Abstract
Cellfree massive multiinput multioutput (MIMO) systems exhibit many characteristics that differ from those of traditional centralized massive MIMO systems, and there are still research gaps in the modeling of cellfree massive MIMO channels. In this paper, a geometrybased stochastic model (GBSM) that combines the doublering model and hemisphere model to comprehensively consider the distribution of scatterers in the environment was proposed for cellfree massive MIMO channels. Combined with the lineofsight (LoS) path, single scattering path, and double scattering path components, the channel matrix between the access points (APs) and the user was derived. The proposed model fully considers geometric parameters such as the arrival/departure direction, elevation angle, time delay, and distance, which can accurately characterize the channel. Then, we proved that the traditional channel model, standard blockfading model, and spatial basis expansion model (SBEM) adopted in a cellfree massive MIMO system could not describe the nonstationarity in the space, time, and frequency domains, whereas the proposed GBSM could. Statistical characteristics of the channel were analyzed, including the space crosscorrelation function (CCF), time autocorrelation function (ACF), Doppler power spectral density (PSD), level crossing rate (LCR), and average fade duration (AFD). Then, we investigated the proposed model by simulating the space CCF, time ACF, Doppler PSD, LCR, and AFD under the conditions of two different scatterer densities. Through simulations and analyses, some new features of cellfree massive MIMO channels were identified, providing a theoretical basis for indepth research on cellfree massive MIMO systems. Finally, the measurementbased scenario and the WINNER II channel model are compared to demonstrate that the GBSM is more practical to characterize real cellfree massive MIMO channels.
1. Introduction
As one of the key technologies of sixthgeneration (6G) mobile communication systems, cellfree massive multiinput multioutput (MIMO) technology is an ideal mobile communication method to further improve spectral efficiency [1, 2]. Cellfree massive MIMO technology introduces a “usercentric” architecture to achieve continuous coverage of a large area, thus breaking through the traditional cellular coverage idea of “cellcentric”, which effectively eliminates the limitations caused by the cell boundary and the static domination of frequency resources. In theory, when the number of antennas on the access point (AP) side is large enough, the total capacity of the system can increase linearly with the increase of the total number of users in the coverage area, which is an ideal mobile communication method to further improve the spectral efficiency.
At present, research on cellfree massive MIMO systems has mainly focused on spectral efficiency and energy efficiency [3–9], system performance analyses [10–14], power and resource optimization [15–18], pilot allocation and channel estimation [2, 19, 20], user achievable rate [21–23], and conjugate beamforming [24–26]. In all the abovementioned analyses, the cellfree massive MIMO channel was mainly modeled using two model types. The first type is the standard blockfading channel model [2–8, 10–26], which consists of channel coefficient, largescale fading including shadow fading and path loss, and smallscale Rayleigh fading obeying an independent and identically standardized normal distribution. It is assumed that smallscale fading is static during each coherence interval and that largescale fading changes slowly. The second type is the spatial basis expansion model (SBEM) [9], which consists of the complex gain of the multiple paths that represent the smallscale Rayleigh fading and largescale fading, as well as the array steering vector with the assumption of parallel waves and ignoring the elevation angle parameter. However, the standard blockfading channel model cannot reflect the nonstationarity of the channel. The assumption of parallel waves in the SBEM is inaccurate since the condition that the distance from the transmitting antenna to the scatterer cluster be far greater than the Rayleigh distance will no longer hold in cellfree massive MIMO systems. Moreover, neither of the abovementioned two models can accurately describe the statistical characteristics of cellfree massive MIMO channels. There are still research gaps in the modeling of cellfree massive MIMO channels, and accurate and efficient channel models are urgently needed for further system design and performance evaluation.
