Spatial Fading Correlation for Semicircular Scattering ...Spatial Fading Correlation for Semicircular Scattering: Angular Spread and Spatial Frequency Approximations Bamrung Tau Sieskul,

10 00

11 01

Spatial Fading Correlation for SemicircularScattering: Angular Spread and Spatial

Frequency Approximations

Bamrung Tau Sieskul, Claus Kupferschmidt, and Thomas Kaiser

in Proc. 3rd International Conference on Communications and Electronics 2010 (ICCE 2010), NhaTrang, Vietnam, August 11-13, 2010, pp. 681-685.

Diese Arbeit wurde finanziell durch das europaische Forschungsprojekt ,,coExisting shortrange radio by advanced Ultra-WideBand radio technology (EUWB)” mit der VertragsnummerFP7-ICT-215669 unterstutzt.

Copyright (c) 2010 IEEE. Personal use of this material is permitted. However, permission touse this material for any other purposes must be obtained from the IEEE by sending a requestto [email protected].

Spatial Fading Correlation for SemicircularScattering: Angular Spread and Spatial Frequency

ApproximationsBamrung Tau Sieskul, Claus Kupferschmidt, and Thomas KaiserInstitute of Communications Technology, Leibniz University Hannover

Appelstraße 9A, 30167 Hannover, Germany, Tel: +49 511 762 2528, Fax: +49 511 762 3030{bamrung.tausieskul, kupfers, thomas.kaiser}@ikt.uni-hannover.de

Abstract—Spatial frequency approximation (SFA) of spatialfading correlation (SFC) is addressed for the case that the exactinfinite summation of Bessel functions is inconvenient or infea-sible. The angular spread is derived for semicircular scattering,especially characterized by uniform, Gaussian, Laplacian, andvon Mises distributions. The semicircular scattering on the range(− 1

2π, 1

2π] happens, e.g., when the antenna is placed on the wall.

In the usual SFA of the SFC, a characteristic function is involvedwith the infinite integration range due to a small angular spreadand a near broadside nominal angle. In this paper, we propose anew SFA of the SFC with a finite integration range. Consideringthe Laplacian angular distribution, numerical examples illustratethat for a moderate angular spread, the new SFA yields higheraccuracy in computing the SFC than the conventional SFA. Forthe von Mises distribution, the new SFA is able to approximatethe SFC, while the ordinary SFA provides discrete solutions,which are unreliable to the SFC approximation.

Index Terms—local scattering, angular distribution, character-istic function.

I. INTRODUCTION

One of the most significant effects in wireless propagationis the local scattering around the transmitter or the receiver. Inmodern wireless communication systems, e.g., ultrawidebandtechnology, a signal is transmitted with a large bandwidth thatrenders fine time resolution. The local scattering thus causesa large number of observable multipath components. As aconsequence, the summation of several paths leads to a con-tinuum or diffuse of the rays [1,2]. In general, the correlationof the impulse responses between the n-th and another n-thantenna elements, denoted by spatial fading correlation (SFC),can be regarded as a link quality. The SFC plays an importantrole in the wireless communications, because the performancemetrics, such as bit error probability [3]–[5], channel capacity[6]–[8] and etc., depend on it. Therefore, the study of theSFC brings the realistic performance analysis up. In literature,several works are devoted to the investigation of the SFC at anantenna array (see, e.g., [1,2,9]–[15]). In [16,17], the SFC ofa circular array is derived from uniform, cosine, and Gaussianangular distributions. The direct computation of the SFC oftenrequires extensive integrations, which are difficult or infeasiblefor an angular distribution whose probability density function(pdf) is complicated. Based on the Taylor-series approximationfrom a small angular spread, spatial frequency approximation(SFA) is considered as an approach to approximate the SFC

(see, e.g., [18] and [19, eq. (2.8)]). It is indicated in [19]that the SFA is accurate, when nominal angle, or mean angle,is near broadside, i.e., close to the perpendicular axis of theantenna array.

In this paper, we provide the exact and approximate ex-pressions of the angular spread in semicircular scattering(− 1

2π,12π]. After transforming the characteristic function of

the SFA represented in spatial frequency domain into thatrepresented in spatial angle domain, we truncate the integrationrange of the characteristic function to cover only the semicir-cular scattering whereby a new SFA is proposed.

A number of notations are invoked as follows. Eφ {·}is the expectation with respect to φ whose pdf is pφ(φ).J0(x) and Jk(x) are the zeroth order and the k-th orderBessel functions of the first kind, respectively. I0(x) andIk(x) are the zeroth order and the k-th order modified Besselfunction of the first kind, respectively. The cumulative densityfunction of the standard Gaussian random variable is definedas Φ(u) = 1√

2π

∫ u

−∞ e−12 v2

dv. The error function erf(z) is

defined as erf(z) = 1√π2∫ z

0e−u2

du. δ(·) is the impulse

symbol defined as δ(x) = 0; x �= 0, and∫∞−∞ δ(x)dx = 1.

The rest of this paper is organized as follows. In Sec. II,the SFC caused by the local scattering is discussed. For thesemicircular scattering, we consider in Sec. III several angulardistributions, angular spread, SFA, characteristic function, andtruncated characteristic function of the angular distributions.In Sec. IV, numerical simulation is performed to illustratethe characteristic of the SFA with the finite integration range.Finally, the results of this paper are summarized in Sec. V.

II. SPATIAL FADING CORRELATION

In a dense object scenario, the local scattering can existin the vicinity of the transmitter and the receiver [20]. For auniform linear array, the time delay at the n-th antenna elementis given by

ψn =1cd(n− 1) sin(φ), (1)

where c is the speed of electromagnetic wave, d is the distancebetween adjacent antenna elements, and φ is the direction ofemitting or incoming ray measured from the perpendicularaxis of the array. The received signal is composed of a large

978-1-4244-7057-0/10/$26.00 ©2010 IEEE216

number of propagating waves along various directions, whichcan be characterized by an angular distribution. The correlationbetween the phase of the received signals at the n-th antennaelement and that at the n-th antenna element can be writtenas [21]

ρn,n = Eφ

{e

1c j2πf0d(n−n) sin(φ)

}, (2)

where f0 is the central frequency.

