New Wideband HAP-MIMO Channels: A 3-D Modeling and Simulation …emichail.weebly.com/uploads/8/8/2/2/8822113/springer_wpc... · 2018. 9. 6. · Wideband HAP-MIMO Channels: A 3-D Modeling

Wireless Pers CommunDOI 10.1007/s11277-013-1311-9

Wideband HAP-MIMO Channels: A 3-D Modeling andSimulation Approach

Emmanouel T. Michailidis · Athanasios G. Kanatas

© Springer Science+Business Media New York 2013

Abstract High-altitude platforms (HAPs) are considered as an alternative technology to pro-vide future generation broadband wireless communications services. This paper proposes athree-dimensional (3-D) geometry-based reference model for wideband HAP multiple-input–multiple-output (MIMO) channels. The statistical properties of the channel are analyticallystudied in terms of the elevation angle of the platform, the antenna arrays configuration, andthe angular, the Doppler and the delay spread. Specifically, the space-time-frequency cor-relation function (STFCF), the space-Doppler power spectrum, and the power space-delayspectrum are derived for a 3-D non-isotropic scattering environment. Finally, a sum-of-sinusoids statistical simulation model for wideband HAP-MIMO channels is proposed. Theresults show that the simulation model accurately and efficiently reproduces the STFCF ofthe reference model. The proposed models provide a convenient framework for the charac-terization, analysis, test, and design of wideband HAP-MIMO communications systems withline-of-sight and non-line-of-sight links.

Keywords High-altitude platform (HAP) · Multiple-input–multiple-output (MIMO)channels · Ricean fading · Channel simulation · Wideband channel · 3-D scattering model

1 Introduction

In recent years, a great interest has been drawn towards the development of high-altitudeplatforms (HAPs), which are expected to be a flexible, rapidly deployable, non-pollutant andcost-effective alternative to satellite and terrestrial communications infrastructures [1,2].The term HAPs defines both aircrafts and airships flying at an altitude 17–22 km above the

E. T. Michailidis · A. G. Kanatas (B)Department of Digital Systems, University of Piraeus, 80 Karaoli & Dimitriou St.,18534 Piraeus, Greecee-mail: [email protected]

E. T. Michailidise-mail: [email protected]

123

E. T. Michailidis, A. G. Kanatas

ground, in the stratosphere. Prospective 4th generation (4G) [3] communications servicesdemand increased bandwidth and enhanced data rates. These services are mainly basedon wideband transmissions and can be potentially upgraded, if multiple-input–multiple-output (MIMO) techniques are used [4,5]. The successful design, optimization, and conciseevaluation of future HAP-MIMO systems require a thorough investigation of the multipathfading channel and its statistical properties. Thus, realistic wideband HAP-MIMO channelmodels are essential.

Wideband effects usually impose an impulse response, which can be modeled by a tappeddelay line (TDL) [6]. In agreement with the TDL concept, channel models for HAP-basedsystems were presented in [7,8]. These models allow for convenient and time-efficient systemanalysis study, when the channel characteristics are dynamic and vary in time. Nevertheless,when terrain and scattering distributions are available, physical-geometrical channel char-acterization is preferred to ensure model accuracy and versatility. A geometry-based single-input single-output (SISO) model for HAPs was proposed in [9]. This model considered anellipsoid as the volume containing all the scatterers. However, the influence of the elevationangle of the platform and the antenna arrays configuration was not considered for the per-formance evaluation. In addition, the scatterers were chosen to be uniformly distributed inspace, which is an assumption that deviates from practical situations.

To the best of the authors’ knowledge, only terrestrial geometry-based wideband MIMOchannel models are available. Two-dimensional (2-D) MIMO models for non-isotropic scat-tering environments were presented in [10–12] and assumed that the radio waves travel onlyhorizontally, which is an acceptable approximation for certain scenarios. Since HAPs areplaced in the stratosphere and mobile users are often located lower than the surroundingscatterers, the waves may travel in both horizontal and vertical planes. Therefore, three-dimensional (3-D) channel modeling is required to ideally characterize the propagation envi-ronment. In [13], a 3-D wideband MIMO model was presented and introduced an elevationangle for the incidence of the radio waves in addition to the azimuth angle. Nevertheless,this model considered that the distance between the user and the scatterers in the azimuthplane is constant, which is unrealistic for HAP-based systems [14]. Moreover, the derivationof this model and its statistical properties was ground on the simplified assumption that themaximum elevation angle of arrival of the scattered waves is smaller than 20◦, which typ-ically represents terrestrial mobile-to-mobile systems [15]. If “above rooftop” propagationis present as in HAP-based communications, the maximum elevation angle approximates90◦ [16].

Recently, the authors proposed a 3-D geometry-based single-bounce (GBSB) referencemodel for HAP-MIMO Ricean channels [14] and suggested that MIMO techniques areapplicable on a single HAP. This model assumes that the propagation delays of all incom-ing scattered waves are approximately equal and small in comparison to the data symbolduration. Notwithstanding, in wideband communications, the data symbol duration is smalland multipath delay spread is introduced, i.e., the propagation delay differences cannot beneglected. In this paper, the aforementioned narrowband model is extended with respect tofrequency-selectivity and a 3-D GBSB reference model for wideband HAP-MIMO chan-nels is derived. The proposed model utilizes carrier frequencies below 10 GHz. Therefore,both line-of-sight (LoS) and non-line-of-sight (NLoS) links may exist, while rain effects areinsignificant [1]. Several parameters are considered, e.g., the elevation angle of the platform,the arrays configuration, the angle, Doppler and delay spread, as well as the scattering dis-tribution. Based on this model, the space-time-frequency correlation function (STFCF), thespace-Doppler power spectrum (SDPS), and the power space-delay spectrum (PSDS) arederived for a 3-D non-isotropic scattering environment. Numerical results demonstrate the

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Wideband HAP-MIMO Channels

theoretical derivations. Although theoretical channel models can be ideally verified throughexperimental real-time field trials, simulation of the radio propagation environment is com-monly used as an alternative approach. Thus, this paper presents a statistical sum-of-sinusoidsbased (SoS) simulation model for wideband HAP-MIMO channels. Simulation results areemployed to analyze the performance of the simulation model and verify its usefulness.

The remainder of the paper is organized as follows. Section 2 introduces the 3-D referencewideband HAP-MIMO channel model and describes its parameters. From this model the inputdelay-spread and time-variant transfer functions are derived in Sect. 3. Section 4 studies thestatistical properties of the reference model and presents numerical results. Section 5 developsthe simulation model and provides simulation results. Finally, conclusions are drawn inSect. 6.

2 A Reference Model for Wideband HAP-MIMO Channels

This paper considers a downlink wideband HAP-MIMO communication channel with nT

transmit and nR receive antenna elements at a stratospheric base station (SBS) and a ter-restrial mobile station (TMS), respectively. All antennas are fixed, omni-directional, and arenumbered as 1 ≤ p ≤ q ≤ nT and 1 ≤ l ≤ m ≤ nR respectively. The complex low-passequivalent time-variant channel response between SBS and TMS is given by h pl (t, τ ) and thecomplex low-pass equivalent input-output relation for the frequency selective fading channelcan be written as

r(t) = H(t, τ ) ∗ s(t)+ n(t), (1)

where H(t, τ ) = [h p,l(t, τ )

]nR×nT

∈ Cn R×nT is the matrix of the input delay-spread func-

tions [6], s(t) ∈ CnT ×1 is the transmitted signal vector, r(t) ∈ C

nR×1 is the received signalvector, n(t) ∈ C

nR×1 is the noise vector, which denotes the additive white Gaussian noise(AWGN) at the receiver branches, and (*) denotes convolution. The entries of the noise vectorare independent and identically distributed (i.i.d.) complex Gaussian random variables withzero-mean and variance N0 where N0 is the noise power spectral density (PSD).

