Research Article A Novel 3D Nonstationary Channel Model...

Research ArticleA Novel 3D Nonstationary Channel Model Based onthe von Mises-Fisher Scattering Distribution

Yuming Bi Jianhua Zhang Ming Zeng Mengmeng Liu and Xiaodong Xu

Key Laboratory of Universal Wireless Communications Beijing University of Posts and TelecommunicationsMinistry of Education Mailbox No 92 Beijing 100876 China

Correspondence should be addressed to Yuming Bi biyuming10507sinacom

Received 26 December 2015 Revised 29 February 2016 Accepted 24 March 2016

Academic Editor David W Matolak

Copyright copy 2016 Yuming Bi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In the last decade the nonstationary properties of channel models have attracted more and more attention for many scenariosthat is vehicle-to-vehicle (V2V) mobile-to-mobile (M2M) and high-speed train (HST) However little research has been doneon the real-physical channel model In this paper we propose a generalized three-dimensional (3D) nonstationary channel modelin which the scatterers are assumed to be distributed around the transmitter (Tx) and receiver (Rx) on a two-sphere model Byemploying the von Mises-Fisher distribution the mean values of the azimuth angle of departure (AAoD) and elevation angle ofdeparture (EAoD) and the azimuth angle of arrival (AAoA) and elevation angle of arrival (EAoA) are tracked by time-variant(TV) Brownian Markov (BM) motion paths which ensure the nonstationarity of the proposed channel model Moreover the TVautocorrelation function (ACF) and Doppler power spectrum density (DPSD) of the proposed nonstationary channel model arecalculated by using signal processing tools for example fast Fourier transform (FFT) and short-time Fourier transform (STFT) Inaddition the simulation results show that the TV scatterer distribution results in a nonstationary nonisotropic channel model andthe proposed model can be employed to simulate the 3D nonstationary channel model

1 Introduction

The transmission channel is one of the most crucial partsin mobile communication systems while the generationof an accurate and effective channel model for testing ofcommunication systems represents a particular challengeThe majority of existing channel models in the literaturerely on the assumption of wide-sense stationary (WSS) anduncorrelated scattering (US) [1 2] Under such an assump-tion different transmission delays stay uncorrelated and thesecond-order channel moments are stationary However thephysical channel in the real world is nonstationary [2ndash4]Furthermore plenty of empirical and analytical reports revealthat theWSSUS assumption is valid only for a short travellingdistance Therefore in order to develop a more accurate andsuitable channel model for the real-physical channel it isessential to study nonstationarity channel modeling whichis a newly explored research field

The channel modeling can be classified into three catego-ries that is deterministic geometry and correlation statistics

[4] In terms of deterministic channel modeling Ai and Hehave applied theirmeasurement data toV2VandHST scenar-ios employing the nonstationary channel modeling In [2 3]the stationary region especially has been extracted accordingto the correlatedmatrix distanceThis is beneficial to elicit theextent of the channel variation if one could reconstruct thenonstationary channel simulators to approximate the real-physical channel For geometrymethods [5] proposed a two-dimensional (2D) evolutionary spectrum approach to modelnonstationary broadband mobile fading channels In [6 7]the nonstationary properties ofmassiveMIMOchannels havebeen characterized by cluster evolution (birth-death process)on both the array and time axes Besides in [8 9] Patzoldsuggested a nonstationary one-ring scattering channel modelby utilizing Brownian randommovement paths of themobilestation and deriving the angle of arrival (AoAs) and angleof departure (AoDs) In addition [10] provided a genericmethod to find the MIMO correlation channel matricesunder nonstationary interferences However on the onehand all methods mentioned above only took into account

Hindawi Publishing CorporationMobile Information SystemsVolume 2016 Article ID 2161460 9 pageshttpdxdoiorg10115520162161460

2 Mobile Information Systems

D

y y998400

x

z z998400

n1 n2

Sn1

VT

OT

RT

Tx Rx

120587 minus 120574T

120573(n1)T

120573(n2)T

120572(n1)T

120572(n2)T

120585n1n2

120585n1 120585n2

Qn1Qn2

Sn2

120573(n1)R

120573(n2)R

120572(n1)R

120572(n2)R

OR 120574R

VR

RR

Figure 1 The proposed 3D two-sphere channel model

the nonstationarity of the vertical plane while they ignoredthe impact of elevation plane on the other hand theirmodeling process required a high computational complexityand lacked flexibility Therefore it is essential to developa simple and feasible 3D channel model for nonstationaryresearch and to extend the 3D nonstationary channel modelto capture the real-physical channel characteristics

In this paper we propose a more general 3D geometricsituation where the scatterers are assumed to be distributedaround the user in a sphere area By employing the vonMises-Fisher scattering distributions the centers of the AoDs andAoAs scattering distribution are tracked by TV Brownianmotion paths [11 12] which could effectively reflect themovement process of the MS Furthermore the ACF ofthe complex channel gain and local PSD of the Dopplerfrequencies are calculated by nonstationary signal processingtools for example STFT and FFT [13 14] In addition theresults show that the time-variation of the scatterer distribu-tion results in a nonstationary nonisotropic channel modeland that the proposed model has the outstanding abilityto describe vital statistical characteristics of nonstationarychannel such as the envelope distribution and the TV ACFand PSD [15] Moreover it also offers the advantages of lowcomputational complexity and easy realization [8]

The remainder of this paper is organized as followsSection 2 discusses a novel 3D channel model based on thegeometrical two-sphere model In Section 3 the von Mises-Fisher (VMF) distribution is employed to describe the 3Dscatter distribution A time-variant Brownianmotion processis provided in Section 4 And then Section 5 proposes a novel3D nonstationary channel model The related simulation andnumerical analysis are presented in Section 6 At last theconclusion is drawn in Section 7

2 A Novel 3D Channel Model

Let us consider a narrowband V2V communication chan-nel with omnidirectional antennas The radio propagation

environment around the transmitter and the receiver is char-acterized by 3D nonisotropic scattering under line-of-sight(LoS) and none line-of-sight (NLoS) conditions Figure 1illustrates the novel 3D geometrical channel model in whichthe local scatterers are modeled on the surface of two spheresof radii 119877

119879and 119877

119877[16] In order to limit the computational

complexity of this model both local scatterers around theTx and Rx are considered Therefore we assume that thereare 119873

1scatterers on the sphere (around the Tx) of radius

119877119879and the 119899

1th (119899

1= 1 2 119873

1) scatterer is denoted by

1198781198991

Similarly around the Rx 1198732scatterers lie on a surface

of sphere of radius 119877119877 and the 119899

2th scatterer is denoted by

1198781198992

The geometric symbols in Figure 1 have the followingmeaning 120572(119899119894)

119877and 120573

(119899119894)

119877denote the main AAoA and main

EAoA of the 119899119894th scatterer respectively In the same way 120572(119899119894)

119879

and 120573(119899119894)119879

denote the main AAoD and main EAoD of the 119899119894th

scatterer respectively and 119863 is the distance between the Txand Rx It is worth noting that the distance 119863 is assumed tobe much larger than the radius 119877

119877or 119877

119879(ie 119877

119879or 119877

119877≪ 119863)

Therefore the channel impulse response (CIR) of the two-sphere model at the carrier frequency 119891

119888can be expressed as

[17]

ℎ (119905) = ℎLOS

(119905) + ℎNLOS

(119905) (1)

where

ℎLOS

(119905) = radic119870

119870 + 1119890minus1198952120587119891c1205911

times 1198901198952120587119891119879max119905 cos(120572LOS

119879minus120574119879)sdotcos120573LOS

119879

times 1198901198952120587119891119877max119905 cos(120572LOS


119877

(2)

Mobile Information Systems 3

The CIR especially of two-sphere model under NLoS con-ditions is constructed as a sum of the single- and double-bounced rays with different energies So it can be rewrittenas

ℎNLOS

(119905) =

2

sum

119894=1

ℎSB119894(119905) + ℎ

DB(119905)

ℎSB119894(119905)

= radic120578SB119894

119870 + 1lim119873119894rarrinfin

119873119894

sum

119899119894=1

1

radic119873119894

119890119895(120593119899119894minus21205871198911198881205912)

times 1198901198952120587119891119879max119905 cos(120572

(119899119894)

119879minus120574119879)sdotcos120573(119899119894)

119879

times 1198901198952120587119891119877max119905 cos(120572

(119899119894)

119877minus120574119877)sdotcos120573(119899119894)

119877

ℎDB

(119905)

= radic120578DB119870 + 1

lim11987311198732rarrinfin

11987311198732

sum

11989911198992=1

1

radic11987311198732

119890119895(12059311989911198992

minus21205871198911198881205913)

times 1198901198952120587119891119879max119905 cos(120572

(1198991)

119879minus120574119879)sdotcos120573(1198991)

119879

times 1198901198952120587119891119877max119905 cos(120572

(1198992)

119877minus120574119877)sdotcos120573(1198992)

119877

(3)

with120572LOS119879

asymp 120573LOS119879

asymp 120573LOS119877

asymp 0

120572LOS119877

asymp 120587

1205911asymp119863

119888

1205912asymp

(120585119899119894

+ 119877119879119877

)

119888

1205913asymp

(12058511989911198992

+ 119877119879+ 119877

119877)

119888

(4)

where SB1and SB

2stand for the subcomponents of single-

bounced rays from the Tx sphere and Rx sphere respectivelyDB means the double-bounced rays 120578SB