III. SEMICIRCULAR SCATTERING

In the semicircular scattering, the azimuth angle lies on φ∈(− 1

2π,12π]. This scenario happens, e.g., when the antenna is

placed on the wall. In order to evaluate the SFC in (2), weneed to consider

∫ 12 π

− 12 πpφ(φ)e

1c j2πf0d(nr−nr) sin(φ)dφ, where

pφ(φ) is the spatial pdf of a certain angular distribution.

A. Angular Distributions

In previous works, some angular distributions have beenconsidered by exploring the geometry [22] or by directlyinferring from several statistical distributions, e.g., cosine dis-tribution [1,23], uniform distribution [9], Gaussian distribution[2,18], Laplacian distribution [24,25], von Mises distribution[26], and etc. To investigate the SFC from an angular profilepoint of view, let us split the azimuth angle φ into

φ = φ+ δφ, (3)

where φ is the nominal angle and δφ is its deviation. As φ isa random variable, the deviation angle δφ remains a randomvariable. For the cosine, uniform, Gaussian, Laplacian, andvon Mises distributions, the pdf can be written respectively as

pδφ(δφ) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1π cc cosn(δφ),

δφ∈(− 12π − φ, 1

2π − φ],n∈{2, 4, 6, . . .};

12√

3σφ, δφ∈(−√3σφ,

√3σφ];

1√2πσφ

cGe− 1

2σ2φ

δ2φ

, δφ∈(− 12π − φ, 1

2π − φ];

1√2σφ

cLe− 1

σφ

√2|δφ|

, δφ∈(− 12π − φ, 1

2π − φ];

12πI0(κ)cvMeκ cos(δφ),

δφ∈(− 12π − φ, 1

2π − φ],κ ≥ 0;

(4)

where σφ is the standard deviation of the underlying standarddistribution, and the normalization constants cc, cG, cL, andcvM are given by

cc =1(n12 n

)2n−1, (5a)

cG =1

Φ(

1σφ

(12π − φ

))− Φ(− 1

σφ

(12π + φ

)) , (5b)

cL =1

1− e− 1√

2σφπ

cosh(

1σφ

√2φ) , (5c)

cvM =1

12 − 1

πI0(κ)2∞∑

k=1

12k−1 (−1)kI2k−1(κ) cos

((2k − 1)φ

) .

(5d)

B. Angular Spread

An important parameter in describing a scattering channelis a statistical value of the deviation of the arrival or departureangles from their nominal angles. From a statistical viewpoint,the angular deviation can be described by its standard devia-tion, which is denoted herein by the angular spread and canbe defined as

σφ =√

Eφ

{(φ− φ)2

}=√

Eδφ{δ2φ}. (6)

In [27], the angular spread is estimated using the measurementresults at 5.2 GHz and found to be 2 to 9 degrees for thedeparture and 2 to 7 degrees for the arrival. For the cosinedistribution, we have from [28, p. 128]

σφ =

√√√√√√√√√

1π2n−1 cc

n∑k=0

1n−2k (−1)n−2k

(nk

)(

1n−2k2φ sin

((n− 2k)φ

)+(

1(n−2k)2 2− (φ2 + 1

4π2))

cos((n− 2k)φ

)).

(7)

Regarding the uniform distribution, we have σφ = σφ. For theGaussian distribution, the integration by parts

∫x2e−ax2

dx =− 1

2axe−ax2

+ 14a

√πa erf(

√ax) (see, e.g., [29, p. 108]) results

in

σφ =

√√√√√σ2φ −

1√2πcGσφ

⎛⎝(

12π + φ

)e− 1

2σ2φ( 1

2 π+φ)2

+(

12π − φ

)e− 1

2σ2φ( 1

2 π−φ)2

⎞⎠

≈ σφ.(8)

By considering the Laplacian distribution, we have from [29,p. 106]

σφ =√cL

√√√√√ σ2φ + e

− 1√2σφ

π((πφ+

√2σφφ

)sinh

(1

σφ

√2φ)

−(

14π

2 + φ2 + 1√2πσφ + σ2

φ

)cosh

(1

σφ

√2φ))

≈ σφ.(9)

For the von Mises distribution, the angular spread can becomputed from (see, e.g., [28, p. 333] and [29, Sec. 2.633])

σφ =√cvM

√√√√√√√√√√√√√√√√√√√

124π

2 + 12 φ

2 + 1I0(κ)2

(∞∑

k2=1

12k2

(−1)k2I2k2(κ)(φ sin

(2k2φ

)+ 1

2k2cos(2k2φ

))

− 1π

∞∑k1=1

12k1−1 (−1)k1I2k1−1(κ)((

14π

2 + φ2 − 1(2k1−1)2 2

)cos((2k1 − 1)φ

)

− 12k1−12φ sin

((2k1 − 1)φ

))).

(10)

217

In what follows, we shall neglect the cosine distribution,since no report ascertains that it conforms with the measure-ment results.

C. Spatial Frequency Approximation

Using the expansions of trigonometry functions (see, e.g.,[30, Sec. 9.1.42-43] and [31, p. 22]), the SFC in (2) can beexpressed as (see, e.g., [19, eq. (2.5)])

ρn,n = J0

(1c2πf0d(n− n)

)

+ 2∞∑

k=1

J2k

(1c2πf0d(n− n)

)ck

+ jJ2k−1

(1c2πf0d(n− n)

)sk,

(11)

where ck and sk are the real and complex sinusoidal coeffi-cients given by

ck =∫ π

−π

pφ(φ) cos(2kφ)dφ, (12a)

sk =∫ π

−π

pφ(φ) sin((2k − 1)φ)dφ. (12b)

To evaluate the SFC in (11), the calculation incurs the inte-grations in (12) and the infinite number of summation termsin (11). Next we consider the SFA, an approximation of theSFC based on the first-order Taylor series expansion.