In a cellfree massive MIMO system, multiple APs are randomly deployed in a coverage area and the distance and scattering environment between users and each AP are different, as shown in Figure 1. The channel modeling assumptions of parallel wave incidence in the farfield and the common parameters between antennas [27, 28] (direction of arrival, direction of wave departure, number of scatterers/clusters, and statistical characteristics of random fading) of traditional massive MIMO systems no longer hold. Compared with a traditional centralized massive MIMO system, cellfree massive MIMO system channel modeling and information theoretical analysis face many practical problems, which are listed as follows. The random layout of APs increases the difficulty of accurately characterizing a cellfree massive MIMO channel. The assumptions of farfield parallel wave incidence, generalized channel stability, and commonparameter characteristics no longer hold. The phase of the channel steering vector no longer has a linear relationship with the antenna number. The angle extension is significantly increased due to the shortening of the userAP distance, and the number of coscattering clusters is significantly reduced. Additionally, it is necessary to accurately characterize and model nonstationary processes such as the time variation in the channel caused by the movement of users/scatterers. The nearfield propagation and environmental evolution characteristics caused by distributed array antennas make the statistical characteristics of cellfree massive MIMO channels significantly different from those of traditional centralized massive MIMO systems. It is necessary to analyze the channel model of a cellfree massive MIMO system from the perspective of multidimensional integration of a timespacedelay antenna to reveal the sparsity and stability characteristics of the channel in a specific domain (space, time, delay, or Doppler) and to explore the common structural characteristics (time and frequency) and spatial distribution structural characteristics of the channel in typical scenarios.
Considering the practical problems of cellfree massive MIMO systems, accurate and efficient channel models urgently need to be established. According to the modeling approach, channel models can be classified into deterministic channel models and stochastic channel models. Deterministic channel models rely on precise information of the propagation environment, while stochastic channel models describe channel parameters using certain probability distributions. Generally, stochastic channel models are mathematically tractable and can be adapted to various scenarios with relatively low accuracy. Among all the stochastic channel models, the geometrybased stochastic model (GBSM) assumes that scatterers are stochastically distributed according to a particular underlying geometry with a regular shape, such as twincluster, ellipse, multiring, twinmultiring, hemisphere, sphere, cylinder, ellipsoid, and semiellipsoid. GBSM has been widely used in a variety of scenarios, such as massive MIMO communications [29–31], vehicletovehicle (V2V) communications [32–34], highspeed train (HST) communications [35, 36], and millimeter wave (mmWave) communications [37–39]. Channel modeling research on cellfree massive MIMO systems is still in its infancy, and it is not known which channel model can accurately describe the channel characteristics of cellfree massive MIMO channels. Since the GBSM can accurately describe the characteristics of the channels in space, time, and delay domains through various parameters, such as geographic location, elevation angle, azimuth angle, and time delay, this paper intends to use the GBSM to establish cellfree massive MIMO channels. Note that the references mentioned above considered the perfect CSI; thus, the perfect CSI was also considered in ours paper. The major contributions and novelties of this paper are summarized as follows: Based on the assumption of a distribution of double rings and hemispheres of scatterers, a GBSM is established for cellfree massive MIMO channels. Compared with the previous models for cellfree massive MIMO channels, the innovation of the GBSM is that it combines both twodimensional (2D) and threedimensional (3D) characteristics, which gives a more accurate description of the channels. The proposed model fully considers the actual channel propagation environment of a cellfree massive MIMO by combining geometric parameters such as the arrival/departure direction, elevation angle, and distance. Based on the assumption of spherical wavefronts, each link from different APs to a specific scatterer cannot be simply approximated as parallel. However, the proposed GBSM can calculate the transmission distance, delay, and Doppler shift of each link. Thus, we can accurately characterize the channel. For the proposed GBSM, we have proved that the traditional channel model, standard blockfading model, and SBEM adopted in a cellfree massive MIMO system could not describe the nonstationarity in the space, time, and frequency domains, whereas the proposed GBSM could. Statistical characteristics are derived and thoroughly investigated, such as the space crosscorrelation function (CCF), time autocorrelation function (ACF), Doppler power spectral density (PSD), level crossing rate (LCR), and average fade duration (AFD). We investigate the proposed model by simulating the space CCF, time ACF, Doppler PSD, LCR, and AFD under the conditions of two different scatterer densities, as well as the influence of antenna spacing and user azimuth angle on the space CCF and time ACF. Through the simulation results and analysis, some new features of cellfree massive MIMO channels are identified. To validate the correctness of the proposed GBSM, the measurementbased scenario and the WINNER II channel model are compared. Simulation results demonstrate that the GBSM was more practical to characterize real cellfree massive MIMO channels.
