A Complete Study of Space-Time-Frequency Statistical Properties of the 6G Pervasive Channel Model

The sixth generation (6G) pervasive channel model (6GPCM) can characterize channels for all spectra from the sub-6 GHz band to the visible light communication (VLC) band and all scenarios, such as maritime, (ultra-)massive multiple-input multiple-output (MIMO), and industrial Internet of things (IIoT) communication scenarios in 6G wireless systems. The unified channel model can enable us to analyze channel statistical properties in systems using different scales of antenna arrays, different frequency bands, and different scenarios with different movement speeds. In this paper, we conduct a complete study on space-time-frequency (STF) statistical properties of the 6GPCM. Mathematical derivations and simulations are provided, including STF correlation function (STFCF), spatial/temporal/frequency correlation functions, angular/Doppler/delay power spectral densities (PSDs), root mean square (RMS) angular/Doppler/delay spreads, coherence distance/time/bandwidth, stationary distance/time/bandwidth, and level-crossing rates (LCRs)/average fade durations (AFDs) in STF domains. In addition, we classify these statistical properties according to their definitions and then reveal the complex relationships between them and channel model parameters. This work will lay a solid foundation and offer useful guidelines for research on 6G wireless communication systems.


I. INTRODUCTION
T HE sixth generation (6G) wireless communication sys- tems are expected to utilize new frequency bands, such as the terahertz (THz) [1] and visible light communication (VLC) [2] bands, and new technologies, such as ultra-massive multiple-input multiple-output (MIMO) [3] and reconfigurable intelligent surface (RIS) [4] to serve space-air-ground-sea integrated communication networks [5].An insightful understanding of the underlying properties of 6G channels between transmitters (Txs) and receivers (Rxs) is a prerequisite for the system design, theoretical analysis, performance evaluation, optimization, and deployment of 6G wireless communication systems [6], [7].
However, there lacks a comprehensive and in-depth analysis of all 6G channel statistical properties.Most of the studies only focus on a portion of these channel properties.For example, authors in [8] introduced and explained correlation functions (CFs) and the corresponding power spectrum densities (PSDs) of a general space-time-frequency (STF) non-stationary channel.Other channel properties, such as the delay spread, Doppler spread, stationary time, were not studied.Note that in this paper, stationary time refers to the stationary interval in the time domain in order to distinguish it from stationary intervals in the space and frequency domains.Also, level crossing rate (LCR) and average fade duration (AFD) of a three dimensional (3D) unmanned aerial vehicle (UAV) channel model in the time domain were investigated in [9].The angular spreads and delay spreads of the urban macro (UMa) channels in two typical street canyon scenarios were investigated based on extensive field measurements and modeling in [10].The authors in [11] discussed and analyzed the channel statistical properties related to directional dispersion and spatial selectivity, such as angular PSD, angular spread, and coherence distance of wireless channel.Their relationships were also analyzed.Moreover, in [12], the authors derived and analyzed the different time-variant channel statistical properties in the high-speed train (HST) tunnel propagation scenario, such as spatial cross-correlation function (SCCF), Doppler PSD, and temporal autocorrelation function (TACF).In [13], the LCRs and AFDs of different UAV scenarios and their impact factors were studied.All these works have only studied part of the channel statistical properties in limited domains or scenarios.Besides, none of these studies has analyzed the relationships 0090-6778 © 2023 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.
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between channel parameters and channel statistical properties, except in very few cases.
The 6G pervasive channel model (6GPCM) [14] was proposed as a 3D cluster-based STF non-stationary geometrybased stochastic model (GBSM) for 6G wireless communication systems.It can be utilized to characterize wireless channels for all scenarios and all spectrum bands from sub-6 GHz to VLC [15] bands.All scenarios include global-coverage scenarios including low earth orbit (LEO) satellite, UAV, and maritime communications [16] scenarios, and full-application scenarios, such as (ultra-)massive MIMO [17], RIS [18], and industrial Internet of things (IIoT) [19] scenarios.Therefore, such a unified channel model has the capability to analyze the impacts of different frequency bands, scenarios, motions of the Tx and/or the Rx and/or scatterers, and scales of antenna arrays on channel statistical properties.However, only part of the statistical properties of the 6GPCM were studied without in-depth analysis in [14].Also, the complicated mapping between the channel statistical properties and different frequency bands, scenarios, and system settings was not revealed.