Lemma 1: (Approximate SFC for Small Angular Spreadand Near Broadside Nominal Angle) For a small value of theangular spread or σφ → 0 and a near broadside nominal angleor |φ| � 1

2π, the SFC between the n-th and the n-th antennaelements can be approximated as

ρn,n ≈ ϕ 1σω

δω((n− n)σω) ej(n−n)ω, (13)

where ω, δω, and σω are given by

ω =1c2πf0d sin(φ), (14a)

δω =1c2πf0d cos(φ)δφ, (14b)

σω =1c2πf0d cos(φ)σφ, (14c)

and ϕ 1σω

δω(·) is the characteristic function of the pdf

p 1σω

δω(v|0; 1) with zero mean and unit variance of the nor-

malized random variable 1σωδω , given by

ϕ 1σω

δω(u) =

1σφ

∫ ∞

−∞p 1

σωδω

(v|0; 1) ejuvdv. (15)

Proof: The derivation of (13) is given in Appendix A.Since φ, δφ, φ, and σφ are the fundamental quantities from

physical angle aspect, the transformed variables ω, δω , ω, andσω in (14) constitute spatial frequency domain. As adoptingthe first-order Taylor series approximation in (24) and theinfinite range approximation in (30), the SFC from (13) iscalled the approximate SFC based on the SFA.

Remark 1: Some notices are summarized as follows.

• The result of (13) is the similar to those in [18] and[19, eq. (2.8)], but different from such works in that theworks in [18] and [19, eq. (2.8)] directly approximate(26) as pδω

(δω|0;σ2ω) ≈ pδφ

(δφ|0;σ2φ). In our derivation,

the change of variable was invoked.• The SFA in Lemma 1 provides a simple form for further

performance analysis. The application of the SFA can befound in e.g. [32].

• However, it is shown in [19, Sec. 2.3] that the errorof the SFA increases with the increase in the nominalangle φ and the angular spread σφ. The cause of theapproximation error is that the absolute value of thenominal angle |φ| should be much less than 1

2π so thatthe approximation of the integration limits in (30) holdsquite well.

D. Characteristic Function

We transform the characteristic function in (13) into

ϕ 1σω

δω(σω(n− n))

=1σφ

∫ ∞

−∞p 1

σωδω

(v|0; 1) ej(n−n)σωvdv

=∫ ∞

−∞pδφ

(δφ|0;σ2φ)e

1c j2πf0d(n−n) cos(φ)δφdδφ

=∫ ∞

−∞pδφ

(δφ|0;σ2φ)e

1cσφ

j(n−n)2πf0d cos(φ)σφδφdδφ

=∫ ∞

−∞pδφ

(δφ|0;σ2φ)e

1σφ

jσω(n−n)δφdδφ.

(16)

The characteristic function in (13) will be evaluated for eachdistribution according to (16) for δφ on the ideal range(−∞,∞).

1) Uniform Distribution: Since δφ for the uniform distri-bution is on the finite range (−√3σφ,

√3σφ], the integration

equivalent to (16) remains

ϕ 1σω

δω(u) =

12√

3σφ

∫ √3σφ

−√3σφ

e1

σφjuδφdδφ

=1√3u

sin(√

3u).

(17)

2) Gaussian Distribution: From [33, p. 65], we have

ϕ 1σω

δω(u) =

1√2πσφ

cG

∫ ∞

−∞e−(

12σ2

φ

δ2φ− 1

σφjuδφ

)dδφ

= cGe− 1

2

σ2φ

σ2φ

u2

.

(18)

3) Laplacian Distribution: One can show that

ϕ 1σω

δω(u) =

1√2σφ

cL

∫ ∞

−∞e− 1

σφ

√2|δφ|+ 1

σφjuδφdδφ

=1

12

σ2φ

σ2φu2 + 1

cL.(19)

218

4) Von Mises Distribution: It can be shown that

ϕ 1σω

δω(u) =

12πI0(κ)

cvM

∫ ∞

−∞eκ cos(δφ)e

1σφ

juδφdδφ

= cvM

(σφδ (u) +

1I0(κ)

∞∑k=1

Ik(κ)(δ

(k +

1σφu

)+ δ

(k − 1

σφu

))).

(20)

E. Truncated Characteristic Function

Other than the ordinary characteristic function in (16), weshall restrict the integration limits of δφ in (16) to be finite onthe semicircular range (− 1

2π − φ, 12π − φ].

1) Gaussian Distribution: When δφ is on the semicircularrange (− 1

2π−φ, 12π−φ], the integration in (16) can be derived

from (see, e.g., [28, p. 109] and [29, p. 108])

ϕ 1σω

δω(u) =

1√2πσφ

cG

∫ 12 π−φ

− 12 π−φ

e− 1

2σ2φ

δ2φ+ 1

σφjuδφ

dδφ

=12cGe

− 12

σ2φ

σ2φ

u2(

erf

(1

2σ2φ

(12π + φ

)+

12√

2

σ2φ

σ2φ

u2

)

+erf

(1

2σ2φ

(12π − φ

)− 1

2√

2

σ2φ

σ2φ

u2

)).

(21)

Note that if the lower limit − 12π − φ in (21) is replaced

by the negative infinity, we have limx→−∞ erf(x) = −1. Ifthe upper limit 1

2π − φ in (21) is replaced by the positiveinfinity, we have limx→∞ erf(x) = 1. Therefore, it results inlimx→∞ 1

2 (erf(x)− erf(−x)) = 1 from which ϕ 1σω

δω(u) in

(21) is equal to ϕ 1σω

δω(u) in (18).