The rest of this paper is organized as follows. Section 2 proposes the GBSM for cellfree massive MIMO channels. The statistical characteristic analysis including the space CCF, time ACF, Doppler PSD, LCR, and AFD are carried out in Section 3. Section 4 presents statistical characteristic simulation investigations, and Section 5 verifies the correctness and practicality of the model. The conclusions are drawn in Section 6.
2. GBSM for CellFree Massive MIMO Channels
2.1. GBSM
We consider a cellfree massive MIMO system with Q APs. Suppose that Q APs are randomly deployed in a certain city area, all of which are equipped with a uniform linear array (ULA). Each AP is equipped with antennas, and the distance between each antenna element is . The farfield plane wavefront assumption in conventional massive MIMO systems is not fulfilled in cellfree massive MIMO systems, as the condition that the distance from the transmitting antenna to the scatterer cluster being far smaller than the Rayleigh distance cannot be upheld. As a result, the wavefront emitted from the nth cluster to the receive array is assumed to be spherical. Then, the angle of arrival (AoA) is no longer linear along the array, and it needs to be computed based on geometrical relationships. Considering the channel between any AP and the user, the signal sent from the AP transmitter is scattered through the lineofsight (LoS) path and the nonlineofsight (NLoS) path to the user. The types of scatterers in the NLoS path can be divided into three categories: scatterers around APs, scatterers around users, and scatterers such as tall buildings and trees. We fully consider the geographic location of the APs and users and the geometric relationship of the scatterer distribution and construct a GBSM with the spherical wave assumption for cellfree massive MIMO channels, as shown in Figure 2. Some of the main parameters and definitions are shown in Table 1. The GBSM needs to be extended for different scenarios to effectively improve the accuracy of channel modeling, that is, the combination of the doublering and hemisphere models. The reason for adopting the doublering and hemisphere models is as follows. This paper considers a cellfree massive MIMO system model in an urban rich scattering scenario. In order to distinguish between nearend scattering and farend scattering, this paper adopts a GBSM combined with a doublering model and a hemisphere model. When the user faces a single AP with a short distance, there are fewer channels reflected by the AP through the higher scatterers, which can be ignored, and the channel with the AP is modeled as a single ring. This is similar to the user and the single ring model is adopted. For the channels between the user and other APs that are far away, the reflection effect of the higher scatterers needs to be considered to form a hemispherical model. Since APs are randomly distributed in the system, some APs are close to the user and the direct path is strong at this time. The scattering of high scatterers can be ignored and only the single ring model is considered. Some APs far away from the user have a weak direct path and can be ignored, while the scattering of high scatterers cannot be ignored, so only the hemispherical model is considered.