2.1 A 3-D Geometrical Two-Concentric-Cylinders Model

The geometrical model proposed in [14] considers that the waves emitted from the SBSantenna elements travel over paths with different lengths and impinge the TMS array ele-ments from different directions due to the 3-D non-isotropic scattering within a cylin-der. This paper proposes a modified two-concentric-cylinders version of this geometry.Figures 1 and 2 show the LoS and NLoS paths of the 3-D model for a 2 × 2 HAP-MIMOchannel, respectively, which is the basis to study uniform linear arrays (ULAs) with arbitrarynumber of antennas. It is assumed that N → ∞ fixed effective local scatterers surroundthe TMS. Then, the nth scatterer is designated by S(n) and its projection to the x-y planeis S(n). By denoting d(a, b) the distance between two points a and b the height of S(n) isd(S(n), S(n)

) = H (n)S ∈ [

HS,min, HS,max], and the distance between S(n) and O ′ (projec-

tion of TMS to the x-y plane) is d(S(n), O ′) = R(n)S ∈ [

RS,min, RS,max], where HS,min =

min{

H (n)S

}, HS,max = max

{H (n)

S

}, RS,min = min

{R(n)S

}, and RS,max = max

{R(n)S

}.

Note that the scatterers occupy the volume between two-concentric-cylinders with heightHc = HS,max − HS,min and radius RS,min and RS,max, respectively. The distance betweenthe coordinate origin O (centre of the projections p and q of the SBS antenna elements p

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Om

p

q

z

y

x

l

TO

Tθ RθRO

Rψ

,maxSH

RγTγ

Tv

Rv

,maxSR

RlLoSaπ −

Tβ

D

,minSR

,minSH

cH

Fig. 1 LoS paths of the 3-D geometrical model for a 2 × 2 HAP-MIMO channel

O O′m

p

q

z

y

x

l

TO

TθRθ

RO

Rψ

,maxSH

RγTγ

Tv

Rv

,maxSR

( )nSβ

( )n

( )nS

( )nSH

( )nSR

( )nTa

( )nRa

D

,minSR

,minSH

cH

Fig. 2 NLoS paths of the 3-D geometrical model for a 2 × 2 HAP-MIMO channel

and q to the x-y plane) and O ′ (lower center of the cylinder) is d(O, O ′) = D, and theheights of the SBS and TMS antenna arrays are d(O, OT ) = HT and d(O, OR) = HR ,respectively. The spacing between two adjacent antenna elements at the SBS and TMS is

denoted by δT and δR , respectively. It is assumed that{

R(n)S , δT

} D, δR

{R(n)S

}, and

HT {

HR, H (n)S

}. In addition, the angles θT and θR represent the orientation of the SBS

and TMS antenna arrays respectively, relative to the x-axis, and the angle ψR describes theelevation angle of the lth TMS antenna element. The symbols a Rl

LoS, a(n)T and a(n)R denotethe azimuth angle of arrival (AoA) of the LoS paths, the azimuth angle of departure (AoD)

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Table 1 Definition of the parameters used in the reference model

D The distance between the centre O of the projections of the SBSantenna elements to the x-y plane and the centre O ′ of thecylinder

δT , δR The spacing between two adjacent antenna elements at the SBS andTMS, respectively

θT , θR The orientation of the SBS and TMS antenna array in the x-y plane(relative to the x-axis) respectively

ΨR The elevation angle of the lth TMS antenna element, relative to thex-y plane

νT , νR The velocities of the SBS and TMS, respectively

γT , γR The moving directions of the SBS and TMS, respectively

fT,max, fR,max The maximum Doppler frequency shift of the SBS and TMS, respectively

βT The elevation angle of the SBS relative to OR

HT , HR The height of the SBS and TMS, respectively

a RlLoS The azimuth angle of arrival of the LoS paths

a(n)T , a(n)R The azimuth angle of arrival and the azimuth angle of departureat/from the nth scatterer, respectively

R(n)S The distance between S(n) and O ′

RS,min, RS,max The minimum and maximum distance between S(n) and O ′,respectively

H (n)S The height of the nth scatterer

HS,min, HS,max The minimum and maximum height of the scatterers, respectively

β(n)S The elevation angle of the nth scatterer relative to OR

βS,min, βS,max The minimum and maximum elevation angle of the nth scattererrelative to OR , respectively

ϕ(n), g(n) The random phase and the amplitude introduced by the nthscatterer, respectively

μ The mean azimuth angle at which the scatterers are distributed inthe x-y plane (von Mises pdf)

k The spread of scatterers around the mean azimuth angle (von Mises pdf)

a The spread of the scatterers around the TMS (hyperbolic pdf)

HS,mean The mean of scatterer’s height (log-normal pdf)

σ The standard deviation of scatterer’s height (log-normal pdf)

of the wave that impinge on the scatterer S(n) and the azimuth AoA of the wave scatteredfrom S(n), respectively. The elevation angle of SBS relative to OR is βT ≈ arctan(HT /D)and the elevation angle of the nth scatterer relative to OR is β(n)S ≈ arctan(H (n)

S /R(n)S ).The proposed geometry enables the determination of the minimum and the maximum val-ues of β(n)S . Specifically, we obtain that βS,min = arctan

(HS,min/RS,max

)and βS,max =

arctan(HS,max/RS,min

).

According to [1,2], aircrafts usually fly on a tight circle, whereas airships can theoreticallystay still and maintain their position. Notwithstanding, stratospheric winds and pressure vari-ations may influence the movement and stability of the platform [17]. This paper considersthat both SBS and TMS may be in motion and contribute to the Doppler spread. Specif-ically, SBS and TMS are moving with speeds νT and νR in the directions determined bythe angles γT and γR, respectively. Table 1 summarizes the model parameters, for ease ofreference.

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3 Input Delay-Spread and Time-Variant Transfer Functions

A HAP-based communication channel is Ricean in its general form [1]. According to the3-D model, the input delay-spread function of the link p − l is a superposition of the LoSand NLoS rays as follows [13]

h pl(t, τ ) = h pl,LoS(t, τ )+ h pl,N LoS(t, τ ), (2)

where

h pl,LoS(t, τ ) =√�pl K pl

K pl + 1ζLoS(t)δ (τ − τLoS) , (3)

h pl,N LoS(t, τ ) =√

�pl

K pl + 1lim

N→∞1√N

N∑

n=1

g(n)e jϕ(n) ζ(n)N LoS(t)δ

(τ − τ

(n)N LoS

), (4)

K pl and �pl denote the Ricean factor (ratio of LoS to NLoS received power) and the trans-mitted power of the subchannel p-l, respectively, and g(n) and ϕ(n) represent the ampli-tude and phase variations introduced by the nth scatterer, respectively. It is assumed that

g(n) is an i.i.d. random variable with limN→∞ N−1 ∑Nn=1 E

[∣∣g(n)∣∣2]

= 1, while ϕ(n) is

a random variable uniformly distributed in the interval [−π, π) and independent fromg(n), a(n)T , a(n)R , R(n)S , H (n)

S , and β(n)S . Moreover, τLoS and τ (n)N LoS are the propagation delays(travel times) of the waves associated with the LoS and NLoS links, respectively, and δdenotes the Dirac delta function. Finally, ζLoS(t) and ζ (n)N LoS(t) are complex exponentialsthat carry phase information and can be expressed as (see Figs. 1 and 2) [14]