2) Laplacian Distribution: When δφ is on the semicircularrange (− 1

2π− φ, 12π− φ], the integration in (16) can be shown

in (22).3) Von Mises Distribution: For the semicircular interval

(− 12π − φ, 1

2π − φ], the finite integration range reads as

ϕ 1σω

δω(u) =

12πI0(κ)

cvM

∫ 12 π−φ

− 12 π−φ

eκ cos(δφ)e1

σφjuδφdδφ

=1πcvMe

− 1σφ

juφ

(1uσφ sin

(1

2σφuπ

)+

1I0(κ)

2

cos(

12σφ

uπ

) ∞∑k=1

11

σ2φu2 − (2k − 1)2

(−1)kI2k−1(κ)

((2k − 1) cos((2k − 1)φ) +

1σφ

ju sin((2k − 1)φ)))

.

(23)

IV. NUMERICAL EXAMPLES

We mainly consider the angular distributions, which aresound to the measurement results, such as the Laplaciandistribution [24,25] and the von Mises distribution [26].

Angular Spread σφ [degree]

Spat

ial

Fadi

ngC

orre

latio

nρ

n,n

SimulationSFA: InfiniteSFA: Finite

15.12 15.14 15.160 2 4 6 8 10 12 14 16 18 20

0.04750.04760.04770.04780.0479

10−2

10−1

100

Fig. 1. The spatial fading correlation ρn,n−1 as a function of the angularspread σφ for the Laplacian distribution with φ = 40◦, Nφ = 106,f0 = 1

2(10.6 + 3.1) GHz, and d = 0.2 m. “Simulation” is calcu-

lated from∣∣ 1

Nφ

Nφ∑m=1

e1cj2πf0d sin(φ+δφm )

∣∣. “SFA: Infinite” is computed

from∣∣ 1

12

σ2φ

σ2φ

σ2ω+1

ejω∣∣. “SFA: Finite” is given by

∣∣ϕ 1σω

δω(σω)ejω

∣∣ with

ϕ 1σω

δω(σω) derived from (22).

In Fig. 1, the SFC ρn,n for the Laplacian angular distributionis shown as a function of the angular spread σφ. From themain figure, it can be seen that i) the SFC of the Laplacianangular distribution with φ = 40◦ monotonically decreaseswith the increase in the angular spread, and ii) the differenceof “Simulation” from both “SFA: Infinite” and “SFA: Finite” isnoticeable for σφ > 13◦. In the corresponding zoomed figure,there exists a difference between the “SFA: Infinite” and the“SFA: Finite” where we can see that the proposed “SFA: Fi-nite” is closer to the “Simulation” than the conventional “SFA:Infinite”. The difference of the approximations results from theintegration in that unlike the infinite range of the integration in(16), the new characteristic function ϕ 1

σωδω

(σω) was truncated

on the semicircular range, i.e., δφ ∈ (− 12π − φ, 1

2π − φ].In Fig. 2, the SFC for the von Mises distribution is shown

as a function of the central frequency f0 for several distancesof antenna element separation d. We can see that the SFCdecreases in large scale with the increase in the central fre-quency and decreases with the increase in the antenna elementseparation distance. The usual SFA with the infinite integrationrange does not appear in the figure, since the solution in (20)has discrete values of its argument. Rather, the new SFA withthe finite integration range can approximate the SFC, whenthe SFC is approximately larger than 0.1. In addition, thenumerical integration of the first equality in (23) is shown tocorrespond to the closed form at the second equality in (23).

V. CONCLUSION

The angular spread is derived for the semicircular scatteringon the range φ∈ (− 1

2π,12π] and especially with the uniform,

Gaussian, Laplacian, and von Mises distributions. This sce-nario happens, e.g., when the antenna is placed on the wall.The SFA of the SFC is considered when the exact infinite

219

ϕ 1σω

δω(u) =

1√2σφ

cL

∫ 12 π−φ

− 12 π−φ

e− 1

σφ

√2|δφ|+ 1

σφjuδφdδφ

=1

σ2φ

σ2φu2 + 2

cL

(2− e

− 1√2σφ

π(− 1

2√

2u sin

(1

2σφ(2φ+ π)u

)e− 1

σφ

√2φ

+1

2√

2u sin

(1

2σφ(2φ− π)u

)e

1σφ

√2φ

+ cos(

12σφ

(2φ+ π)u)

e− 1

σφ

√2φ

+ cos(

12σφ

(2φ− π)u)

e1

σφ

√2φ

+ j(− 1

2√

2u cos

(1

2σφ(2φ+ π)u

)e− 1

σφ

√2φ

+1

2√

2u cos

(1

2σφ(2φ− π)u

)e

1σφ

√2φ − sin

(1

2σφ(2φ+ π)u

)e− 1

σφ

√2φ − sin

(1

2σφ(2φ− π)u

)e

1σφ

√2φ)))

.

(22)

Central Frequency f0 [Hz]

Spat

ial

Fadi

ngC

orre

latio

nρ

n,n

SimulationSFA: InfiniteSFA: FiniteSFA: Numerical Finite

d = 10 m

d = 1 m d = 0.1 m

106 108 1010 101210−4

10−3

10−2

10−1

100

Fig. 2. The spatial fading correlation ρn,n−1 as a function of thecentral frequency f0 for the von Mises distribution with φ = 20◦,Nφ = 106, κ = 1, and N∞ = 100. “Simulation” is calcu-

lated by∣∣ 1

Nφ

Nφ∑m=1

e1cj2πf0d sin(φ+δφm )

∣∣. “SFA: Infinite” is computed by∣∣ϕ 1σω

δω(σω)ejω

∣∣ with ϕ 1σω

δω(σω) derived from (20). “SFA: Finite” is

given by∣∣ϕ 1

σωδω

(σω)ejω∣∣ with ϕ 1

σωδω

(σω) derived from (23). “SFA:

Numerical Finite” is the numerical integration of the first equality in (23).

summation of Bessel functions in (11) is inconvenient orinfeasible. The conventional SFA of the SFC with the infiniteintegration range is derived. We also have proposed its coun-terpart as the SFA of the SFC with the finite integration range.Numerical examples illustrate that for the moderate angularspread in the Laplacian distribution, the new SFA can providehigher accuracy in computing the SFC than the usual SFA.For the von Mises distribution, the conventional SFA causesthe discrete values of the SFC, which cannot approximate theactual SFC, whereas the new SFA can approximate the SFC.The results in this work can be applied to various communities,e.g., channel modeling, channel measurement, estimation ofspatial channel parameters, bit error performance analysis, andtransmission rate analysis.