To fill the above gap, in this paper we provide a complete study of channel statistical properties of the 6GPCM with both mathematical derivations and simulations.We classify them according to the space/time/frequency domains and relationships among their calculation methods.Also, the impacts of different frequency bands, scenarios, and system settings on statistical properties are thoroughly investigated.The novelties and main contributions of this paper are as follows:  3) The relationships between channel model parameters and all channel statistical properties at different frequency bands in different scenarios are thoroughly investigated and revealed by both theoretical derivations and simulation analysis.The rest of this paper is organized as follows.In Section II, the 6GPCM [14] and its parameters are briefly reviewed.Theoretical derivations, relationships, and classifications for channel statistical properties of the 6GPCM are provided in Section III.The theoretical/simulation results are presented and compared with the corresponding measurement results in Section IV.Finally, conclusions are drawn in Section V.

II. 6GPCM
The illustration of the 6GPCM is shown in Fig. 1.For clarity, the figure shows only the nth (n = 1, . . ., N qp (t)) clusters, where N qp (t) is the number of clusters between A T p and A R q .Here, A T p represents the pth antenna element of the Tx antenna array and A R q represents the qth antenna element of the Rx antenna array.Besides, Fig. 1 shows a multi-bounce propagation phenomenon where the nth cluster is essentially a cluster pair including the first-bounce cluster C A n and the last-bounce cluster C Z n .The propagation between C A n and C Z n is a virtual link [20].When the delay of the virtual link is zero, C A n and C Z n become the same and the multi-bounce cluster degenerates into a single-bounce cluster.Also, there are M n (t) rays in the nth cluster at time instant t.Moreover, the dotted-line components in the figure represent movements of Tx, Rx, and clusters in the channel, since the 6GPCM supports multi-mobility.Other parameters are defined in Table I.

A. Channel Impulse Response (CIR)
The complex channel matrix of the 6GPCM can be expressed as  [14] where P L, SH, BL, W E, and AL are path loss, shadowing, blockage loss [21], weather effect loss [22], and atmospheric gas absorption loss [23], respectively.We focus on analyzing the statistical properties of small-scale fading (SSF) in this paper.SSF in radio frequency channels can be represented as H s = [h qp,fc (t, τ )] M R ×M T , where the CIR h qp,fc (t, τ ) can be calculated as Here, K(t) is the Ricean K-factor at time instant t, h L qp,fc (t, τ ) and h N qp,fc (t, τ ) are the CIRs of the line-of-sight (LoS) component and the non-LoS (NLoS) component as and P qp,mn,fc (t) In ( 3) and ( 4), {•} T denotes the transposition operation, F p(q),fc,V and F p(q),fc,H stand for the antenna patterns.Also, κ mn (t) and µ are the cross polarization power ratio and the co-polar imbalance [8], respectively.In addition, θ V V mn , θ V H mn , θ HV mn , θ HH mn , θ V V L and θ HH L are the initial phases, which are random variables uniformly distributed over (0, 2π].Moreover, F r represents Faraday rotation matrix in a LEO satellite scenario [24].In the derivations of channel statistical properties, we set F r = 1.Additionally, P qp,mn,fc (t) and τ qp,mn (t) are the time-varying power and delay of the mth ray in the nth path between A T p and A R q , respectively.In multi-bounce paths, τ qp,mn (t) = (d T mn (t) + d R mn (t))/c + τmn (t), where τmn is the delay of the virtual link between C A mn and C Z mn .Also, τ L qp (t) is the LoS-path time delay at time t, which can be calculated as τ L qp (t) = D qp (t)/c, where c is the speed of light.Then we can calculate the power P qp,mn,fc (t) using the delay [14].
For the positions of scatterers, they are modeled by the ellipsoid Gaussian scattering distribution [8]  , respectively.Using the geographical positions of Tx, Rx, and scatterers, the delays, angles, and powers of the rays in the clusters can be obtained.For convenience, we have omitted the time variable t, but note that these variables are all time-varying.Furthermore, the 6GPCM supports birth-death processes of clusters in STF domains [14].The number of clusters at time instant t can be calculated as where N surv (t) is the number of survived clusters at time instant t.It is determined by the survival probability P surv (δ p , δ q , ∆t BD , ∆f BD ), given by P surv (δ p , δ q , ∆t BD , ∆f BD ) = P T surv (∆t BD , δ p ) P R surv (∆t BD , δ q ) P surv (∆f BD ) .(6) Here, ∆t BD and ∆f BD are integer multiples of ∆t and ∆f , respectively.Calculations and typical values of (6) can be found in [25] and [26].Additionally, N new (t) is the newly generated clusters at time instant t.It follows the Poisson distribution with mean value E (N new ), given by where λ G is the generation (birth) rate of clusters and λ R is the recombination (death) rate of clusters.Their typical values can be found in [8].

