CN107425895B - Actual measurement-based 3D MIMO statistical channel modeling method - Google Patents

Actual measurement-based 3D MIMO statistical channel modeling method Download PDF

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CN107425895B
CN107425895B CN201710476525.6A CN201710476525A CN107425895B CN 107425895 B CN107425895 B CN 107425895B CN 201710476525 A CN201710476525 A CN 201710476525A CN 107425895 B CN107425895 B CN 107425895B
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张阳
李晨
庞立华
栾英姿
李�杰
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Guangzhou Its Communication Equipment Co ltd
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Abstract

The invention belongs to the technical field of wireless communication, and discloses a 3D MIMO statistical channel modeling method based on actual measurement, which can effectively and accurately reflect a real three-dimensional channel environment and improve the precision of a channel model; extracting statistical characteristics of large-scale parameters through external field measurement, generating a cross-correlation matrix of the large-scale parameters to solve the problem that the matrix is not positive in the existing model, statistically modeling pitch angle expansion of a 3D MIMO channel by using a linear model, increasing the dependency relationship between vertical domain angle expansion and distance, and randomly generating a sub-path azimuth angle and a sub-path pitch angle with interdependence through mixing Von Mises Fisher distribution; and determining each characterization parameter of the channel model according to the statistical analysis of the external field measurement to generate a 3D MIMO channel coefficient. The invention expands the application of the 3D MIMO channel model and provides a powerful tool for accurately and efficiently evaluating the related algorithm of the 3D MIMO system.

Description

Actual measurement-based 3D MIMO statistical channel modeling method
Technical Field
The invention belongs to the technical field of wireless communication, and particularly relates to a measured-based 3D MIMO statistical channel modeling method.
Background
The wireless channel is the basis of research on the design of a new generation of wireless communication system, a new communication technology algorithm and the like, so that the accurate channel modeling is very important. At present, most of statistical channel models based on geometric architectures are two-dimensional, and the traditional urban macro cell wireless channel modeling requirements are met. As the channel modeling scenarios are refined, the two-dimensional channel model cannot accurately describe the wireless channels of various practical scenarios. Therefore, in order to describe the channel characteristics more objectively and accurately, a three-dimensional multiple-input multiple-output (3D MIMO) channel model is proposed. The existing standardized channel models such as SCM, WINNER and QuaDRiGa models all provide a modeling method and thought for MIMO channel 3D. Compared with the traditional 2D MIMO, the largest change of 3D MIMO is the utilization of the space pitch dimension information. Two important angle parameters, namely a pitch arrival angle (EoA) and a pitch departure angle (EoD), are modeled in the channel model, so that the objective and real wireless propagation environment is well reflected, and therefore, the channel model is widely concerned and researched. The 3D MIMO obtains more spatial degrees of freedom by utilizing a channel vertical domain, improves the throughput of a system, obviously improves the average and edge spectrum efficiency of a cell and reduces multi-user interference. Due to its advantages, 3D MIMO has penetrated into various mainstream wireless communication systems (e.g., LTE-advanced, and future 5G standards).
At present, the 3D MIMO channel modeling has a preliminary progress, and simultaneously, some problems also exist: corresponding 3D MIMO channel parameter parts in a WINNER + model are obtained through literature investigation and are not subjected to complete external field measurement, so that some errors can occur in the established channel model; the vertical domain angular characteristics of the channel are very important for 3D MIMO channels, and experiments observe that pitch angle spread depends on the distance between a Base Station (BS) and a user terminal (UE), but most existing channel models do not account for the distance dependency characteristics of the pitch-off angle spread ESD and the pitch-arrival angle spread ESA; most 3D MIMO channel models take the generation process of the azimuth angle and the pitch angle into consideration without researching the cross correlation between the azimuth angle and the pitch angle, so that the established channel model is not accurate enough and is not consistent with the characteristics of the actual propagation environment. Therefore, it is important to model the 3d mimo channel completely and accurately. In addition, the wireless channel can be modeled by statistical methods, and can be mainly classified into the following categories: statistical models based on geometric architectures (GBSMs), statistical models based on Correlations (CBSMs) and Parametric Stochastic Models (PSMs). With a predefined distribution of scatterers in the propagation environment, the GBSM can be constructed based on the basic laws of reflection, refraction, diffraction, and scattering of electromagnetic waves. CBSM statistically characterizes MIMO channel characteristics by introducing cross-correlation between channel matrix elements. The PSM characterizes multipath components (MPCs) by using characteristic parameters such as delay, power and the angle of arrival or departure of the multipath. Based on a PSM framework, the 3D MIMO statistical channel model based on actual measurement redefines a statistical characterization method of part of parameters in the model and gives a recommended value based on the actual measurement of an external field.