119894

and 120578DB are thefactors of normalized power which specify the amountpowers of single- and double-bounced rays contribute to thetotal normalized power Herein 119870 is the Rician factor and 119888

denotes the speed of light and 120593119899119894

and 12059311989911198992

are independentand identically distributed (iid) random variables withuniform distribution over [minus120587 120587) [17] In addition119891

119879max and119891119877max represent the maximum Doppler frequencies of the

transmitter and the receiver respectively 120574119879and 120574

119877stand

for moving direction of the Tx and Rx respectively Thestatistical properties and performances of the proposed 3DV2V channel model will be analyzed as follows

21 Envelope and Phase Multiple uncorrelated fading pro-cesses bring the complex envelope which can be written as

ℎ (119905) = ℎ119894 (119905) + 119895ℎ

119902 (119905) (5)

Considering the distribution of the channel envelope 120585(119905) =|ℎ(119905)| and phase 120601(119905) = argℎ(119905) their correspondingprobability density functions (PDF) can be written as [6]

119901120585 (119911) =

119911

1205752

0

119890minus(1199112+1198962

0)21205752

0 sdot 1198680(119911119896

0

1205752

0

) (6)

119901120601(120601) =

119890minus1198962

021205752

0

2120587

sdot 1 +1198960

1205750

sdot radic120587

2cos (120601 minus arg ℎLOS

(119905))

(7)

where 119911 represents the amplitude variable the symbol 1198960is

defined as 1198960= radic119870(119870 + 1) and 119868

0(sdot) is the zeroth-order

modified Bessel function of the first kind while 12057520stands for

the mean power of the channel

22 ACF and PSD The normalized ACF between any twocomplex fading envelopes is defined as

119903ℎℎ (120591) =

119864 [ℎ (119905) sdot ℎlowast(119905 minus 120591)]

radic119864 [|ℎ (119905)|]2sdot 119864 [|ℎ (119905 minus 120591)|]

2

(8)

where (sdot)lowast is the complex conjugate operation and 119864[sdot] rep-resents the statistical expectation operator Then we obtainthe ACF of the proposed channel model Taking the LoS andNLoS components into account the equation of the ACF canbe written as

119903ℎℎ

= 119903LOSℎℎ

(120591) +

2

sum

119894=1

119903NLOSSB119894

(120591) + 119903NLOSDB (120591) (9)

where

119903LOSℎℎ

(120591) =119870

119870 + 1119890119895(2120587120582)sdot2119863

sdot 1198901198952120587120591(119891

119879max cos 120574119879minus119891119877max cos 120574119877)

119903NLOSSB119894

(120591) =

120578SB119894

119870 + 1∬

120587

minus120587

[1198901198952120587120591(119891

119879max119880(119894)+119891119877max119881

(119894))]

sdot 119891 (120572(119899119894)

119879119877 120573

(119899119894)

119879119877) 119889 (120572

(119899119894)

119879119877 120573

(119899119894)

119879119877)

119903NLOSDB (120591) =

120578DB119870 + 1

intintintint

120587

minus120587

[1198901198952120587120591(119891

119879max119880DB+119891119877max119881

DB)]

sdot 119891 (120572(1198991)

119879 120573

(1198991)

119879)

sdot 119891 (120572(1198992)

119877 120573

(1198992)

119877) 119889 (120572

(1198991)

119879 120573

(1198991)

119879) 119889 (120572

(1198992)

119877 120573

(1198992)

119877)

(10)

where 119894 means there are two subcomponents for single-bounced rays For example 119894 = 1 means the single-bouncedrays from the Tx sphere Similarly 119894 = 2 means thesingle-bounced rays from the Rx sphere And 119891(120572

(119899119894)

119879119877 120573

(119899119894)

119879119877)

stands for the joint azimuth and elevation angle distributionfunction of the 119899

119894th scatterers on Tx or Rx According to


the mathematical law of triangle functions the geometricrelationships between each parameter are defined as

119880(1)

= cos (120572(1198991)119879

minus 120574119879) sdot cos (120573(1198991)

119879)

119881(1)

= cos (120572(1198991)119877

minus 120574119877) sdot cos (120573(1198991)

119877)

120572(1198991)

119877asymp 120587 minus

119877119879sdot cos120573(1198991)

119879

119863sdot sin (120572(1198991)

119879)

120573(1198991)

119877= arccos(

119863 minus 119877119879cos120573(1198991)

119879sdot cos120572(1198991)

119879

1205851198991

)

1205851198991

= radic1198762

1198991

+ 1198772

119879sin2120573(1198991)

119879

1198761198991

= 119863 + 119877 sdot cos (120573(1198991)119877

) cos (120572(1198991)119877

)

119880(2)

= cos (120572(1198992)119879

) sdot cos (120573(1198992)119879

)

119881(2)

= cos (120572(1198992)119877

) sdot cos (120573(1198992)119877

)

(11)

120572(1198992)

119879asymp119877119877sdot cos120573(1198992)

119877

119863sdot sin (120572(1198992)

119877) (12)

120573(1198992)

119879= arccos(

119863 + 119877119877cos120573(1198992)

119877sdot cos120572(1198992)

119877

1205851198992

) (13)

1205851198992

= radic1198762

1198992

+ 1198772

119877sin2120573(1198992)

119877 (14)

1198761198992

asymp 119863 + 119877119877cos120573(1198992)

119877cos120572(1198992)

119877 (15)

119880DB

= cos (120572(1198991)119879

) sdot cos (120573(1198991)119879

) (16)

119881DB

= cos (120572(1198992)119877

) sdot cos (120573(1198992)119877

) (17)

The DPSD has been used for nonisotropic scatteringchannels and the theoretical expression DPSD is

119878ℎℎ(119891) = int

infin

minusinfin

119903ℎℎ (120591) 119890

minus1198952120587119891120591119889120591 (18)

3 von Mises-Fisher Distribution

TheVMF distribution [18 19] is a close model for directionaldata distributed uniformly with rotational symmetry on theunit hypersphere 119878

119901minus1 When 119901 is 3 the ordinary sphereusually corresponds to the set of all points embedded in theEuclidean space 119877

3 The vectorΩ stands for any direction ofscatterers on the unit sphere surface and it can be describedin Cartesian coordinates as

Ω = [cos120573 cos120572 cos120573 sin120572 sin120573]119879 (19)

where [sdot]119879 represents the transpose operation It is obviousthat 120573 and 120572 denote the coelevation and azimuth angles ofvector Ω respectively Then the general form of the VMFdistribution can be written as

119891119901(Ω 120583 119896) =

(1198962)119889

Γ (119889 + 1) sdot 119868119889 (119896)exp (119896120583119879Ω) (20)

where 119896 controls the concentration of the distribution aboutthe mean direction vector In particular when 119896 = 0 thedistribution is isotropic and when 119896 rarr infin the scatter-ing becomes a point source on the surface Furthermoreparameter 119889 = 1199012 minus 1 From a physical point of viewthe direction spread represents the degree of direction fromthe arrival subpaths of a cluster to the mean center of theAoA Therefore the mean center value of the AoA indicatesthe mean direction vector from the MS to the cluster anda subpath shows a multiple within a cluster of scatterersFor the ordinary sphere the VMF PDF to characterize thedistribution of effective scatterers can be defined as

119891 (120572 120573)

=119896

4120587 sinh (119896)

sdot exp 119896 [cos1205730cos120573 cos (120572 minus 120572

0) + sin120573

0sin120573]

(21)

where1205720and120573

0represent themean values of the azimuth and

elevation angle respectively minus120587 le 1205720le 120587 0 le 120573

0le 120587

4 TV Brownian Random Process

In this section we will bring the temporal Brownian randomprocess [17] to simulate a nonstationary TV 3D channelmodel A standard BM process 119861(119905) 119905 isin [0 119879] is aWiener process in which the increments satisfy a normaland independent distribution Besides it needs to satisfy thefollowing conditions

(1) 119861(0) = 0(2) When 0 le 119894 le 119895 le 119879 the increment 119861(119895)minus119861(119894) obeys

the normal distribution with zero mean and variance119895 minus 119894 that is 119861(119895) minus 119861(119894) sim 119873(0 119895 minus 119894)

(3) When 0 le 119898 le 119899 le 119894 le 119895 le 119879 the increments119861(119898)minus119861(119899) and 119861(119895) minus119861(119894) are statistically indepen-dent

Due to the fact that the same statistical properties impactthe receiver between the AoA distribution motion and theMS randommovement this paper proposes that the variationof the AoA distribution can be used for simulating a 3Dnonstationary channel To model the motion process of theAoA distribution in the 3D plane we provide a path modelwith BM movement components along the elevation andazimuth planeThe fluctuations of the motion path especiallyare modeled by two independent temporal BM processes119861(120572)

119905and 119861(120573)

119905 Therefore the motion path can be modeled as

119875 (120572119905 120573

119905) =

120572119905 (119905) = 120572

119904+ 119908119905 sdot Δ119905 + Δ119905

2sdot 119861

(120572)

119905

120573119905 (119905) = 120573

119904+ 119908119905 sdot Δ119905 + Δ119905

2sdot 119861

(120573)

119905

(22)

where 119908 stands for the random drift parameter used tocontrol the behavior of the deterministic drift degrees alongeach axis and Δ119905 represents the degree of randomness of thepath Meanwhile Δ119905 = 1119908 120572