B. Channel Transfer Function (CTF)
We can get the CTF by taking the Fourier transform of the CIR in (2) over τ as where and Note that, when large bandwidth in the millimeter wave (mmWave) band or THz band is used, we need to multiply the power by f fc γm n to mimic the frequency-dependent property in the frequency domain, where γ mn is the frequency-dependent factor [27].

III. STATISTICAL PROPERTIES OF THE 6GPCM
In general, we introduce channel statistical properties in the sequence of space/angle domain, time/Doppler frequency domain, and frequency/delay domain.This section is divided into six sub-sections from Section III-A to Section III-F corresponding to six red dashed line boxes in Fig. 2, respectively.Firstly, we can get the STF non-stationary CF using the CTFs of channels in Section III-A.Then, space-or time-or frequency-domain correlation function can be obtained from the general STF non-stationary CF by imposing the intervals in the other two domains to 0. Afterwards, the PSDs, RMS spreads, and coherence variables can be calculated from the CFs for different domains.These statistical properties in different domains are described in Section III-B, Section III-C, and Section III-D, respectively.In order to understand the stationarity regions, stationary distance/time/bandwidth are investigated in Section III-E.In addition, we can obtain the amplitude probability density function (PDF) and phase PDF using the CTF of the channel.Finally, in Section III-F we study LCRs and corresponding AFDs in space, time, and frequency domains based on their amplitude PDF and the joint PDF of the amplitude and the amplitude slope.

A. STF Non-Stationary CF
STF non-stationary CF measures the correlation between two channels at different time instants and different frequencies between different antenna elements.The STF non-stationary CF between H qp (t, f ) and H q p(t+∆t, f +∆f ) is defined as where E [•] is the statistical average, (•) * is the complex conjugation operation, ∆r T is the antenna element spacing between A T p to A T p and ∆r R is the antenna element spacing between A R q to A R q , as ∆r T = δ p − δ p and ∆r R = δ q − δ q .By substituting (8) into (11), the STF non-stationary CF can be rewritten as where R L qp,q p t, f ; ∆r T , ∆r R , ∆t, ∆f is the STF non-stationary CF of LoS component.It can be calculated as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where d qp (t)/d q p(t) represents the travel distance from A T p /A T p to A R q /A R q of the LoS component.However, the calculation is complicated and the details can be referred to [14].
Also, R N qp,q p t, f ; ∆r T , ∆r R , ∆t, ∆f is the STF non-stationary CF of NLoS components.It can be calculated as Similarly, calculations of d qp,mn (t) and d q p,mn (t) of NLoS components can be referred to [14].
A channel is STF non-stationary if the STFCF is not only a function of ∆r T , ∆r R , ∆t, and ∆f , but also relies on the antenna element indexes p/q, time instant t, and frequency point f .As shown in Fig. 3, when the channel is wide-sense stationary (WSS) in the space domain, the space WSS STFCF can be written as R(t, f ; ∆r T , ∆r R , ∆t, ∆f ).When the channel is WSS in the time domain, the time WSS STFCF can be written as R qp,q p(f ; ∆r T , ∆r R , ∆t, ∆f ).Also, when the channel is WSS in the frequency domain, the frequency Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.WSS STFCF can be written as R qp,q p(t; ∆r T , ∆r R , ∆t, ∆f ).In addition, we can further assume that the channel is WSS in two domains, or in other words the channel is non-stationary in the third domain.For example, if the channel is WSS in the space-time domains, the space-time WSS STFCF can be written as R(f ; ∆r T , ∆r R , ∆t, ∆f ).If the channel is WSS in the time-frequency domains, the time-frequency WSS STFCF can be written as R qp,q p(∆r T , ∆r R , ∆t, ∆f ).Also, if the channel is WSS in the space-frequency domains, the space-frequency WSS STFCF can be written as R(t; ∆r T , ∆r R , ∆t, ∆f ).Finally, if the channel is WSS in the STF domains, the STFCF can be written as R(∆r T , ∆r R , ∆t, ∆f ).In the following subsections, unless otherwise specified, we assume that the channel is non-stationary in the STF domains.