In summary, the problems of the prior art are as follows: the problem that cross-correlation matrices of large-scale parameters in existing models are not positive, most channel models do not describe the distance dependence characteristics of elevation departure angle spread ESD and elevation arrival angle spread ESA, and existing 3D MIMO channel models take the generation processes of azimuth angles and pitch angles into consideration without researching the cross-correlation between the azimuth angles and the pitch angles.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a measured-based 3D MIMO statistical channel modeling method.
The invention is realized in this way, a 3D MIMO statistical channel modeling method based on actual measurement, which extracts the statistic of large-scale parameters through the measurement of an external field channel and generates a cross-correlation matrix of the large-scale parameters; the pitch angle expansion of a 3DMIMO channel is statistically modeled by using a linear model, and the dependency relationship between the vertical domain angle expansion and the distance in large-scale parameters is shown; introducing mixed VMF distribution to depict the interdependence of a plurality of clusters of azimuth angles and pitch angles and generating an arrival angle and a departure angle of a 3D space;
further, the method for modeling the 3D MIMO statistical channel based on the actual measurement comprises the following steps:
step one, carrying out channel measurement activity, obtaining channel impulse response through channel measurement, and extracting channel multipath components according to the channel impulse response;
step two, modeling to generate large-scale parameters LSPs and generating a cross-correlation matrix thereof by using a circular filtering method, wherein the seven large-scale parameters are respectively as follows: delay Spread (DS), Shadow Fading (SF), azimuth departure Angle Spread (ASD), azimuth arrival Angle Spread (ASA), pitch departure angle spread (ESD), pitch arrival angle spread (ESA), and rice K factor (K);
seven large-scale parameters can be modeled as lognormal distribution, and the generated LSPs are as follows:
Figure BDA0001328394340000031
wherein s is a large-scale parameter vector, mu and sigma are mean and standard deviation vectors of lognormal distribution, which can be obtained by performing statistical analysis on external field measurement data,
Figure BDA0001328394340000032
is a parameter vector describing the correlation between large-scale parameters,
Figure BDA0001328394340000033
a is a cross-correlation matrix of seven large-scale parameters, ξ is generated by using a circular filtering method by using a correlation distance, and in the case of not obtaining the correlation distance, the matrix is randomly generated by a Gaussian independent same distribution variable with a mean value of 0 and a variance of 1.
For different large-scale parameters on the same link, the correlation coefficients of the two different large-scale parameters are as follows:
Figure BDA0001328394340000034
where ρ isxyThe correlation coefficient, C, of the large scale parameter x, yxyIs the covariance of the large-scale parameter x, y, Cxx，CyyThe variances of the large-scale parameters x and y are respectively;
thirdly, statistically modeling the pitch angle expansion of the 3D MIMO channel by using a linear model, and showing the dependence relationship between the vertical domain angle expansion and the distance in the large-scale parameters;
step four, introducing mixed Von Mises Fisher distribution modeling to generate an arrival angle and a departure angle of a 3D space; in the 3D MIMO statistical channel model, the number of angle parameters is four, namely the azimuth angle and the pitch angle of the receiving end and the transmitting end;
fifthly, determining each characterization parameter of the model according to the statistical analysis of the channel measurement; a cross-correlation matrix comprising LSPs; the slope and intercept of the linear model; cluster number, cluster spread, azimuth and pitch in the hybrid VMF distribution;
step six, generating channel coefficients;
(1) setting an initial random phase, and setting the random initial phase under four polarization modes (vv, vh, hv, hh) for the mth minor diameter in the nth cluster
Figure BDA0001328394340000041
The initial phase is uniformly distributed within (-pi, pi);
(2) determining a steering vector and a Doppler frequency of the antenna array; the channel response matrix of the nth cluster from the s-th transmitting antenna to the u-th receiving antenna can be obtained according to the following formula:
Figure BDA0001328394340000042
wherein Hu,s,n(t) is the channel coefficient matrix of the nth cluster, and each cluster has M rays, PnIs the power of the nth cluster, F is the field pattern of the transmit or receive antenna in either horizontal or vertical polarization,
Figure BDA0001328394340000043
is the azimuth angle of arrival phin,m,AoAAnd perpendicular angle of arrival thetan,m,EoANormalized angle vector of (2) is obtained by the following equation
Figure BDA0001328394340000044
Figure BDA0001328394340000045
Figure BDA0001328394340000046
Is the azimuth departure angle phin,m,AoDAnd a vertical departure angle thetan,m,EoDNormalized angle vector of (2) is obtained by the following equation
Figure BDA0001328394340000047
Figure BDA0001328394340000048
Figure BDA0001328394340000049
Position vectors for the receiving antenna u and the transmitting antenna s, respectively; kappan,mIs the cross-polarization power ratio in the linear range; lambda [ alpha ]0Is the wavelength of the carrier frequency; doppler frequency component vn,mDerived from the angle of arrival (AoA, EoA) and the velocity vector v of the UE.