ζLoS(t) = e− j2πλ

d(p,l)e j2π t(FT,LoS + FR,LoS), (5)

ζ(n)N LoS(t) = e− j2π

λ

[d(

p,S(n))+ d

(S(n),l

)]e j2π t(FT,N LoS + FR,N LoS), (6)

where

d (p, l) ≈ (D − DT1 + DR1

)/ cosβT ,

d(

p, S(n))

≈(

D − DT1 − DT2 R(n)S sin a(n)R

)/ cosβT ,

d(

S(n), l)

≈ R(n)S

cosβ(n)S

− DR1 cosα(n)R cos β(n)S − DR2 sin β(n)S − DR3 sin α(n)R cosβ(n)S ,

FT,LoS = fT,max cos(π − aRl

LoS − γT

),

FR,LoS = fR,max cos(

aRlLoS − γR

),

FT,N LoS ≈ fT,max

(R(n)S sin γT sin a(n)R /D + cos γT

),

FR,N LoS = fR,max cos(

a(n)R − γR

)cosβ(n)S , (7)

j2 = −1, DT1 = 0.5 (nT + 1 − 2p) δT cos θT , DT2 = 0.5 (nT + 1 − 2p) δT sin θT /D,DR1 = 0.5 (nR + 1 − 2l) δR cos θR cosψR, DR2 = 0.5 (nR + 1 − 2l) δR sinψR, DR3 =0.5× (nR + 1 − 2l) δR sin θR cosψR . Moreover, fT,max = νT /λ and fR,max = νR/λ denotethe maximum Doppler shifts of SBS and TMS, respectively.

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From Fig. 1, since HT HR the propagation delay from the SBS to the TMS, i.e., theminimum propagation delay, is given by

τLoS = τmin ≈ D/ (c0 cosβT ) , (8)

where c0 is the speed of light. From Fig. 2, since HS,max HT , and R(n)S D, using thecosine law, the travel time of a wave originated from SBS, reflected from S(n) and receivedby TMS can be expressed as

τ(n)N LoS ≈

√

D2 +(

R(n)s

)2 − 2DR(n)S cos(π − a(n)R

)

c0 cosβT+ R(n)S

c0 cosβ(n)S

. (9)

Using the approximation√

1 + x ≈ 1 + x/2 for small x , (9) becomes

τ(n)N LoS ≈ τLoS + R(n)S cos a(n)R

c0 cosβT+ R(n)S

c0 cosβ(n)S

. (10)

Using (10) and considering that cos (arctan x) = 1/√

1 + x2, cos a(n)R = 1, β(n)S ≈arctan

(H (n)

S /R(n)S

), R(n)S = RS,max, and H (n)

S = HS,max, the maximum possible propa-

gation delay can be derived as follows

τmax ≈ τLoS + RS,max

c0 cosβT+√

R2S,max + H2

S,max

c0. (11)

Note that τLoS is common to all paths and can be neglected, without loss of generality. Then,the relative minimum and maximum propagation delays, and the relative propagation delayassociated with the NLoS link are given, respectively, by τr,LoS = τr,min = 0, τr,max ≈τmax − τLoS, τ

(n)r,N LoS ≈ τ

(n)N LoS − τLoS . From (11), considering that HS,max and τr,max are

known, RS,max can be easily determined. Although values of τr,max for terrestrial systems arewell known, there are no measurements available for HAP-based systems. Estimated valuesof τr,max achieved with conventional aircrafts are in the order of hundreds of nanoseconds,which are much smaller than the corresponding values in terrestrial systems [18].

Finally, the time-variant transfer function is the Fourier transform (FT) of the input delay-spread function [19] and can be written as

Tpl (t, f ) = τ{h pl (t, τ )

} = Tpl,LoS (t, f )+ Tpl,N LoS (t, f ) , (12)

where the LoS and NLoS components of the time-variant transfer function are, respectively,

Tpl,LoS (t, f ) =√�pl K pl

K pl + 1ζLoS (t) e− j2π f τLoS , (13)

Tpl,N LoS (t, f ) =√

�pl

K pl + 1lim

N→∞1√N

N∑

n=1

g(n)e jϕ(n) ζ(n)N LoS (t) e− j2π f τ (n)N LoS . (14)

4 Statistical Properties of 3-D Wideband HAP-MIMO Channels

Considering a 3-D non-isotropic scattering environment, we first derive the STFCF of thecomplex faded envelope. Then, using the STFCF, we derive the SDPS and the PSDS.

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4.1 Space-Time-Frequency Correlation Function

The normalized STFCF between two time-variant transfer functions Tpl (t, f ) , andTqm (t, f ) is defined as [6,11,20]

Rpl,qm (δT , δR,�t,� f ) = E[Tpl (t, f )∗ Tqm (t +�t, f +� f )

]

√�pl�qm

, (15)

where (·)∗ denotes complex conjugate operation and E [·] is the statistical expectation oper-ator. Since the number of local scatterers is infinite, central limit theorem implies thatTpl,N LoS (t, f ) is a complex Gaussian random process with zero-mean. Then, (15) can bewritten as follows [14]

Rpl,qm (δT , δR,�t,� f ) = RLoSpl,qm (δT , δR,�t,� f )+ RN LoS

pl,qm (δT , δR,�t,� f ) , (16)

where RLoSpl,qm (δT , δR,�t,� f ) and RN LoS

pl,qm (δT , δR,�t,� f ) denote the STFCF of the LoSand the NLoS components, respectively. Using (5), (7), (8), (13), and (15), and the approxi-mate relation aRl

LoS = aRmLoS ≈ π for δT D, the STFCF of the LoS component is derived

as follows

RLoSpl,qm (δT , δR,�t,� f ) ≈

√K pl Kqm(

K pl + 1) (

Kqm + 1)e

j2π[(q−p)δT cos θT −(m−l)δR cos θR cosψR ]λ cosβT

×e j2π�t( fT,max cos γT − fR,max cos γR)e− j2π� f D

c0 cosβT . (17)

Moreover, using (6), (7), (10), (14), and (15), the STFCF of the NLoS component can bewritten as

RN LoSpl,qm (δT , δR,�t,� f ) ≈ 1

√(K pl + 1

) (Kqm + 1

) limN→∞

1

N

N∑

n=1

E

[∣∣∣g(n)

∣∣∣2

ej2π(q−p)δT cos θT

λ cosβT

×ej2π(q−p)δT sin θT R(n)S sin a(n)R

λD cosβT ej2π(m−l)δR sinψR sin

[arctan

(H(n)S /R(n)S

)]

λ

×ej2π(m−l)δR cos θR cosψR cos a(n)R cos

[arctan

(H(n)S /R(n)S

)]

λ ej2π(m−l)δR sin θR cosψR sin a(n)R cos

[arctan

(H(n)S /R(n)S

)]

λ

×ej2π�t fT,max

(R(n)S sin γT sin a(n)R

D +cos γT

)

ej2π�t fR,max cos

(a(n)R −γR

)cos[arctan

(H (n)

S /R(n)S

)]

×e− j2π� f

[D

c0 cosβT+ R(n)S cos a(n)R

c0 cosβT+ R(n)S

c0 cos[arctan

(H(n)S /R(n)S

)]

]⎤

⎥⎦ . (18)

The infinite number of scatterers enables the replacement of the discrete variablesa(n)R , R(n)S , and H (n)

S with the continuous random variables aR, RS, and HS with joint prob-

ability density function (pdf) f (aR, RS, HS). According to Fig. 2, a(n)R , R(n)S , and H (n)S are

independent. Thus, the joint pdf f (aR, RS, HS) can be decomposed to f (aR) f (RS) f (HS)

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and the STFCF of the NLoS component becomes

RN LoSpl,qm (δT , δR,�t,� f )

≈ 1√(

K pl + 1) (

Kqm + 1)

HS,max∫

0

RS,max∫

0

π∫

−πf (aR) f (RS) f (HS) e

j2π(q−p)δT cos θTλ cosβT

×ej2π(q−p)δT sin θT RS sin aR

λD cosβT ej2π(m−l)δR sinψR sin[arctan(HS/RS)]

λ

×ej2π(m−l)δR cos θR cosψR cos aR cos[arctan(HS/RS)]

λ ej2π(m−l)δR sin θR cosψR sin aR cos[arctan(HS/RS)]

λ

×ej2π�t fT,max

(RS sin γT sin aR

D +cos γT

)

e j2π�t fR,max cos(aR−γR) cos[arctan(HS/RS)]

×e− j2π� f

[D

c0 cosβT+ RS cos aR

c0 cosβT+ RS

c0 cos[arctan(HS/RS)]

]

daRd RSd HS . (19)

In agreement with the narrowband model in [14], the empirically justified in urban andsuburban environments von Mises pdf [21,22] is used to characterize aR and is defined as

faR (aR) = ek cos(aR−μ)

2π I0 (k), (20)

where I0 (·) is the zeroth-order modified Bessel function of the first kind, μ ∈ [−π, π ] isthe mean angle at which the scatterers are distributed in the x-y plane, and k ≥ 0 controlsthe spread around the mean. In addition, the following truncated hyperbolic pdf is used tocharacterize RS

fT,RS (RS) = fRS

(RS∣∣0 < RS ≤ RS,max

)

FRS

(RS,max

)− FRS

(RS,min

) , (21)

where

fRS


) = a

tanh(a RS,max

)cosh2 (a RS)

, (22)

FRS


) = tanh (a RS)

tanh(a RS,max

) (23)

are the experimentally verified hyperbolic pdf [23,24] and the hyperbolic cumulative dis-tribution function (cdf), respectively, and a ∈ (0, 1) controls the spread of the scatterersaround the TMS. Note that increasing a reduces the spread of the pdf of RS . Furthermore,the following truncated log-normal pdf is adopted to characterize HS

fT,HS (HS) = fHS

(HS

∣∣0 < HS ≤ HS,max)

FHS

(HS,max

)− FHS

(HS,min

) , (24)

where

fHS

(HS

∣∣0 < HS ≤ HS,max) = e

[− 1

2σ2 ln2(

HSHS,mean

)]

HSσ√

2π, (25)

FHS

(HS

∣∣0 < HS ≤ HS,max) = 0.5 erfc

[−ln

(HS/HS,mean

)/(σ√

2)]

(26)

are the experimentally validated log-normal pdf [25,26] and the log-normal cdf, respec-tively, HS,mean and σ are the mean and standard deviation of HS, respectively, and

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erfc (x) = (2/

√π) ∫∞

x e−u2du is the complementary error function. Finally, using the equal-

ity∫ π−π ea sin(c)+b cos(c)dc = 2π I0

(√a2 + b2

)[27, eq. 3.338-4], and (19)–(26), the STFCF

of the NLoS component becomes

RN LoSpl,qm (δT , δR,�t,� f ) =

HS,max∫

HS,min

RS,max∫

RS,min

w1 I0

(√w2

2 + w23

)d RSd HS, (27)

where w1, w2, and w3 are, respectively

w1 = a · ej2π

((q−p)δT cos θT

λ cosβT+ (m−l)δR sinψR sin[arctan(HS/RS)]

λ

)

[tanh

(a RS,max

)− tanh(a RS,min

)] [FHS

(HS,max

)− FHS

(HS,min

)]

×e

[− 1

2σ2 ln2(

HSHS,mean

)]

e j2π�t fT,max cos γT e− j2π

(� f D

c0 cosβT+ � f RS

c0 cos[arctan(HS/RS)]

)

HS cosh2 (a RS) σ√

2π I0 (k)√(

K pl + 1) (

Kqm + 1) , (28)

w2 = j2π (q − p) δT sin θT RS

λD cosβT+ j2π�t fT,max sin γT RS

D

+ j2π (m − l) δR sin θR cosψR cos [arctan (HS/RS)]

λ+ j2π�t fR,max sin γR cos [arctan (HS/RS)] + k sinμ, (29)

w3 = j2π (m − l) δR cos θR cosψR cos [arctan (HS/RS)]

λ+ j2π�t fR,max cos γR cos [arctan (HS/RS)]

− j2π� fRS

c0 cosβT+ k cosμ. (30)

The double integral in (27) has to be evaluated numerically, since there is no closed-form solution. Finally, the STFCF between two time-variant transfer functions Tpl (t, f ) ,and Tqm (t, f ) becomes a summation of the STFCFs of the LoS and the NLoS componentsdefined in (17) and (27)–(30), respectively.

4.2 Space-Doppler Power Spectrum

Considering a wide-sense stationary uncorrelated scattering (WSSUS) time-varying HAP-MIMO channel, the SDPS is derived by applying the FT to the space-time correlation function(STCF), i.e., Rpl,qm (δT , δR,�t,� f = 0) , in (16). Other approaches for determining theDoppler spectrum of terrestrial and satellite systems by assuming 3-D scattering were previ-ously proposed in [13,28–31].

The SDPS of the LoS component is obtained by calculating the FT of the STCF of theLoS component in (17) as follows

SLoSpl.qm (δT , δR, ν) = �t

{RLoS

pl,qm (δT , δR,�t,� f = 0)}

=√

K pl Kqm(K pl + 1

) (Kqm + 1

)ej

2π[(q−p)δT cos θT −(m−l)δR cos θR cosψR ]λ cosβT

×δ (ν + fT,max cos γT − fR,max cos γR), (31)

where ν is the Doppler frequency shift relative to the carrier frequency.

123


“Appendix 1” shows that the SDPS of the NLoS component can be obtained as

SN LoSpl,qm (δT , δR, ν)

=x1∫

HS,min

R,max∫

x2

A′e j2πs2(ν− fT,max cos γT )

π fR,max cos [arctan (HS/RS)]

√

1 −(

ν− fT,max cos γTfR,max cos[arctan(HS/RS)]

)2

× cos

⎡

⎣2πs3 fR,max cos [arctan (HS/RS)]

√

1 −(

ν − fT,max cos γT

fR,max cos [arctan (HS/RS)]

)2⎤

⎦ d RSd HS,

(32)

where

A′ = ej2π

((q−p)δT cos θT


λ

)

e

[− 1

2σ2 ln2(

HSHS,mean

)]


2π I0 (k)√(

K pl + 1) (

Kqm + 1)

× a[FHS (x1)− FHS

(HS,min

)] [tanh

(a RS,max

)− tanh (ax2)] , (33)

x1 ={

HS,max, 0 ≤ ∣∣ν − fT,max cos γT∣∣ ≤ νmin

RS,min B, νmin <∣∣ν − fT,max cos γT

∣∣ ≤ νmax, (34)

x2 ={

RS,min, 0 ≤ ∣∣ν − fT,max cos γT∣∣ ≤ νmin

HS,max/B, νmin <∣∣ν − fT,max cos γT

∣∣ ≤ νmax, (35)

B = tan

(

arccos

(∣∣ν − fT,max cos γT∣∣

fR,max

))

(36)

and νmin and νmax are defined in (65) and (67), respectively

4.3 Power Space-Delay Spectrum

Considering a WSSUS frequency-varying HAP-MIMO channel, the PSDS is derived byapplying the inverse Fourier transform (IFT) to the space-frequency correlation function(SFCF) in (16) i.e., Rpl,qm (δT , δR,�t = 0,� f ).