119904and 120573

119904are the start points

of the process along each axis According to the proof of the


BM paths model in [17] it is not difficult to find that themean value of each angle always depends on time 119905 and theACF is also determined by a function of the time difference 120591Consequently we can conclude that the BM path model is anonstationary process More details about Brownian randomprocess can be found in [17]

5 The Nonstationary Properties ofthe Proposed Model

In this section we consider the TV CIRs ACF and DPSDof the nonstationary process At first we introduce the BMprocess to the VMF scattering distribution and the TV-VMFPDF can be rewritten as

119891 (120572(119899119894)

119879119877(119905) 120573

(119899119894)

119879119877(119905)) =

119896

4120587 sinh (119896)sdot exp 119896

sdot [cos120573(119899119894)1198791198770

(119905) sdot cos120573(119899119894)119879119877

cos (120572(119899119894)119879119877

minus 120572(119899119894)

1198791198770(119905))

+ sin120573(119899119894)1198791198770

(119905) sdot sin120573(119899119894)119879119877

]

(23)

Then the TV CIRs of the proposed model in (1)ndash(3) areobtained when the AoA and AoD become TV variables dueto the random motion of the scatterers Furthermore theenvelopePDFof themodified simulationmodel is the same asthat in (6) and the phase PDF is also presented by (7) becauseboth of them are unrelated with TV frequenciesWe considerthe TV ACF and DPSD of the nonstationary process [20] as

119903ℎℎ(119905 120591) = 119903

LOSℎℎ

(119905 120591) +

2

sum

119894=1

119903NLOSSB119894

(119905 120591) + 119903NLOSDB (119905 120591) (24)

where

119903LOSℎℎ

(119905 120591) =119870

119870 + 1119890119895(2120587120582)sdot2119863

sdot 1198901198952120587120591(119891


119903NLOSSB119894

(119905 120591) =

120578SB119894

119870 + 1∬

120587

minus120587

[1198901198952120587120591(119891

119879max119880(119894)+119891119877max119881

(119894))]

sdot 119891 (120572(119899119894)

119879119877(119905)

120573(119899119894)

119879119877(119905)) 119889 (120572

(119899119894)

119879119877(119905) 120573

(119899119894)

119879119877(119905))

119903NLOSDB (119905 120591) =

120578DB119870 + 1

intintintint

120587

minus120587

[1198901198952120587120591(119891

119879max119880DB+119891119877max119881

DB)]

sdot 119891 (120572(1198991)

119879(119905) 120573

(1198991)

119879(119905)) 119891 (120572

(1198992)

119877(119905)

120573(1198992)

119877(119905)) 119889 (120572

(1198991)

119879(119905) 120573

(1198991)

119879(119905)) 119889 (120572

(1198992)

119877(119905) 120573

(1198992)

119877(119905))

(25)

In particular the parameters in (24)-(25) are given inSection 2 Note that the AoAs (120572

119877(119905) 120573

119877(119905)) and AoDs

(120572119879(119905) 120573

119879(119905)) are applied together with the TV variables [18]

It is noteworthy that the definition of the TV ACF keepsthe symmetrical delay characteristic 120591 119903

ℎℎ(119905 120591) = 119903

ℎℎ(119905 minus 120591)

and 119903ℎℎ(119905 120591) is the real function if ℎ(119905) is a real process which

minus3 minus2 minus1 0 1 2 30

02040608

112141618

2D v

on M

ises P

DF

p(120572)

k = 1

k = 5

k = 10

k = 20

Figure 2 2D VM PDF

results in the Fourier transform having the symmetrical andreal characteristics Beyond the TV DPSD the short-timeDPSD is widely used to analyze the nonstationary process[17] which is the amplitudersquos square of the signalrsquos STFT[21 22]

STℎℎ(119891 119905) = int

infin

minusinfin

119903ℎℎ(119905 120591) 119890

minus1198952120587119891120591sdot 119908 (119905 minus 120591) 119889120591 (26)

where (119905 minus 120591) is the analytic window slipping with time Thenonstationary process can be viewed as stationary within theanalysis window

6 Simulation Results and Numerical Analysis

In this section simulations are carried out to illustrate thenonstationary properties of our proposed 3D TV channelmodel based on the VMF scatterer distribution The impactof the model parameters on the VMF scatterer distributionis investigated first Then the nonstationary properties ofthe proposed two-sphere channel model are evaluated andanalyzed in terms of the AoAs and AoDs motion path at theTx and Rx the TV ACF of the complex channel gain and theTV DPSD

Firstly we consider the performance of the VMF scat-terers distribution From Figure 2 it is not difficult to findthat the parameter 119896 controls the shape of the distributionMeanwhile Figure 4 shows the VMF PDF in 3D coordinatesby setting the mean angles 120572

0= 90

∘ and 1205730= 45

∘ It isclear that the two-dimensional (2D) von Mises (VM) PDFis derived from the 3D VMF PDF for azimuth angle 120572 with120573 = 0

∘ Furthermore the scatterer distribution based on theVMF PDF has been shown in Figure 3 Assuming scattererdistribution is dependent on the axis of symmetry specifiedby the 119911-axis it can be seen that the larger the value of 119896 thehigher the density near the 119911-axis Especially when 119896 = 0 thedistribution is isotropic

Secondly Figure 5 illustrates a realization of the temporalBM motion path where 119896 = 1 119908 = 10 and Δ119905 = 001As seen from this figure the trajectories of 120572

(119894)

0(119905) and

120573(119894)

0(119905) are varying with time respectively The properties of


K = 0

y-axis

y-axis

y-axis

y-axis

x-axis x-axis

x-axisx-axis

z-axis z-axis

z-axisz-axis

K = 5

K = 15 K = 30

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

Figure 3 Effect of the concentration parameter

0

05

1

15

3D V

MF

PDF

120572

minus120587minus21205873

minus12058730

0

1205873120587321205873

21205873

120587

120587

120573

Figure 4 3D VMF PDF

0 200 400 600 800 1000 0 200 400 600 800minus12

minus1minus08minus06minus04minus02

00204

t (s)

Bt

120572(n1)T0

(t)

120572(n2)T0

(t)

120573(n1)T0

(t)

120573(n2)T0

(t)

Figure 5 The simulation of Brownian paths

the BM process prove that the proposed path results in anonstationary nonisotropic channel model

Thirdly Figures 6ndash8 show the proposed TV 3Dmodels indetail for each statistical value The simulation settings are asfollows 119891

119888= 59GHz 119863 = 300m 119891

119879max = 119891119877max = 570Hz

119877 = 25m 120574119879

= 120574119877

= 0∘ 120572(1198991)

1198770= 1478

∘ 120573(1198991)1198770

= 172∘

120572(1198992)

1198790= 316

∘ and 120573(1198992)

1198790= 1716

∘ Considering the vehiculartraffic density (VTD) based on [4] we have 119870 = 3876 119896 =

36 120578SB1

= 0625 120578SB2

= 0225 and 120578DB = 015 for low and119870 = 0156 119896 = 06 120578SB

1

= 015 120578SB2

= 0225 and 120578DB = 0625

for high VTD scenarios because both parameters 119870 and 119896

are related to the distribution of scatterers (normally largervalues of 119896 corresponding to less dense moving vehicles)

Figure 6 validates the absolute values of the temporalACF for the proposed 3D channel model in both low andhigh VTD scenarios It can be observed that the temporalACF is affected by the VTD namely the temporal ACF inlow VTD scenarios is always higher than that in high VTDscenario Additionally Figures 7 and 8 depict the envelopeand the phase PDF of 3D channel model determined by theparameters 119870 and 119896 respectively

Finally we assume that the mean elevation and azimuthangles are moving along the TV-BM motion paths shown inFigure 5 and we set the maximum Doppler frequency of thetransmitter and receiver as 570Hz Meanwhile the effect ofthe proposed TV-BM motion path on the scattering modelis the same as the one in which the vehicle moves randomlyin different directions on the AoAs and AoDs Therefore asshown in Figures 9 and 12 the absolute value of the resultinglocal ACF ((24) and (25)) is illustrated According to thesefigures it is manifest that the shapes of the ACF change withdifferent values of time which is due to the nonstationarityof the TV AoAs and AoDs motion processes It has beenproven that the AoAs and AoDs motion processes based on


5 10 15 20 25 30 35 40010203040506070809

1

Tem

pora

l ACF

Low VTDHigh VTD

Time difference 120591 (s)

Figure 6The absolute values of the temporal ACF for the proposed3D TV channel model

minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 50

00501

01502

02503

03504

z

K = 05

K = 1K = 2

PDFph(z)

Figure 7 The PDF of amplitude

the TV-BM path model are first-order nonstationary in [17]For the same reason the TV PSD which is derived from thelocal ACF by utilizing the FFT with respect to time delay120591 is illustrated in Figures 10 and 13 for low and high VTDscenarios respectively Both of them show that the similar U-shape of the PSD varies with time 119905 And this is another way todisplay the nonstationary properties of the proposed channelmodel In addition the short-time (ST) DPSD ST

ℎℎ(119891 119905)

(see (26)) is shown in Figures 11 and 14 Compared withthe FFT DPSD and ST DPSD we can conclude that thenonstationary properties of them are very similar to eachother This means the STFT has the ability to process thenonstationary channel model In addition by comparing thenonstationary properties of the channel model in the lowand high VTD scenarios it can be found that the DPSDderived from the low VTD scenarios is closer to the idealJakes DPSD (U-shape spectrum)That is because the effect ofnonstationary properties of channel model is more obviousin the high VTD assumption scenarios