Figure BDA00013283943400000410
Wherein
Figure BDA00013283943400000411
When an LOS path exists, calculating the channel coefficient of the LOS path:
Figure BDA0001328394340000051
further, the specific process of generating the LSPs statistical parameters and the cross-correlation matrix in the second step is as follows:
(1) preliminary transform domain large scale parameters (TLSPs) are generated in the transform domain.
Figure BDA0001328394340000052
Obey Gaussian distribution, in
Figure BDA0001328394340000053
Mapping to obtain siBefore the start of the operation of the device,
Figure BDA0001328394340000054
to be mixed with
Figure BDA0001328394340000055
The association is carried out in such a way that,
Figure BDA0001328394340000056
is a transform domain large scale parameter corresponding to other LSPs or other links. The generation method of TLSPs is different for different network layouts, considering two communication networks as follows:
a) the link is from one BS to multiple UEs
UE coordinate is (x)1,y1),···(xk,yk) Generating a system grid, generating seven Gaussian random variables for each node, respectively corresponding to 7 TLSPs, and finding out the positions loc1 & cndot & cnk of k users in grid points; generate autocorrelation filter responses for the seven TLSPs:
Figure BDA0001328394340000057
wherein λ ismIs the autocorrelation distance of each LSP; d is coordinate extension value in the system grid; filtering seven Gaussian random variables in each node by using a filter, and recording 7 groups of data with the filtered positions of loc1 & cndot & cnk as TLSPs of the k links;
b) the link is one BS to one UE
Directly generating seven Gaussian random variables as TLSPs of the link;
(2) adding cross-correlation between TLSPs
Obtaining 7 transform domain large-scale parameters ξ of each link, where the cross-correlation matrix is a (7 × 7), and the final TLSPs is:
Figure BDA0001328394340000061
(3) conversion from TLSPs to LSPs
Figure BDA0001328394340000062
Further, the pitch angle expansion of the 3D MIMO channel is statistically modeled by using a linear model in the third step, which shows the dependency relationship between the vertical domain angle expansion and the distance in the large-scale parameter:
the angular characteristics of the vertical domain depend on the distance between the BS and the UE, and the corresponding angular spread is modeled as a lognormal random distribution:
Figure BDA0001328394340000063
fig. 2 shows the relationship between the vertical domain angle spread and the distance of the 3D MIMO channel under LOS condition.
The dependence of ESD and ESA on the relevant distance is represented by a linear model:
μ＝λd+η；
where λ and η are linear function coefficients and d is the distance between the BS and the UE in meters.
Further, mixed Von Mises Fisher distribution modeling is introduced in the fourth step to generate an arrival angle and a departure angle of the 3D space; in the 3D MIMO statistical channel model, the total number of angle parameters is four, namely the azimuth angle and the pitch angle of the receiving end and the transmitting end. Taking AoD and EoD of the BS side as an example, the generation method is as follows, AoA and EoA of the UE side can be generated using a similar method. The method comprises the following specific steps:
(1) based on external field measurement, fitting a pitch angle and a fitting azimuth angle by adopting truncated Laplace distribution; calculating inverse gaussians and inverse laplacian functions by using the cluster power and the respective angular spread as input to obtain AoD and EoD;
the azimuth angle power spectrum PAS follows truncated Gaussian distribution and is divided into cluster power PnAnd root mean square angular spread σASDTo generate a random angle AoD:
Figure BDA0001328394340000064
wherein sigmaASDIs the azimuth departure angle spread from step two, the constant C is a scale factor related to the cluster number, which depends on the Rice K factor in the case of LOS, with CLOSInstead of:
CLOS＝C·(1.1035-0.028K-0.002K2+0.0001K3)；
the power spectrum PAS of the pitch angle obeys the truncated Laplace distribution and is divided by the cluster power PnAnd root mean square angular spread σESDGenerating a random angle EoD:
Figure BDA0001328394340000071
wherein sigmaESDIs the angular spread of the pitch from step two, C is a scale factor related to the number of clusters, in the case of LOS depends on the Rice K factor, with CLOSInstead of:
CLOS＝C·(1.3086+0.0339K-0.0077K2+0.0002K3)；
(2) randomly pairing the AoD and the EoD of n clusters generated based on external field measurement to form n groups of AoD and EoD angle sets which serve as the average pitch angle and azimuth angle of rays in the clusters to generate the average wave sending/arrival angle vector of the appointed cluster; the average transmission/arrival angle vector of a cluster is represented by a unit vector Δ, Δ ═ sin θocosφosinθosinφocosθo]TWherein thetaoAnd phioRespectively serving as the average pitch angle and the azimuth angle of rays in the cluster;
