A General 3D Non-Stationary Wireless Channel Model for 5G and Beyond
Ji Bian, Cheng-Xiang Wang, Xiqi Gao, Xiaohu You, Minggao Zhang
IIEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2020 1
A General 3D Non-Stationary Wireless ChannelModel for 5G and Beyond
Ji Bian,
Member, IEEE , Cheng-Xiang Wang,
Fellow, IEEE , Xiqi Gao,
Fellow, IEEE , Xiaohu You,
Fellow, IEEE ,and Minggao Zhang
Abstract —In this paper, a novel three-dimensional (3D) non-stationary geometry-based stochastic model (GBSM) for the fifthgeneration (5G) and beyond 5G (B5G) systems is proposed.The proposed B5G channel model (B5GCM) is designed tocapture various channel characteristics in (B)5G systems such asspace-time-frequency (STF) non-stationarity, spherical wavefront(SWF), high delay resolution, time-variant velocities and direc-tions of motion of the transmitter, receiver, and scatterers, spatialconsistency, etc. By combining different channel properties into ageneral channel model framework, the proposed B5GCM is ableto be applied to multiple frequency bands and multiple scenar-ios, including massive multiple-input multiple-output (MIMO),vehicle-to-vehicle (V2V), high-speed train (HST), and millimeterwave-terahertz (mmWave-THz) communication scenarios. Keystatistics of the proposed B5GCM are obtained and comparedwith those of standard 5G channel models and correspondingmeasurement data, showing the generalization and usefulness ofthe proposed model.
Index Terms —3D space-time-frequency non-stationary GBSM,massive MIMO, mmWave-THz, high-mobility, multi-mobilitycommunications.
I. I
NTRODUCTION
The growing requirement of high data rate transmissioncaused by the popularization of wireless services and applica-tions results in a spectrum crisis in current sub-6 GHz bands.To address this challenge, the fifth generation (5G)/beyond5G (B5G) wireless communication systems will transmit datausing millimeter wave (mmWave)/terahertz (THz) bands in
Manuscript received January 31, 2020; revised August 11, 2020 andDecember 12, 2020; accepted December 23, 2020. The authors would like toacknowledge the support from the National Key R&D Program of China underGrant 2018YFB1801101, the National Natural Science Foundation of China(NSFC) under Grant 61960206006, the Shandong Provincial Natural ScienceFoundation for Young Scholars of China under Grant ZR2020QF001, TaishanScholar Program of Shandong Province, the Frontiers Science Center forMobile Information Communication and Security, the High Level Innovationand Entrepreneurial Research Team Program in Jiangsu, the High LevelInnovation and Entrepreneurial Talent Introduction Program in Jiangsu, theResearch Fund of National Mobile Communications Research Laboratory,Southeast University, under Grant 2020B01, the Fundamental Research Fundsfor the Central Universities under Grant 2242020R30001, and the EU H2020RISE TESTBED2 project under Grant 872172.J. Bian is with School of Information Science and Engineering, Shan-dong Normal University, Jinan, Shandong, 250358, China (e-mail: [email protected]).C.-X. Wang (corresponding author), X. Q. Gao and X. H. You are with theNational Mobile Communications Research Laboratory, School of InformationScience and Engineering, Southeast University, Nanjing, 210096, China, andalso with the Purple Mountain Laboratories, Nanjing, 211111, China (email: { chxwang, xqgao, xhyu } @seu.edu.cn).M. G. Zhang is with Shandong Provincial Key Lab of Wireless Communica-tion Technologies, School of Information Science and Engineering, ShandongUniversity, Qingdao, Shandong, 266237, China (e-mail: [email protected]). multiple propagation scenarios, e.g., high-speed train (HST)and vehicle-to-vehicle (V2V) scenarios [1]. The short wave-lengths of mmWave-THz bands make it possible to deploylarge antenna arrays with high beamforming gains that canovercome the severe path loss [2]. Revolutionary technologiesemployed in 5G and B5G wireless communication systemssuch as massive multiple-input multiple-output (MIMO), HST,V2V, and mmWave-THz communications introduce new chan-nel properties, such as spherical wavefront (SWF), spatialnon-stationarity, oxygen absorption, etc. These in turn willset new requirements to standard (B)5G channel models, i.e.,supporting multiple frequency bands and multiple scenarios,as follows [3], [4]:1) Multiple frequency bands, covering sub-6 GHz,mmWave, and THz bands;2) Large bandwidth, e.g., 0.5–4 GHz for mmWave bandsand 10 GHz for THz bands;3) Massive MIMO scenarios: spatial non-stationarity andSWF;4) HST scenarios: temporal non-stationarity including pa-rameters’ drifting and clusters’ appearance and disappear-ance over time;5) V2V scenarios: temporal non-stationarity and multi-mobility, i.e., the transmitter (Tx), receiver (Rx), andscatterers may move with time-variant velocities andheading directions;6) Three-dimensional (3D) scenarios, especially for indoorand outdoor small cell scenarios;7) Spatial consistency scenarios, i.e., closely located linksexperience similar channel statistical properties.In massive MIMO communications, the large arrays makethe channel spatially non-stationary, which means channelparameters and statistical properties vary along array axis [4].For example, measurement results in [5] and [6] showed thatthe angles of multipath components (MPCs) drift across thearray, justifying the SWF assumption of the channel. Thisimplies that travel distances from every Tx antenna elementto the Rx/scatterers at each time instant (if temporal non-stationarity is considered) have to be calculated, resulting inhigh model complexity [7]–[9]. Measurements in [10] and[11] revealed that the mean delay, delay spread, and clusterpower can vary across the large array. Here, a cluster is agroup of MPCs having similar properties in delay, power, andangles. Note that the cluster power variation over array hasnot been considered in most massive MIMO channel models[7]–[9]. Furthermore, when large arrays are adopted, clusters a r X i v : . [ ee ss . SP ] J a n EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2020 2 illustrate a partially visible property. Some clusters are visibleover the whole array, while other clusters can only interactwith part of the array [6], [12]. In [13] and [14], the partiallyvisibility of clusters was modeled by introducing the conceptof “BS-visibility region (VR)”. Other researches such as [7]and [15] described the partial visibility of clusters using birth-death or Markov processes. In general, how to efficiently andsynthetically model the non-stationarities in the time and spacedomains has to be solved in the (B)5G channel modeling.In V2V channels, channel parameters and statistical proper-ties are time-varying caused by the motions of the Tx, Rx, andscatterers [16]. Besides, channel measurements showed thatclusters of V2V channels can exhibit a birth-death behavior,i.e., appear, exist for a time period, and then disappear [4].Large numbers of V2V channel models were designed as puregeometry-based stochastic model (pure-GBSM) [17], [18].The scatterers of those models were assumed to be locatedon regular shapes, which are less versatile than those ofthe WINNER/3GPP models [19], [20]. More realistic V2Vchannel models, e.g., [21] and [22], were developed basedon channel measurements. However, the motions of scattererswere neglected. Furthermore, the variations of velocity andtrajectory of the Tx/Rx were not taken into account.The HST channels share some similar properties withV2V channels, e.g., large Doppler shifts and temporal non-stationarity. Widely used standard channel models, e.g., WIN-NER II [23] and IMT-Advanced channel models [24] can beapplied to HST scenarios where the velocity of train can beup to 350 km/h. However, those models are developed ontemporal wide-sense stationary (WSS) assumption. The HSTchannel model in [25] was developed based on IMT-Advancedchannel model [24] by taking into account the time-varyingangles and cluster evolution. More general HST channel modelwas proposed in [26], which extended the elliptical model byassuming velocity and moving direction variations. However,the above-mentioned models are two-dimensional (2D) andcan only be applied to the scenarios where the transceiverand scatterers are sufficiently far