7 Conclusion

In this paper a novel 3D nonstationary channel modelfor V2V is proposed By employing the proposed temporal

00102030405060708

K = 05

K = 1K = 2

120579


PDFp120579(120579

)

Figure 8 The PDF of phase

Abso

lute

val

ues o

f ACF

1

09

08

07

06

05

04

03

02

01

Time T (s)

1000750

500250 0

Time difference 120591 (ms)0

1020

3040

Figure 9 Time-variant ACF

DPS

D

12

10

8

6

4

2

0

Time (s)

1000

500

0

Frequency f (Hz)minusfmminusfm2 0

fm2fm

Figure 10 Time-variant DPSD (FFT for high VTD)

8 Mobile Information SystemsD

PSD

12

10

8

6

4

2

0

Time (s)

1000750

500250

0Frequency f

(Hz)

minusfmminusfm2

0fm2

fm

Figure 11 Time-variant DPSD (STFT for high VTD)

Tem

pora

l ACF

1

09

08

07

06

05

Time T (s)

1000

500

0Time difference 120591 (ms)

0

20

40

Figure 12 Time-variant ACF (low VTD)

DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f

(Hz)

minusfm

minusfm20

fm2fm

Figure 13 Time-variant DPSD (FFT for low VTD)

DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f (Hz)

minusfmminusfm2

0fm2

fm

Figure 14 Time-variant DPSD (STFT for low VTD)

BM process with the VMF PDF scatterer distribution wederive a nonstationary nonisotropic channel model whichcan be applied to simulate the 3D channels in real timeMoreover the dynamic changes of the local ACF and PSDhave been provided Lastly the nonstationary properties ofthe proposed channel model are verified by simulation Inparticular we have proven that the STFT is also valid foranalyzing nonstationary channel models

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported in part by China Impor-tant National Science and Technology Specific Projects(no 2013ZX03001020-002) by the National Key Technol-ogy Research and Development Program of China (no2012BAF14B01) by the National Natural Science Foundationof China (no 61171105 and no 61322110) by the 863 ProgramProject (no 2015AA01A703) and by the Doctor FundingProgram (no 201300051100013)

References

[1] O Renaudin V-M Kolmonen P Vainikainen and C Oest-ges ldquoNon-stationary narrowband MIMO inter-vehicle channelcharacterization in the 5-GHz bandrdquo IEEE Transactions onVehicular Technology vol 59 no 4 pp 2007ndash2015 2010

[2] O Renaudin V-M Kolmonen P Vainikainen and C Oest-ges ldquoWideband measurement-based modeling of inter-vehiclechannels in the 5-GHz bandrdquo IEEE Transactions on VehicularTechnology vol 62 no 8 pp 3531ndash3540 2013

[3] R He Z Zhong B Ai J Ding Y Yang and A F MolischldquoShort-term fading behavior in high-speed railway cuttingscenario measurements analysis and statistical modelsrdquo IEEETransactions on Antennas and Propagation vol 61 no 4 pp2209ndash2222 2013


[4] A F Molisch F Tufvesson J Karedal and C F Mecklen-brauker ldquoA survey on vehicle-to-vehicle propagation channelsrdquoIEEE Wireless Communications vol 16 no 6 pp 12ndash22 2009

[5] Z Chen Q Wang D Wu and P Fan ldquoTwo-dimensional evo-lutionary spectrum approach to nonstationary fading channelmodelingrdquo IEEE Transactions on Vehicular Technology vol 65no 3 pp 1083ndash1097 2016

[6] Y Yuan C-X Wang X Cheng B Ai and D I LaurensonldquoNovel 3D geometry-based stochastic models for non-isotropicMIMO vehicle-to-vehicle channelsrdquo IEEE Transactions onWireless Communications vol 13 no 1 pp 298ndash309 2014

[7] S Wu C-X Wang E-H M Aggoune M M Alwakeel andY He ldquoA non-stationary 3-D wideband twin-cluster model for5Gmassive MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 32 no 6 pp 1207ndash1218 2014

[8] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013

[9] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011

[10] K Guan Z Zhong B Ai and T Kurner ldquoDeterministicpropagation modeling for the realistic high-speed railway envi-ronmentrdquo in Proceedings of the IEEE 77th Vehicular TechnologyConference (VTC rsquo13) pp 1ndash5 Dresden Germany June 2013

[11] D P Gaillot E Tanghe P Stefanut et al ldquoAccuracy of specularpath estimates with ESPRIT and RiMAX in the presence ofmeasurement-based diffuse multipath componentsrdquo in Pro-ceedings of the 5th European Conference on Antennas andPropagation (EUCAP rsquo11) pp 3619ndash3622 April 2011

[12] A G Zajic and G L Stuber ldquoThree-dimensional modelingsimulation and capacity analysis of space-time correlatedmobile-to-mobile channelsrdquo IEEE Transactions on VehicularTechnology vol 57 no 4 pp 2042ndash2054 2008

[13] L Chang J Zhang X Li and B Liu ldquoChannel estimation andperformance analysis for MIMO-OFDM in doubly-selectivechannelsrdquo in Proceedings of the 15th International Symposium onWireless PersonalMultimedia Communications (WPMC rsquo12) pp505ndash509 September 2012

[14] A Borhani and M Paetzold ldquoModelling of non-stationarymobile radio channels using two-dimensional Brownianmotion processesrdquo in Proceedings of the International Confere-nce on Advanced Technologies for Communications (ATC rsquo13)pp 241ndash246 Ho Chi Minh City Vietnam October 2013

[15] A Borhani and M Patzold ldquoA unified disk scattering modeland its angle-of-departure and time-of-arrival statisticsrdquo IEEETransactions on Vehicular Technology vol 62 no 2 pp 473ndash485 2013

[16] W Chen J Zhang Z Liu and Y Bi ldquoA geometrical-based 3Dmodel for fixed MIMO BS-RS channelsrdquo in Proceedings of theIEEE 26th Annual International Symposium on Personal Indoorand Mobile Radio Communications (PIMRC rsquo15) pp 502ndash506Hong Kong August 2015

[17] A Borhani and M Patzold ldquoA highly flexible trajectory modelbased on the primitives of brownian fieldsmdashpart I fundamentalprinciples and implementation aspectsrdquo IEEE Transactions onWireless Communications vol 14 no 2 pp 770ndash780 2015

[18] K Mammasis P Santi and A Goulianos ldquoA three-dimensionalangular scattering response including path powersrdquo IEEE

Transactions on Wireless Communications vol 11 no 4 pp1321ndash1333 2012

[19] K Mammasis R W Stewart and J S Thompson ldquoSpatialfading correlation model using mixtures of von mises fisherdistributionsrdquo IEEE Transactions on Wireless Communicationsvol 8 no 4 pp 2046ndash2055 2009

[20] H-P Lin M-J Tseng and F-S Tsai ldquoA non-stationary hiddenMarkov model for satellite propagation channel modelingrdquo inProceedings of the IEEE 56th Vehicular Technology Conference(VTC-Fall rsquo02) pp 2485ndash2488 Vancouver Canada September2002

[21] A G Zajic and G L Stuber ldquoThree-dimensional modeling andsimulation of wideband MIMO mobile-to-mobile channelsrdquoIEEE Transactions onWireless Communications vol 8 no 3 pp1260ndash1275 2009

[22] X Cheng Q Yao C-X Wang et al ldquoAn improved parametercomputationmethod for aMIMOV2V rayleigh fading channelsimulator under non-isotropic scattering environmentsrdquo IEEECommunications Letters vol 17 no 2 pp 265ndash268 2013

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks


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ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


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Journal of

Journal of

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Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

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Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



D

y y998400

x

z z998400

n1 n2

Sn1

VT

OT

RT

Tx Rx

120587 minus 120574T

120573(n1)T

120573(n2)T

120572(n1)T

120572(n2)T

120585n1n2

120585n1 120585n2

Qn1Qn2

Sn2

120573(n1)R

120573(n2)R

120572(n1)R

120572(n2)R

OR 120574R

VR

RR

Figure 1 The proposed 3D two-sphere channel model

the nonstationarity of the vertical plane while they ignoredthe impact of elevation plane on the other hand theirmodeling process required a high computational complexityand lacked flexibility Therefore it is essential to developa simple and feasible 3D channel model for nonstationaryresearch and to extend the 3D nonstationary channel modelto capture the real-physical channel characteristics

In this paper we propose a more general 3D geometricsituation where the scatterers are assumed to be distributedaround the user in a sphere area By employing the vonMises-Fisher scattering distributions the centers of the AoDs andAoAs scattering distribution are tracked by TV Brownianmotion paths [11 12] which could effectively reflect themovement process of the MS Furthermore the ACF ofthe complex channel gain and local PSD of the Dopplerfrequencies are calculated by nonstationary signal processingtools for example STFT and FFT [13 14] In addition theresults show that the time-variation of the scatterer distribu-tion results in a nonstationary nonisotropic channel modeland that the proposed model has the outstanding abilityto describe vital statistical characteristics of nonstationarychannel such as the envelope distribution and the TV ACFand PSD [15] Moreover it also offers the advantages of lowcomputational complexity and easy realization [8]

The remainder of this paper is organized as followsSection 2 discusses a novel 3D channel model based on thegeometrical two-sphere model In Section 3 the von Mises-Fisher (VMF) distribution is employed to describe the 3Dscatter distribution A time-variant Brownianmotion processis provided in Section 4 And then Section 5 proposes a novel3D nonstationary channel model The related simulation andnumerical analysis are presented in Section 6 At last theconclusion is drawn in Section 7