(3) modeling 3D angle joint distribution of a single cluster by utilizing Von Mises Fisher distribution, describing cluster angle expansion of azimuth angle and pitch angle, and obtaining azimuth departure angle phi of the mth sub-diameter of the nth clustern,mAnd a pitch departure angle thetan,m(ii) a According to the VMF distribution, the correlation between the azimuth angle and the pitch angle can be represented;
the probability density function of the VMF distribution is expressed as: f. ofp(Ω；Δ,κ)＝Cp(κ)exp(κΔTΩ)sinθ；
Figure BDA0001328394340000072
Wherein [ sin θ cos φ sin θ sin φ cos θ]TRepresenting any one of the wave-emitting/wave-reaching directions on the unit sphere, and theta is a pitch angleAnd phi is the azimuth angle. Delta is the average wave sending/wave reaching angle vector of the cluster, namely the direction of the cluster center; the convergence parameter k describes the degree of diffusion of cluster wave-sending/arrival angles, the larger k is, the more concentrated the cluster angles are, the anisotropy is obtained, and when k is 0, the isotropic scattering occurs in the cluster angles; as shown in fig. 4, the effect of the convergence parameter κ on the VMF distribution was observed when the z-axis was taken as the axis of symmetry. I isd(kappa) is a modified Bessel function of the first type, having an order d, and
Figure BDA0001328394340000073
p-3 in a 3D spatial scene;
from (2) the cluster mean wave sending/arrival angle vector Δ, Δ ═ sin θocosφosinθosinφocosθo]TAnd κ is modeled as a lognormal distribution, the Probability Density Function (PDF) of the VMF distribution is rewritten as:
fp(θ,φ|θo,φo,κ)＝Cp(κ)exp{κ[sinθosinθcos(φ-φo)+cosθocosθ]}sinθ；
wherein ΔTThe inner product of sum Ω is reduced to scalar form, ΔTΩ＝sinθosinθcos(φ-φo)+cosθocosθ。
From the PDF of the VMF distribution, the edge probability density functions of φ and θ can be calculated. Thus producing azimuth and pitch angles with angular spread within the cluster, resulting in azimuth departure angles phi for the respective sub-diametersn,mAnd a pitch departure angle thetan,m。
And then the cross correlation between the azimuth angle and the pitch angle is illustrated by the VMF distribution, and it can be seen that the PDF of the VMF depends on the rotational symmetry axis delta and the convergence parameter kappa. At thetao,φoThe PDF is re-discussed by taking 0 as an example, where the rotational symmetry axis is the z-axis and Δ is [ 001 ═ in]TPDF of VMF distribution is
Figure BDA0001328394340000081
The edge PDF of θ is expressed as
Figure BDA0001328394340000082
The edge PDFs for phi obey a uniform distribution, denoted as
Figure BDA0001328394340000083
In this case, the pitch angle and the azimuth angle are independently distributed. But the average directional vector theta of the vast majority of clusters in a practical propagation environmento,φoThe rotational symmetry axis of ≠ 0,0, Δ points beyond the z-axis. The pitch and azimuth angles are correlated.
It is worth noting that there are a large number of scatterers in a wireless channel, and the channel is multi-clustered, so that a mixed VMF distribution is required to characterize the actual propagation characteristics. Constructing a mixed VMF distribution involves: the angle should be attributed to which cluster and the optimum number of clusters. A clustering algorithm is needed to determine the angles and the number of clusters. A soft expectation maximization algorithm may be used to cluster angles and determine the number of clusters.
Another object of the present invention is to provide a three-dimensional channel model established by the measured-based 3D MIMO statistical channel modeling method.
The invention has the advantages and positive effects that: the method comprises the steps of extracting statistics of large-scale parameters (LSPs) through external field measurement, generating a cross-correlation matrix of the large-scale parameters to solve the problem that the matrix is not definite in the existing model, increasing pitch angle expansion related to distance, introducing mixed Von Mises Fisher distribution to describe the mutual dependence relationship between an azimuth angle and a pitch angle, and improving the precision of the current standardized 3D channel model. Based on a PSM framework, the 3D MIMO statistical channel model based on actual measurement redefines a statistical characterization method of part of parameters in the model and gives a recommended value based on the actual measurement of an external field.