APPENDIX APROOF OF LEMMA 1

For a small value of the angular spread σφ, a spatialfrequency is approximated as (see, e.g., [18], [19, Sec. 2.2.2],

and [25])

ω =1c2πf0d

(sin(φ) cos(δφ) + cos(φ) sin(δφ)

)

≈δφ→0

1c2πf0d

(sin(φ) + δφ cos(φ)

)= ω + δω.

(24)

From (14b), we can write

σ2ω = Eδω

{(ω − ω)2

}

= Eδφ

{(1c2πf0dδφ cos(φ)

)2}

=(

1c2πf0d cos(φ)

)2

σ2φ.

(25)

Using the result of (25), the change of random variableprovides

pδω(δω|0;σ2

ω) = pδφ(δφ|0;σ2

φ)∣∣∣∣ ddδω

δφ

∣∣∣∣= pδφ

(δφ|0;σ2φ)∣∣∣∣ 1

1c2πf0d cos(φ)

∣∣∣∣=σφ

σωpδφ

(δφ|0;σ2φ),

(26)

and

p 1σω

δω

(1σωδω|0; 1

)= pδω

(δω|0;σ2ω)

∣∣∣∣∣∣d

d(

1σωδω

)δω∣∣∣∣∣∣

= σωpδω(δω|0;σ2

ω).

(27)

For a small angular spread, we can approximate

ρn,n =∫ 1

2 π

− 12 π

pφ(φ)e1c j2πf0d(n−n) sin(φ)dφ

≈δφ→0

e1c j2πf0d(n−n) sin(φ)

∫ 12 π−φ

− 12 π−φ

pδφ(δφ|0;σ2

φ)

e1c j2πf0d(n−n)δφ cos(φ)dδφ.

(28)

220

It can be further shown that

ρn,n =σω

σφej(n−n)ω

∫ 1c 2πf0d( 1

2 π−φ) cos(φ)

1c 2πf0d(− 1

2 π−φ) cos(φ)

pδω(δω|0;σ2

ω)

ej(n−n)δωdδω

=σω

σφej(n−n)ω

∫ 1cσω

2πf0d( 12 π−φ) cos(φ)

1cσω

2πf0d(− 12 π−φ) cos(φ)

pδω

(δω|0;σ2

ω

)

ej(n−n)( 1σω

δω)σωσωd(

1σωδω

)

=1σφ

ej(n−n)ω

∫ 1σφ

( 12 π−φ)

− 1σφ

( 12 π+φ)

p 1σω

δω

(1σωδω|0; 1

)

ej(n−n)( 1σω

δω)σωd(

1σωδω

).

(29)

If φ is not close to − 12π and 1

2π, we can approximate

ρn,n ≈ 1σφ

ej(n−n)ω

∫ limσφ→0

1σφ

( 12 π−φ)

limσφ→0

− 1σφ

( 12 π−φ)

p 1σω

δω

(1σωδω|0; 1

)

ej(n−n)( 1σω

δω)σωd(

1σωδω

)

=1σφ

ej(n−n)ω

∫ ∞

−∞p 1

σωδω

(1σωδω|0; 1

)

ej(n−n)( 1σω

δω)σωd(

1σωδω

)

= ϕ 1σω

δω(σω(n− n)) ej(n−n)ω.

(30)

REFERENCES

[1] W. C. Y. Lee, “Effects on correlation between two mobile radio basestation antennas,” IEEE Trans. Commun., vol. 21, no. 11, pp. 1214–1224,Nov. 1973.

[2] F. Adachi, M. T. Feeney, A. G. Williamson, and J. D. Parsons, “Cross-correlation between the envelopes of 900 MHz signals received at amobile radio base station site,” IEE Proc.–Part F: Commun., Radar andSignal Process., vol. 133, no. 6, pp. 506–512, Oct. 1986.

[3] J. Luo, J. R. Zeidler, and S. McLaughlin, “Performance analysis ofcompact antenna arrays with MRC in correlated Nakagami fadingchannels,” IEEE Trans. Veh. Technol., vol. 50, no. 1, pp. 267–277, Oct.2001.

[4] A. Maaref and S. Aıssa, “On the error probability performance oftransmit diversity under correlated Rayleigh fading,” in Proc. IEEERadio Wirel. Conf., Atlanta, GA, Sept. 2004, pp. 275–278.

[5] J.-A. Tsai, R. M. Buehrer, and B. D. Woerner, “BER performance of auniform circular array versus a uniform linear array in a mobile radioenvironment,” IEEE Trans. Wireless Commun., vol. 3, no. 3, pp. 695–700, May 2004.

[6] D.-S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fadingcorrelation and its effect on the capacity of multielement antennasystems,” IEEE Trans. Commun., vol. 48, no. 3, pp. 502–513, Mar. 2000.

[7] M. K. Ozdemir, E. Arvas, and H. Arslan, “Dynamics of spatial correla-tion and implications on MIMO systems,” IEEE Commun. Mag., vol. 42,no. 6, pp. S14–S19, June 2004.

[8] C. Cozzo and B. L. Hughes, “Space diversity in presence of discretemultipath fading channel,” IEEE Trans. Commun., vol. 10, no. 51, pp.1629–1632, Oct. 2004.

[9] J. Salz and J. H. Winters, “Effects of fading correlation on adaptivearrays in digital mobile radio,” IEEE Trans. Veh. Technol., vol. 43, no. 4,pp. 1049–1057, Nov. 1994.