away. In [27], a 3D non-stationary HST channel model was proposed, which canonly be used in tunnel scenarios. In [28], a tapped delayline model was presented for various HST scenarios, e.g.,viaduct and cutting. The fidelity of the model relies on themodel parameters obtained from ray-tracing, resulting in highcomputation complexity.For the mmWave-THz communications, high frequenciesresult in large path loss and render the mmWave-THz propa-gation susceptible to blockage effects and oxygen absorption[3]. Compared with sub-6 GHz bands, the mmWave-THzchannels are sparser [29]. High delay resolution is requiredin channel modeling due to the large bandwidth. Rays withina cluster can have different time of arrival, leading to unequalray powers [20]. Besides, as the relative bandwidth increases,the channel becomes non-stationary in the frequency domain,which means the uncorrelated scattering (US) assumption maynot be fulfilled [30]. Furthermore, directional antennas areoften deployed to overcome the high attenuation at mmWave-THz frequency bands. In [31], a 2D mmWave V2V channelmodel was proposed. The influence of directional antennas was represented by eliminating the clusters outside the main lobeof antenna patterns. MmWave-THz channel models shouldfaithfully recreate the spatial, temporal and frequency char-acteristics for every single ray, such as the models in [32] and[33]. However, those models are oversimplified and cannotsupport time evolution since the model parameters are time-invariant.Apart from modeling channel characteristics in variousscenarios, another challenge for (B)5G channel modeling liesin how to combine those channel characteristics into a generalmodeling framework. Standard channel models aim to solvethis problem. In order to achieve smooth time evolution,the COST 2100 channel model introduces the “VR”, whichindicates whether a cluster can be “seen” from the mobilestation (MS). The clusters are considered to physically existin the environments and do not belong to a specific link. Thus,closely located links can experience similar environments,justifying the spatial consistency of the model. However, theCOST 2100 channel model can only support sub-6 GHzbands. Furthermore, massive MIMO and dual-mobility werenot supported. The QuaDriGa [34] channel model is extendedfrom 3GPP TR36.873 [35] and support frequencies over 0.45–100 GHz. The model parameters were generated based onspatially correlated random variables. Links located nearbyshare similar channel parameters and therefore supports spatialconsistency. However, the complexities of those models arerelatively high and the dual-mobility was neglected. The 3GPPTR38.901 [20] channel model extended the 3GPP TR36.873by supporting the frequencies over 0.5–100 GHz. The oxygenabsorption and blockage effect for the mmWave bands weremodeled. However, for high-mobility scenarios, the clustersfade in and fade out were not considered, which results inlimited ability for capturing time non-stationarity. Besides,the dual-mobility and SWF cannot be supported. Note thatall the above-mentioned standard channel models assumedconstant cluster power for different antenna elements and didnot consider scatterers movements.A GBSM called more general 5G channel model(MG5GCM) aims at capturing various channel properties in5G systems [9]. Based on a general model framework, themodel can support various communication scenarios. However,the azimuth angles, elevation angles, and travel distancesbetween Tx/Rx and scatterers were generated independently.The model can only evolve along the time and array axes,and neglected the non-stationary properties in the frequencydomain. The locations of scatterers were implicitly determinedby the angles and delays, which makes the model difficult toachieve spatial consistency. Besides, it neglected the dynamicvelocity and direction of motion of the Tx, Rx, and scatterers.Through above analysis, we find that none of these standardchannel models can meet all the 5G channel modeling require-ments. Considering these research gaps, this paper proposesa general (B)5G channel model (B5GCM) towards multiplefrequency bands and multiple scenarios. The contributions ofthis paper are listed as follows:1) The system functions, correlation functions (CFs), andpower spectrum densities (PSDs) of space-time-frequency(STF) stationary and non-stationary channels are pre-
EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2020 3 sented. A general 3D STF non-stationary ultra-widebandchannel model is developed. By setting appropriate chan-nel model parameters, the model can support multiplefrequency bands and multiple scenarios.2) A highly accurate approximate expression of the 3D SWFis proposed, which is more scalable and efficient than thetraditional modeling method and can capture spatial andtemporal channel non-stationarities. The cluster evolu-tions along the time and array axes are jointly consideredand simulated using a unified birth-death process. Thesmooth power variation over the large arrays is modeledby a 2D spatial lognormal process.3) The novel ellipsoid Gaussian scattering distribution isproposed which can jointly describe the azimuth angles,elevation angles, and distances from the Tx(Rx) to thefirst(last)-bounce scatterers. By tracking the locations ofthe Tx, Rx, and scatterers, the spatial consistency of thechannel is supported.4) The proposed B5GCM takes into account a multi-mobility communication environment, where the Tx, Rx,and scatterers can change their velocities and movingdirections.5) Key statistics including local STF-CF, local spatial-Doppler PSD, local Doppler spread, and array coherencedistance are derived and compared with standard 5Gchannel models and the corresponding measurement data.The remainder of this paper is organized as follows. InSection II, the system, correlation, and spectrum functionsof the STF non-stationary channel model are presented. Theproposed B5GCM is described in Section III. Statistics of theB5GCM are investigated in Section IV. In Section V, numericaland simulation results are provided and discussed. Conclusionsare drawn in Section VI.II. S
YSTEM F UNCTIONS , CF S , AND
PSD
S OF S PACE ,T IME , AND F REQUENCY N ON -S TATIONARY C HANNELS
Wireless channels can be described through system func-tions, CFs, and PSDs. The time-frequency selectivity anddelay-Doppler dispersion of wireless channels have been stud-ied in [36] and [37]. In this paper, the non-stationarity ofchannels in the space domain is considered, which is anessential property for massive MIMO channels. The basicfunction characterizing the wireless channel is the space andtime-varying channel impulse response (CIR) h ( r, t, τ ) . It ismodeled as a stochastic process on space r , time t , and delay τ . Note that the space domain indicates the region where r is confined along a linear antenna array. Taking the Fouriertransform of h ( r, t, τ ) with respect to (w.r.t.) τ results in space-and time-varying transfer function, which is given by H ( r, t, f ) = (cid:90) h ( r, t, τ ) e − j πτf d τ. (1)The space and time-varying transfer function characterizes thespace, time, and frequency selectivity of wireless channels.Moreover, taking the Fourier transform of h ( r, t, τ ) w.r.t. r and t results in the spatial-Doppler Doppler delay spread function, i.e., s ( (cid:36), ν, τ ) = (cid:90) (cid:90) h ( r, t, τ ) e − j π ( r(cid:36)λ − + tν ) d r d t. (2)Here, λ is the wavelength, (cid:36) is the spatial-Doppler frequencyvariable defined as (cid:36) = Ω · ˜Ω , where Ω is an angle unitvector of the departure/arrival waves, ˜Ω indicates the antennaarray orientation [38]. The space variable r and the spatial-Doppler variable (cid:36) are Fourier transformation pair. Since r has a distance unit, the unit of (cid:36) must be its reciprocal,i.e., per normalized distance (w.r.t. λ ). The terminology (cid:36) stems from the fact that Ω · ˜Ω is the Doppler shift of awave with direction Ω impinging on an antenna which moveswith λ m/s in direction ˜Ω . The spatial-Doppler Dopplerdelay spread function characterizes the dispersions of wirelesschannels in the spatial-Doppler frequency, Doppler frequency,and delay domains. Similarly, by taking the Fourier transformof h ( r, t, τ ) w.r.t. r , t , and/or τ , totally eight system functionscan be obtained [39]. Based on those formulas, e.g., h ( r, t, τ ) , H ( r, t, f ) , and s ( (cid:36), ν, τ ) , the six-dimensional (6D) CFs arederived as R h ( r, t, τ ; ∆ r, ∆ t, ∆ τ )= E { h ( r, t, τ ) h ∗ ( r − ∆ r, t − ∆ t, τ − ∆ τ ) } (3) R H ( r, t, f ; ∆ r, ∆ t, ∆ f )= E { H ( r, t, f ) H ∗ ( r − ∆ r, t − ∆ t, f − ∆ f ) } (4) R s ( (cid:36), ν, τ ; ∆ (cid:36), ∆ ν, ∆ τ )= E { s ( (cid:36), ν, τ ) s ∗ ( (cid:36) − ∆ (cid:36), ν − ∆ ν, τ − ∆ τ ) } (5)where E {·} indicates ensemble average and ( · ) ∗ stands forcomplex conjugation, ∆ r , ∆ t , and ∆ f are space, time, andfrequency lags, respectively. A channel is STF non-stationaryif R H ( r, t, f ; ∆ r, ∆ t, ∆ f ) is not only a function of ∆ r , ∆ t ,and ∆ f , but also relies on r , t , and f . Its simplification, i.e.,WSS over r , t , and f is widely used in the existing channelmodels. However, it is valid only if the channel satisfies certainconditions. For example, when the distance from the Tx to theRx (or a cluster) is less than the Rayleigh distance, i.e., L λ ,where L denotes the aperture size of the antenna array and λ isthe carrier wavelength, the spatial WSS condition is fulfilled[8]. The temporal WSS assumption is valid as long as thechannel stationary interval is larger than the observation time.Finally, when the relative bandwidth of the channel is small(typically less than 20% of the carrier frequency), the channelbecomes WSS in the frequency domain. Considering the STFWSS conditions, the 6D CFs in (3)-(5) can be reduced to R h ( r, t, τ ; ∆ r, ∆ t, ∆ τ ) = S h ( τ ; ∆ r, ∆ t ) δ (∆ τ ) (6) R H ( r, t, f ; ∆ r, ∆ t, ∆ f ) = R H (∆ r, ∆ t, ∆ f ) (7) R s ( (cid:36), ν, τ ; ∆ (cid:36), ∆ ν, ∆ τ )= C s ( (cid:36), ν, τ ) δ (∆ (cid:36) ) δ (∆ ν ) δ (∆ τ ) (8)where δ ( · ) is the Dirac function, S h ( τ ; ∆ r, ∆ t ) is space timecorrelation, delay PSD and C s ( (cid:36), ν, τ ) is spatial-DopplerDoppler delay PSD. Fig. 1 shows the complete CFs and theirsimplifications by the STF WSS assumption.Another important function is the STF-varying spatial-Doppler Doppler delay PSD, since it plays a central role in EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2020 4
Fig. 1. The relationship among CFs for STF-WSS and STF-non-WSSchannels. deriving other correlation/spectrum functions. For example,(3)–(5) are written as R h ( r, t, τ ; ∆ r, ∆ t, ∆ τ )= (cid:90) (cid:90) (cid:90) C s ( r, t, f ; (cid:36), ν, τ ) e j π ( (cid:36) ∆ rλ − + ν ∆ t + f ∆ τ ) d (cid:36) d ν d f (9) R H ( r, t, f ; ∆ r, ∆ t, ∆ f )= (cid:90) (cid:90) (cid:90) C s ( r, t, f ; (cid:36), ν, τ ) e j π ( (cid:36) ∆ rλ − + ν ∆ t − τ ∆ f ) d (cid:36) d ν d τ (10) R s ( (cid:36), ν, τ ; ∆ (cid:36), ∆ ν, ∆ τ )= (cid:90) (cid:90) (cid:90) C s ( r, t, f ; (cid:36), ν, τ ) e j π ( − ∆ (cid:36)rλ − − ∆ νt +∆ τf ) d r d t d f. (11)The STF-varying spatial-Doppler PSD, Doppler PSD, anddelay PSD can be obtained by integrating C s ( r, t, f ; (cid:36), ν, τ ) over other two dispersion domains, and are expressed as G s ( r, t, f ; (cid:36) ) = (cid:90) (cid:90) C s ( r, t, f ; (cid:36), ν, τ )d ν d τ (12) Q s ( r, t, f ; ν ) = (cid:90) (cid:90) C s ( r, t, f ; (cid:36), ν, τ )d (cid:36) d τ (13) P s ( r, t, f ; τ ) = (cid:90) (cid:90) C s ( r, t, f ; (cid:36), ν, τ )d (cid:36) d ν. (14)The STF-varying spatial-Doppler PSD, Doppler PSD, anddelay PSD describe the average power distribution at space r ,time t , and frequency f over the spatial-Doppler, Doppler, anddelay domains, respectively. Note that STF-varying delay PSD P s ( r, t, f ; τ ) is also called STF-varying power delay profile(PDP).III. T HE
3D N ON -S TATIONARY U LTRA -W IDEBAND M ASSIVE
MIMO GBSMLet us consider a massive MIMO communication system.As is shown in Fig. 2, large uniform linear arrays (ULAs)with antenna spacings δ T and δ R are deployed at the Tx andRx, respectively. Symbol β T ( R ) A is the tilt angle of Tx(Rx)antenna array in the xy plane, β T ( R ) E is the elevation angleof the Tx(Rx) antenna array relative to the xy plane. Forclarity, only the n th ( n = 1 , ..., N qp ( t ) ) cluster is shown in Fig. 2. A 3D non-stationary ultra-wideband massive MIMO GBSM. this figure. The n th path is represented by one-to-one pairclusters, i.e., C An at the Tx side and the cluster C Zn at theRx side. N qp ( t ) is the total number of paths in the linkbetween the p th ( p = 1 , ..., M T ) Tx antenna A Tp and the q th ( q = 1 , ..., M R ) Rx antenna A Rq at time instant t . Thepropagation between C An and C Zn is abstracted by a virtuallink [23]. There could be other clusters between C An and C Zn ,introducing more than two reflections/interactions between theTx and Rx. When the delay of the virtual link is zero, themulti-bounce rays reduce to single-bounce rays. In this model,the Tx, Rx, and clusters are allowed to change their velocitiesand trajectories. The movements of the Tx, Rx, C An , and C Zn are described by the speeds v X ( t ) , travel azimuth angles α XA ( t ) , and travel elevation angles α XE ( t ) , respectively. Thesuperscript X ∈ { T, R, A n , Z n } denotes the Tx, Rx, C An ,and C Zn , respectively. The azimuth angle of departure (AAoD)and elevation AoD (EAoD) of the m th ray in C An transmittedfrom A T are denoted by φ TA,m n and φ TE,m n , respectively.Similarly, φ RA,m n and φ RE,m n stand for the azimuth angle ofarrival (AAoA) and elevation AoA (EAoA) of the m th ray inthe C Zn impinging on A R , respectively. The EAoD, EAoA,AAoD, and AAoA of the line-of-sight (LoS) path are denotedby φ TE,L , φ RE,L , φ TA,L , and φ RA,L , respectively. The distancesof A T – S Am n , S Zm n – A R , and A T – A R are denoted by d Tm n , d Rm n , and D , respectively, where S A ( Z ) m n ( m n = 1 , ..., M n )is the m th scatterer in C A ( Z ) n . Note that the above-mentioneddeparture/arrival angles and the distances d Tm n , d Rm n , and D are the initial values at time t . The time-variation of theproposed model is described in the remainder of this section.The parameters in Fig. 2 are defined in Table I. A. Channel Impulse Response
Considering small-scale fading, path loss, shadowing, oxy-gen absorption, and blockage effect, the complete channelmatrix is given by H = [ P L · SH · BL · OL ] · H s , where P L denotes the path loss. Widely used path loss model canbe found in [40], which has been recommended as the pathloss model for 5G systems. SH denotes the shadowing andis modeled as lognormal random variables [40]. The blockageloss BL caused by humans and vehicles is taken from [3].The oxygen absorption loss OL for mmWave and THz com-munications can be found in [41] and [42], respectively. Notethat P L , SH , BL , and OL are in power level, which can betransformed into the corresponding dB values as
10 log ( α ) ,where α ∈ { P L, SH, BL, OL } . EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2020 5
TABLE I. SUMMARY OF KEY PARAMETER DEFINITIONS. A Tp , A Rq The p th Tx antenna element and the q th Rx antenna element, respectively δ T , δ R Antenna spacings of the Tx and Rx arrays, respectively β T ( R ) A , β T ( R ) E Azimuth and elevation angles of the Tx(Rx) antenna array, respectively C An , C Zn The first- and last-bounce clusters of the n th path, respectively v T ( t ) , v R ( t ) , v A n ( t ) , v Z n ( t ) Speeds of the Tx, Rx, C An , and C Zn , respectively α TA ( t ) , α RA ( t ) , α A n A ( t ) , α Z n A ( t ) Travel azimuth angles of the Tx, Rx, C An , and C Zn , respectively α TE ( t ) , α RE ( t ) , α A n E ( t ) , α Z n E ( t ) Travel elevation angles of the Tx, Rx, C An , and C Zn , respectively φ TA,m n , φ TE,m n AAoD and EAoD of the m th ray in C An transmitted from A T at initial time, respectively φ RA,m n , φ RE,m n AAoA and EAoA of the m th ray in C Zn impinging on A R at initial time, respectively φ TA,L , φ TE,L
AAoD and EAoD of the LoS path transmitted from A T at initial time, respectively φ RA,L , φ RE,L