B. SCCF, Coherence Distance, Angular PSD, and RMS Angular Spread 1) SCCF: The STF non-stationary CF can be decomposed into SCCF R qp,q p t, f ; ∆r T , ∆r R by setting ∆t = 0 and ∆f = 0.It is important to mention that use the same transmitting antenna element A T p , i.e., ∆r T = 0, resulting in the SCCF R qp,q t, f ; ∆r R at the Rx side.Also, use the same receiving antenna element A R q , i.e., ∆r R = 0, resulting in the SCCF R qp, p t, f ; ∆r T at the Tx side.Here, SCCF at the Tx side R qp, p t, f ; ∆r T is used by setting ∆t = 0, ∆f = 0, q = q, and ∆r R = 0 in (11) as an example for analysis.It can be obtained as Following some simplifications [8] and assuming that the environment is static, d qp,mn (t) − d q p,mn (t) can be got as where ϑ T is the angle between the Tx and the departure wave of C A mn .It is determined by Similarly, the SCCF observed at the Rx side R qp,q t, f ; ∆r R can also be obtained.
2) Coherence Distance: The coherence distance on the antenna array axis is the minimum antenna element spacing within which the SCCF equals to the threshold c thresh D ∈ [0, 1].The Tx-side coherence distance is defined as Similarly, the Rx-side coherence distance D R qp,q (t, f ) based on the Rx-side SCCF R qp,q t, f ; ∆r R can also be obtained.
3) Angular PSD: Spatial-Doppler PSD S qp,q p t, f ; ϖ T , ϖ R is defined as the Fourier transform of SCCF R qp,q p t, f ; ∆r T , ∆r R with respect to (w.r.t.) the interval ∆r T and ∆r R .Here, spatial-Doppler PSDs at the Tx Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
and Rx sides, i.e., S qp, p t, f ; ϖ T and S qp,q t, f ; ϖ R , can be obtained as where ϖ T (ϖ R ) is the spatial-Doppler frequency variable.In the above expressions, ϖ T (R) is defined as ) , where Ω T (R) is the unit vector of the AoD (AoA) wave and β T (R) is the orientation of the antenna array at the Tx (Rx) side [11].

4) RMS Angular Spread:
The RMS angular spread at the Tx side, which is the AoD spread, is the square root of the second-order central moment of angular PSD at the corresponding side, defined as (21) where S qp, p (t, f ; ϕ AoD ) is the Tx-side AoD PSD.Here, u ϕAoD,qp, p (t, f ) is the average AoD and is also the first-order moment of Tx-side AoD PSD.In the same way, we can get the average AoA u ϕAoA,qp,q (t, f ) and the Rx-side RMS AoA spread σ ϕAoA,qp,q (t, f ), which are the first-order moment and the square root of the second-order central moment of Rx-side AoA PSD S qp,q (t, f ; ϕ AoA ).
C. TACF, Coherence Time, Doppler PSD, and RMS Doppler Spread 1) TACF: TACF measures the time correlation of the channel.Let ∆r T = 0, ∆r R = 0, and ∆f = 0, i.e., using the same antenna elements at both Tx and Rx sides, in (11), we can obtain the TACF R qp (t, f ; ∆t) as Here, R L qp (t, f ; ∆t) is the TACF of the LoS component and is given by Also, the TACF of the NLoS component R N qp (t, f ; ∆t) can be calculated as Mn(t) m=1 P qp,mn (t, f ) where d qp,mn (t) − d qp,mn (t + ∆t) in ( 24) can be approximated as 2) Coherence Time: Coherence time is the minimum time interval that fulfils the condition that the TACF equals to the threshold c thresh T ∈ [0, 1], as 3) Doppler PSD: Doppler PSD, which measures the power distribution over the Doppler frequency axis, is the Fourier transform of TACF R qp (t, f ; ∆t) w.r.t. the time interval ∆t.It can be calculated as where υ is the Doppler frequency.Besides, the time-varying Doppler frequency of the mth ray in the nth cluster from A T p to A R q at t can be calculated [14] as 4) RMS Doppler Spread: The motions of Tx, Rx, and scatterers in the channel cause the channel dispersion in the Doppler frequency domain, and the multipath propagation causes channel dispersions in the time and angular domains.RMS Doppler spread, RMS delay spread, and RMS angular spread are often used to measure these dispersions.RMS Doppler spread can be calculated as the square root of the second-order central moment of Doppler PSD, as where u υ,qp (t, f ) is the average Doppler shift and is also the first-order moment of the Doppler PSD.Since the value of coherence time is the reciprocal of the value of the RMS Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Doppler spread [28], we only use one of them as an example to analyze the impact of coherence time and RMS Doppler spread.Similarly, the coherence distance and RMS angular spread are a pair, so are the coherence bandwidth and RMS delay spread.Consequently, we only analyze the impact of RMS Doppler spread, RMS Delay spread, and RMS angular spread in Section IV.D. FCF, Coherence Bandwidth, Delay PSD, and RMS Delay Spread 1) FCF: FCF measures the frequency correlation of the channel.We can reduce the STF non-stationary CF to FCF R qp (t, f ; ∆f ) by setting ∆r T = 0, ∆r R = 0, and ∆t = 0 in (11) as 2) Coherence Bandwidth: The coherence bandwidth is the minimum frequency intervals that fulfills the condition that the FCF of a channel equals to the threshold c thresh B ∈ [0, 1].It can be written as 3) Delay PSD: Delay PSD gives the channel PSD as a function of the time delay τ .Also, it is the inverse Fourier transform of FCF R qp (t, f ; ∆f ) w.r.t.∆f .It can be obtained as This can be calculated as Note that the two expressions of delay PSD can be derived from each other.Here, P qp,mn (t, f ) is affected by the STF evolution of the clusters in space, time, and frequency domains, and will further affect the delay PSD. 4) RMS Delay Spread: Similar to the RMS Doppler spread, the RMS delay spread is defined as the square root of the second-order central moment of the delay PSD.It can be calculated as where u τ,qp (t, f ) is the average delay and is also the first-order moment of the delay PSD.