For scatterers around APs and users where the elevation angle/depression angle changes are relatively small, we assume that both categories are distributed on a single ring. Since the scatterers around tall buildings and trees are relatively concentrated, the 3D distribution characteristics of APs are particularly prominent. Hemisphere models need to be used to model cellfree massive MIMO channels. We take the first antenna element of the AP transmitter and the user as the centers to establish a doublering model with radii of and and establish a hemisphere model with the two centers as diameter . We set the layout direction of the ULA to the axis. Suppose that scatterers are distributed on a circle centered on the AP and that scatterers are distributed on a circle centered on the user. The moving speed of the user is , and the moving azimuth angle is . Suppose that stationary scatterers are distributed in the hemisphere area. Note that the GBSM proposed in this paper includes 2D and 3D characteristics. Among them, the doublering model represents the 2D distribution of scatterers and the hemispherical model represents the 3D distribution of scatterers. The fading channel matrix from all the APs to the user is described by an matrix of complex fading envelopes, which can be represented as follows.where denotes the channel from the qth AP pth antenna to the user. Note that includes the LoS path component and the NLoS path component, where the NLoS path component consists of the single scattering path component passing through and the double scattering path component passing through . The channel of can be represented as follows:
The channel of LoS path component is written as follows:
The channel of the single scattering component is written as follows:
The channel of the double scattering component is written as follows:where denotes the Rice factor. denotes the largescale fading that consists of path loss and shadowed fading. denotes distancedependent path loss, which can be expressed as , where denotes the minimum propagation distance, denotes the path loss at the minimum propagation distance, and denotes the path loss index and depends on the specific propagation environment. denotes the shadowed fading. The measured results show that the lognormal distribution can be used to describe shadowed fading, and the corresponding probability density function can be expressed as , where and denote the mean power and variance of shadowed fading, respectively. denotes the maximum Doppler frequency caused by the user, and and denote the normalized power coefficient that satisfies , specifying how much the single and doublebounced rays contribute to the total scattered power. The phases and are independent and identically distributed (i.i.d.) random variables that satisfy uniform distributions over . Note that represents the type of single scattering, i.e., the single scattering path from the transmitter AP through the scatterer to the user. For the double scattering path, only the path through the two singlering scatterers to the user is considered, i.e., the path. Different paths have different delays to users. To accurately characterize the channel model, it is necessary to calculate the delays of each path. The time delay of the LoS path, the single scattering path, and the double scattering path can be represented as , , and , respectively. According to the geometric relationship, the abovementioned distance parameters can be obtained as , , and .
The parameter definitions that appear above are shown in Table 1, where . The expressions of distance parameters and are given by (6) and (7), respectively.
Proof. See Appendix A.
For a hemisphere, if is expressed directly in terms of the departure of the AP and the arrival angle of the user, the formula is very complicated. Therefore, the position of the scattering cluster, i.e., the intersection of the departure angle and the arrival angle, is used to express . We take the center of the sphere as the coordinate origin to construct a Cartesian coordinate system. The 3D coordinates of the scattering clusters distributed in the hemisphere can be expressed as , the coordinates of the qth AP pth antenna can be expressed as , and the coordinates of the user can be expressed as . According to the above coordinates, the expressions of and are given by (8) and (9), respectively.
Proof. See Appendix B.Note that the azimuth angle of departure (AAoD) of each single scattering path corresponds to the azimuth angle of arrival (AAoA) with respect to the geometric relationship, and the elevation angle of departure (EAoD) and the elevation angle of arrival (EAoA) have a similar relationship. Due to this correspondence, the unknown variable groups and are reduced by two, i.e., we need to know only or to know the other parameters. According to the geometric relationship, the available angle parameters are as , , , , , and .
The double scattering path can be randomly generated and the abovementioned correspondence does not exist. At this point, the channel model between the AP and user is completed. The specific process to establish a GBSM is given in Algorithm 1. Note that the GBSM proposed for cellfree massive MIMO channels can be degenerated into a centralized massive MIMO channel model. When there exists only 1 AP, i.e., , the channel matrix in (1) can be reexpressed as , which denotes the channel matrix of the transmitter with antennas and denotes the channel from each antenna to the user. When , is equivalent to the SBEM of a centralized massive MIMO as shown in [9], , where denotes the complex gain of the path that represents the smallscale Rayleigh fading, denotes the largescale fading, and denotes the array steering vector and . Although the degeneration model can characterize centralized massive MIMO channels, there are still some differences compared with the multipath model . The multipath model assumes parallel waves and the phase differences of the subantennas are all . However, each subantenna of the degeneration model has an independent and accurate expression by substituting into a specific position. It takes into account the phase difference caused by the distance and the Doppler frequency shift, which is a more accurate channel model.