2 A Novel 3D Channel Model

Let us consider a narrowband V2V communication chan-nel with omnidirectional antennas The radio propagation

environment around the transmitter and the receiver is char-acterized by 3D nonisotropic scattering under line-of-sight(LoS) and none line-of-sight (NLoS) conditions Figure 1illustrates the novel 3D geometrical channel model in whichthe local scatterers are modeled on the surface of two spheresof radii 119877

119879and 119877

119877[16] In order to limit the computational

complexity of this model both local scatterers around theTx and Rx are considered Therefore we assume that thereare 119873

1scatterers on the sphere (around the Tx) of radius

119877119879and the 119899

1th (119899

1= 1 2 119873

1) scatterer is denoted by

1198781198991

Similarly around the Rx 1198732scatterers lie on a surface

of sphere of radius 119877119877 and the 119899

2th scatterer is denoted by

1198781198992

The geometric symbols in Figure 1 have the followingmeaning 120572(119899119894)

119877and 120573

(119899119894)

119877denote the main AAoA and main

EAoA of the 119899119894th scatterer respectively In the same way 120572(119899119894)

119879

and 120573(119899119894)119879

denote the main AAoD and main EAoD of the 119899119894th

scatterer respectively and 119863 is the distance between the Txand Rx It is worth noting that the distance 119863 is assumed tobe much larger than the radius 119877

119877or 119877

119879(ie 119877

119879or 119877

119877≪ 119863)

Therefore the channel impulse response (CIR) of the two-sphere model at the carrier frequency 119891

119888can be expressed as

[17]

ℎ (119905) = ℎLOS

(119905) + ℎNLOS

(119905) (1)

where

ℎLOS

(119905) = radic119870

119870 + 1119890minus1198952120587119891c1205911

times 1198901198952120587119891119879max119905 cos(120572LOS


119879

times 1198901198952120587119891119877max119905 cos(120572LOS


119877

(2)



ℎNLOS

(119905) =

2

sum

119894=1

ℎSB119894(119905) + ℎ

DB(119905)

ℎSB119894(119905)

= radic120578SB119894

119870 + 1lim119873119894rarrinfin

119873119894

sum

119899119894=1

1

radic119873119894

119890119895(120593119899119894minus21205871198911198881205912)

times 1198901198952120587119891119879max119905 cos(120572

(119899119894)

119879minus120574119879)sdotcos120573(119899119894)

119879

times 1198901198952120587119891119877max119905 cos(120572

(119899119894)

119877minus120574119877)sdotcos120573(119899119894)

119877

ℎDB

(119905)

= radic120578DB119870 + 1


11987311198732

sum

11989911198992=1

1

radic11987311198732

119890119895(12059311989911198992

minus21205871198911198881205913)

times 1198901198952120587119891119879max119905 cos(120572

(1198991)

119879minus120574119879)sdotcos120573(1198991)

119879

times 1198901198952120587119891119877max119905 cos(120572

(1198992)

119877minus120574119877)sdotcos120573(1198992)

119877

(3)

with120572LOS119879

asymp 120573LOS119879

asymp 120573LOS119877

asymp 0

120572LOS119877

asymp 120587

1205911asymp119863

119888

1205912asymp

(120585119899119894

+ 119877119879119877

)

119888

1205913asymp

(12058511989911198992

+ 119877119879+ 119877

119877)

119888

(4)

where SB1and SB



119894



and 12059311989911198992




119877stand



ℎ (119905) = ℎ119894 (119905) + 119895ℎ

119902 (119905) (5)


119901120585 (119911) =

119911

1205752

0

119890minus(1199112+1198962

0)21205752

0 sdot 1198680(119911119896

0

1205752

0

) (6)

119901120601(120601) =

119890minus1198962

021205752

0

2120587

sdot 1 +1198960

1205750

sdot radic120587


(119905))

(7)







119903ℎℎ (120591) =



2

(8)


119903ℎℎ

= 119903LOSℎℎ

(120591) +

2

sum

119894=1

119903NLOSSB119894

(120591) + 119903NLOSDB (120591) (9)

where

119903LOSℎℎ

(120591) =119870

119870 + 1119890119895(2120587120582)sdot2119863

sdot 1198901198952120587120591(119891


119903NLOSSB119894

(120591) =

120578SB119894

119870 + 1∬

120587

minus120587

[1198901198952120587120591(119891

119879max119880(119894)+119891119877max119881

(119894))]

sdot 119891 (120572(119899119894)

119879119877 120573

(119899119894)

119879119877) 119889 (120572

(119899119894)

119879119877 120573

(119899119894)

119879119877)

119903NLOSDB (120591) =

120578DB119870 + 1

intintintint

120587

minus120587

[1198901198952120587120591(119891

119879max119880DB+119891119877max119881

DB)]

sdot 119891 (120572(1198991)

119879 120573

(1198991)

119879)

sdot 119891 (120572(1198992)

119877 120573

(1198992)

119877) 119889 (120572

(1198991)

119879 120573

(1198991)

119879) 119889 (120572

(1198992)

119877 120573

(1198992)

119877)

(10)


(119899119894)

119879119877 120573

(119899119894)

119879119877)





119880(1)

= cos (120572(1198991)119879

minus 120574119879) sdot cos (120573(1198991)

119879)

119881(1)

= cos (120572(1198991)119877

minus 120574119877) sdot cos (120573(1198991)

119877)

120572(1198991)


119877119879sdot cos120573(1198991)

119879

119863sdot sin (120572(1198991)

119879)

120573(1198991)

119877= arccos(

119863 minus 119877119879cos120573(1198991)

119879sdot cos120572(1198991)

119879

1205851198991

)

1205851198991

= radic1198762

1198991

+ 1198772

119879sin2120573(1198991)

119879

1198761198991

= 119863 + 119877 sdot cos (120573(1198991)119877

) cos (120572(1198991)119877

)

119880(2)

= cos (120572(1198992)119879

) sdot cos (120573(1198992)119879

)

119881(2)

= cos (120572(1198992)119877

) sdot cos (120573(1198992)119877

)

(11)

120572(1198992)

119879asymp119877119877sdot cos120573(1198992)

119877

119863sdot sin (120572(1198992)

119877) (12)

120573(1198992)

119879= arccos(

119863 + 119877119877cos120573(1198992)

119877sdot cos120572(1198992)

119877

1205851198992

) (13)

1205851198992

= radic1198762

1198992

+ 1198772

119877sin2120573(1198992)

119877 (14)

1198761198992

asymp 119863 + 119877119877cos120573(1198992)

119877cos120572(1198992)

119877 (15)

119880DB

= cos (120572(1198991)119879

) sdot cos (120573(1198991)119879

) (16)

119881DB

= cos (120572(1198992)119877

) sdot cos (120573(1198992)119877

) (17)


119878ℎℎ(119891) = int

infin

minusinfin

119903ℎℎ (120591) 119890

minus1198952120587119891120591119889120591 (18)







119891119901(Ω 120583 119896) =

(1198962)119889

Γ (119889 + 1) sdot 119868119889 (119896)exp (119896120583119879Ω) (20)


119891 (120572 120573)

=119896

4120587 sinh (119896)


0) + sin120573

0sin120573]

(21)

where1205720and120573



0le 120587







119905and 119861(120573)


119875 (120572119905 120573

119905) =

120572119905 (119905) = 120572

119904+ 119908119905 sdot Δ119905 + Δ119905

2sdot 119861

(120572)

119905

120573119905 (119905) = 120573

119904+ 119908119905 sdot Δ119905 + Δ119905

2sdot 119861

(120573)

119905

(22)


119904and 120573







119891 (120572(119899119894)

119879119877(119905) 120573

(119899119894)

119879119877(119905)) =

119896

4120587 sinh (119896)sdot exp 119896

sdot [cos120573(119899119894)1198791198770

(119905) sdot cos120573(119899119894)119879119877

cos (120572(119899119894)119879119877

minus 120572(119899119894)

1198791198770(119905))

+ sin120573(119899119894)1198791198770

(119905) sdot sin120573(119899119894)119879119877

]

(23)


119903ℎℎ(119905 120591) = 119903

LOSℎℎ

(119905 120591) +

2

sum

119894=1

119903NLOSSB119894

(119905 120591) + 119903NLOSDB (119905 120591) (24)

where

119903LOSℎℎ

(119905 120591) =119870

119870 + 1119890119895(2120587120582)sdot2119863

sdot 1198901198952120587120591(119891


119903NLOSSB119894

(119905 120591) =

120578SB119894

119870 + 1∬

120587

minus120587

[1198901198952120587120591(119891

119879max119880(119894)+119891119877max119881

(119894))]

sdot 119891 (120572(119899119894)

119879119877(119905)

120573(119899119894)

119879119877(119905)) 119889 (120572

(119899119894)

119879119877(119905) 120573

(119899119894)

119879119877(119905))

119903NLOSDB (119905 120591) =

120578DB119870 + 1

intintintint

120587

minus120587

[1198901198952120587120591(119891

119879max119880DB+119891119877max119881

DB)]

sdot 119891 (120572(1198991)

119879(119905) 120573

(1198991)

119879(119905)) 119891 (120572

(1198992)

119877(119905)

120573(1198992)

119877(119905)) 119889 (120572

(1198991)

119879(119905) 120573

(1198991)

119879(119905)) 119889 (120572

(1198992)

119877(119905) 120573

(1198992)