The relative PSDS of the LoS component can be obtained by calculating the IFT of theSFCF of the LoS component in (17) as follows

P LoSpl,qm (δT , δR, τr ) = −1

� f

{RLoS

pl,qm (δT , δR,�t = 0,� f )}

=√

K pl Kqm(K pl + 1

) (Kqm + 1

)ej

2π [q−p]δT cos θT −(m−l)δR cos θR cosψRλ cosβT δ (τr ) ,

(37)

where τr = τ − τLoS ∈ [0, τr,max]

is the relative propagation delay.

123


“Appendix 2” shows that the relative PSDS of the NLoS component can be obtained asfollows

P N LoSpl,qm (δT , δR, τr ) =

y2∫

y1

RS,max∫

y3

A′′e j2πp2(τr −p4) cos

(2πp3

√p2

1 − (τr − p4)2)

π

√p2

1 − (τ − p4)2

d RSd HS,

(38)

where

A′′ = ej2π

((q−p)δT cos θT


λ

)

e

[− 1

2σ2 ln2(

HSHS,mean

)]


2π I0 (k)√(

K pl + 1) (

Kqm + 1)

× a[FHS (y2)− FHS (y1)

] [tanh

(a RS,max

)− tanh (ay3)] , (39)

y1 ={

HS,min, 0 ≤ τr ≤ τ2√tan2 βT R2

S,min − 2c0τr secβT RS,min + c20τ

2r , τr ≤ τ2 ≤ τr,max

, (40)

y2 ={√

tan2 βT R2S,min − 2c0τr secβT RS,min + c2

0τ2r , 0 ≤ τr ≤ τ2

HS,max, τr ≤ τ2 ≤ τr,max, (41)

y3 =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

−c0τr secβT +√

c20τ

2r +H2

S,max tan2 βT

tan2 βT, 0 ≤ τr ≤ τ1

RS,min, τ1 ≤ τr ≤ τ2

c0τr secβT −√

c20τ

2r +H2

S,min tan2 βT

tan2 βT, τr ≤ τ2 ≤ τr,max

. (42)

4.4 Numerical Results

This section demonstrates the theoretical derivations described above. Figures 3, 4, 5, 6 depictthe absolute space-time-frequency correlation for a 3-D non-isotropic scattering environment.Unless indicated otherwise, the values of the model parameters used to obtain the curvesare nT = nR = 2, K pl = Kqm = 3 dB,HT = 20 km, βT = 60◦, θT = 45◦, θR =30◦, ψR = 15◦, k = 3, μ = 60◦, RS,min = 7 m,RS,max = 140 m, HS,min = 5 m, HS,max =60 m, fT,max = 200 Hz, fR,max = 100 Hz, γT = 30◦, and γR = 45◦. Moreover, we considera typical densely built-up district (London, UK [25]) to be the scattering region, i.e., thesurrounding buildings act as scatterers, and we set HS,mean = 17.6 m and σ = 0.31. Inaddition, we assume that a = 0.01, which corresponds to a reasonable average distancebetween the TMS and the effective scatterer of approximately 60 m [14].

Figure 3 shows the absolute frequency correlation function (FCF)∣∣Rpl,qm

(δT = 0, δR=0,

�t = 0,� f)∣∣ of a HAP-SISO, i.e., nT = nR = 1, Rayleigh channel for different βT . One

observes that the correlation increases as βT decreases. From Fig. 3, the coherence bandwidthBc can also be estimated. By assuming that the coherence bandwidth is the bandwidth overwhich the absolute frequency correlation is above 0.5 [32], Bc is approximately 0.8, 0.7, 0.5,and 0.3 MHz, when βT = 50◦, βT = 60◦, βT = 70◦, and βT = 80◦, respectively.

The shape of the absolute SFCFs∣∣Rpl,qm (δT , δR = 0,�t = 0,� f )

∣∣ and∣∣Rpl,qm

(δT =0,

δR,�t = 0,� f)∣∣ of a HAP-MIMO channel is illustrated in Figs. 4 and 5, while

Fig. 6 depicts the shape of the absolute time-frequency correlation function (TFCF)

123


0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

Frequency Separation Δf (MHz)

Freq

uenc

y C

orre

latio

n

βT

=70o

βT

=50o

βT

=60o

βT

=80o

Fig. 3 Frequency correlation of a HAP-SISO Rayleigh channel for various elevation angles of the platform

∣∣Rpl,qm (δT = 0, δR = 0,�t,� f )∣∣ of a HAP-SISO channel. Both Rayleigh and Ricean fad-

ing conditions are considered in all 3-D figures.Figure 7 shows the SDPSs of a HAP-SISO Rayleigh channel and a 2 × 2 HAP-MIMO

Rayleigh channel for a 3-D non-isotropic scattering environment. The values of the modelparameters used are K pl = Kqm = 0, HT = 20 km, βT = 60◦, δT = 80 λ, δR = λ, θT =θR = 45◦, ψR = 30◦, a = 0.01, RS,min = 15 m, RS,max = 150 m, HS,min = 5 m, HS,max =40 m, HS,mean = 17.6 m, σ = 0.31, k = 3, μ = 0◦, fT,max = 0 Hz, fR,max = 150 Hzand γR = 30◦. Then, we obtain that βS,min = arctan

(HS,min/RS,max

) ≈ 2◦ and βS,max =arctan

(HS,max/RS,min

) ≈ 70◦. A normalized Doppler frequency νn = ν/νmax is used,such that −1 ≤ νn ≤ 1. The results show that the SDPSs are similar to the typical U-shaped spectrum of fixed-to-mobile cellular channels. However, the SDPSs do not have anyundesirable discontinuities1 and exhibit finite2 values for the maximum Doppler frequencies.

Finally, Fig. 8 depicts the relative PSDSs of a HAP-SISO Rayleigh channel and a 2 × 2HAP-MIMO Rayleigh channel for a 3-D non-isotropic scattering environment. The values ofthe model parameters used are K pl = Kqm = 0, HT = 20 km, βT = 60◦, δT = 80 λ, δR =λ, θT = θR = 45◦, ψR = 10◦, a = 0.01, RS,min = 12 m, RS,max = 140 m, HS,min =6 m, HS,max = 50 m, HS,mean = 17.6 m, σ = 0.31, k = 8, and μ = 90◦. A normalizedrelative propagation delay τr,n = τr/τr,max is used, such that 0 ≤ τr,n ≤ 1.

5 A 3-D Statistical Model for Wideband HAP-MIMO Channels

Theoretical models can be ideally verified through experimental real-time field trials. How-ever, simulation of the radio propagation environment is commonly used as an alternative,

1 The Doppler spectrum obtained from the Aulin’s 3-D model is constant for fmax cosβmax ≤ |ν| ≤ fmax,where fmax and βmax are the maximum Doppler frequency and the maximum elevation angle of the scatteredwaves, respectively [28,30].2 The Doppler spectrum obtained from the Clarke’s 2-D model becomes infinite at the maximum Dopplerfrequency [33].