AAoA and EAoA of the LoS path impinging on A R at initial time, respectively d Tm n , d Rm n Distance from A T ( A R ) to C An ( C Zn ) via the m th ray at initial time d Tp,m n ( t ) , d Rq,m n ( t ) Distance from A Tp ( A Rq ) to C An ( C Zn ) via the m th ray at time instant tD Distance from A T to A R at initial time D qp ( t ) Distance from A Tp to A Rq at time instant t The small-scale fading is represented as a complex matrix H s = [ h qp ( t, τ )] M R × M T , where h qp ( t, τ ) is the CIR between A Tp and A Rq and expressed as the summation of the LoS andnon-LoS (NLoS) components, i.e., h qp ( t, τ ) = (cid:114) K R K R + 1 h Lqp ( t, τ ) + (cid:114) K R + 1 h Nqp ( t, τ ) (15)where K R is the K-factor. The NLoS components h Nqp ( t, τ ) can be written as h Nqp ( t, τ ) = N qp ( t ) (cid:88) n =1 M n (cid:88) m =1 (cid:20) F q,V ( φ RE,m n , φ RA,m n ) F q,H ( φ RE,m n , φ RA,m n ) (cid:21) T · (cid:34) e jθ V Vmn (cid:112) µκ − m n e jθ V Hmn (cid:112) κ − m n e jθ HVmn √ µe jθ HHmn (cid:35) (cid:20) F p,V ( φ TE,m n , φ TA,m n ) F p,H ( φ TE,m n , φ TA,m n ) (cid:21) · (cid:113) P qp,m n ( t ) e j πf c τ qp,mn ( t ) · δ ( τ − τ qp,m n ( t )) (16)where {·} T denotes transposition, f c is the carrier frequency, F p ( q ) ,V and F p ( q ) ,H are the antenna patterns of A Tp ( A Rq ) forvertical and horizontal polarizations, respectively. Note thatthe proposed propagation channel model is designed to beantenna independent, which means different antenna patternscan be applied without modifying the basic model framework.Symbol κ m n stands for the cross polarization power ratio[20], µ is co-polar imbalance, θ V Vm n , θ V Hm n , θ HVm n , and θ HHm n areinitial phases with uniform distribution over (0 , π ] , P qp,m n ( t ) and τ qp,m n ( t ) are the powers and delays of the m th ray inthe n th cluster between A Tp and A Rq at time t , respectively.Considering large sizes of antenna array and high-mobilityscenarios, the non-stationarities on time axis and array axishave to be considered. The number of clusters N qp ( t ) , thepower of ray P qp,m n ( t ) , and the propagation delay τ qp,m n ( t ) are modeled as space and time-dependent. The propagationdelay τ qp,m n ( t ) is determined as τ qp,m n ( t ) = d qp,m n ( t ) /c + ˜ τ m n . (17)Here, c denotes the speed of light, ˜ τ m n indicates the delayof the link between S Am n and S Zm n , and is modeled as ˜ τ m n =˜ d m n /c + τ C, link , where ˜ d m n is the distance of S Am n – S Zm n , τ C, link is a non-negative variable randomly generated according toexponential distribution [43]. The travel distance d qp,m n ( t ) is expressed as d qp,m n ( t ) = (cid:107) (cid:126)d Tp,m n ( t ) (cid:107) + (cid:107) (cid:126)d Rm n ,q ( t ) (cid:107) , where (cid:107)·(cid:107) stands for the Frobenius norm, (cid:126)d Tp,m n ( t ) and (cid:126)d Rm n ,q ( t ) are thevector from A Tp to S Am n and the vector from A Rq to S Zm n attime t , respectively. Since the symmetry of the propagation,only the first-bounce propagation between A Tp and S Am n isdescribed. For the sake of clarity, Fig. 3 shows the projectionof propagation between the Tx and S Am n on the xy plane.Considering the time-varying speeds and trajectories of theTx and C An , (cid:126)d Tp,m n ( t ) is calculated as (cid:126)d Tp,m n ( t ) = (cid:126)d Tm n − [ (cid:126)l Tp + (cid:90) t (cid:126)v T ( t ) − (cid:126)v A n ( t )dt] (18)where (cid:126)d Tm n = d Tm n cos( φ TE,m n ) cos( φ TA,m n )cos( φ TE,m n ) sin( φ TA,m n )sin( φ TE,m n ) T (19) (cid:126)l Tp = δ p cos( β TE ) cos( β TA )cos( β TE ) sin( β TA )sin( β TE )] T (20) (cid:126)v T ( t ) = v T ( t ) cos (cid:0) α TE ( t ) (cid:1) cos (cid:0) α TA ( t ) (cid:1) cos (cid:0) α TE ( t ) (cid:1) sin( α TA ( t ))sin (cid:0) α TE ( t ) (cid:1) T (21) (cid:126)v A n ( t ) = v A n ( t ) cos( α A n E ( t )) cos( α A n A ( t ))cos( α A n E ( t )) sin( α A n A ( t ))sin( α A n E ( t ))] T . (22)Here δ p = ( p − δ T , indicates the distance of A Tp – A T .For most cases, the Tx, Rx, and scatterers move in the xy plane. For conciseness, we use v T = (cid:107) (cid:126)v T − (cid:126)v A n (cid:107) and α T = arg { (cid:126)v T − (cid:126)v A n } to denote the relative speed and angle of motionof the Tx w.r.t. C An , respectively, where arg {·} calculates theargument of a 2D vector. When the Tx, Rx and clusters movewith constant speeds along straight trajectories, by extendingthe 2D parabolic wavefront [44] into 3D time non-stationarycase, travel distance d Tp,m n ( t ) = (cid:107) (cid:126)d Tp,m n ( t ) (cid:107) is approximated EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2020 6
Fig. 3. The projection of the propagation between the Tx and S Am n on the xy plane. as d Tp,m n ( t ) ≈ d Tm n − cos( ω Tp ) v T t − cos( ϑ T ) δ p (cid:124) (cid:123)(cid:122) (cid:125) WSS PWF approximation + sin ( ϑ T ) δ p d Tm n (cid:124) (cid:123)(cid:122) (cid:125) SWF term + sin ( ω Tp )( v T t ) d Tm n − cos( ϑ T ) δ p ] (cid:124) (cid:123)(cid:122) (cid:125) non-WSS term (23)where ϑ T is the angle between the the transmit antenna arrayand the m th ray in the n th cluster transmitted from A T , andis calculated as cos( ϑ T ) = cos( φ TE,m n ) cos( β TE ) cos( β TA − φ TA,m n )+ sin( φ TE,m n ) sin( β TE ) . (24)In (23), ω Tp stands for the angle from moving direction of theTx to the m th ray of the n th cluster transmitted from A Tp , andcan be determined as cos( ω Tp ) = d Tm n cos( α T − φ TA,m n ) cos( φ TE,m n ) − δ p cos( α T − β TA ) cos( β TE )[( d Tm n ) − d Tm n δ p cos( ϑ T ) + δ p ] / . (25) Equation (23) gives an efficient and scalable approach formodeling the 3D SWF under time non-stationary assumption.The first term in (23) gives the travel distance of A Tp – S Am n link based on plane wavefront (PWF) and temporal WSSassumptions. The second and third terms account for the non-stationary properties of the channel in the space and timedomains, respectively. Under certain conditions, the traveldistance in (23) can be further simplified.
1) Case I: non-WSS & PWF:
When small antenna arraysare used, i.e., δ p (cid:28) d Tm n , the angle ω Tp becomes constant fordifferent antenna elements and (25) reduces to cos( ω T ) = cos( α T − φ TA,m ) cos( φ TE,m ) . (26)Note that the subscript “ p ” has been omitted for convenience.The SWF term in (23) tends to zero and d Tp,m n ( t ) reduces to d Tp,m n ( t ) ≈ d Tm n − cos( ω T ) v T t − cos( ϑ T ) δ p + sin ( ω T )( v T t ) d Tm n − cos( ϑ T ) δ p ] . (27)
2) Case II: WSS & SWF:
For slow-moving scenarios orshort time periods, i.e. v T t (cid:28) d Tm n , the non-WSS term in(23) tends to zero, which makes the model stationary over thetime. The travel distance in (23) reduces to d Tp,m n ( t ) ≈ d Tm n − cos( ω Tp ) v T t − cos( ϑ T ) δ p + sin ( ϑ T ) δ p d Tm n . (28)
3) Case III: WSS & PWF:
When both time WSS and PWFconditions are fulfilled. The angle ω Tp is simplified accordingto (26). The travel distance in (23) reduces to a fundamentalexpression, which can be found in most of the existing channelmodels as [19], [23], [24], [35] d Tp,m n ( t ) ≈ d Tm n − cos( ω T ) v T t − cos( ϑ T ) δ p . (29)For the Rx side, (cid:126)d Rm n ,q ( t ) , ϑ R , and ω Rq are obtained byreplacing the superscript “ T ” and subscript “ p ” with “ R ” and“ q ” in (18)–(29), respectively. Here, we briefly discuss theinfluence of the Doppler shifts on the proposed model. In (16)the phase rotation associated with time-varying travel distanceis given as ϕ qp,m n ( t ) = 2 πf c τ qp,m n ( t ) , and is decomposedas ϕ qp,m n ( t ) = 2 πf c ( (cid:107) (cid:126)d Tp,m n ( t ) + (cid:126)d Rq,m n ( t ) (cid:107) /c + ˜ τ m n ) . TheDoppler shift can be estimated by f m n ( t ) = d ϕ qp,mn ( t )d t and is time-varying. Considering the WSS & SWF casein (28), the phase rotation can be further expressed as ϕ qp,m n ( t ) = πλ (cid:0) d Tp,m n + ˜ τ m n c + d Tm n ,q (cid:1) − πt ( f Tm n + f Rm n ) ,where d Tp,m n + ˜ τ m n c + d Tm n ,q accounts for the distance of A Tp – S Am n – S Zm n – A Rq link, f Tm n = v T λ cos ω Tp and f Rm n = v R λ cos ω Rq are the Doppler shifts caused by the movement of the Txrelative to C An and the movement of the Rx relative to C Zn ,respectively. Finally, the LoS component in (15) is calculatedas h Lqp ( t, τ ) = (cid:20) F q,V ( φ RE,L , φ
RA,L ) F q,H ( φ RE,L , φ
RA,L ) (cid:21) T · (cid:34) e jθ V VL − e jθ HHL (cid:35) · (cid:20) F p,V ( φ TE,L , φ
TA,L ) F p,H ( φ TE,L , φ