2.2. Conversion of a Discrete Angle into a Continuous Angle
Since it is assumed that the number of scatterers is infinite, i.e., , the model proposed in (2) is actually a mathematical reference model. Due to infinite complexity, these conditions cannot be achieved in practice. However, as described in Reference [40], the reference model can be used for theoretical analysis, the design of communication systems, and the realizable simulation models with reasonable complexity. For our reference model, we assume that is a finite value and convert the discrete angle value into a continuous angle value with a certain probability density distribution. Many different distributions have been proposed to characterize the angular distribution, such as the uniform, Gaussian, wrapped Gaussian, and heartshaped probability density function (PDF) distributions [41]. Here, we use the von Mises–Fisher (vMF) PDF distribution [42], which can effectively simulate the distribution of scatterers in 3D space, and the expression is as follows.where denotes the hyperbolic sine function and and denote the random variables of the elevation angle and azimuth angle, respectively. and denote the average values of the elevation angle and azimuth angle, respectively. When the scatterers are distributed on a double ring, the elevation angle tends to be ignored. At this time, only the distribution of the azimuth angle is considered. The vMF distribution should simplify to the von Mises distribution by integrating over the angle , and the probability density function is . Note that in the two single scattering paths, due to the low height of the surrounding scatterers, the influence of the elevation angle is ignored, i.e., , . The calculation of the available angle parameters indicates that due to the corresponding relationship between the azimuth angles and , the 2D variable can be converted into a probability density function that needs to consider only the onedimensional variable . In the double scattering path , the azimuth angles of the AP and user are independent of each other. Therefore, the probability density function products of and should be considered. In the single scattering path, since the scatterers are distributed on the hemisphere, the joint probability density distribution function of the elevation angle and the azimuth angle should be considered.
3. Statistical Characteristic Analysis
Since the smallscale fading of the wireless channel causes the received signal to fluctuate randomly, it is necessary to use statistical characteristics to describe the fading characteristics of the wireless channel. Statistical characteristics are used to better understand the nonstationary channel behavior. Statistical feature parameters mainly include firstorder statistical properties probability density function (PDF), and secondorder statistical properties, including space CCF, time ACF, DPSD, LCR, and AFD. The firstorder statistical parameters cannot fully describe the characteristics of the wireless channel, so it is necessary to use the secondorder statistical parameters to supplement the characteristics of the wireless channel.
3.1. SpaceTime Correlation Function (ST CF)
The ST CF can describe the variation in the channel in time and space and indicate the characteristics of the channel [43]. Therefore, to further verify the performance of the proposed GBSM, it is necessary to investigate the ST CF. The ST CF between any two channels and can be represented as follows:where denotes the complex conjugate operation, is the statistical expectation operator, , and . Note that (11) is a function of space separation caused by different APs and antennas and time separation . With the assumption that the LoS path and NLoS path components are independent [43], we can rewrite formula (11) as the sum of the correlation of the LoS path, single scattering path, and double scattering path components, which can be given by the following equation:
For the LoS path component, the azimuth angle and the elevation angle can be considered. Incorporating formula (12), the ST CF of the LOS path component is represented as follows:
Here, denotes the time delay and , . The ST CF of a single scattering component through and scatterers, i.e., the paths and are represented as follows:where and . The ST CF of the single scattering component through scatterers, i.e., the path , is represented as (15):where
The ST CF of the double scattering component through scatterers, i.e., the path , is represented as follows:where and .
3.2. Space CCF
For formula (12), we set the delay factor , and we can obtain the space CCF.
The space CCF of the LoS path can be obtained as follows:
The space CCF of a single scattering component through and scatterers, i.e., the paths and are represented as follows:
The space CCF of a single scattering component through scatterers, i.e., the path is represented as (21), and the double scattering component through scatterers is represented as formula (22).
3.3. Time ACF
By setting the indexes and in formula (12), the time ACF is represented as follows:
The time ACF of the LoS path is represented as follows:
The time ACF of a single scattering component through and scatterers, i.e., the paths and , is represented as formula (25). The single scattering component through scatterers, i.e., the path is represented as formula (26), and the double scattering component through path is represented as formula (27).
3.4. Doppler PSD Function
Applying the Fourier transform to the spacetime function in (12) in terms of , we can obtain the corresponding Doppler PSD function as follows:where denotes the Doppler frequency, which satisfies .