123


01

23

45

200

150

100

50

00

0.2

0.4

0.6

0.8

1

Δf (MHz)δΤ /λ

Spac

e-Fr

eque

ncy

Cor

rela

tion

01

23

45

200

150

100

50

0

0.7

0.8

0.9

1

Δf (MHz)δT/λ

Spac

e-Fr

eque

ncy

Cor

rela

tion

(a)

(b)

Fig. 4 Transmit space-frequency correlation of a 2 × 2 HAP-MIMO a Rayleigh and b Ricean channel

cost-effective, and time-saving approach to the test, optimization, and performance eval-uation of wireless communications systems. Many different methods have been adoptedfor the simulation of fading channels. The SoS principle introduced by Rice [34] has beenwidely accepted by academia and industry as an adequate basis for the design of simulationmodels due to its reasonably low computational costs. Two main categories of SoS-basedsimulation models are reported in the literature, the deterministic simulation models [35]and the statistical simulation models [36,37]. This section proposes a statistical simula-tion model for wideband HAP-MIMO channels, under the framework of the referencemodel.

The reference model for wideband HAP-MIMO channels assumes an infinite number ofscatterers. Although this model is practically non-realizable, it can be used to derive sim-ulation models, which have similar statistical properties as the reference model and utilizea finite (preferably small) number of scatterers. Assuming a 3-D non-isotropic scatteringenvironment, the following time-variant transfer function of the sub-channel p − l is pro-posed [13].

123


01

23

45

3

2

1

00

0.2

0.4

0.6

0.8

1

Δf (MHz)δ

R/λ

Spac

e-Fr

eque

ncy

Cor

rela

tion

01

23

45

3

2

1

00.2

0.4

0.6

0.8

1

Δf (MHz)δ

R/λ

Spac

e-Fr

eque

ncy

Cor

rela

tion

(a)

(b)

Fig. 5 Receive space-frequency correlation of a 2 × 2 HAP-MIMO a Rayleigh and b Ricean channel

T ′pl (t, f ) = Tpl,LoS (t, f )+ T ′

pl,N LoS (t, f ) , (43)

where

T ′pl,N LoS (t, f ) =

√�pl

K pl + 1

1√N1 N2 N3

N1∑

n1=1

N2∑

n2=1

N3∑

n3=3

e jϕ(n1 ,n2 ,n3)

×e− j2πλ

[d(

p,S(n1 ,n2 ,n3))+d

(S(n1 ,n2 ,n3),l

)]

ej2π t

(F ′

T,LoS+F ′R,N LoS

)

e− j2π f τ(n1 ,n2 ,n3)

N LoS (44)

is the NLoS component of the time-variant transfer function, n1 ∈ {1, . . . N1} , n2 ∈{1, . . . N2} , n3 ∈ {1, . . . N3} , N1 N2 N3 = N is the total finite number of scatterers,Tpl,LoS (t, f ) is defined in (13), ϕ(n1,n2,n3) ∈ [−π, π) is the random phase, and

123


01

23

45

0.1

0.05

00

0.2

0.4

0.6

0.8

1

Δf (MHz)Δt (sec)

Tim

e-Fr

eque

ncy

Cor

rela

tion

01

23

45

0.1

0.05

00.4

0.6

0.8

1

Δf (MHz)Δt (sec)

Tim

e-Fr

eque

ncy

Cor

rela

tion

(a)

(b)

Fig. 6 Time-frequency correlation of a HAP-SISO a Rayleigh and b Ricean channel

d(

p, S(n1,n2,n3))

≈(

D − DT1 − DT2 Rn2S sin a(n1)

R

)/ cosβT , (45)

d(

S(n1,n2,n3), l)

≈ R(n2)S / cos

[arctan

(H (n3)

S /R(n2)S

)]

−DR1 cosα(n1)R cos

[arctan

(H (n3)

S /R(n2)S

)]

−DR2 sin[arctan

(H (n3)

S /R(n2)S

)]

−DR3 sin α(n1)R cos

[arctan

(H (n3)

S /R(n2)S

)], (46)

F ′T,N LoS = fT,max

(R(n2)

S sin γT sin a(n1)R /D + cos γT

), (47)

F ′R,N LoS = fR,max cos

(a(n1)

R − γR

)cos

[arctan

(H (n3)

S /R(n2)S

)], (48)

τ(n1,n2,n3)N LoS ≈ τLoS + R(n2)

S cos a(n1)R

c0 cosβT+ R(n2)

S

c0 cos[arctan

(H (n3)

S /R(n2)S

)] . (49)

123


-1 -0.5 0 0.5 1-80

-60

-40

-20

0

20

Normalized Doppler Frequency

Spac

e-D

oppl

er P

ower

Spe

ctru

m (

dB)

HAP-MIMO

HAP-SISO

Fig. 7 The SDPSs of a 2 × 2 HAP-MIMO Rayleigh channel and a HAP-SISO Rayleigh channel

0 0.2 0.4 0.6 0.8 1-25

-20

-15

-10

-5

0

5

10

Normalized Propagation Delay

Rel

ativ

e Po

wer

Spa

ce-D

elay

Spe

ctru

m (

dB) HAP-MIMO

HAP-SISO

Fig. 8 The relative PSDSs of a 2 × 2 HAP-MIMO Rayleigh channel and a HAP-SISO Rayleigh channel

Note that the input delay-spread function of the simulation models is obtained as the IFT of

the time-variant transfer function in (43), i.e., h′pl (t, π) = −1

f

{T ′

pl (t, f )}

.

Providing that a sufficient number of scatterers is used, i.e., N1 N2 N3 ≥ 20, T ′pl,N LoS (t, f )

and T ′qm,N LoS (t, f ) are close to a low-pass zero-mean complex Gaussian process and the

STFCF between T ′pl (t, f ) , and T ′

qm (t, f ) is given by

R′pl,qm (δT , δR,�t,� f ) = RLoS

pl,qm (δT , δR,�t,� f )+ R′N LoSpl,qm (δT , δR,�t,� f ) , (50)

123


Where

R′N LoSpl,qm (δT , δR,�t,� f ) ≈ 1

N1 N2 N3

√(K pl + 1

) (Kqm + 1

)

N1∑

n1=1

N2∑

n2=1

N3∑

n3=1

ej

2π(q−p)δT cos θTλ cosβT

×ej2π(q−p)δT sin θT R

(n2)S sin a

(n1)R

λD cosβT ej2π(m−l)δR sinψR sin

[arctan

(H(n3)S /R

(n2)S

)]

λ

×e

j2π(m−l)δR cos θR cosψR cos a(n1)R cos

[arctan

(H(n3)S /R

(n2)S

)]

λ

×e

j2π(m−l)δR sin θR cosψR sin a(n1)R cos

[arctan

(H(n3)S /R

(n2)S

)]

λ

×ej2πτ�t

(F ′

T,N LoS+F ′R,N LoS

)

e

− j2π� f

⎡

⎣ Dc0 cosβT

+ R(n2)S cos a

(n1)R

c0 cosβT+ R

(n2)S

c0 cos

[arctan

(H(n3)S /R

(n2)S

)]

⎤

⎦

(51)

is the STFCF of the NLoS component and RLoSpl,qm (δT , δR,�t,� f ) is defined in (17). To

successfully approximate the statistical properties of the reference model, a(n1)R , R(n2)

S , and

H (n3)S must be properly determined.This section proposes a statistical simulation model. The statistical properties of this model

are stochastic for each simulation trial and converge to the theoretical statistical propertiesafter averaging over a sufficient number of simulation trials for an arbitrary finite number ofscatterers, i.e., for any {N1, N2, N3} ≥ 1. Based on [13,36] and considering that δ, ζ , andξ are independent random variables uniformly distributed in the interval [0,1), we generatethe random variables as

a(n1)R = F−1

aR

(n1 + δ − 1

N1

), (52)

R(n2)S = F−1

T,RS

(n2 + ζ − 1

N2

), (53)