Compared with the existing 3D MIMO channel statistical modeling method, the method can more effectively and accurately reflect the real three-dimensional channel environment, extracts LSPs through actual channel measurement statistics and obtains the positive cross-correlation matrix thereof; expressing the relation between the characteristic parameters of the channel vertical domain and the BS-UE distance by using a linear model; the hybrid VMF distribution characterizes the interdependence of multiple clusters of azimuth and pitch angles. The model can generate a channel impulse response with good statistical characteristics. The invention expands the research and application of the 3D MIMO channel model and provides a powerful tool for accurately and efficiently evaluating the related algorithm of the 3D MIMO system.
Drawings
Fig. 1 is a flow chart of a method for statistical channel modeling based on measured 3D MIMO.
Fig. 2 is a schematic diagram of the relationship between the vertical domain angle spread and the distance of the 3D channel in the LOS path.
Fig. 3 is a schematic diagram showing dependence of EAS and 2DBS-UE distance under LOS and NLOS propagation conditions according to an embodiment of the present invention.
FIG. 4 is a graphical illustration of the effect of the convergence parameter κ on the VMF distribution when the z-axis is taken as the axis of symmetry.
FIG. 5 is a site map and related street view of a survey activity provided by an embodiment of the invention.
FIG. 6 is a schematic diagram of an angular power spectrum generated by a hybrid VMF distribution in a measurement path 5(LOS) provided by an embodiment of the present invention.
Fig. 7 is a schematic diagram of validity verification of the 3D MIMO statistical channel modeling method according to the embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The application of the principles of the present invention will now be described in further detail with reference to the accompanying drawings.
As shown in fig. 1, the statistical channel modeling method provided by the present invention needs to establish a 3D MIMO statistical channel model based on actual measurement, and specifically includes the following steps:
step one, carrying out channel measurement activity, obtaining channel impulse response through channel measurement, and extracting channel multipath components according to the channel impulse response.
The high and new technology industry development area in the Western-Ann state carries out outfield channel measurement activities, strong static scattering objects exist around a measurement route, and the surrounding environment is a typical urban macro cell (UMa) propagation scene. Experimental data is recorded at several different locations with and without LOS propagation, capturing the radio channel as fully as possible. An XDU channel detector is adopted in the channel measuring activity, and the wireless channel detection is realized by transmitting periodic spread spectrum signals to collect time, frequency and space domain information. For the antenna array, the transmitter is provided with a planar antenna array; a crown antenna array is employed at the receiver in order to identify all incoming wave paths from omni-directional space. The horizontal and vertical spacing between adjacent elements of the two antenna arrays is 0.5 wavelengths. The transmitter is placed at the edge of the roof of the building, about 40m, at a height well above the surroundings, while the receiver moves along a pre-planned measurement path. Therefore, only downlink measurements are made during the measurement activity. The BS location, UE movement route and corresponding street view in the channel measurement activity are shown in fig. 5.
And step two, modeling to generate large-scale parameter LSPs and generating a cross-correlation matrix thereof by using a circular filtering method. The seven large-scale parameters are respectively: delay Spread (DS), Shadow Fading (SF), azimuth departure Angle Spread (ASD), azimuth arrival Angle Spread (ASA), pitch departure angle spread (ESD), pitch arrival angle spread (ESA), and the rice K factor (K).
Seven large-scale parameters can be modeled as lognormal distribution, and the generated LSPs are as follows:
Figure BDA0001328394340000101
wherein s is a large-scale parameter vector, mu and sigma are mean and standard deviation vectors of lognormal distribution, which can be obtained by performing statistical analysis on external field measurement data,
Figure BDA0001328394340000102
is a parameter vector describing the correlation between large-scale parameters,
Figure BDA0001328394340000103
a is a cross-correlation matrix of seven large-scale parameters, ξ is generated by using a circular filtering method by using a correlation distance, and is randomly generated by a Gaussian independent same distribution variable with a mean value of 0 and a variance of 1 under the condition that the correlation distance is not obtained.
In the whole channel, there is correlation between the large-scale parameters, and for different large-scale parameters on the same link, it can be described by using a cross-correlation coefficient matrix, and the correlation coefficients of two different large-scale parameters are as follows:
Figure BDA0001328394340000111
where ρ isxyThe correlation coefficient is a large-scale parameter x, y; cxyIs the covariance of the large scale parameters x, y; cxx，CyyThe variance of the large scale parameters x, y, respectively.