3.5. Envelope LCR and AFD
The LCR and AFD are secondorder statistical parameters that describe how fast the fading of a wireless channel changes. The LCR refers to the rate at which the amplitude of a fading signal crosses a specified level with a positive or negative slope in a unit time. Using the traditional PDFbased method [44], we derive the expression of the LCR for cellfree massive MIMO channels of the LoS path as follows:where is the hyperbolic cosine function, is the error function,, , , , and .
The LCR for NLoS conditions can be obtained from (8) by setting . The AFD is defined as the mathematical expectation of a duration lower than the specified level in a unit time [45], and the AFD of the LoS path can be written as follows:where denotes the Marcum Q function. , where is the zerothorder modified Bessel function of the first kind. We can obtain the closedform solution of the Marcum Q function as follows:
Proof. See Appendix C
For NLoS conditions, the AFD in (30) simplifies to .
4. Statistical Characteristic Simulation and Discussion
In this section, we first compare the GBSM with the standard blockfading channel model and the SBEM as shown in Figure 3. We can conclude from the simulation result that both the standard blockfading channel model and the SBEM could not describe the nonstationarity in the frequency domain since the normalized Doppler PSD does not change with the frequency domain but is always a discrete point. The same result appeared in the space and time domains. However, the proposed GBSM could describe the nonstationarity of cellfree massive MIMO channels. Moreover, based on the statistical characteristic analysis of the channel, the proposed GBSM is simulated by the space CCF, time ACF, Doppler PSD function, envelope LCR, and AFD. In addition, the effects of the antenna space and azimuth angle on the channel are analyzed in detail. The following parameters [46] are chosen for our simulations. , , , , , , , , , , , . , , , , , and can be obtained by geometric relationship in Section 2.
Note that the value of is related to the distribution of scatterers. Generally, the smaller the value of is, the greater the density of scatterers distributed around the user and AP. The value of is related to the power intensity of the LoS component. It can be considered that the higher the density of the scatterer is, the smaller the value of because under this condition. The NLoS component accounts for a larger proportion, while the LoS component accounts for a smaller proportion. This paper simulates two scattering environments, namely, lowdensity and highdensity scenarios, and analyzes the proposed model. represents the number of finite scatterers, which have different values in different density scenarios. The values of each parameter are shown in Table 2.
4.1. Space CCF
Figure 4 analyzes the relationship between the space CCF and the index of antennas with different antenna spacings and scatterer densities, i.e., low density and high density. It is obvious that the space CCF varies significantly in different scattering environments. In the highdensity scatterer scenario, that is, in a rich scattering environment, the space correlation between antennas is relatively small. The reason is that the higher the scatterer density, the more spatial diversity the massive MIMO channel has. In addition, the antenna spacing has an impact on the space CCF. The space CCF is always lower when than when . Increasing the antenna spacing causes the CCF to weaken.
4.2. Time ACF
Figure 5 analyzes the variation between the time ACF and with different antenna spacings and scatterer densities. It is obvious that the time ACF varies significantly in different scattering environments. Combined with Figure 4, we can see that both the space CCF and time ACF are relatively small and that the values remain approximately unchanged in a rich scattering environment. It can be considered to conform to widesense stationarity (WSS) in a certain period of time under highdensity conditions. Similar to the space CCF, increasing the antenna spacing causes the ACF to weaken. The time ACF is always lower when than when , which indicates that time ACF can be reduced by increasing the antenna spacing.
Since the value of Rice factor is related to the power intensity of the LoS component, Figure 6 analyzes the impact of on the statistics of ACFs for the LoS scenario. The simulation results indicate that when Rice factor is smaller, the LoS path occupies a smaller space and the ACF of the channel is weaker. The simulation results demonstrate that the cellfree massive MIMO system has more favorable propagation in a rich scattering environment.