H (n3)S = F−1

T,HS

(n3 + ξ − 1

N3

). (54)

The function F−1aR(·) denotes the inverse function of the von Mises cdf and can be numerically

evaluated as shown in [38]. Moreover,

FT,RS (RS) = FRS (RS)

FRS

(RS,max

)− FRS

(RS,min

) (55)

is a truncated hyperbolic cdf applicable for the range RS,min ≤ RS ≤ RS,max and FRS (RS)

is defined in (23). Finally,

FT,HS (HS) = FHS (HS)

FHS

(HS,max

)− FHS

(HS,min

) (56)

is a truncated log-normal cdf applicable for the range HS,min ≤ HS ≤ HS,max and FHS (HS)

is defined in (26).The STFCF R′′pl,qm (δT , δR,�t,� f ) of the statistical model is obtained

after averaging R′pl,qm (δT , δR,�t,� f ) over Ntrials simulation trials as follows

R′′pl,qm (δT , δR,�t,� f ) = 1

Ntrials

Ntrials∑

ntrials=1

R′pl,qm (δT , δR,�t,� f ). (57)

123


The statistical model requires an average procedure and the generation of three randomvariables. Hence, the number of the scatterers and the number of simulation trials should becarefully chosen to keep the complexity to a minimum.

5.1 Simulation Results

This section evaluates the performance of the statistical simulation model for widebandHAP-MIMO channels. The values of the model parameters used are nT = nR = 2, K pl =Kqm = 0 dB,� f = 180 Hz, HT = 20 km, βT = 60◦, δT = 50 λ, δR = λ/2, θT =45◦, θR = 45◦, ψR = 30◦, k = 2, μ = 0◦, a = 0.01, RS,min = 10 m, RS,max =150 m, HS,min = 6 m, HS,max = 60 m, HS,mean = 17.6 m, σ = 0.31, fT,max =50 Hz, fR,max = 100 Hz, γR = 0◦, and γR = 45◦. Moreover, a normalized samplingperiod fR,maxTS = 0.01 is used, where TS is the sampling period.

Figure 9 compares the absolute STFCFs of the reference model and the statistical simula-tion model. One observes that the statistical model almost perfectly fits the theoretical STFCFfor N1 = 10, N2 = 7, N3 = 3, and Ntrials = 30 for the time delay range 0 ≤ �t (sec) ≤ 0.1,which is typically of interest for many communications systems. We also obtain that the rootmean square error (RMSE) between the absolute STFCFs of the reference model and thestatistical model is approximately 0.005 for Ntrials = 30 and 0.003 for Ntrials = 50. Thus,increasing the number of simulation trials, improves the performance of the statistical model.Nevertheless, adequate statistics can be achieved with only 10 simulation trials. Then, theRMSE is equal to approximately 0.008.

Finally, Table 2 investigates the accuracy of the statistical model for 0 ≤ �t (sec) ≤0.1, Ntrials = 30, and different values of the model parameters that control the location ofthe scatterers within the cylinder. The accuracy evaluation is realized in terms of the RMSEbetween the absolute STFCFs of the simulation model and the reference model. One observesthat the resulting error is negligible in the scenarios simulated.

6 Conclusion

In this paper, a theoretical model for wideband Ricean HAP-MIMO channels has been pro-posed, where both the HAP and the TMS are in motion and equipped with ULA antennas.In particular, a recently proposed narrowband 3-D HAP-MIMO channel model has beenextended to the wideband channel scenario with respect to frequency selectivity. Severalparameters related to the physical properties of wideband stratospheric communicationshave been considered, such as the propagation time delays, the Doppler and delay spread, thearray configuration, the elevation angle of the platform and the distribution of the scatterers,in order to properly and thoroughly characterize the wideband HAP-MIMO channels. Fromthis model the channel statistics have been analytically studied. Specifically, the STFCF, theSDPS, and the PSDS have been derived, under a 3-D non-isotropic scattering environment.Numerical results have been provided to demonstrate the theoretical derivations. Based onthe SoS principle, a statistical simulation model for wideband HAP-MIMO channels hasbeen developed. The statistical properties of this simulation model have been verified bycomparison with the corresponding statistical properties of the reference model. The resultshave shown that the proposed simulation model satisfactorily approximate the STFCF ofthe reference model and has nearly the same statistics as the reference model, while keepsthe complexity to a minimum. To the best of the authors’ knowledge, there are no experi-mental data available in the literature to fully verify the theoretical results. Thus, the need

123


0 0.02 0.04 0.06 0.08 0.10.2

0.3

0.4

0.5

0.6

0.7

0.8

Δt(sec)

Spac

e-T

ime-

Freq

uenc

y C

orre

latio

n

Reference Model

Statistical Model, N1=10, N

2=7, N

3=3, N

trials=30

Fig. 9 The absolute STFCFs of the reference model and the statistical simulation model

Table 2 Accuracy evaluation ofthe statistical simulation model

Values of the model parameters RMSE

k 0 5.8 × 10−3

2 5.0 × 10−3

8 4.2 × 10−3

μ 0◦ 5.0 × 10−3

45◦ 4.7 × 10−3

90◦ 4.3 × 10−3

α 0.005 5.9 × 10−3

0.01 5.0 × 10−3

0.03 3.8 × 10−3

HS,mean 7.5 m 4.9 × 10−3

17.6 m 5.0 × 10−3

22.5 m 5.2 × 10−3

for experimental verification data is pointed out to confirm the validity of this referencewideband HAP-MIMO channel model. The theoretical and empirical results can easily becompared, since the proposed model is flexible and applicable to a wide range of propa-gation environments, i.e., one may choose proper values for the model parameters to fit aparticular environment. Overall, the proposed models provide helpful insights and guidelinesfor analyzing, designing, optimizing, and testing emerging wideband HAP-MIMO mobilecommunications systems, under LoS and NLoS conditions.

Acknowledgements This work has been co-financed by the European Union (European Social Fund—ESF)and Greek national funds through the Operational Program “Education and Lifelong Learning” of the NationalStrategic Reference Framework (NSRF)—Research Funding Program THALES MIMOSA (MIS: 380041).Investing in knowledge society through the European Social Fund.

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Appendix 1: The SPDS of the NLoS Component

The SDPS of the NLoS component is obtained by calculating the FT of the STCF of theNLoS component in (27)–(30)

SN LoSpl,qm (δT , δR, ν) = �t

{RN LoS

pl,qm (δT , δR,�t,� f = 0)}. (58)

Considering that � f = 0, the Bessel function I0

(√w2

2 + w23

)of (27) can be written as

I0

(√w2

2 + w23

)=

� f =0J0

[s1

√(�t + s2)

2 + s23

], (59)

where

s1 = 2π fR,max cos [arctan (HS/RS)] ,

s2 = q1 + q2 − q3

fR,max cos [arctan (HS/RS)],

s3 = q4 + q5 − q6

fR,max cos [arctan (HS/RS)],

q1 = (q − p) δT sin θT RS sin γR

λD cosβT,

q2 = (m − l) δR cosψR cos (θR − γR) cos [arctan (HS/RS)]

λ,

q3 = jk cos (μ− γR)

2π,

q4 = (q − p) δT sin θT RS cos γR

λD cosβT,

q5 = (m − l) δR cosψR sin (θR + γR) cos [arctan (HS/RS)]

λ,

q6 = jk sin (μ+ γR)

2π. (60)

Using (59) and (60), (58) becomes

SN LoSpl,qm (δT , δR, v)

=HS,max∫

HS,min

RS,max∫

RS,min

A

∞∫

−∞e− j2π�t(v− fT,max cos γT ) J0

[s1

√(�t + s2)