The WINNER + model generates the mapping of the cross-correlation large-scale parameters by using a circular filtering method, and the QuaDRiGa model adopts the same method, but simultaneously considers the diagonal direction, expands the algorithm generated by mapping, and further supports the autocorrelation characteristic of the large-scale parameters when the UE moves diagonally. By
Figure BDA0001328394340000112
It can be known that in order to calculate
Figure BDA0001328394340000113
The cross-correlation matrix a must guarantee positive certainty. In the WINNER + model, the positive nature of the cross-correlation matrix A of the 3D channel cannot be guaranteed, so that the cross-correlation matrix of the LSPs is obtained through actual channel measurement, all characteristic values of the matrix are positive, and the positive nature of the A is guaranteed. Table 1 shows the cross-correlation of the vertical domain parameters compared to the WINNER + model.
TABLE 1 comparison of Cross-correlation of vertical Domain parameters with WINNER + model
Figure BDA0001328394340000114
Figure BDA0001328394340000121
The cross-correlation matrix of seven large-scale parameters is obtained based on the external field measurement, and the specific values are shown in table 2. The cross-correlation information of the vertical domain is extracted and generated according to data statistics of actual external field measurement, other parameters follow the data of a WINNER + model, and finally a 3D large-scale parameter cross-correlation matrix A (LOS/NLOS) under the scene of the urban macro cell is obtained.
Table 23D Large Scale parameter Cross-correlation matrix A (LOS/NLOS)
DS ASD ESD ASA ESA SF K
DS
1 0.4/0.4 -0.3/-0.3 0.8/0.6 0.13/-0.05 -0.4/-0.4 -0.4/NaN
ASD 0.4/0.4 1 0.27/0.43 0.3/0.4 0.44/-0.2 -0.5/-0.6 0.1/NaN
ESD -0.3/-0.3 0.27/0.43 1 -0.28/0 0.08/-0.01 0/0 0.22/NaN
ASA 0.8/0.6 0.3/0.4 -0.28/0 1 0.3/0 -0.5/-0.3 -0.2/NaN
ESA 0.13/-0.05 0.44/-0.2 0.08/-0.01 0.3/0 1 -0.8/-0.5 0/NaN
SF -0.4/-0.4 -0.5/-0.6 0/0 -0.5/-0.3 -0.8/-0.5 1 0.3/NaN
K -0.4/NaN 0.1/NaN 0.22/NaN -0.2/NaN 0/NaN 0.3/NaN 1
Step three, statistically modeling the pitch angle expansion of the 3D MIMO channel by using a linear model, and showing the dependence relationship between the vertical domain angle expansion and the distance: the angular characteristics of the vertical domain depend on the distance between the BS and the UE, and the corresponding angular spread (EAS) is modeled as a lognormal random distribution.
Figure BDA0001328394340000122
As shown in fig. 2, the angular spread of the vertical domain of the channel is related to the BS-UE related distance. A dual path downlink propagation scenario with one LOS path and another NLOS path due to ground reflections. The dependence of ESD and ESA on the relevant distance is represented by a linear model:
μ＝λd+η；
where λ and η are linear function coefficients, the recommended values for λ and η for ESD and ESA for LOS or NLOS paths have been summarized in Table 6, and d is the distance between the BS and the UE.
The ESD and ESA versus distance characteristics are shown in fig. 3. It can be seen that ESA has a more pronounced distance dependence than ESD in the LOS case. Specifically, the BS-UE distance is between 50m and 350m, and the linear fit of the ESA decreases approximately at a rate of 0.04 degrees/meter. ESD and ESA are relatively small when the BS-UE distance is less than 50m in external field measurements. This is due to two practical limitations: (1) the bottom of the receiving device may block some of the reflected paths with large angles of incidence, resulting in a smaller ESA; (2) the exit path with large angle is located outside the main lobe of the antenna field pattern resulting in less ESD.
Step four, introducing mixed Von Mises Fisher distribution modeling to generate an arrival angle and a departure angle of a 3D space; in the 3D MIMO statistical channel model, the total number of angle parameters is four, namely the azimuth angle and the pitch angle of the receiving end and the transmitting end. Taking AoD and EoD of the BS side as an example, the generation method is as follows, AoA and EoA of the UE side can be generated using a similar method. The method comprises the following specific steps:
(1) based on external field measurement, truncated Laplace distribution is adopted to fit a pitch angle and truncated Gaussian distribution to fit an azimuth angle. The cluster powers and their respective angular spreads are used to calculate the inverse gaussian and inverse laplacian functions, resulting in AoD and EoD.