E. Stationary Distance/Time/Bandwidth
The stationary time is the maximum time duration during which the channel can be seen as a WSS channel in the time domain.We can use several methods to calculate the stationary time, such as correlation matrix distance (CMD) [29], local region of stationarity (LRS) [30], and spectral divergence (SD) [31].Here, we use the LRS method and consider stationary time as the longest time interval during which the correlation coefficient of two delay PSDs exceeds a specified threshold.The correlation coefficient of the two delay PSDs can be obtained as Therefore, the stationary time can be calculated as Here, c thresh_IT is usually set as 0.8 [32].
The definition of stationary time in time domain can be extended to the space and frequency domains.For example, [34] proposed the correlation coefficient of two channels in STF domains, which is defined as the normalized ACF of the delay PSDs of two channels, given by R Λqp,q p t, f ; ∆r T , ∆r R , ∆t, ∆f = S qp (t, f ; τ ) S q p (t + ∆t, f + ∆f ; τ ) dτ max S qp (t, f ; τ ) 2 dτ, S q p(t + ∆t, f + ∆f ; τ ) 2 dτ .
In the time domain, substituting p = p, q = q, namely ∆r T = 0, ∆r R = 0, and ∆f = 0 into (37), and then the stationary time can be obtained.It is consistent with (36).
In the space domain, substituting ∆t = 0, ∆f = 0, and ∆r R = 0 into (37), we can get the Tx-side correlation coefficient as Then the Tx-side stationary distance can be written as Similarly, we can get the Rx-side stationary distance I R s,qp,q (t, f ).Also, substituting ∆t = 0, ∆r T = 0, and ∆r R = 0 into (37), the correlation coefficient in the frequency domain can be calculated as Then, we can obtain the stationary bandwidth as

F. LCRs and AFDs in STF Domains
Based on the CTF in ( 8), the amplitude PDFs and phase PDFs of the received complex envelope in space, time, and frequency domains can be obtained.In this subsection, for a better understanding, we assume that the t and f in ( 8) are constants when we discuss the space-domain amplitude PDF, phase PDF, LCR, and AFD.We assume p, q, and f in ( 8) are constants when we analyze these statistical properties in the time domain.Also, we assume p, q, and t in ( 8) are constants when we deal with frequency-domain statistical properties.Therefore, the signal amplitude and phase are ξ(t) and θ(t), and the amplitude PDF and phase PDF will accordingly be p ξ (ξ t , t) and p θ (θ t , t) in the time domain.Then, we can get the time-domain LCR and AFD according to the amplitude PDF and joint PDF of the amplitude and the amplitude slope.
The temporal LCR [33] describes the time-variance of a fading channel by determining the rate of change of the signal amplitude in time.It is defined as the average number of crossings per second at which a fading signal amplitude passing through a certain signal level L t in the positive (or negative) direction.Mathematically, it is given as where ξ = dξ/dt is the derivative of time (amplitude slope) and p ξ ξ (L t , ξ) is the joint PDF p ξ ξ (ξ, ξ) of ξ and ξ that the signal crosses the level L t .Using results in [35], N ξ (L t , t) can be derived as where cosh(•) is the hyperbolic cosine function and erf(•) is the error function.Here, χ(t) and b l (t) (l = 0, 1, 2) can be obtained as and Here, we assume that the p, q, and f are constants in the calculation.Hence the TACF for the NLoS components can be written as R N (t; ∆t).Although the LCR is usually calculated in the time domain, it can be extended to the frequency and space domains [36], [37].The frequency LCR N ξ (L f , f ), Tx-side spatial LCR N ξ,q (L s , r T ), and Rx-side spatial LCR N ξ,p (L s , r R ) refer to the average times the signal crosses a given threshold in the positive or negative direction per frequency or distance unit.Here, r T (r R ) is the space-domain variable observed at the Tx (Rx) side.N ξ (L f , f ), N ξ,q (L s , r T ), and N ξ,p (L s , r R ) can be obtained by replacing R N (t; ∆t) with R N (f ; ∆f ), R N qp, p(∆r T ), and R N qp,q (∆r R ) in (45).
The temporal AFD is defined as the average time the signal amplitude is below a certain threshold value L t .Mathematically, it is where p ξ (ξ t , t) is the PDF of the signal amplitude.Also, frequency AFD (average fading bandwidth) T ξ (L f , f ) and spatial AFDs (average fading distances) at the Tx and Rx sides T ξ,q (L s , r T ) and T ξ,p (L s , r R ) can be defined as the average bandwidth and the average distances at both Tx and Rx sides signal remains below the threshold value.In summary, temporal LCR and AFD (frequency LCR and AFD, spatial LCR and AFD) can characterize the time-variance (frequencyvariance, space-variance) of a fading channel.

IV. RESULTS AND ANALYSIS
In this section, analytical results, simulation results, and the corresponding measurement results for the channel statistical properties will be discussed and compared.The analytical results were obtained through the plot of the theoretical derivation results of the statistical properties in Section IV, while the simulation results were obtained by simulating the corresponding statistical properties of the channel model based on the CIR or the Fourier transform of the CIR.Good consistency of the analytical result and the simulation result can illustrate the correctness of both mathematical derivations of the model and simulation.Then, the analytical/simulation results fit measurement data well, showing the validity of the proposed channel model in terms of the corresponding channel statistical property.
The 6GPCM provides different parameter sets for different frequency bands and scenarios.Those parameters can be adjusted based on measurement data when there is channel measurement.Firstly, user defined parameters, such as carrier frequency, antenna related parameters, positions of the Tx/Rx, and velocities of the Tx/Rx/clusters are set on the basis of real measurements.In the simulation, the uniform linear arrays (ULAs) are employed at both the Tx and Rx sides.Then, parameters that have no/neglected effect on the observed statistical property according to theoretical derivation results in Section IV are set to constants.After most parameters are SCCFs at frequencies with fixed antenna spacing and half-wavelength antenna spacing at the Tx side determined, some parameters such as scatterers distribution related parameters are optimized using the minimum mean square error (MMSE) criterion when suitable channel measurement data are used.Otherwise, parameters are chosen to be the same as those of 6GPCM in [14].
Fig. 4 (a) and Fig. 4 (b) show the SCCFs of the channels at different carrier frequencies with fixed and half-wavelength antenna spacing at the Tx side, respectively.Among them, we set the antenna spacing as half wavelength of the minimum carrier frequency for the fixed antenna spacing, and half wavelength of carrier frequencies for the relative antenna spacing.Also, analytical results, simulation results, and the measurement results are compared.Firstly, good consistencies between analytical/simulation results with measurement results in Fig. 4 (a) and Fig. 4 (b) verify the correctness of the 6GPCM.Here, the measurement was conducted in a static court yard scenario at 2.6 GHz with 128-elements Tx antenna array [38], i.e., f c = 2.6 GHz, M T = 128, and v T = v R = 0 m/s.Also, we set Rx-side channel model parameters which have no/negligible effect on Tx-side SCCF to constants, i.e., m and so on.Then we adjust parameters related to cluster distribution at the Tx side, i.e., mean(d T mn ), min(d T mn ), σ T x , σ T y , and σ T z .Finally, after optimal parameters searching using the MMSE criterion, we set mean(d T mn ) = 60 m, min(d T mn ) = 10 m, σ T x = 7 m, σ T y = 12 m, σ T z = 9 m, which are reasonable in the court yard scenario.After fitting with measurement data, we simulated the effects of different carrier frequencies (2.6 GHz at sub-6 GHz band, 26 GHz at the mmWave band, and 300 GHz at the THz band) on SCCF.As shown in Fig. 4 (a), the SCCF of the channel decreases with the increase of carrier frequency.In addition, the SCCF of the channel increases with the increase of carrier frequency in Fig. 4 (b), as the increased frequency reduces the spacing between antennas, resulting in the increase of the SCCF.Note that, when analyzing the influence of carrier frequency on the SCCF in Table II later, we use the half-wavelength antenna spacing.