TA,L ) (cid:21) exp { j [2 πf c τ Lqp ( t )] } δ ( τ − τ Lqp ( t )) (30)where θ V VL and θ HHL are random phases with uniform distribu-tion over (0 , π ] , τ Lqp ( t ) are space and time-variant propagationdelay of the LoS path, and determined as τ Lqp ( t ) = D qp ( t ) /c ,where D qp ( t ) = (cid:107) (cid:126)D qp ( t ) (cid:107) is the distance between A Tp and A Rq . The vector (cid:126)D qp ( t ) is calculated as (cid:126)D qp ( t ) = (cid:126)D + (cid:126)l Rq − (cid:126)l Tp + (cid:90) t (cid:126)v R ( t ) − (cid:126)v T ( t )dt (31)where (cid:126)D = [ D, , . When the Tx and Rx travel in the hori-zontal plane with constant velocity, D qp ( t ) can be determinedas [ D qp ( t )] = [ D + cos( α R ) v R t − cos( α T ) v T t − cos( β TA ) cos( β TE ) δ p + cos( β RA ) cos( β RE ) δ q ] + [sin( α R ) v R t − sin( α T ) v T t − cos( β TE ) sin( β TA ) δ p + cos( β RE ) sin( β RA ) δ q ] + [sin( β TE ) δ p − sin( β RE ) δ q ] . (32) EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2020 7
The space and time-varying transfer function H qp ( t, f ) iscalculated as the Fourier transform of h qp ( t, τ ) w.r.t. τ , i.e., H qp ( t, f ) = (cid:114) K R K R + 1 H Lqp ( t, f ) + (cid:114) K R + 1 H Nqp ( t, f ) . (33)For simplicity, we use omnidirectional antennas and considervertical polarization. The LoS and NLoS components of thetransfer function are written as H Lqp ( t, f ) = exp { j πτ Lqp ( t )( f c − f ) } (34) H Nqp ( t, f ) = N qp ( t ) (cid:88) n =1 M n (cid:88) m =1 (cid:113) P qp,m n ( t ) × exp { j [ θ m n + 2 πτ qp,m n ( t )( f c − f )] } . (35)For the case when the system bandwidth is relatively large,e.g., B/f c > , the frequency dependence of the channelcannot be neglected and the US assumption may not befulfilled [30]. A typical approach for the non-US assumption isto model the path gain as frequency-dependent [33]. The NLoScomponents of the space and time-varying transfer function isrewritten as H Nqp ( t, f ) = N qp ( t ) (cid:88) n =1 M n (cid:88) m =1 (cid:113) P qp,m n ( t )( ff c ) γ mn × exp { j [ θ m n + 2 πτ qp,m n ( t )( f c − f )] } (36)where γ m n is a environment-dependent random variable. B. Space and Time-Varying Ray Power
For most of the standard 5G channel models, e.g., [9], [20],and [41], the cluster powers are constant for different antennaelements, which may be inconsistent with the measurementresults [10]. Based on [20], the ray power between A Tp and A Rq at time t is given as P (cid:48) qp,m n ( t ) = exp (cid:18) − τ qp,m n ( t ) r τ − r τ DS (cid:19) − Zn · ξ n ( q, p ) (37)where Z n is the per cluster shadowing term in dB, DS is the root mean square (RMS) delay spread, r τ denotesthe delay distribution proportionality factor and determinedas the ratio of the standard deviation of the delays to theRMS delay spread [20]. The smooth power variations overthe transmit and receive arrays can be simulated by a 2Dspatial lognormal process ξ n ( q, p ) , and can be calculated as ξ n ( q, p ) = 10 [ µ n ( q,p )+ σ n · s n ( q,p )] / where µ n ( q, p ) is thelocal mean and s n ( q, p ) is a 2D Gaussian process, whichaccount for the path loss and shadowing along the large arrays,respectively. The final ray powers are obtained by normalizing P (cid:48) qp,n m ( t ) as P qp,m n ( t ) = P (cid:48) qp,m n ( t ) (cid:14) N qp ( t ) (cid:88) n =1 M n (cid:88) m n =1 P (cid:48) qp,m n ( t ) . (38)The space and time-varying ray power can be simplifiedunder certain condition. For example, when conventional an-tenna array is employed at the Rx, the 2D spatial lognormal process ξ n ( q, p ) reduces to one-dimensional (1D) process ξ n ( p ) . Imposing ξ n ( q, p ) = 1 indicates farfield condition isfulfilled at both ends. Furthermore, if the delays within acluster are unresolvable, the cluster power can be generatedby replacing τ qp,n m ( t ) with cluster delay τ qp,n ( t ) , where τ qp,n ( t ) = [ (cid:80) M n m n =1 τ qp,m n ( t )] /M n . The ray powers withina cluster are equally determined as P (cid:48) qp,m n ( t ) = 1 M n exp (cid:18) − τ qp,n ( t ) r τ − r τ DS (cid:19) − Zn · ξ n ( q, p ) . (39)Finally, the ray powers are normalized as (38). C. Unified Space-Time Evolution of Clusters
Channel measurements have shown that in high-mobilityscenarios, e.g., V2V and HST scenarios, clusters exhibit abirth-death behavior over time [21]. In massive MIMO com-munication systems, similar properties can be observed onthe array axis [6]. Here, the space-time cluster evolutions aremodeled in a uniform manner. For the Tx side, the probabilityof a cluster remains over time interval ∆ t and antenna elementspacing δ p can be calculated as P T remain (∆ t, δ p )= exp (cid:16) − λ R [( ε T ) + ( ε T ) − ε T ε T cos( α TA − β TA )] (cid:17) . (40)The process is described by the cluster generation rate λ G and cluster recombination (disappearance) rate λ R , which canbe estimated as in [45]. Note that λ G and λ R are relatedto characteristics of scenarios and antenna patterns. In (40), ε T = δ p cos( β TE ) /D Ac and ε T = v T ∆ t/D Sc characterize theposition differences of transmit antenna element on array andtime axes, respectively. Symbols D Ac and D Sc are scenario-dependent correlation factors in the array and time domains,respectively. Typical values of D Ac and D Sc such as 10 m and30 m can be chosen, which are the same order of correlationdistances in [9], [20].For the Rx side, the probability of a cluster exist over timeinterval ∆ t and element spacing δ q , i.e., P R remain (∆ t, δ q ) , iscalculated similarly. Since each antenna element has its ownobservable cluster set, only a cluster can be seen by at leastone Tx antenna and one Rx antenna, it can contribute to thereceived power. Therefore, the joint probability of a clusterexist over ∆ t and δ q is calculated as P remain (∆ t, δ p , δ q ) = P T remain (∆ t, δ p ) · P R remain (∆ t, δ q ) . (41)The mean number of newly generated clusters is obtained by E { N new } = λ G λ R [1 − P remain (∆ t, δ p , δ q )] . (42) D. Ellipsoid Gaussian Scattering Distribution
The Gaussian scatter density model (GSDM) has widelybeen used in channel modeling for various communicationscenarios and validated by the measurement data [46], [47].In GSDM, the scatterers are gathered around their centerand usually modeled by certain shapes, e.g., discs in 2D
EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2020 8 models and spheres in 3D models [48]. However, channelmeasurements indicate that the spatial dispersions of scattererswithin a cluster, which can be described by cluster angularspread (CAS), cluster elevation spread (CES), and clusterdelay spread (CDS), are usually unequal [3], [19]. Basedon the aforementioned assumption, the positions of scattererscentering on the origin of coordinates are modeled as p ( x (cid:48) , y (cid:48) , z (cid:48) ) = exp (cid:16) − x (cid:48) σ DS − y (cid:48) σ AS − z (cid:48) σ ES (cid:17) (2 π ) / σ DS σ AS σ ES (43)where σ DS , σ AS , and σ ES are the standard derivations of theGaussian distributions and characterize the CDS, CAS, andCES, respectively. The scatterers centering around the spher-ical coordinates ( d, φ E , φ A ) can be obtained by shifting theabove-mentioned cluster using the following transformation xyz = cos( φ A ) − sin( φ A ) 0sin( φ A ) cos( φ A ) 00 0 1 · cos( φ E ) 0 − sin( φ E )0 1 0sin( φ E ) 0 cos( φ E ) · x (cid:48) − dy (cid:48) z (cid:48) (44)where d denotes the distance from the Tx/Rx to the center ofthe cluster, φ E and φ A are the mean values of the elevationangles and azimuth angles of scatterers, respectively. Note thatthe orientation of the cluster toward the Tx/Rx is constantthrough the aforementioned transformation, which ensuresthe values of CDS, CAS, and CES remain unchanged. Bysubstituting x = d cos( φ E ) cos( φ A ) , y = d cos( φ E ) sin( φ A ) ,and z = d sin( φ E ) in to (44), after some manipulations, theangle distance joint distribution can be obtained as p ( d, φ E , φ A ) = | J ( x (cid:48) , y (cid:48) , z (cid:48) ) | · p ( x (cid:48) , y (cid:48) , z (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) x (cid:48) = d [cos( φ E ) cos( φ E ) cos( φ A − φ A ) + sin( φ E ) sin( φ E )] − dy (cid:48) = d cos( φ E ) sin( φ A − φ A ) z (cid:48) = d [sin( φ E ) cos( φ E ) − cos( φ A − φ A ) cos( φ E ) sin( φ E )] (45)where J ( x (cid:48) , y (cid:48) , z (cid:48) ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂x (cid:48) ∂d ∂x (cid:48) ∂φ E ∂x (cid:48) ∂φ A ∂y (cid:48) ∂d ∂y (cid:48) ∂φ E ∂y (cid:48) ∂φ A ∂z (cid:48) ∂d ∂z (cid:48) ∂φ E ∂z (cid:48) ∂φ A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − d cos( φ E ) . (46)By imposing ˜ τ m n = 0 in (17), the multi-bounce rays reduceto single-bounce rays. The delay angle joint distribution for thecluster can be obtained based on (45) by transform parameter d into τ , i.e., p ( τ, φ XE , φ XA ) = | J ( d X ) | · p ( d X , φ XE , φ XA ) (47)where d X = ( τ c ) − D τ c − D cos( φ XE ) cos( φ XA ) (48) J ( d X ) = d d X d τ = c [ D + ( τ c ) − Dcτ cos( φ XE ) cos( φ XA )]2[ τ c − D cos( φ XE ) cos( φ XA )] (49) (a) −50050 −100 0 10001234 AAoD, φ TA,m n (deg)EAoD, φ TE,m n (deg) F r equen cy o f o cc u rr en c e (b) Fig. 4. (a) Theoretical and (b) simulated ellipsoid Gaussian scatteringdistribution ( σ DS = 8 , σ AS = 10 , σ ES = 6 , d ∼ N (100 , m). and X ∈ { T, R } . The angular parameters and travel distancesof the first- and last-bounce propagations becomes interdepen-dent. The relationship between them can be expressed as tan( φ TA ) = d R cos( φ RE ) sin( φ RA ) D + d R cos( φ RE ) cos( φ RA ) (50) tan( φ TE ) = d R sin( φ RE ) { D + [ d R cos( φ RE )] + 2 D · d R cos( φ RE ) cos( φ RA ) } / (51) d T = [ D + ( d R ) + 2 D · d R cos( φ RE ) sin( φ RA )] / . (52)Note that the subscripts m n are omitted for clarity. Unlikethe WINNER/3GPP channel models [19], [20], [35], wherethe clusters are randomly generated for every link, in theproposed model, the clusters are assumed to physically existin the propagation environments. Thus, spatial consistency canbe achieved based on the locations of clusters. This makes itpossible to prediction channel state information (CSI) basedon the channel associated with nearby users or in the previoustime snapshots. For instance, in HST scenarios, different trainstravel to the same location of the track will see similar envi-ronments and hence have similar channel behaviors. Channelcan be estimated from the communication process of last trainsor nearby remote radio heads (RRHs) [49].Fig. 4 shows the theoretical and simulated ellipsoid Gaus-sian scattering distribution. The mean angles, i.e., φ E and φ A are obtained from [20] in urban micro-cell scenario,NLoS case. The distances between the Tx and the center ofthe first-bounce cluster follows a Gaussian distribution, i.e., N (100 , m. The simulated result is obtained using MonteCarlo method and a total of 500 rays are generated. A goodconsistency between theoretical and simulated results can beobserved.By adjusting the model parameters or components, theproposed model can be applied to various scenarios. Let’stake mmWave-THz scenario as an example. The path loss,shadowing, oxygen absorption, and blockage attenuation com-ponents can be replaced with those at mmWave-THz bands.The sparsity of MPCs can be represented by adjusting thenumber of clusters and the number of rays within a cluster.The remarkable birth-death behaviour of clusters over timeresulting from large propagation loss can be modeled byincreasing the cluster generation rate λ G and recombinationrate λ R . The antenna patterns in (16) can be replaced with EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2020 9 those of high-directional antennas, which are often used inmmWave-THz communications.IV. S
TATISTICAL P ROPERTIES
A. Local STF-CF
The local STF-CF between H qp ( t, f ) and H ˜ q ˜ p ( t − ∆ t, f − ∆ f ) is defined as R qp, ˜ q ˜ p ( t, f ; ∆ r, ∆ t, ∆ f ) = E { H qp ( t, f ) H ∗ ˜ q ˜ p ( t − ∆ t, f − ∆ f ) } . (53)By substituting (33) into (53), the STF-CF is further writtenas R qp, ˜ q ˜ p ( t, f ; ∆ r, ∆ t, ∆ f ) = K R K R + 1 R Lqp, ˜ q ˜ p ( t, f ; ∆ r, ∆ t, ∆ f )+ 1 K R + 1 N qp ( t ) (cid:88) n =1 R Nqp, ˜ q ˜ p,n ( t, f ; ∆ r, ∆ t, ∆ f ) (54)where the LoS and NLoS components of the STF-CF can beobtained as R Lqp,q ˜ p ( t, f ; ∆ r, ∆ t, ∆ f ) = [ P Lqp ( t ) P L ˜ q ˜ p ( t − ∆ t )] × e j π ( fc − f ) λfc [ d Lqp ( t ) − d L ˜ q ˜ p ( t − ∆ t )] − j π ∆ fλfc d L ˜ q ˜ p ( t − ∆ t ) (55) R Nqp,q ˜ p,n ( t, f ; ∆ r, ∆ t, ∆ f ) = P remain (∆ t, ∆ r ) · E { M n (cid:88) m n =1 a m n e j π ( fc − f ) λfc [ d qp,mn ( t ) − d ˜ q ˜ p,mn ( t − ∆ t )] − j π ∆ fλfc d ˜ q ˜ p,mn ( t − ∆ t ) } (56)where ∆ r = { ∆ r T , ∆ r R } , ∆ r T = δ ˜ p − δ p , ∆ r R = δ ˜ q − δ q , a m n = [ P qp,m n ( t ) P ˜ q ˜ p,m n ( t − ∆ t )] [ f ( f − ∆ f ) f c ] γ mn , P remain (∆ t, ∆ r ) is the joint probability of a cluster survivesfrom t − ∆ t to t on time axis and from A Tp to A T ˜ p and from A Rq to A R ˜ q on array axes.For the stationary case, i.e., the model is stationary over r , t , and f . We further assume that the delays within a clusterare irresolvable. By setting P remain (∆ t, ∆ r ) = 1 , γ m n = 0 , a m n = 1 /M n and removing the SWF and non-WSS terms in(23), the NLoS components of STF-CF reduces to R Nqp, ˜ q ˜ p,n ( t, f ; ∆ r, ∆ t, ∆ f ) = E { M n (cid:88) m n =1 M n · e j π ( fc − f ) λfc [cos ϑ R ∆ r R +cos ϑ T ∆ r T − (cos ω T v T +cos ω R v R )∆ t ] · e − j π ∆ fλfc d ˜ q ˜ p,mn ( t − ∆ t ) } . (57)Note that the CF in this case is still STF-dependent. Byimposing ∆ f = 0 , i.e., removing the frequency selectivity, theCF becomes WSS in the space and time domains, i.e., onlyrelies on ∆ r and ∆ t . Similarly, the CF is WSS in the frequencydomain when the time selectivity and space selectivity areignored, i.e., setting ∆ t = 0 , ∆ r = 0 . B. Local Spatial-Doppler PSD
The local spatial-Doppler PSD can be obtained as theFourier transform of spatial CF R q,p ˜ p,n ( t, f ; ∆ r, , w.r.t. ∆ r . For the Tx side, the local spatial-Doppler PSD is obtainedas G q,p ˜ p,n ( t, f ; (cid:36) ) = (cid:90) R q,p ˜ p,n ( t, f ; ∆ r T , , e − j πλ (cid:36) ∆ r T d∆ r T . (58)Note that ∆ r R = 0 indicates two links share the same receiveantenna. The local spatial-Doppler PSD in (58) describes thedistribution of average power on the spatial-Doppler frequencyaxis at antenna A Tp , time t , and frequency f . C. Local Doppler Spread
The instantaneous frequency provides a measure of theenergy distribution of a signal over the frequency domain andis important for signal recognition, estimation, and modeling.The instantaneous frequency, which is given by the instanta-neous Doppler frequency, is estimated as d ϕ ( t )2 π d t , where ϕ ( t ) accounts for the phase change of the channel [50]. Basedon (16), the instantaneous Doppler frequency of the proposedmodel is given by ν qp,m n ( t ) = λ d[ d Tp,mn ( t )+ d Rmn,q ( t )]d t , and isfurther expressed as ν qp,m n ( t ) = − v T λ cos( ω Tp ) − v R λ cos( ω Rq )+ sin ( ω Tp )( v T ) tλ ( d Tm n − cos( ϑ T ) δ p ) + sin ( ω Rq )( v R ) tλ ( d Rm n − cos( ϑ R ) δ q ) . (59)Note that the instantaneous Doppler frequency varies with timecaused by the movements of the Tx, Rx, and scatterers. Theadvantage of (59) w.r.t. other channel models such as [17] and[9] lies in that the Doppler frequency can be written as thesummation of two components. The first two terms of (59) arethe conventional Doppler frequency expression in stationarycase. The last two terms of (59) accounts for the time-variationof the Doppler frequency in the non-stationary case. Finally,the local Doppler spread can be calculated as B (2) qp ( t ) = (cid:0) E [ ν qp,m n ( t ) ] − E [ ν qp,m n ( t )] (cid:1) . (60) D. Array Coherence Distance
As a counterpart of coherence time and coherence band-width, the array coherence distance is the minimum antennaelement spacing during which the spatial CF equals to a giventhreshold c thresh . The transmit antenna array coherence distancecan be expressed as [38] I p ( c thresh ) = min { (cid:12)(cid:12) ∆ r T (cid:12)(cid:12) : (cid:12)(cid:12) R q,p ˜ p,n ( t, f ; ∆ r T , , (cid:12)(cid:12) = c thresh } (61)where c thresh ∈ [0 , . The receive antenna array coherence dis-tance can be calculated similarly. In (61), small values of c thresh results in the minimum distance between two antenna elementsover which the channels can be considered as independent.However, larger values of c thresh provide the maximum antennaspacing within which the channels do not change significantly. EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2020 10
Time difference, t (s) -3 Lo c a l t e m po r a l C F Analytical, t=0sSimulation, t=0sAnalytical, t=1sSimulation, t=1sAnalytical, t=2sSimulation, t=2s (a)