The azimuth angle power spectrum PAS follows truncated Gaussian distribution and is divided into cluster power PnAnd root mean square angular spread σASDTo generate a random angle AoD:
Figure BDA0001328394340000131
wherein sigmaASDIs the azimuth departure angle spread from step two, and the constant C is a scaling factor related to the number of clusters, as shown in table 3.
TABLE 3C relationship to Cluster number
Number of clusters 4 5 8 10 11 12 14 15 16 20
C 0.779 0.860 1.018 1.090 1.123 1.146 1.190 1.211 1.226 1.289
In the case of LOS, depending on the Leise K factor, with CLOSInstead of:
CLOS＝C·(1.1035-0.028K-0.002K2+0.0001K3)；
the power spectrum PAS of the pitch angle obeys the truncated Laplace distribution and is divided by the cluster power PnAnd root mean square angular spread σESDGenerating a random angle EoD:
Figure BDA0001328394340000132
wherein sigmaESDIs the pitch-off angle spread from step two, and C is a scaling factor related to the number of clusters, as shown in table 4.
TABLE 4C relationship to Cluster number
Number of clusters 12 19 20
C 1.104 1.184 1.178
In the case of LOS, depending on the Leise K factor, with CLOSInstead of:
CLOS＝C·(1.3086+0.0339K-0.0077K2+0.0002K3)；
(2) randomly pairing the AoD and the EoD of n clusters generated based on external field measurement to form an angle set of n groups of AoD and EoD, and generating the angle set as the average pitch angle and azimuth angle of rays in the clustersThe mean launch/arrival angle vector for the cluster is specified. The average transmission/arrival angle vector of a cluster is represented by a unit vector Δ, Δ ═ sin θocosφosinθosinφocosθo]TWherein thetaoAnd phioAs the mean pitch and azimuth of the rays in the cluster, respectively.
(3) Modeling 3D angle joint distribution of a single cluster by utilizing Von Mises Fisher distribution (VMF), describing cluster angle expansion of azimuth angle and pitch angle, and obtaining azimuth departure angle phi of mth sub-diameter of the nth clustern,mAnd a pitch departure angle thetan,m. The VMF distribution may characterize the correlation between azimuth and pitch.
The probability density function of the VMF distribution is expressed as:
fp(Ω；Δ,κ)＝Cp(κ)exp(κΔTΩ)sinθ；
Figure BDA0001328394340000141
wherein [ sin θ cos φ sin θ sin φ cos θ]TRepresenting any wave sending/arrival direction (the antenna array element is positioned at the center of the sphere) on the unit sphere, theta is a pitch angle, and phi is an azimuth angle. Δ is the average launch/arrival angle vector of the cluster in (2), i.e., the orientation of the cluster center. The convergence parameter κ describes the degree of spread of the cluster launch/arrival angles, the larger κ, the more concentrated the cluster angles and becomes anisotropic, and the cluster angles appear isotropic scattering when κ is 0. As shown in fig. 4, the effect of the convergence parameter κ on the VMF distribution was observed when the z-axis was taken as the axis of symmetry. I isd(kappa) is a modified Bessel function of the first type, having an order d, and
Figure BDA0001328394340000142
in a 3D spatial scene p is 3.
From (2) the cluster mean wave sending/arrival angle vector Δ, Δ ═ sin θocosφosinθosinφocosθo]TAnd κ is modeled as a lognormal distribution, the Probability Density Function (PDF) of the VMF distribution can be rewritten as:
fp(θ,φ|θo,φo,κ)＝Cp(κ)exp{κ[sinθosinθcos(φ-φo)+cosθocosθ]}sinθ；
wherein ΔTThe inner product of sum Ω is reduced to scalar form, ΔTΩ＝sinθosinθcos(φ-φo)+cosθocosθ。
From the PDF of the VMF distribution, the edge probability density functions of φ and θ can be calculated. Thus producing azimuth and pitch angles with angular spread within the cluster, resulting in azimuth departure angles phi for the respective sub-diametersn,mAnd a pitch departure angle thetan,m。
And then the cross correlation between the azimuth angle and the pitch angle is illustrated by the VMF distribution, and it can be seen that the PDF of the VMF depends on the rotational symmetry axis delta and the convergence parameter kappa. At thetao,φoThe PDF is re-discussed by taking 0 as an example, where the rotational symmetry axis is the z-axis and Δ is [ 001 ═ in]TThe PDF of the VMF distribution is:
Figure BDA0001328394340000151
the edge PDF of θ is expressed as
Figure BDA0001328394340000152
The edge PDFs for phi obey a uniform distribution, denoted as
Figure BDA0001328394340000153
In this case, the pitch angle and the azimuth angle are independently distributed. But the average directional vector theta of the vast majority of clusters in a practical propagation environmento,φoThe rotational symmetry axis of ≠ 0,0, Δ points beyond the z-axis. The pitch and azimuth angles are correlated.