Coherence distances with different distances from Tx to scatterers are as shown in Fig. 5. Here, we use analytical results, which can be calculated according to (18).Since we can obtain the coherence distance based on SCCF, we utilize the similar parameter values as SCCFs in Fig. 4 except for otherwise specified.Note that the coherence distances are normalized w.r.t.antenna spacing.The larger the distance between Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
the scatterers in a cluster and the antenna array, the larger the coherence distance will be, which is consistent with (16).
In ultra-massive MIMO scenarios, due to the spherical wavefront property, the angular PSDs of different antenna elements are different, as illustrated in Fig. 6, reflecting non-stationarity in the space domain.
The TACF characterizes the time-varying property of a channel, which is determined by the moving speeds of Tx, Rx, and scatterers in the channel according to (22) and (25).Note that the velocity includes both magnitude and direction of the velocity.The TACFs with different speeds of Rx in the channel is shown in Fig. 7 (a).The measurement [17] was conducted by our group at outdoor UMa scenario in NLoS condition.According to the channel measurement configuration, we set , and α R E = 0. Here, TACF is a time-domain channel statistical property, so we use the same antenna element pair at the Tx and Rx sides, i.e., M T = M R = 1.Some parameters were chosen by fitting channel measurement data such as λ G , λ R , σ T x , σ R x , σ T y , σ R y , σ T z , and σ R z .The rest parameters were the same as Table IV in [14].In terms of different speeds, we use 2.5 m/s to simulate pedestrian speed, 10 m/s to simulate driving speed in urban area, 20 m/s to simulate driving speed on highway, and 100 m/s to simulate high-speed train speed.The result shows that the theoretical result, simulation result, and measurement result match very well when the speed of Rx is 2.5 m/s, illustrating the correctness of the model.Also, with the increase of speed, TACF decreases, which is consistent with the theoretical derivation.Moreover, the birth-death process of clusters and the power components in (22) show that TACF is also affected by environmental factors.What's more, we can get the coherence time according to (26) directly.The smaller the TACF, the smaller the corresponding coherence time will be, under the same conditions.
The Doppler PSD characterizes the average power distribution at the Doppler frequency and the Doppler frequency shift of the carrier frequency is caused by the movements of Tx, Rx, and scatterers, as indicated by (28).We also use the same parameters as TACFs in Fig. 7 (a) since we can get Doppler PSD by the Fourier transform of TACF.Fig. 7 (b) shows the Doppler PSDs with different moving speeds of the Rx.We can conclude that the larger the Doppler frequency shift is, the larger the Doppler frequency range of the average power distribution and the smoother the change of Doppler PSD will be.Doppler spread describes the dispersion degree of the Doppler PSD.The greater the Doppler frequency shift caused by motion, the greater the Doppler spread will be.Due to the page limitation, we will not show the simulation results of the coherence time and the RMS Doppler spread here, but from ( 26) and ( 29), we know that when TACF decreases, the coherence time will decrease, and the corresponding RMS Doppler spread will increase, as shown in Table II.Fig. 8 shows the FCFs and the values of them are different at different carrier frequencies.The FCF measures the frequency correlation of the channel.It can be seen from ( 30) that FCF of a channel is mainly affected by the value of ⃗ d qp,mn (t), namely the instantaneous total distance between the Tx, scatterers, and the Rx.Therefore, the distance between the Tx and the Rx, the range of cluster distribution, and the distribution of scatterers within clusters can significantly affect the value of FCF.Furthermore, in the large bandwidth scenarios, channels   are non-stationary in the frequency domain [34], that is, the value of FCF is related not only to the frequency interval, but also to the carrier frequency value.
Fig. 9 (a) and Fig. 9 (b) are RMS delay spreads with different clutter densities and different scatterer densities within a cluster in the IIoT scenario.The channel measurements [39] were carried out at 5.85 GHz at light clutter and dense clutter two sub scenarios in the NLoS condition.In the simulation, we set the initial number of clusters N c (t 0 ) to 15 and 20 in sparse and dense clutter environments [19], respectively.Also, most parameters were randomly generated according to the 3GPP TR 38.901 channel model [25] for industrial factory scenario.Then, we fine-tuned the parameters about distributions of scatterers based on channel measurement result, such as σ T x , σ T y , and σ T z at the Tx side and σ R x , σ R y , and σ R z at the Rx side.Since the parameters can significantly affect the RMS delay spread according to theoretical derivation results in Section IV.Fig. 9 (a) shows that the channel in dense clutter environment has a larger RMS delay spread.Fig. 9 (b) shows the influence of the dispersion degree of the scatterers in the cluster under the same number of scatterers.The more discrete the scatterer distribution or the more dispersed the value of delay is, the greater the RMS delay spread will be.In (34), we know that RMS delay spread measures the dispersion degree of power in time delay, i.e., τ qp,mn (t), whereas the distribution of τ qp,mn (t) is directly affected by multipath.Thus, the sparse clutter and dense clutter environments in IIoT scenarios at the same frequency are used as examples to illustrate the impact of obstacle density in the environment on RMS delay spread.