119877(119905))

(25)


119877(119905) 120573

119877(119905)) and AoDs

(120572119879(119905) 120573



ℎℎ(119905 120591) = 119903

ℎℎ(119905 minus 120591)



02040608

112141618

2D v

on M

ises P

DF

p(120572)

k = 1

k = 5

k = 10

k = 20

Figure 2 2D VM PDF


STℎℎ(119891 119905) = int

infin

minusinfin

119903ℎℎ(119905 120591) 119890

minus1198952120587119891120591sdot 119908 (119905 minus 120591) 119889120591 (26)





0= 90

∘ and 1205730= 45




(119894)

0(119905) and

120573(119894)



K = 0

y-axis

y-axis

y-axis

y-axis

x-axis x-axis

x-axisx-axis

z-axis z-axis

z-axisz-axis

K = 5

K = 15 K = 30

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1


0

05

1

15

3D V

MF

PDF

120572


minus12058730

0

1205873120587321205873

21205873

120587

120587

120573

Figure 4 3D VMF PDF

0 200 400 600 800 1000 0 200 400 600 800minus12


00204

t (s)

Bt

120572(n1)T0

(t)

120572(n2)T0

(t)

120573(n1)T0

(t)

120573(n2)T0

(t)




119888= 59GHz 119863 = 300m 119891

119879max = 119891119877max = 570Hz

119877 = 25m 120574119879

= 120574119877

= 0∘ 120572(1198991)

1198770= 1478

∘ 120573(1198991)1198770

= 172∘

120572(1198992)

1198790= 316

∘ and 120573(1198992)

1198790= 1716


36 120578SB1

= 0625 120578SB2


1

= 015 120578SB2

= 0225 and 120578DB = 0625






5 10 15 20 25 30 35 40010203040506070809

1

Tem

pora

l ACF

Low VTDHigh VTD




00501

01502

02503

03504

z

K = 05

K = 1K = 2

PDFph(z)



ℎℎ(119891 119905)


7 Conclusion


00102030405060708

K = 05

K = 1K = 2

120579


PDFp120579(120579

)


Abso

lute

val

ues o

f ACF

1

09

08

07

06

05

04

03

02

01

Time T (s)

1000750

500250 0


1020

3040


DPS

D

12

10

8

6

4

2

0

Time (s)

1000

500

0


fm2fm



PSD

12

10

8

6

4

2

0

Time (s)

1000750

500250

0Frequency f

(Hz)

minusfmminusfm2

0fm2

fm


Tem

pora

l ACF

1

09

08

07

06

05

Time T (s)

1000

500


0

20

40


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f

(Hz)

minusfm

minusfm20

fm2fm


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f (Hz)

minusfmminusfm2

0fm2

fm



Competing Interests


Acknowledgments


References
































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





ℎNLOS

(119905) =

2

sum

119894=1

ℎSB119894(119905) + ℎ

DB(119905)

ℎSB119894(119905)

= radic120578SB119894

119870 + 1lim119873119894rarrinfin

119873119894

sum

119899119894=1

1

radic119873119894

119890119895(120593119899119894minus21205871198911198881205912)

times 1198901198952120587119891119879max119905 cos(120572

(119899119894)

119879minus120574119879)sdotcos120573(119899119894)

119879

times 1198901198952120587119891119877max119905 cos(120572

(119899119894)

119877minus120574119877)sdotcos120573(119899119894)

119877

ℎDB

(119905)

= radic120578DB119870 + 1


11987311198732

sum

11989911198992=1

1

radic11987311198732

119890119895(12059311989911198992

minus21205871198911198881205913)

times 1198901198952120587119891119879max119905 cos(120572

(1198991)

119879minus120574119879)sdotcos120573(1198991)

119879

times 1198901198952120587119891119877max119905 cos(120572

(1198992)

119877minus120574119877)sdotcos120573(1198992)

119877

(3)

with120572LOS119879

asymp 120573LOS119879

asymp 120573LOS119877

asymp 0

120572LOS119877

asymp 120587

1205911asymp119863

119888

1205912asymp

(120585119899119894

+ 119877119879119877

)

119888

1205913asymp

(12058511989911198992

+ 119877119879+ 119877

119877)

119888

(4)

where SB1and SB



119894



and 12059311989911198992




119877stand



ℎ (119905) = ℎ119894 (119905) + 119895ℎ

119902 (119905) (5)


119901120585 (119911) =

119911

1205752

0

119890minus(1199112+1198962

0)21205752

0 sdot 1198680(119911119896

0

1205752

0

) (6)

119901120601(120601) =

119890minus1198962

021205752

0

2120587

sdot 1 +1198960

1205750

sdot radic120587


(119905))

(7)







119903ℎℎ (120591) =



2

(8)


119903ℎℎ

= 119903LOSℎℎ

(120591) +

2

sum

119894=1

119903NLOSSB119894

(120591) + 119903NLOSDB (120591) (9)

where

119903LOSℎℎ

(120591) =119870

119870 + 1119890119895(2120587120582)sdot2119863

sdot 1198901198952120587120591(119891


119903NLOSSB119894

(120591) =

120578SB119894

119870 + 1∬

120587

minus120587

[1198901198952120587120591(119891

119879max119880(119894)+119891119877max119881

(119894))]

sdot 119891 (120572(119899119894)

119879119877 120573

(119899119894)

119879119877) 119889 (120572

(119899119894)

119879119877 120573

(119899119894)

119879119877)

119903NLOSDB (120591) =

120578DB119870 + 1

intintintint

120587

minus120587

[1198901198952120587120591(119891

119879max119880DB+119891119877max119881

DB)]

sdot 119891 (120572(1198991)

119879 120573

(1198991)

119879)

sdot 119891 (120572(1198992)

119877 120573

(1198992)

119877) 119889 (120572

(1198991)

119879 120573

(1198991)

119879) 119889 (120572

(1198992)

119877 120573

(1198992)

119877)

(10)


(119899119894)

119879119877 120573

(119899119894)

119879119877)





119880(1)

= cos (120572(1198991)119879

minus 120574119879) sdot cos (120573(1198991)

119879)

119881(1)

= cos (120572(1198991)119877

minus 120574119877) sdot cos (120573(1198991)

119877)

120572(1198991)


119877119879sdot cos120573(1198991)

119879

119863sdot sin (120572(1198991)

119879)

120573(1198991)

119877= arccos(

119863 minus 119877119879cos120573(1198991)

119879sdot cos120572(1198991)

119879

1205851198991

)

1205851198991

= radic1198762

1198991

+ 1198772

119879sin2120573(1198991)

119879

1198761198991

= 119863 + 119877 sdot cos (120573(1198991)119877

) cos (120572(1198991)119877

)

119880(2)

= cos (120572(1198992)119879

) sdot cos (120573(1198992)119879

)

119881(2)

= cos (120572(1198992)119877

) sdot cos (120573(1198992)119877

)

(11)

120572(1198992)

119879asymp119877119877sdot cos120573(1198992)

119877

119863sdot sin (120572(1198992)

119877) (12)

120573(1198992)

119879= arccos(

119863 + 119877119877cos120573(1198992)

119877sdot cos120572(1198992)

119877

1205851198992

) (13)

1205851198992

= radic1198762

1198992

+ 1198772

119877sin2120573(1198992)

119877 (14)

1198761198992

asymp 119863 + 119877119877cos120573(1198992)

119877cos120572(1198992)

119877 (15)

119880DB

= cos (120572(1198991)119879

) sdot cos (120573(1198991)119879

) (16)

119881DB

= cos (120572(1198992)119877

) sdot cos (120573(1198992)119877

) (17)


119878ℎℎ(119891) = int

infin

minusinfin

119903ℎℎ (120591) 119890

minus1198952120587119891120591119889120591 (18)







119891119901(Ω 120583 119896) =

(1198962)119889

Γ (119889 + 1) sdot 119868119889 (119896)exp (119896120583119879Ω) (20)


119891 (120572 120573)

=119896

4120587 sinh (119896)


0) + sin120573

0sin120573]

(21)

where1205720and120573



0le 120587







119905and 119861(120573)


119875 (120572119905 120573

119905) =

120572119905 (119905) = 120572

119904+ 119908119905 sdot Δ119905 + Δ119905

2sdot 119861

(120572)

119905

120573119905 (119905) = 120573

119904+ 119908119905 sdot Δ119905 + Δ119905

2sdot 119861

(120573)

119905

(22)


119904and 120573







119891 (120572(119899119894)

119879119877(119905) 120573

(119899119894)

119879119877(119905)) =

119896

4120587 sinh (119896)sdot exp 119896

sdot [cos120573(119899119894)1198791198770

(119905) sdot cos120573(119899119894)119879119877

cos (120572(119899119894)119879119877

minus 120572(119899119894)

1198791198770(119905))

+ sin120573(119899119894)1198791198770

(119905) sdot sin120573(119899119894)119879119877

]

(23)


119903ℎℎ(119905 120591) = 119903

LOSℎℎ

(119905 120591) +

2

sum

119894=1

119903NLOSSB119894

(119905 120591) + 119903NLOSDB (119905 120591) (24)

where

119903LOSℎℎ

(119905 120591) =119870

119870 + 1119890119895(2120587120582)sdot2119863

sdot 1198901198952120587120591(119891


119903NLOSSB119894

(119905 120591) =

120578SB119894

119870 + 1∬

120587

minus120587

[1198901198952120587120591(119891

119879max119880(119894)+119891119877max119881

(119894))]

sdot 119891 (120572(119899119894)