[10] J. S. Sadowsky and V. Kafedziski, “On the correlation and scatteringfunctions of the WSSUS channel for mobile communications,” IEEETrans. Veh. Technol., vol. 47, no. 1, pp. 270–282, Feb. 1998.

[11] P. C. Eggers, “Angular propagation descriptions relevant for base stationadaptive antenna operations,” Wireless Person. Commun., vol. 11, no. 3,pp. 3–29, Oct. 1999.

[12] J.-A. Tsai, R. M. Buehrer, and B. D. Woerner, “Spatial fading correlationfunction of circular antenna arrays with Laplacian energy distribution,”IEEE Commun. Lett., vol. 6, no. 5, pp. 425–434, May 2002.

[13] P. D. Teal, T. D. Abhayapala, and R. A. Kennedy, “Spatial correlationfor general distributions of scatterers,” IEEE Signal Processing Lett.,vol. 9, no. 10, pp. 305–308, Oct. 2002.

[14] P. De Doncker, “Spatial correlation functions for fields in three-dimensional Rayleigh channels,” Progress In Electromagnetics Research(PIER), vol. 40, pp. 55–69, 2003.

[15] V. I. Piterbarg and K. T. Wong, “Spatial-correlation-coefficient at thebasestation, in closed-form explicit analytic expression, due to het-erogeneously poisson distributed scatterers,” IEEE Antennas WirelessPropagat. Lett., vol. 4, pp. 385–388, 2005.

[16] J.-A. Tsai and B. D. Woerner, “The fading correlation function of acircular antenna array in mobile radio environment,” in Proc. IEEEGlobal Telecommun. Conf., vol. 5, San Antonio, TX, Nov. 2001, pp.3232–3236.

[17] J. Zhou, K. Ishizawa, S. Sasaki, S. Muramatsu, H. Kikuchi, andY. Onozato, “Generalized spatial correlation equations for antenna arraysin wireless diversity reception exact and approximate analyses,” in Proc.Int. Conf. Neural Netw. Signal Process., vol. 1, Nanjing, China, Dec.2003, pp. 180–184.

[18] T. Trump and B. Ottersten, “Estimation of nominal direction of arrivaland angular spread using an array of sensors,” Signal Process., no. 1-2,pp. 57–69, Apr. 1996.

[19] M. Bengtsson, “Antenna array signal processing for high rank datamodels,” Ph.D. dissertation, Royal Institute of Technology, Stockholm,Sweden, Dec. 1999.

[20] D. Astely, “Spatial and spatio-temporal processing with antenna arraysin wireless systems,” Ph.D. dissertation, Royal Institute of Technology,Stockholm, Sweden, June 1999.

[21] W. C. Jakes, Ed., Microwave Mobile Communications, 2nd ed. Piscat-away, NJ: Wiley-IEEE Press, 1994.

[22] T. Fulghum and K. Molnar, “The Jakes fading model incorporatingangular spread for a disk of scatters,” in Proc. IEEE Veh. Technol. Conf.(VTC’98), vol. 1, Ottawa, Ont., Canada, May 1998, pp. 489–493.

[23] R. G. Vaughan, “Pattern translation and rotation in uncorrelated sourcedistributionsfor multiple beam antenna design,” IEEE Trans. AntennasPropagat., vol. 46, no. 7, pp. 982–990, July 1998.

[24] Q. H. Spencer, B. D. Jeffs, M. A. Jensen, and A. L. Swindlehurst,“Modeling the statistical time and angle of arrival characteristics of anindoor multipath channel,” IEEE J. Select. Areas Commun., vol. 18,no. 3, pp. 347–360, May 2000.

[25] T. L. Fulghum, K. J. Molnar, and A. Duel-Hallen, “The Jakes fadingmodel for antenna arrays incorporating azimuth spread,” IEEE Trans.Veh. Technol., vol. 51, no. 5, pp. 968–977, Sept. 2002.

[26] A. Abdi, J. A. Barger, and M. Kaveh, “A parametric model for thedistribution of the angle of arrival and the associated correlation functionand power spectrum at the mobile station,” IEEE Trans. Veh. Technol.,vol. 51, no. 3, pp. 425–434, May 2002.

[27] N. Czink, E. Bonek, X. Yin, and B. Fleury, “Cluster angular spreadsin a MIMO indoor propagation environment,” in Proc. IEEE Int. Symp.Personal, Indoor, Mobile Radio Commun. 2005, vol. 1, Berlin, Germany,Sept. 2005, pp. 664–668.

[28] W. Grobner and N. Hofreiter, Integraltafel: Erster Teil—UnbestimmteIntegrale. Wien: Springer-Verlag, 1965, Vierte, verbesserte Auflage.

[29] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, andProducts, 7th ed. San Diego, CA: Elsevier, 2007.

[30] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functionswith Formulas, Graphs, and Mathematical Tables. New York, NY:Dover Publications, 1972, tenth printing with corrections.

[31] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed.Cambridge, UK: Cambridge at the University Press, 1944.

[32] O. Besson and P. Stoica, “Decoupled estimation of DOA and angularspread for a spatially distributed source,” IEEE Trans. Signal Processing,vol. 48, no. 7, pp. 1872–1882, July 2000.

[33] W. Grobner and N. Hofreiter, Integraltafel: Zweiter Teil—BestimmteIntegrale. Wien: Springer-Verlag, 1966, Vierte, verbesserte Auflage.

221

10 00

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Spatial Fading Correlation for SemicircularScattering: Angular Spread and Spatial

Frequency Approximations

Bamrung Tau Sieskul, Claus Kupferschmidt, andThomas Kaiser

Institute of Communications TechnologyLeibniz University of Hannover

Germany

August 12, 2010

Bamrung Tau Sieskul12.08.2010 17:00-17:20 | Session III: Signal Process. & App. | ICCE 2010 | Nha Trang, Vietnam

1/37

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Outline

1. IntroductionI ProblemI Research gap

2. Spatial fading correlationI Time delay across antenna elementsI Spatial fading correlationI Angular distributions

I Angular distributionsI Angular spread

I Spatial fading correlation in terms of Bessel functionsI Spatial frequency approximation

I Spatial frequency approximationI Characteristic function

3. Conclusion


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Outline






3. Conclusion


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Introduction

I Local scattering causes multipath propagation.