4.3. Doppler PSD
Figure 7 analyzes the relationship between the PSD and the Doppler frequency in the LoS path and NLoS path under different scattering scenarios. The simulation results show that in both the LoS and NLoS paths, the Doppler PSD has a peak and tends to be evenly distributed on the sides. In a rich scattering environment, the PSD in the highdensity scenario is larger than that in a lowdensity scenario. The underlying physical reason is that in the highdensity scenario, the received power is mainly obtained by the NLoS path, including all directions reflected by the scatterers distributed on the circle and hemisphere. In contrast, the LoS path obtains a higher PSD in a lowdensity scattering environment since the received power comes mainly from specific directions identified by LoS components.
4.4. Envelope LCR and AFD
Figure 8 analyzes the relationship between the envelope LCR and the envelope level for different azimuth angles and scatterer densities. Again, the density of the scatterers significantly affects the envelope LCR for cellfree MIMO channels. This indicates that the LCR is higher when the channel is in a rich scattering environment. Additionally, the azimuth angle of the user has little influence on the LCR.
Figure 9 analyzes the relationship between the envelope AFD and the envelope level with different azimuth angles and scatterer densities. Unlike the LCR, the AFD tends to be small when the channel is in a rich scattering environment. Additionally, the azimuth angle of the user has little influence on the AFD.
5. Validation of the GBSM
Note that Section 4 considered several features of cellmassive MIMO channel based on reasonable assumptions. We would utilize some relevant channel measurement data [47] about cellfree massive MIMO channel to claim that the proposed GBSM model is accurate and efficient in this Section. Figure 10 indicates the practical measurement scenario where the user trajectory consists of regions R4 and R5. The AP is blocked by the tall building, dividing the user trajectory into NLOS regions. Three different AP array configurations are implemented with a significant variation in the array aperture and antenna element spacing. Specifically, the AP array I denotes a single AP with a linear antenna array of 32 elements aligned horizontally and spaced with frequency of GHz. The AP array II denotes two APs with two antenna arrays of 16 elements, and the two APs are distributed with a distance of 44.6 m between them. The AP array III denotes 32 APs with a single antenna element horizontally aligned but spaced with a distance of from one AP to the next.
To validate the correctness of the proposed GBSM, the measurementbased scenario [47] and the WINNER II channel model [48] are introduced in this section. The specific details of the two comparation scenarios are described as follows. The measurementbased scenario for cellfree massive MIMO channel in reference [47] is performed in an outdoor environment with a 32element polarized virtual linear array covering both LoS and NLoS scenarios. As the reference described, it is the worldfirst fully parallel and coherent widely distributed array channel sounding measurements in a high mobility scenario. Here, we utilize the user’s trajectory of R4 in [47], which can be equivalent to the NLoS path in ours paper. The WINNER II channel model [48] follows a geometrybased stochastic channel modeling approach, which is based on statistical distributions extracted from channel measurement. Here, we utilize the urban microcell scenario in the WINNER II model. In the urban microcell scenario, the height of both the antenna at the BS and at the MS is assumed to be well below the tops of surrounding buildings. The propagation environment of the urban microcell scenario is similar to the cellfree channel in ours paper.
We evaluate the delay spread and the Doppler spread as defined in reference [47] on each individual AP antenna from a transmitter for the three considered AP array configurations along the user trajectory.
5.1. The Delay Spread
The time delay spread of the proposed GBSM, the practical measurement data, and the WINNER II channel model are compared in Figure 11, Figure 12, and Figure 13 with configurations AP array I, AP array II, and AP array III, respectively. It is clear that the delay spread of the proposed GBSM aligns well with the practical measured data, which reflects the advantages of the GBSM proposed in this paper. However, the WINNER II channel model deviates greatly from the practical measurement data with the user trajectory. The reason may be the WINNER II channel model cannot accurately describe the propagation environment of cellfree massive MIMO channels while the proposed GBSM could.
5.2. The Doppler Spread
The Doppler spread of the proposed GBSM, the practical measurement data, and the WINNER II channel model are compared in Figures 14–16 with configurations AP array I, AP array II, and AP array III, respectively. It is clear that the Doppler spread of the proposed GBSM aligns well with the practical measured data, which demonstrates that the GBSM was more practical to characterize real cellfree massive MIMO channels. However, the WINNER II channel model deviates greatly from the practical measurement data with the user trajectory.