2 + s23

]d�td RSd HS,

(61)

where

A = ej2π

((q−p)δT cos θT


λ

)

e

[− 1

2σ2 ln2(

HSHS,mean

)]


2π I0 (k)√(

K pl + 1) (

Kqm + 1)

× a[FHS

(HS,max

)− FHS

(HS,min

)] [tanh

(a RS,max

)− tanh(a RS,min

)] . (62)

It is well known that e± j x = cos x ± j sin x, the integral of the product of an odd function,i.e., sin x , and an even function, i.e., cos x and J0(x), from −∞ to ∞ is equal to zero, the

123


product of two even functions is an even function, and the integral of an even function from−∞ to ∞ is twice the integral from 0 to ∞. Under these considerations and using the

equality∫∞

0 J0

(a√

x2 + z2)

cos (bx) dx = cos(

z√

a2 − b2)/√

a2 − b2 [27, eq. 6.677-3],

(61) becomes

SN LoSpl,qm (δT , δR, ν)

=HS,max∫

HS,min

RS,max∫

RS,min

Ae j2πs2(ν− fT,max cos γT )

π fR,max cos [arctan (HS/RS)]

√

1 −(

ν− fT,max cos γTfR,max cos[arctan(HS/RS)]

)2

× cos

⎡

⎣2πs3 fR,max cos [arctan (HS/RS)]

√

1 −(

ν − fT,max cos γT

fR,max cos [arctan (HS/RS)]

)2⎤

⎦ d RSd HS .

(63)

The double integral in (61) has to be evaluated numerically, since there is no closed-formsolution. One observes that (61) is applicable for the range

0 ≤ ∣∣ν − fT,max cos γT∣∣ ≤ fR,max cos [arctan (HS/RS)]

⇒ 0 ≤ ∣∣ν − fT,max cos γT∣∣ ≤ νmin, (64)

where

νmin = minHS ,RS

{fR,max cos [arctan (HS/RS)]

}

= fR,max cos[arctan

(HS,max/RS,min

)]. (65)

However, the SDPS should be sketched for the range

0 ≤ ∣∣ν − fT,max cos γT∣∣ ≤ νmax, (66)

where

νmax = maxHS ,RS

{fR,max cos [arctan (HS/RS)]

}

= fR,max cos[arctan

(HS,min/RS,max

)]. (67)

Hence, the SDPS of the NLoS component can be written as in (32)–(36).

Appendix 2: The Relative PSDS of the NLoS Component

The relative PSDS of the NLoS component can be obtained by calculating the IFT of theSFCF of the NLoS component in (27)–(30)

P N LoSpl,qm (δT , δR, τr ) = −1

� f

{RN LoS

pl,qm (δT , δR,�t = 0,� f )}. (68)

Considering that �t = 0, the Bessel function I0

(√w2

2 + w23

)of (27) can be written as

I0

(√w2

2 + w23

)=

�t=0J0

[2πp1

√(� f − p2)

2 + p23

], (69)

123


where

p1 = RS

c0 cosβT,

p2 = (m − l) δR cos θR cosψRc0 cosβT cos [arctan (HS/RS)]

λRS− jkc0 cosμ cosβT

2πRS,

p3 = (q − p) δT sin θT c0

λD+ (m − l) δR sin θR cosψRc0 cosβT cos [arctan (HS/RS)]

λRS

− jkc0 sinμ cosβT

2πRS. (70)

Using (69) and (70), (68) becomes

P N LoSpl,qm (δT , δR, τr )

=HS,max∫

HS,min

RS,max∫

RS,min

A

∞∫

−∞e j2π� f (τr −p4) J0

(2πp1

√(� f − p2)

2 + p23

)d� f d RSd HS, (71)

where

p4 = RS

c0 cos [arctan (HS/RS)]. (72)

Using the equality [27, eq. 6.677-3] and after extensive calculations, the relative PSDS ofthe NLoS component is derived as follows

P N LoSpl,qm (δT , δR, τr )

=HS,max∫

HS,min

RS,max∫

RS,min

Ae j2πp2(τr −p4) cos

(2πp3

√p2

1 − (τr − p4)2)

π

√p2

1 − (τr − p4)2

d RSd HS . (73)

The double integral in (73) has to be evaluated numerically, since there is no closed-formsolution. One observes that (73) is applicable for the range

|τr − p4| ≤ p1

⇒ p4 − p1 ≤ τr ≤ p4 + p1

⇒ τ1 ≤ τr ≤ τ2, (74)

where

τ1 = maxHS ,RS

{p4 − p1}

=√

R2S,min + H2

S,max

c0− RS,min

c0 cosβT, (75)

τ2 = minHS ,RS

{p4 + p1}

=√

R2S,min + H2

S,min

c0+ RS,min

c0 cosβT. (76)

123


However, the relative PSDS should be sketched for the range

0 ≤ τr ≤ τr,max, (77)

where τr,max ≈ τmax − τLoS and τLoS and τmax are defined in (8) and (11), respectively.Hence, the relative PSDS of the NLoS component can be written as in (38)–(42).

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Author Biographies

Emmanouel T. Michailidis was born in Athens, Greece, in 1980. Hereceived the Ph.D. degree in broadband wireless communications fromthe University of Piraeus, Piraeus, Greece, in 2011. Since 2012, he hasbeen a Postdoctoral Researcher in satellite communications with theDepartment of Digital Systems, University of Piraeus. Since 2007, hehas been a Laboratory Instructor with the Department of ElectronicsEngineering, Technological Educational Institute (TEI) of Piraeus. Hiscurrent research interests include channel characterization, modeling,and simulation for future wireless and satellite communication systemsand new transmission schemes for cooperative and multiple-input-multiple-output (MIMO) systems. Dr. Michailidis received the BestPaper Award at The Second International Conference on Advances inSatellite and Space Communications (SPACOMM) in 2010. He serveson the editorial board of the International Journal on Advances inTelecommunications and participates in the European Cooperation inScience and Technology Action on Cooperative Radio Communica-tions for Green Smart Environments (COST Action IC1004).

Athanasios G. Kanatas received the Diploma in Electrical Engineer-ing from the National Technical University of Athens (NTUA), Greece,in 1991, the M.Sc. degree in Satellite Communication Engineeringfrom the University of Surrey, Surrey, UK in 1992, and the Ph.D.degree in Mobile Satellite Communications from NTUA, in February1997. From 1993 to 1994 he was with National Documentation Cen-ter of National Research Institute. In 1995 he joined SPACETEC Ltd.In 1996 he joined the Mobile Radio-Communications Laboratory asa research associate. From 1999 to 2002 he was with the Institute ofCommunication & Computer Systems. In 2000 he became a memberof the Board of Directors of OTESAT S.A. In 2002 he joined the Uni-versity of Piraeus where he is a Professor in the Department of DigitalSystems. From 2008 to 2009 he has served as Member of the Senateof University of Piraeus. From 2007 to 2009 he served as Greek Dele-gate to the Mirror Group of the Integral Satcom Initiative. His currentresearch interests include the development of new digital techniques for

wireless and satellite communications systems, channel characterization, simulation, and modeling for futuremobile, mobile satellite and wireless communication systems, antenna selection and RF preprocessing tech-niques, new transmission schemes for MIMO systems, and energy efficient techniques for Wireless SensorNetworks. He has published more than 120 papers in international Journals and conference proceedings. Hehas been a Senior Member of IEEE since 2002. From 1999 to 2009 he chaired the IEEE ComSoc Chapterof the Greek Section.

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