I There exists a correlation between two antenna elements,namely spatial fading correlation (SFC).

I The SFC plays an important role in wireless communications,such as

I probability of bit error rate,I channel capacity,I and etc.


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Outline






3. Conclusion


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Problem

I The direct computation of the SFC requires extensiveintegrals.

I The SFC can be approximated by truncating the first-orderTaylor series expansion

I around the spatial frequency angleI for a small angular spread.

I The approximation is entitled spatial frequency approximation(SFA).


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Outline






3. Conclusion


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Research Gap

I In the traditional SFA (see, e.g., [Trump and Ottersten 1996]1

and [Bengtsson 1999]2), the characteristic function is involvedwith the integration with infinite range.

I In this work, we truncate the integration range of thecharacteristic function to a finite range on semicircularscattering (−1

2π,12π].

1[Trump and Ottersten 1996] T. Trump and B. Ottersten, “Estimation of nominal direction of arrival and

angular spread using an array of sensors,” Signal Process., no. 1-2, pp. 57-69, Apr. 1996.2[Bengtsson 1999] M. Bengtsson, “Antenna array signal processing for high rank data models, Ph.D.

dissertation, Royal Institute of Technology, Stockholm, Sweden, Dec. 1999.


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Outline






3. Conclusion


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Outline






3. Conclusion


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Time Delay across Antenna Elements

I For a uniform linear array, the time delay at the n-th antennaelement is given by

ψn =1

cd(n− 1) sin(φ), (1)

where

I c is the speed of electromagnetic wave,I d is the distance between two adjacent antenna elements,I φ is the direction of arrival or emitting ray measured from the

perpendicular axis of the array.


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Outline






3. Conclusion


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Spatial Fading Correlation

I The correlation between the complex envelope of the signalfrom the n-th antenna element and another n-th antennaelement is given by

ρn,n = Eφ

{e−

1cj2πf0d(n−n) sin(φ)

}. (2)

I For the semicircular scattering (−12π,

12π], the SFC can be

calculated from

ρn,n =

∫ 12π

− 12πpφ(φ)e−

1cj2πf0d(n−n) sin(φ)dφ, (3)

where pφ(φ) is the probability density function (pdf) of theangle of arrival or departure.


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Outline






3. Conclusion


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Outline






3. Conclusion


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Angular Distributions

I For the cosine, uniform, Gaussian, Laplacian, and von Misesdistributions, the pdf can be written respectively as

pδφ(δφ) =

1π cc cosn(δφ),

δφ∈(−12π − φ,

12π − φ],

n∈{2, 4, 6, . . .};1

2√

3σφ, δφ∈(−

√3σφ,

√3σφ];

1√2πσφ

cGe− 1

2σ2φ

δ2φ, δφ∈(−1

2π − φ,12π − φ];

1√2σφ

cLe− 1σφ

√2|δφ|

, δφ∈(−12π − φ,

12π − φ];

12πI0(κ)cvMeκ cos(δφ),

δφ∈(−12π − φ,

12π − φ],

κ ≥ 0.

(4)


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Outline






3. Conclusion


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Angular Spread

I An important parameter in describing a scattering channel is astatistical value of the deviation of the arrival or departureangles from their nominal angles.

I From a statistical viewpoint, the angular spread can bedescribed as

σφ =√

Eφ{

(φ− φ)2}

=√

Eδφ{δ2φ}. (5)

I In [Czink et al. 2005]3, the angular spread is estimated usingthe measurement results at 5.2 GHz and found to be 2 to 9degrees for the departure and 2 to 7 degrees for the arrival.

3[Czink et al. 2005] N. Czink, E. Bonek, X. Yin, and B. Fleury, “Cluster angular spreads in a MIMO indoor

propagation environment,” in Proc. IEEE Int. Symp. Personal, Indoor, Mobile Radio Commun. 2005 (PIMRC2005), vol. 1, Berlin, Germany, Sept. 2005, pp. 664-668.


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Laplacian Distribution: Angular Spread

I For the Laplacian distribution, we have from [Gradshteyn andRyzhik 2007, p. 106]4

σφ =√cL

√√√√√ σ2φ + e

− 1√2σφ

π ((πφ+

√2σφφ

)sinh

(1σφ

√2φ)

−(

14π

2 + φ2 + 1√2πσφ + σ2

φ

)cosh

(1σφ

√2φ))

≈ σφ.(6)

4[Gradshteyn and Ryzhik 2007] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products,

7th ed. San Diego, CA: Elsevier, 2007.


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Laplacian Distribution: Numerical Example (I/III)

Standard Deviation σφ [degree]

AngularSpreadσφ[degree]

φ = 0◦: Simulationφ = 0◦: Derivationφ = 0◦: Approximationφ = 20◦: Simulationφ = 20◦: Derivationφ = 20◦: Approximationφ = 40◦: Simulationφ = 40◦: Derivationφ = 40◦: Approximation

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

Figure: The angular spread σφ as a function of the standard deviationσφ. with φmax = 1

2π −√

3σφ,max = 90◦ −√

3 · 20◦ = 55.3590◦.


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Laplacian Distribution: Numerical Example (II/III)

Nominal Angle φ [degree]

AngularSpreadσφ[degree]

σφ = 0.1◦: Simulationσφ = 0.1◦: Derivationσφ = 0.1◦: Approximationσφ = 10◦:Simulationσφ = 10◦: Derivationσφ = 10◦: Approximationσφ = 20◦: Simulationσφ = 20◦: Derivationσφ = 20◦: Approximation

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Figure: The angular spread σφ as a function of the nominal angle φ withNφ = 105 and σφ,max = 1√

3

(12π − φmax

)= 1√

3(90◦ − 50◦) = 23.0940◦.