6. Conclusion
In this paper, we proposed a GBSM for cellfree massive MIMO channels that combined the doublering model and hemisphere model to comprehensively consider the distribution of scatterers in the environment. Combined with the LoS path, single scattering path, and double scattering path component, the channel matrix between APs and the user is derived. The proposed model fully considers geometric parameters such as the arrival/departure direction, elevation angle, and distance and can accurately characterize the channel. We have proved that the traditional channel model, standard blockfading model, and SBEM adopted in a cellfree massive MIMO system could not describe the nonstationarity in the space, time, and frequency domains, whereas the proposed GBSM could. The statistical characteristics of the channel, including the space CCF, time ACF, Doppler PSD, LCR, and AFD, are analyzed and simulated. By comparing these simulation results, we can see that the scattering environment, antenna space, and azimuth angle have an impact on all channel statistical properties. Specifically, the space CCF, time ACF, Doppler PSD, and AFD are relatively small in a rich scattering environment, while the LCR is relatively large. Since the space CCF and time ACF are relatively small and the values remain approximately unchanged in a rich scattering environment, it can be considered to conform to the WSS condition in a certain period of time. Moreover, as the antenna spacing and user azimuth angle decrease, the space CCF and time ACF gradually weaken. In summary, the simulations and analyses clearly showed that a rich scattering environment always obtains better channel performance than a lowdensity scattering case. To validate the correctness of the proposed GBSM, the measurementbased scenario and the WINNER II channel model were compared. Simulation results demonstrated that the GBSM was more practical to characterize real cellfree massive MIMO channels. Our research work can be considered theoretical guidance for establishing more purposeful cellfree massive MIMO measurement campaigns in the future [49].
Appendix
A. Calculation of the Distance Parameters in the DoubleRing Model
In the proposed geometrybased stochastic channel model, there exists a geometric relationship in the single scattering path. As shown in Figure 17, the parameters of distance and angle are marked and we can obtain the distance of each path through the geometric relationship.
For the single scattering path , , where , . Combining the abovementioned formulas, we can obtain . , where , , , and . Combining the abovementioned formulas, we can obtain the expression of as (6). For the single scattering path , , , where , . Combining the abovementioned formulas, we can obtain . For the double scattering path , and . Combining the abovementioned formulas, we can obtain the expression of as formula (7).
B. Calculation of the Distance Parameters in the Hemisphere Model
As shown in Figure 18, the parameters of distance and angle are marked. First, we establish three parallel coordinate systems with the AP, user, and center of the sphere to express the elevation angle and azimuth angle. Then, the coordinates of the AP, user, and scatterer can be obtained, that is, , , and .
Thus,
Since , , where denotes the modulo operation. Combining the abovementioned formulas, we can obtain the expression of and as (8) and (9), respectively.
C. Numerical Integration Algorithm of the Marcum Q Function
The definition of the Q function is as follows:where . Substituting into (C.2) and exchanging the order of integration, we can express the Marcum Q function as an integral in the following form.where
Formula (C.3) has the following closedform solution:where is the error remainder function and . Substituting formula into (C.5) (C.2) and combining (C.2), we can obtain the following equation:
Part of the paper has been published in a conference version, entitled “3D Stochastic Geometry Channel Model of CellFree Massive MIMO System” [50], published in 2020 International Conference on Wireless Communications and Signal Processing (WCSP), and we have improved the conference version by adding new content to ensure the innovation of this paper.
Data Availability
No data were used to support this study.
Disclosure
A part of the paper has been published in a conference version, entitled “3D Stochastic Geometry Channel Model of CellFree Massive MIMO System”, published in 2020 International Conference on Wireless Communications and Signal Processing (WCSP).
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under National Natural Science Foundation of China Grant 62271503, Grant 62071485, Grant 61901519, and Grant 62001513 and in part by the Basic Research Project of Jiangsu Province under Grant BK 20192002 and the Natural Science Foundation of Jiangsu Province under Grants BK20201334 and BK20200579.