Frequency difference, f (MHz) Lo c a l f r equen cy C F Analytical, 58 GHzSimulation, 58 GHzAnalytical, 60 GHzSimulation, 60 GHZAnalytical, 62 GHzSimulation, 62 GHz (b)
Antenna index, p Lo c a l s pa t i a l C F B5GCM, AnalyticalB5GCM, SimulationMeasurement3GPP TR38.901 (c)Fig. 5. Local temporal, frequency, and spatial CFs of the B5GCM,and measurement data in [5] ( v T ( R ) = 10 m/s σ DS = 6 . m, σ AS = 11 . m, σ ES = 9 . m). V. R
ESULTS AND A NALYSIS
In this section, results of important statistics of the B5GCMare presented. Some of the statistics including spatial CF,cluster VR length, cluster power variation, Doppler spread,and delay spread are compared with the corresponding channelmeasurement data. In the simulation, the parameters such ascarrier frequency, antenna height, Tx-Rx separation, and veloc-ity of the Tx/Rx are set according to the corresponding mea-surements. Only a small number of parameters, such as λ R , D Ac , σ n , σ AS , σ ES , and σ DS , which differentiate the proposedmodel as compared to conventional ones, are determined byfitting statistical properties to the channel measurement data.The rest of parameters are randomly generated according to the3GPP TR38.901 channel model. Unless otherwise noted, thefollowing parameters are used for simulation: f c = 2 . GHz, M T = 128 , M R = 1 , β TA = π/ , β TE = 0 , µ = 1 , D = 100 m, λ R = 6 . /m, λ G = 81 . /m, D Ac = 9 . m, σ n = 0 . . In addition, and half-wave dipole antennas withvertical polarization are adopted in simulations.For the parameters listed above, the local temporal, fre-quency, and spatial CFs of the proposed model are shownin Fig. 5. Specifically, the local temporal CFs at 0 s, 1 sand 2 s are shown in Fig. 5(a). Note that the analyticalresults are generated by imposing ∆ r = 0 and ∆ f = 0 in (53). The simulation results are obtained based on two VR length on the Tx array, (m) CD F o f t he V R l eng t h Simulation, R = 4/mSimulation, R = 6.79/mSimulation, R = 12/mMeasurement Fig. 6. CDF of the VR length of the proposed B5GCM and themeasurement data in [12] ( λ R = 6 . /m, λ G = 81 . /m, D Ac =9 . m, σ n = 0 . ). channel transfer functions separated by different time. Thetime-variations of temporal CFs result from the motions ofthe Tx, Rx and the survival probability of the cluster, whichmake the model non-stationary in the time domain. Fig. 5(b)presents the frequency CFs of the B5GCM. The frequencyCFs vary with frequency due to the frequency dependenceof the path gains, indicating the frequency non-stationarityof the proposed model. Fig. 5(c) provides the comparison ofthe local spatial CFs of the B5GCM, 3GPP TR38.901 [20],and the measurement data [5]. Note that the space differenceshave been normalized w.r.t. antenna spacing. The measurementwas carried out at 2.6 GHz in a court yard scenario, where a7.3 m 128-element virtual ULA is used. The antenna formingthe virtual ULA is spaced at half-wavelength and illustratesomnidirectional pattern in the azimuth plane. The result showsthat the spatial CFs of the B5GCM provide a better consistencywith the measurement data than those of the 3GPP model.This is because the 3GPP model neglected the effect of SWF.Besides, the non-stationarity over large antenna array wasneglected.The simulated cumulative distribution functions (CDFs) ofVR length and slope of cluster power variation on the arrayare presented in Figs. 6 and 7, respectively. The measurementused for comparison was carried out at 2.6 GHz in a campusscenario, where an omnidirectional antenna moves along a railwith a half-wavelength spacing, constituting a 128-elementvirtual ULA [12]. The model parameters were chosen byminimizing the error norm ε = (cid:80) m =1 w m E {| ˆ F m − F m ( P ) | } ,where ˆ F m and F m are the measured and derived statistics,respectively, w m is the weight of the m th error norm andsatisfying w + w = 1 , P = { λ R , D Ac , σ n } is parameter setto be jointly optimized. Note that we impose λ G /λ R = 12 toensure a constant cluster number along the array. It is foundthat λ R = 6 . /m, D Ac = 9 . m, and σ n = 0 . can bechosen as a good match. The results show that increasingthe cluster recombination rate leads to a shorter VR, whichindicates a larger spatial non-stationarity. Furthermore, results EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2020 11 -4 -3 -2 -1 0 1 2 3 4
Slope of cluster power variation, (dB/m) CD F o f s l ope o f c l u s t e r po w e r v a r i a t i on B5GCM, n = 0.02B5GCM, n = 0.054B5GCM, n = 0.1MG5GCMMeasurement Fig. 7. CDF of the slopes of cluster power variations of the proposedB5GCM, MG5GCM in [9], and the measurement data in [12] ( λ R =6 . /m, λ G = 81 . /m, D Ac = 9 . m, σ n = 0 . ). -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Spatial-Doppler frequency, -18-16-14-12-10-8-6-4-20 N o r m a li z ed s pa t i a l - D opp l e r PS D ( d B ) SWF, A T1 SWF, A
T100
SWF, A
T200
PWF, A T1 PWF, A
T100
PWF, A
T200
Fig. 8. The simulated normalized spatial-Doppler PSDs of theB5GCM at A T , A T , and A T using SWF and PWF assumptions( φ TA = 2 π/ , φ TE = π/ , λ R = 6 . /m, λ G = 81 . /m, D Ac = 9 . m, σ n = 0 . ). in Fig. 7 suggest that large values of σ n can increase the clusterpower variation over the array. However, the channel modelin [9] assumed that the cluster powers are constant along thearray, which may underestimate the spatial non-stationarity ofmassive MIMO channels.Fig. 8 shows the simulated normalized spatial-DopplerPSDs, which are obtained according to (58). For the SWFcase, the variations of spatial-Doppler PSDs along the transmitarray are caused by the large size of the transmit antennaaperture. However, for the PWF case, the values of spatial-Doppler PSDs are constant over the array, which may resultin inaccurate performance estimations of massive MIMO sys-tems. Besides, Fig. 9 shows the simulated array coherencedistance based on (61). Note that the coherence distances havebeen normalized w.r.t. antenna spacing. The results indicatethat the spatial non-stationarity, which is caused by SWF and Threshold, c
Thresh N o r m a li z ed c ohe r en c e d i s t an c e o v e r t he T x a rr a y β TA = 90 ◦ β TA = 80 ◦ β TA = 70 ◦ d T =200 md T =100 m Fig. 9. Simulated coherence distances of the B5GCM over array( φ TA = π/ , φ TE = π/ , λ R = 6 . /m, λ G = 81 . /m, D Ac = 9 . m, σ n = 0 . ). Effective speed, v eff (m/s) Lo c a l D opp l e r s p r ead , ( H z ) B5GCM, AnalyticalB5GCM, SimulationMeasurement
Fig. 10. Doppler spread of the B5GCM versus effective speed andthe measurement data in [51] ( f c = 5 . GHz, σ DS = 81 . m, σ AS = 88 . m, σ ES = 72 . m, α T = 0 , α R = π , v A n = 0 m/s, v Z n = 2 m/s, α Z n = 0 ). cluster array evolution, is affected by both the array orientationand the Tx/Rx-cluster separation. The channel has a shorterarray coherence distance when distance from the Tx/Rx to thecluster decreases. Furthermore, increasing the angles betweenarray orientation and rays can lead to a larger array coherencedistance.Fig. 10 compares the Doppler spread of the B5GCM withthe measurement data [51]. The Doppler spread is obtainedaccording to (60). The channel measurement was conductedat 5.9 GHz in highway, rural, and suburban environments.The x -axis of this figure is the effective speed, which isdefined as v eff = [( v T ) + ( v R ) ] . A good consistencyamong the simulated, analytical results, and the correspondingmeasurement data can be observed. Noting that the Dopplerspread illustrates a nonzero value when v eff = 0 . It stems from EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2020 12