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Outline






3. Conclusion


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Spatial Fading Correlation in Terms of Bessel FunctionsI Using the expansions of the trigonometric functions, we can

derive

ρn,n = J0

(1

c2πf0d(n− n)

)+ 2

∞∑k=1

J2k

(1

c2πf0d(n− n)

)ck

+ jJ2k−1

(1

c2πf0d(n− n)

)sk,

(7)

where ck and sk are the sinusoidal coefficients given by

ck =

∫ 12π

− 12πpφ(φ) cos(2kφ)dφ, (8a)

sk =

∫ 12π

− 12πpφ(φ) sin((2k − 1)φ)dφ. (8b)


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Outline






3. Conclusion


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Outline






3. Conclusion


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Spatial Frequency ApproximationI For a small angular spread σφ → 0 and a near broadside

nominal angle |φ| � 12π, the SFC can be approximated as

ρn,n ≈ ϕ 1σωδω

((n− n)σω) ej(n−n)ω, (9)

where ω, δω, σω, and ϕ 1σωδω

(·) are given by

ϕ 1σωδω

(u) =1

σφ

∫ ∞−∞

p 1σωδω

(v|0; 1) ejuvdv, (10a)

δω =1

c2πf0d cos(φ)δφ, (10b)

σω =1

c2πf0d cos(φ)σφ, (10c)

ω =1

c2πf0d sin(φ). (10d)


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Outline






3. Conclusion


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Characteristic FunctionI We transform the characteristic function in (9) into

ϕ 1σωδω

(σω(n− n)) =1

σφ

∫ ∞−∞

p 1σωδω

(v|0; 1) ej(n−n)σωvdv

=

∫ ∞−∞

pδφ(δφ|0;σ2φ)e

1σφ


(11)

I The characteristic function with the angular truncation isgiven by

ϕ 1σωδω

(σω(n− n)) =

∫ 12π

− 12πpδφ(δφ|0;σ2

φ)e1σφ


(12)


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Laplacian Distribution: Characteristic Function

I One can show that

ϕ 1σωδω

(u) =1

12

σ2φ

σ2φu2 + 1

cL. (13)


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Laplacian Distribution: Truncated Characteristic Function

ϕ 1σωδω

(u)

=1

σ2φ

σ2φu2 + 2

cL

(2− e

− 1√2σφ

π(− 1

2√

2u sin

(1

2σφ(2φ+ π)u

)e−w

+1

2√

2u sin

(1

2σφ(2φ− π)u

)ew + cos

(1

2σφ(2φ+ π)u

)e−w

+ cos

(1

2σφ(2φ− π)u

)ew + j

(− 1

2√

2u cos

(1

2σφ(2φ+ π)u

)e−w

+1

2√

2u cos

(1

2σφ(2φ− π)u

)ew − sin

(1

2σφ(2φ+ π)u

)e−w

− sin

(1

2σφ(2φ− π)u

)ew)))

, w =1

σφ

√2φ.


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Laplacian Distribution: Numerical Example (III/III)

Angular Spread σφ [degree]

SpatialFadingCorrelationρn,n

Simulation

SFA: Infinite

SFA: Finite

15.12 15.14 15.16

0 2 4 6 8 10 12 14 16 18 20

0.0475

0.0476

0.0477

0.0478

0.0479

10−2

10−1

100

Figure: The spatial fading correlation ρn,n−1 as a function of the angularspread σφ for the Laplacian distribution with φ = 40◦, Nφ = 106,f0 = 1

2 (10.6 + 3.1) GHz, and d = 0.2 m.


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Von Mises Distribution: Characteristic Function

I It can be shown that

ϕ 1σωδω

(u) = cvM

(σφδ (u) +

1

I0(κ)

∞∑k=1

Ik(κ)

(δ

(k +

1

σφu

)+ δ

(k − 1

σφu

))).

(14)


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Von Mises Distribution: Truncated Characteristic Function

I For the semicircular interval (−12π − φ,

12π − φ], the finite

integration range reads as

ϕ 1σωδω

(u) =1

πcvMe

− 1σφ

juφ

(1

uσφ sin

(1

2σφuπ

)+

1

I0(κ)2

cos

(1

2σφuπ

) ∞∑k=1

11σ2φu2 − (2k − 1)2

(−1)kI2k−1(κ)

((2k − 1) cos((2k − 1)φ) +

1

σφju sin((2k − 1)φ)

)).

(15)


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Von Mises Distribution: Numerical Example

Central Frequency f0 [Hz]

SpatialFadingCorrelationρn,n

Simulation

SFA: Infinite

SFA: Finite

SFA: Numerical Finite

d = 10 m

d = 1 m d = 0.1 m

106 108 1010 101210−4

10−3

10−2

10−1

100

Figure: The spatial fading correlation ρn,n−1 as a function of the centralfrequency f0 for the von Mises distribution with φ = 20◦, Nφ = 106,κ = 1, and N∞ = 100.


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Outline






3. Conclusion


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Conclusion

I The angular spread is derived for the semicircular scatteringon the range (−1

2π,12π] for the uniform, Gaussian, Laplacian,

and von Mises distributions.

I We have proposed the counterpart of the usual SFA as theSFA of the SFC with the finite integration range.

I For the moderate angular spread in the Laplacian distribution,the new SFA can provide higher accuracy in computing theSFC than the usual SFA.

I For the von Mises distribution, the conventional SFA causesthe discrete values of the SFC, which cannot approximate theactual SFC, whereas the new SFA can approximate the SFC.


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Thank you for your attention


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Documents

Spatial Fading Correlation for Semicircular Scattering ...Spatial Fading Correlation for Semicircular Scattering: Angular Spread and Spatial Frequency Approximations Bamrung Tau Sieskul,