119879119877(119905)

120573(119899119894)

119879119877(119905)) 119889 (120572

(119899119894)

119879119877(119905) 120573

(119899119894)

119879119877(119905))

119903NLOSDB (119905 120591) =

120578DB119870 + 1

intintintint

120587

minus120587

[1198901198952120587120591(119891

119879max119880DB+119891119877max119881

DB)]

sdot 119891 (120572(1198991)

119879(119905) 120573

(1198991)

119879(119905)) 119891 (120572

(1198992)

119877(119905)

120573(1198992)

119877(119905)) 119889 (120572

(1198991)

119879(119905) 120573

(1198991)

119879(119905)) 119889 (120572

(1198992)

119877(119905) 120573

(1198992)

119877(119905))

(25)


119877(119905) 120573

119877(119905)) and AoDs

(120572119879(119905) 120573



ℎℎ(119905 120591) = 119903

ℎℎ(119905 minus 120591)



02040608

112141618

2D v

on M

ises P

DF

p(120572)

k = 1

k = 5

k = 10

k = 20

Figure 2 2D VM PDF


STℎℎ(119891 119905) = int

infin

minusinfin

119903ℎℎ(119905 120591) 119890

minus1198952120587119891120591sdot 119908 (119905 minus 120591) 119889120591 (26)





0= 90

∘ and 1205730= 45




(119894)

0(119905) and

120573(119894)



K = 0

y-axis

y-axis

y-axis

y-axis

x-axis x-axis

x-axisx-axis

z-axis z-axis

z-axisz-axis

K = 5

K = 15 K = 30

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1


0

05

1

15

3D V

MF

PDF

120572


minus12058730

0

1205873120587321205873

21205873

120587

120587

120573

Figure 4 3D VMF PDF

0 200 400 600 800 1000 0 200 400 600 800minus12


00204

t (s)

Bt

120572(n1)T0

(t)

120572(n2)T0

(t)

120573(n1)T0

(t)

120573(n2)T0

(t)




119888= 59GHz 119863 = 300m 119891

119879max = 119891119877max = 570Hz

119877 = 25m 120574119879

= 120574119877

= 0∘ 120572(1198991)

1198770= 1478

∘ 120573(1198991)1198770

= 172∘

120572(1198992)

1198790= 316

∘ and 120573(1198992)

1198790= 1716


36 120578SB1

= 0625 120578SB2


1

= 015 120578SB2

= 0225 and 120578DB = 0625






5 10 15 20 25 30 35 40010203040506070809

1

Tem

pora

l ACF

Low VTDHigh VTD




00501

01502

02503

03504

z

K = 05

K = 1K = 2

PDFph(z)



ℎℎ(119891 119905)


7 Conclusion


00102030405060708

K = 05

K = 1K = 2

120579


PDFp120579(120579

)


Abso

lute

val

ues o

f ACF

1

09

08

07

06

05

04

03

02

01

Time T (s)

1000750

500250 0


1020

3040


DPS

D

12

10

8

6

4

2

0

Time (s)

1000

500

0


fm2fm



PSD

12

10

8

6

4

2

0

Time (s)

1000750

500250

0Frequency f

(Hz)

minusfmminusfm2

0fm2

fm


Tem

pora

l ACF

1

09

08

07

06

05

Time T (s)

1000

500


0

20

40


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f

(Hz)

minusfm

minusfm20

fm2fm


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f (Hz)

minusfmminusfm2

0fm2

fm



Competing Interests


Acknowledgments


References
































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119880(1)

= cos (120572(1198991)119879

minus 120574119879) sdot cos (120573(1198991)

119879)

119881(1)

= cos (120572(1198991)119877

minus 120574119877) sdot cos (120573(1198991)

119877)

120572(1198991)


119877119879sdot cos120573(1198991)

119879

119863sdot sin (120572(1198991)

119879)

120573(1198991)

119877= arccos(

119863 minus 119877119879cos120573(1198991)

119879sdot cos120572(1198991)

119879

1205851198991

)

1205851198991

= radic1198762

1198991

+ 1198772

119879sin2120573(1198991)

119879

1198761198991

= 119863 + 119877 sdot cos (120573(1198991)119877

) cos (120572(1198991)119877

)

119880(2)

= cos (120572(1198992)119879

) sdot cos (120573(1198992)119879

)

119881(2)

= cos (120572(1198992)119877

) sdot cos (120573(1198992)119877

)

(11)

120572(1198992)

119879asymp119877119877sdot cos120573(1198992)

119877

119863sdot sin (120572(1198992)

119877) (12)

120573(1198992)

119879= arccos(

119863 + 119877119877cos120573(1198992)

119877sdot cos120572(1198992)

119877

1205851198992

) (13)

1205851198992

= radic1198762

1198992

+ 1198772

119877sin2120573(1198992)

119877 (14)

1198761198992

asymp 119863 + 119877119877cos120573(1198992)

119877cos120572(1198992)

119877 (15)

119880DB

= cos (120572(1198991)119879

) sdot cos (120573(1198991)119879

) (16)

119881DB

= cos (120572(1198992)119877

) sdot cos (120573(1198992)119877

) (17)


119878ℎℎ(119891) = int

infin

minusinfin

119903ℎℎ (120591) 119890

minus1198952120587119891120591119889120591 (18)







119891119901(Ω 120583 119896) =

(1198962)119889

Γ (119889 + 1) sdot 119868119889 (119896)exp (119896120583119879Ω) (20)


119891 (120572 120573)

=119896

4120587 sinh (119896)


0) + sin120573

0sin120573]

(21)

where1205720and120573



0le 120587







119905and 119861(120573)


119875 (120572119905 120573

119905) =

120572119905 (119905) = 120572

119904+ 119908119905 sdot Δ119905 + Δ119905

2sdot 119861

(120572)

119905

120573119905 (119905) = 120573

119904+ 119908119905 sdot Δ119905 + Δ119905

2sdot 119861

(120573)

119905

(22)


119904and 120573







119891 (120572(119899119894)

119879119877(119905) 120573

(119899119894)

119879119877(119905)) =

119896

4120587 sinh (119896)sdot exp 119896

sdot [cos120573(119899119894)1198791198770

(119905) sdot cos120573(119899119894)119879119877

cos (120572(119899119894)119879119877

minus 120572(119899119894)

1198791198770(119905))

+ sin120573(119899119894)1198791198770

(119905) sdot sin120573(119899119894)119879119877

]

(23)


119903ℎℎ(119905 120591) = 119903

LOSℎℎ

(119905 120591) +

2

sum

119894=1

119903NLOSSB119894

(119905 120591) + 119903NLOSDB (119905 120591) (24)

where

119903LOSℎℎ

(119905 120591) =119870

119870 + 1119890119895(2120587120582)sdot2119863

sdot 1198901198952120587120591(119891


119903NLOSSB119894

(119905 120591) =

120578SB119894

119870 + 1∬

120587

minus120587

[1198901198952120587120591(119891

119879max119880(119894)+119891119877max119881

(119894))]

sdot 119891 (120572(119899119894)

119879119877(119905)

120573(119899119894)

119879119877(119905)) 119889 (120572

(119899119894)

119879119877(119905) 120573

(119899119894)

119879119877(119905))

119903NLOSDB (119905 120591) =

120578DB119870 + 1

intintintint

120587

minus120587

[1198901198952120587120591(119891

119879max119880DB+119891119877max119881

DB)]

sdot 119891 (120572(1198991)

119879(119905) 120573

(1198991)

119879(119905)) 119891 (120572

(1198992)

119877(119905)

120573(1198992)

119877(119905)) 119889 (120572

(1198991)

119879(119905) 120573

(1198991)

119879(119905)) 119889 (120572

(1198992)

119877(119905) 120573

(1198992)

119877(119905))

(25)


119877(119905) 120573

119877(119905)) and AoDs

(120572119879(119905) 120573



ℎℎ(119905 120591) = 119903

ℎℎ(119905 minus 120591)



02040608

112141618

2D v

on M

ises P

DF

p(120572)

k = 1

k = 5

k = 10

k = 20

Figure 2 2D VM PDF


STℎℎ(119891 119905) = int

infin

minusinfin

119903ℎℎ(119905 120591) 119890

minus1198952120587119891120591sdot 119908 (119905 minus 120591) 119889120591 (26)





0= 90

∘ and 1205730= 45




(119894)

0(119905) and

120573(119894)



K = 0

y-axis

y-axis

y-axis

y-axis

x-axis x-axis

x-axisx-axis

z-axis z-axis

z-axisz-axis

K = 5

K = 15 K = 30

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1


0

05

1

15

3D V

MF

PDF

120572


minus12058730

0

1205873120587321205873

21205873

120587

120587

120573

Figure 4 3D VMF PDF

0 200 400 600 800 1000 0 200 400 600 800minus12


00204

t (s)

Bt

120572(n1)T0

(t)

120572(n2)T0

(t)

120573(n1)T0

(t)

120573(n2)T0

(t)




119888= 59GHz 119863 = 300m 119891

119879max = 119891119877max = 570Hz

119877 = 25m 120574119879

= 120574119877

= 0∘ 120572(1198991)

1198770= 1478

∘ 120573(1198991)1198770

= 172∘

120572(1198992)

1198790= 316

∘ and 120573(1198992)