Fig. 11. A snapshot of the APS of AoD of the B5GCM ( λ R =6 . /m, λ G = 81 . /m, D Ac = 9 . m, σ n = 0 . ). the extra Doppler shifts due to the motion of scatterers, andcannot be obtained by the models assuming static clusters [19],[20], [35].The simulated angle power spectrum (APS) of AoD ofthe B5GCM is shown in Fig. 11. The result is obtainedusing the multiple signal classification (MUSIC) algorithm[52]. A sliding window consisting of 12 consecutive antennasis shifted along the array in order to capture the channelnon-stationaries in the space domain. Besides, the birth-deathprocess of clusters along the transmit array can be seen. Someclusters with strong powers are observable along the wholearray. Other weak power clusters only appear to part of thearray. The power of clusters vary smoothly over the array canbe observed due to the spatial lognormal process. Moreover,the angles of rays experience linear drifts along the arraycaused by the nearfield effects, which has been validated byseveral channel measurement campaigns [5], [6].The CDF of the RMS delay spreads of the B5GCM andthe measurement data in [53] are compared in Fig. 12. Themeasurements were conducted at 58 GHz in three indoorscenarios, i.e., Lecture room (Room G), Laboratory room(Room H), and Lecture room (Room F). Both the Tx andRx antennas are equipped with motionless bicone antennas.The different RMS delay spreads for the three propagationenvironments are caused by different distributions of scattererswithin cluster. Good agreements between the results of theB5GCM and measurement data show the usefulness of theproposed model. VI. C ONCLUSIONS
This paper has proposed a novel 3D STF non-stationaryGBSM for 5G and B5G wireless communication systems.The proposed model is applicable to various communicationscenarios, e.g., massive MIMO, HST, V2V, and mmWave-THzcommunication scenarios. Important (B)5G channel character-istics have been integrated, including SWF, cluster power vari-ation over array, Doppler shifts caused by motion of scatterers,time-variant velocity and trajectory, and spatial consistency. −8 RMS delay spread, τ (s) CD F o f R M S de l a y s p r ead B5GCMMeasurementG H F
Fig. 12. RMS delay spread statistics of the proposed B5GCM andmeasurement data in [53] ( f c = 58 GHz, D = 3 m, d T ∼ N (3 , . m, Room G: σ DS = 1 . m, σ AS = 1 . m, σ ES = 1 . m, Room H: σ DS = 2 . m, σ AS = 1 . m, σ ES = 1 . m, Room F: σ DS = 3 . m, σ AS = 2 . m, σ ES = 1 . m. ) Note that the above-mentioned channel characteristics havenot been fully considered in the current 5G channel models,e.g., MG5GCM [9], 3GPP TR38.901 [20], and IMT-2020channel models [41]. Furthermore, this paper has presenteda general modeling framework. The model can reduce to avariety of simplified channel models according to channelproperties of specific scenarios, or be applied to new com-munication scenarios by setting appropriate model parameters.Key statistics of the proposed model have been derived, someof which have been validated by measurement data, illustratingthe generalization and usefulness of the proposed model.R
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Ji Bian (M’20) received the B.Sc. degree in elec-tronic information science and technology fromShandong Normal University, Jinan, China, in 2010,the M.Sc. degree in signal and information process-ing from Nanjing University of Posts and Telecom-munications, Nanjing, China, in 2013, and the Ph.D.degree in information and communication engineer-ing from Shandong University, Jinan, China, in2019. From 2017 to 2018, he was a visiting scholarwith the School of Engineering and Physical Sci-ences, Heriot-Watt University, Edinburgh, U.K. Heis currently a lecturer with the School of Information Science and Engineering,Shandong Normal University, Jinan, China. His research interests include 6Gchannel modeling and wireless big data.
Cheng-Xiang Wang (S’01-M’05-SM’08-F’17) re-ceived the BSc and MEng degrees in Commu-nication and Information Systems from ShandongUniversity, China, in 1997 and 2000, respectively,and the PhD degree in Wireless Communicationsfrom Aalborg University, Denmark, in 2004.He was a Research Assistant with the HamburgUniversity of Technology, Hamburg, Germany, from2000 to 2001, a Visiting Researcher with SiemensAG Mobile Phones, Munich, Germany, in 2004, anda Research Fellow with the University of Agder,Grimstad, Norway, from 2001 to 2005. He has been with Heriot-Watt Uni-versity, Edinburgh, U.K., since 2005, where he was promoted to a Professorin 2011. In 2018, he joined Southeast University, China, as a Professor. Heis also a part-time professor with the Purple Mountain Laboratories, Nanjing,China. He has authored four books, three book chapters, and more than 410papers in refereed journals and conference proceedings, including 24 HighlyCited Papers. He has also delivered 22 Invited Keynote Speeches/Talks and7 Tutorials in international conferences. His current research interests includewireless channel measurements and modeling, 6G wireless communicationnetworks, and applying artificial intelligence to wireless communicationnetworks.Prof. Wang is a Member of the Academia Europaea (The Academy ofEurope), a fellow of the IET, an IEEE Communications Society DistinguishedLecturer in 2019 and 2020, and a Highly-Cited Researcher recognized byClarivate Analytics, in 2017-2020. He is currently an Executive Edito-rial Committee member for the IEEE TRANSACTIONS ON WIRELESSCOMMUNICATIONS. He has served as an Editor for nine internationaljournals, including the IEEE TRANSACTIONS ON WIRELESS COMMUNI-CATIONS from 2007 to 2009, the IEEE TRANSACTIONS ON VEHICULARTECHNOLOGY from 2011 to 2017, and the IEEE TRANSACTIONS ONCOMMUNICATIONS from 2015 to 2017. He was a Guest Editor for the IEEEJOURNAL ON SELECTED AREAS IN COMMUNICATIONS, Special Issueon Vehicular Communications and Networks (Lead Guest Editor), SpecialIssue on Spectrum and Energy Efficient Design of Wireless CommunicationNetworks, and Special Issue on Airborne Communication Networks. He wasalso a Guest Editor for the IEEE TRANSACTIONS ON BIG DATA, SpecialIssue on Wireless Big Data, and is a Guest Editor for the IEEE TRANS-ACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING,Special Issue on Intelligent Resource Management for 5G and Beyond. Hehas served as a TPC Member, TPC Chair, and General Chair for over 80international conferences. He received twelve Best Paper Awards from IEEEGLOBECOM 2010, IEEE ICCT 2011, ITST 2012, IEEE VTC 2013-Spring,IWCMC 2015, IWCMC 2016, IEEE/CIC ICCC 2016, WPMC 2016, WOCC2019, IWCMC 2020, and WCSP 2020.
Xiqi Gao (S’92-AM’96-M’02-SM’07-F’15) re-ceived the Ph.D. degree in electrical engineeringfrom Southeast University, Nanjing, China, in 1997.He joined the Department of Radio Engineering,Southeast University, in April 1992. Since May2001, he has been a professor of information sys-tems and communications. From September 1999to August 2000, he was a visiting scholar at Mas-sachusetts Institute of Technology, Cambridge, andBoston University, Boston, MA. From August 2007to July 2008, he visited the Darmstadt Universityof Technology, Darmstadt, Germany, as a Humboldt scholar. His currentresearch interests include broadband multi-carrier communications, MIMOwireless communications, channel estimation and turbo equalization, andmulti-rate signal processing for wireless communications. From 2007 to 2012,he served as an Editor for the IEEE TRANSACTIONS ON WIRELESSCOMMUNICATIONS. From 2009 to 2013, he served as an AssociateEditor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING. From2015 to 2017, he served as an Editor for the IEEE TRANSACTIONS ONCOMMUNICATIONS.Dr. Gao received the Science and Technology Awards of the State EducationMinistry of China in 1998, 2006 and 2009, the National TechnologicalInvention Award of China in 2011, and the 2011 IEEE CommunicationsSociety Stephen O. Rice Prize Paper Award in the field of communicationstheory.
Xiaohu You (SM’11-F’12) has been working withNational Mobile Communications Research Labora-tory at Southeast University, where now he holds therank of director and professor. He has contributedover 300 IEEE journal papers and 3 books in theareas of signal processing and wireless communi-cations. From 1999 to 2002, he was the PrincipalExpert of the C3G Project. From 2001–2006, hewas the Principal Expert of the China National 863Beyond 3G FuTURE Project. Since 2013, he hasbeen the Principal Investigator of China National863 5G Project.Professor You served as the general chairs of IEEE WCNC 2013, IEEEVTC 2016 Spring and IEEE ICC 2019. Now he is Secretary General of theFuTURE Forum, vice Chair of China IMT-2020 (5G) Promotion Group, viceChair of China National Mega Project on New Generation Mobile Network.He was the recipient of the National 1st Class Invention Prize in 2011, andhe was selected as IEEE Fellow in same year.