𝜌ℎ𝑣 𝑑𝑠𝑝   𝜆 

2

( 𝑑 𝑠𝑝 2 −2 𝜋 𝜆

)2

Σ𝑣 𝑑𝑐

sin 𝛽

Σ𝑣

sin 𝛽

2

( 𝑑 −2 𝜋 𝑠𝑝

)2

[

( 𝑑 𝑠𝑝 −2 𝜋 𝜆

Σ𝑣 𝑑

sin 𝛽

)2 ]

vv

Σℎ 𝑑𝑐

sin 𝛽

)2

(22) (23)

[ ( 𝑑𝑠𝑝 −2 𝜋 𝜆

Σℎ 𝑑

sin 𝛽

)2 ]

Besides Figures 3 and 2’s validation of the derived formulas in (21) and (24) by empirical data, (21) and (24) exhibit ( these)following intuitively appealing qualitative trends.   𝑑𝑠𝑝 Σ 𝜌 𝜆𝑑𝑐 , 𝛽  decreases quasi-exponentially with

0.8

sp

Spatial Correlation Coefficient, |ρ (d /λ)|

( 𝑑 𝑠𝑝 −2 𝜋 𝜆

(21)

C. The Derived Formulas’ General Qualities

Empirical data Proposed Model

0.7

0.6

1) increasing sin2 𝛽, as the cluster impinges from a spatial direction more to the antenna-array’s axis. ) ( perpendicular

0.5

𝑑

2

𝑠𝑝 as the base-station’s antennas are 2) increasing 𝜆 spaced farther apart, except for 𝛽 ≈ 0 when the mobile aligns along( the)antenna-array axis.

0.4

0.3

0.2

2

0

0.5

1

1.5 2 2.5 3 Inter−Sensor Separation in Wavelengths, dsp/λ

3.5

4

4.5

1 Empirical data Proposed Model

3) increasing Σ𝑑𝑐𝑣 as the first cluster’s azimuth-angular spread appears larger to the base-station, except for 𝛽 ≈ 0 when the mobile aligns along the axis. ) ( antenna-array Similar considerations apply for cluster.

0.9

Spatial Correlation Coefficient, |ρhh(dsp/λ)|

)2

𝑐 𝑐 (1 − 𝛾)2 𝑠𝑣,𝑣 𝑠ℎ,𝑣 𝑎2𝑣 𝑒 + 𝛾 2 𝑠ℎ,ℎ 𝑠𝑣,ℎ 𝑎2ℎ 𝑒 √ √ (24) (1 − 𝛾)2 (𝑠𝑣,𝑣 𝑎𝑣 )2 + 𝛾 2 (𝑠𝑣,ℎ 𝑎ℎ )2 (1 − 𝛾)2 (𝑠ℎ,𝑣 𝑎𝑣 )2 + 𝛾 2 (𝑠ℎ,ℎ 𝑎ℎ )2

0.9

Σℎ 𝑑𝑐

2

for the second

0.8

IV. C ONCLUSION

0.7

0.6

0.5

0.4

0.3

0.2

0.1

(b)

sin 𝛽

𝜆 𝑑𝑐 (1 − 𝛾)2 (𝑠ℎ,𝑣 𝑎𝑣 )2 𝑒 + 𝛾 2 (𝑠ℎ,ℎ 𝑎ℎ )2 𝑒 ≈ (1 − 𝛾)2 (𝑠ℎ,𝑣 𝑎𝑣 )2 + 𝛾 2 (𝑠ℎ,ℎ 𝑎ℎ )2  ) (   𝑑𝑠𝑝  = 𝜌𝑣ℎ 𝜆 

1

(a)

Σℎ 𝑑𝑐

(1 − 𝛾) (𝑠𝑣,𝑣 𝑎𝑣 ) 𝑒 + 𝛾 (𝑠𝑣,ℎ 𝑎ℎ ) 𝑒 (1 − 𝛾)2 (𝑠𝑣,𝑣 𝑎𝑣 )2 + 𝛾 2 (𝑠𝑣,ℎ 𝑎ℎ )2

≈

≈

0.1

( 𝑑 𝑠𝑝 2 −2 𝜋 𝜆

0

0.5

1

1.5 2 2.5 3 Inter−Sensor Separation in Wavelengths, dsp/λ

3.5

4

  ( ) ( ) 𝑑𝑠𝑝  𝑑𝑠𝑝    Fig. 3. The 𝜌𝑣𝑣 𝜆  formulas in (21) and the 𝜌ℎℎ 𝜆  formula in (22), calibrated together to the empirical data in Figure 3 of [1] with the 𝑣 ℎ base-station at 100 meters aboveground. Σ = 0.13594, Σ = 1.2749, 𝑑𝑐 𝑑𝑐 𝑎𝑣 = 0.37883, 𝑠𝑣,𝑣 = 0.43052, 𝑠𝑣,ℎ = 0.53651, 𝛽 = 20.015∘ , 𝛾 = 0.37292.

collected by a base-station receiver at 100 or 30 meters high. Figures 3 and 2 show the proposed model’s efficacy to model real-world spatial correlation as function of the transmitting and the receiving antennas’ polarizations.

This paper is first in the open literature to account for the transmitting and receiving antennas’ polarizations, in geometric-model-based derivation of truly closed-form expressions for the spatial correlation coefficient across a receiving antenna-array. The derived formulas are explicitly in terms of (1) the inter-antenna spacing between two receiving-antennas, and (2) the two receiving-antenna’s linear polarizations. These formulas are simple in mathematical form and fit some empirical data well. R EFERENCES [1] H. Xu, M. Gans, D. Chizhik, J. Ling, P. Wolniansky, and R. Valenzuela, “Spatial and temporal variations of MIMO channels and impacts on capacity," in Proc. IEEE International Conf. Commun., vol. 1, pp. 262266, 2002. [2] T.-A. Chen, M. P. Fitz, W.-Y. Kuo, M. D. Zoltowski, and J. H. Grimm, “A space-time model for frequency nonselective Rayleigh fading channels with applications to space-time modems," IEEE J. Sel. Areas Commun., vol. 18, no. 7, pp. 1175-1190, July 2000. [3] V. I. Piterbarg and K. T. Wong, “Spatial-correlation-coefficient at the basestation, in closed-form explicit analytic expression, due to heterogeneously poisson distributed scatterers," IEEE Antennas Wireless Propagation Lett., vol. 4, pp. 385-388, 2005.

Authorized licensed use limited to: University of Waterloo. Downloaded on January 18, 2010 at 22:44 from IEEE Xplore. Restrictions apply.