In addition, Fig. 10 shows different RMS delay spreads at different frequency bands, i.e., 5.3 GH, 58 GHz, and 300 GHz.Here, the measurement data were obtained in a laboratory room using 58 GHz carrier frequency [40].We can see that the measurement result and simulation result fit well, which Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Fig. 10.
RMS delay spreads at different frequency bands shows the accuracy of the model.Also, the simulation results show that, with the increase of frequency band, the value of RMS delay spread decreases.Fig. 11 compares the analytical results with the measurement result for the LCRs in STF domains.The measurement data shown in Fig. 11 (b) was obtained in [41], which was conducted in the HST scenario at 930.2 MHz and the speed of the HST is about 300 km/h.The match between the analytical result and the measurement result verifies the correctness of the 6GPCM.Also, Fig. 11 (a) is the LCR in the space domain.We can see that, the elevation angle of the antenna array at the Tx side affects the Tx-side spatial LCR.Here, we increase this elevation angle to make the angle between the antenna array and the departure wave larger, which will increase the LCR.Moreover, Fig. 11 (b) shows the LCR in the time domain at a HST scenario.As the speed decreases, the LCR decreases.The results are consistent with the theoretical derivation results in (42).In addition, LCR in the frequency domain in Fig. 11 (c) shows that as the dispersion of clusters increase, the frequency LCR increases.
In summary, relationships between statistical properties and the channel parameters are given in Table II.In the table, DS means the delay spread, ASD means the azimuth spread of departure, ESD means the elevation spread of departure, "↑" means increase, "↓" means decrease, and "-" means no impact or negligible impact.Note that Table II only considers the Tx side.The Rx side is similar and is not discussed here.
In addition, channel parameters are seriously affected by the environment, i.e., parameters extracted from channel measurement.For example, authors in [42] conducted measurements in five different factory halls and found that the material of the wall and the density and the positions of objects in the factory halls have a great influence on the number of multipath components in the channel at IIoT scenarios.In [43], the authors carried out the channel measurements in wooded, rural, suburban, and urban, four different environments mimicking satellite scenarios and draw the conclusion that the RMS delay spread values for suburb and urban environments are larger than those for wooded and rural environments.Also, channel parameters are affected by different frequency bands.For instance, channels at the THz band normally have a larger number of multipath components than channels at sub-6 GHz and mmWave frequency bands, since the wavelength of the THz wave is at the same order as the roughness of objects in the environment, so that there is a larger proportion of diffuse rays at THz frequency band [34].Further, channel parameters will affect channel statistical properties and communication system performance as mentioned above.

V. CONCLUSION
In this paper, we have conducted a complete study of STF statistical properties of the 6GPCM, which characterized wireless channels for all spectra from the sub-6 GHz to VLC bands and all scenarios such as (ultra-)massive MIMO, HST, IIoT, and UAV scenarios in 6G wireless communication systems.Also, mathematical expressions have been derived for those statistical properties, i.e., STFCF, SCCF, TACF, FCF, angular/Doppler/delay PSDs, RMS angular/Doppler/delay spreads, coherence distance/time/bandwidth, stationary distance/time/bandwidth, and LCRs/AFDs in STF domains.Then, we have classified them according to their relationships.Also, we have explored and summarized the mapping between channel model parameters and these statistical properties at different frequency bands and scenarios through theoretical derivations and simulations.It has been shown that the space/angle-domain statistical properties and the frequency/delay-domain statistical properties are more affected by the carrier frequency, orientation of the antenna array, and propagation scenario.The moving speeds of Tx, Rx, and scatterers in the channel have a significant impact on channel statistical properties in the time/Doppler frequency domain.The work lays a solid foundation for 6G channel model standardization and system design.

TABLE I DEFINITIONS
OF MAIN CHANNEL PARAMETERS

TABLE II RELATIONSHIPS
BETWEEN STATISTICAL PROPERTIES AND THE CHANNEL PARAMETERS AT THE TX SIDE