1198790= 1716


36 120578SB1

= 0625 120578SB2


1

= 015 120578SB2

= 0225 and 120578DB = 0625






5 10 15 20 25 30 35 40010203040506070809

1

Tem

pora

l ACF

Low VTDHigh VTD




00501

01502

02503

03504

z

K = 05

K = 1K = 2

PDFph(z)



ℎℎ(119891 119905)


7 Conclusion


00102030405060708

K = 05

K = 1K = 2

120579


PDFp120579(120579

)


Abso

lute

val

ues o

f ACF

1

09

08

07

06

05

04

03

02

01

Time T (s)

1000750

500250 0


1020

3040


DPS

D

12

10

8

6

4

2

0

Time (s)

1000

500

0


fm2fm



PSD

12

10

8

6

4

2

0

Time (s)

1000750

500250

0Frequency f

(Hz)

minusfmminusfm2

0fm2

fm


Tem

pora

l ACF

1

09

08

07

06

05

Time T (s)

1000

500


0

20

40


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f

(Hz)

minusfm

minusfm20

fm2fm


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f (Hz)

minusfmminusfm2

0fm2

fm



Competing Interests


Acknowledgments


References
































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







119891 (120572(119899119894)

119879119877(119905) 120573

(119899119894)

119879119877(119905)) =

119896

4120587 sinh (119896)sdot exp 119896

sdot [cos120573(119899119894)1198791198770

(119905) sdot cos120573(119899119894)119879119877

cos (120572(119899119894)119879119877

minus 120572(119899119894)

1198791198770(119905))

+ sin120573(119899119894)1198791198770

(119905) sdot sin120573(119899119894)119879119877

]

(23)


119903ℎℎ(119905 120591) = 119903

LOSℎℎ

(119905 120591) +

2

sum

119894=1

119903NLOSSB119894

(119905 120591) + 119903NLOSDB (119905 120591) (24)

where

119903LOSℎℎ

(119905 120591) =119870

119870 + 1119890119895(2120587120582)sdot2119863

sdot 1198901198952120587120591(119891


119903NLOSSB119894

(119905 120591) =

120578SB119894

119870 + 1∬

120587

minus120587

[1198901198952120587120591(119891

119879max119880(119894)+119891119877max119881

(119894))]

sdot 119891 (120572(119899119894)

119879119877(119905)

120573(119899119894)

119879119877(119905)) 119889 (120572

(119899119894)

119879119877(119905) 120573

(119899119894)

119879119877(119905))

119903NLOSDB (119905 120591) =

120578DB119870 + 1

intintintint

120587

minus120587

[1198901198952120587120591(119891

119879max119880DB+119891119877max119881

DB)]

sdot 119891 (120572(1198991)

119879(119905) 120573

(1198991)

119879(119905)) 119891 (120572

(1198992)

119877(119905)

120573(1198992)

119877(119905)) 119889 (120572

(1198991)

119879(119905) 120573

(1198991)

119879(119905)) 119889 (120572

(1198992)

119877(119905) 120573

(1198992)

119877(119905))

(25)


119877(119905) 120573

119877(119905)) and AoDs

(120572119879(119905) 120573



ℎℎ(119905 120591) = 119903

ℎℎ(119905 minus 120591)



02040608

112141618

2D v

on M

ises P

DF

p(120572)

k = 1

k = 5

k = 10

k = 20

Figure 2 2D VM PDF


STℎℎ(119891 119905) = int

infin

minusinfin

119903ℎℎ(119905 120591) 119890

minus1198952120587119891120591sdot 119908 (119905 minus 120591) 119889120591 (26)





0= 90

∘ and 1205730= 45




(119894)

0(119905) and

120573(119894)



K = 0

y-axis

y-axis

y-axis

y-axis

x-axis x-axis

x-axisx-axis

z-axis z-axis

z-axisz-axis

K = 5

K = 15 K = 30

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1


0

05

1

15

3D V

MF

PDF

120572


minus12058730

0

1205873120587321205873

21205873

120587

120587

120573

Figure 4 3D VMF PDF

0 200 400 600 800 1000 0 200 400 600 800minus12


00204

t (s)

Bt

120572(n1)T0

(t)

120572(n2)T0

(t)

120573(n1)T0

(t)

120573(n2)T0

(t)




119888= 59GHz 119863 = 300m 119891

119879max = 119891119877max = 570Hz

119877 = 25m 120574119879

= 120574119877

= 0∘ 120572(1198991)

1198770= 1478

∘ 120573(1198991)1198770

= 172∘

120572(1198992)

1198790= 316

∘ and 120573(1198992)

1198790= 1716


36 120578SB1

= 0625 120578SB2


1

= 015 120578SB2

= 0225 and 120578DB = 0625






5 10 15 20 25 30 35 40010203040506070809

1

Tem

pora

l ACF

Low VTDHigh VTD




00501

01502

02503

03504

z

K = 05

K = 1K = 2

PDFph(z)



ℎℎ(119891 119905)


7 Conclusion


00102030405060708

K = 05

K = 1K = 2

120579


PDFp120579(120579

)


Abso

lute

val

ues o

f ACF

1

09

08

07

06

05

04

03

02

01

Time T (s)

1000750

500250 0


1020

3040


DPS

D

12

10

8

6

4

2

0

Time (s)

1000

500

0


fm2fm



PSD

12

10

8

6

4

2

0

Time (s)

1000750

500250

0Frequency f

(Hz)

minusfmminusfm2

0fm2

fm


Tem

pora

l ACF

1

09

08

07

06

05

Time T (s)

1000

500


0

20

40


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f

(Hz)

minusfm

minusfm20

fm2fm


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f (Hz)

minusfmminusfm2

0fm2

fm



Competing Interests


Acknowledgments


References
































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




K = 0

y-axis

y-axis

y-axis

y-axis

x-axis x-axis

x-axisx-axis

z-axis z-axis

z-axisz-axis

K = 5

K = 15 K = 30

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1

1

1 1

06

06 06

02

02 02

minus02

minus02 minus02

minus06

minus06 minus06

minus1minus1 minus1


0

05

1

15

3D V

MF

PDF

120572


minus12058730

0

1205873120587321205873

21205873

120587

120587

120573

Figure 4 3D VMF PDF

0 200 400 600 800 1000 0 200 400 600 800minus12


00204

t (s)

Bt

120572(n1)T0

(t)

120572(n2)T0

(t)

120573(n1)T0

(t)

120573(n2)T0

(t)




119888= 59GHz 119863 = 300m 119891

119879max = 119891119877max = 570Hz

119877 = 25m 120574119879

= 120574119877

= 0∘ 120572(1198991)

1198770= 1478

∘ 120573(1198991)1198770

= 172∘

120572(1198992)

1198790= 316

∘ and 120573(1198992)

1198790= 1716


36 120578SB1

= 0625 120578SB2


1

= 015 120578SB2

= 0225 and 120578DB = 0625






5 10 15 20 25 30 35 40010203040506070809

1

Tem

pora

l ACF

Low VTDHigh VTD




00501

01502

02503

03504

z

K = 05

K = 1K = 2

PDFph(z)



ℎℎ(119891 119905)


7 Conclusion


00102030405060708

K = 05

K = 1K = 2

120579


PDFp120579(120579

)


Abso

lute

val

ues o

f ACF

1

09

08

07

06

05

04

03

02

01

Time T (s)

1000750

500250 0


1020

3040


DPS

D

12

10

8

6

4

2

0

Time (s)

1000

500

0


fm2fm



PSD

12

10

8

6

4

2

0

Time (s)

1000750

500250

0Frequency f

(Hz)

minusfmminusfm2

0fm2

fm


Tem

pora

l ACF

1

09

08

07

06

05

Time T (s)

1000

500


0

20

40


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f

(Hz)

minusfm

minusfm20

fm2fm


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f (Hz)

minusfmminusfm2

0fm2

fm



Competing Interests


Acknowledgments


References
































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




5 10 15 20 25 30 35 40010203040506070809

1

Tem

pora

l ACF

Low VTDHigh VTD




00501

01502

02503

03504

z

K = 05

K = 1K = 2

PDFph(z)



ℎℎ(119891 119905)


7 Conclusion


00102030405060708

K = 05

K = 1K = 2

120579


PDFp120579(120579

)


Abso

lute

val

ues o

f ACF

1

09

08

07

06

05

04

03

02

01

Time T (s)

1000750

500250 0


1020

3040


DPS

D

12

10

8

6

4

2

0

Time (s)

1000

500

0


fm2fm



PSD

12

10

8

6

4

2

0

Time (s)

1000750

500250

0Frequency f

(Hz)

minusfmminusfm2

0fm2

fm


Tem

pora

l ACF

1

09

08

07

06

05

Time T (s)

1000

500


0

20

40


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f

(Hz)

minusfm

minusfm20

fm2fm


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f (Hz)

minusfmminusfm2

0fm2

fm



Competing Interests


Acknowledgments


References
































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




PSD

12

10

8

6

4

2

0

Time (s)

1000750

500250

0Frequency f

(Hz)

minusfmminusfm2

0fm2

fm


Tem

pora

l ACF

1

09

08

07

06

05

Time T (s)

1000

500


0

20

40


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f

(Hz)

minusfm

minusfm20

fm2fm


DPS

D

40

30

20

10

0

Time (s)

1000

500

0Frequency f (Hz)

minusfmminusfm2

0fm2

fm



Competing Interests


Acknowledgments


References
































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Research Article A Novel 3D Nonstationary Channel Model...

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Transcript of Research Article A Novel 3D Nonstationary Channel Model...