It is worth noting that there are a large number of scatterers in a wireless channel, and the channel is multi-clustered, so that a mixed VMF distribution is required to characterize the actual propagation characteristics. Constructing a mixed VMF distribution involves: the angle should be attributed to which cluster and the optimum number of clusters. A clustering algorithm is needed to determine the angles and the number of clusters. A soft expectation maximization algorithm may be used to cluster angles and determine the number of clusters. The BS side in the measurement path 5(LOS) determines angle clustering based on outfield measurement activity using a soft expectation maximization algorithm, the generated parameters are listed in table 5, and the values in the last column of the table define the prior probability of each direction vector generated by that particular VMF distribution. And the corresponding angular power spectrum (PAS) is represented in fig. 6 by the mixed VMF distribution.
Table 5 parameters extracted from the measurement path 5(LOS) BS side using the soft expectation maximization algorithm
ID θo([°]) φo([°]) κ Prior probability
1 118 73 543 0.217
2 90 117 612 0.331
3 79 143 183 0.09
4 84 131 129 0.024
5 92 89 347 0.121
6 82 54 331 0.078
7 88 126 277 0.027
8 110 82 94 0.112
It can be seen from fig. 6 that the PAS of clusters 1, 2 and 5 are more distinct than other clusters due to the common influence of the convergence parameter k and the prior probability, so that a specific scene can be well described by 6 or even 3 clusters. It can be assumed that the dominant path of cluster 2 is from a LOS path, verified by comparing the average direction angle derived based on site map information with the geometrically calculated average direction angle. Furthermore, it is observed from table 5 that the convergence parameter κ for cluster 2 is the largest because the azimuth and elevation spread of the LOS path is relatively small. In general, the cluster angle spread is generally more spread out on the UE side and under the NLOS path. Furthermore, it can be seen that the average direction vector of all clusters points out of the z-axis, so azimuth and elevation angles are typically correlated in a real propagation environment.
The cross-correlation matrix of LSPs, the slope lambda and intercept η of the linear model, and the cluster number, cluster expansion, azimuth angle and pitch angle in the mixed VMF distribution.
After the measurement data are obtained, removing an antenna directional diagram from the measurement result by utilizing a space alternating generalized expectation maximization (SAGE) algorithm to obtain pure wireless transmission characteristics; deconvoluting the detection signal waveform by adopting an improved CLEAN algorithm to extract a pure CIR; finally, 12 channel characteristic parameters including power, delay, Doppler frequency, wave sending/wave reaching angle of the sub-path, channel delay, angle spread and the like are extracted. The channel measurement activity is performed around 2.6GHz, but these extracted characteristic parameters are equally applicable to 3D MIMO CIRs at carrier frequencies of 2-6GHz, and these parameters have no significant frequency dependence in a particular band.
Table 6 summarizes the statistics of the channel parameters extracted from the outfield measurement data in the urban macro cell scenario, which are mainly reflected in the vertical domain and the delay domain. The path loss and cross polarization power ratio (XPR) are well studied in WINNER series channel models, and the information in the models is used, so that complete model parameters are obtained.
Table 6 channel model parameters extracted from the UMa scenario
Figure BDA0001328394340000171
Figure BDA0001328394340000181
And step six, generating channel coefficients.
(1) An initial random phase is set. Setting random initial phase under four polarization modes (vv, vh, hv, hh) for the mth minor diameter in the nth cluster
Figure BDA0001328394340000182
The initial phase is uniformly distributed within (-pi, pi). Under LOS condition, only the initial phase of vv and hh polarization needs to be calculated
Figure BDA0001328394340000183
(2) A steering vector and a doppler frequency of the antenna array are determined.
The channel response matrix of the nth cluster from the s-th transmitting antenna to the u-th receiving antenna can be obtained according to the following formula:
Figure BDA0001328394340000184
wherein Hu,s,n(t) is the channel coefficient matrix of the nth cluster, and each cluster has M rays, PnIs the power of the nth cluster, F is the field pattern of the transmit or receive antenna in either horizontal or vertical polarization,
Figure BDA0001328394340000185
is the azimuth angle of arrival phin,m,AoAAnd perpendicular angle of arrival thetan,m,EoANormalized angle vector of (2) is obtained by the following equation
Figure BDA0001328394340000186
Figure BDA0001328394340000187
Where n represents a cluster and m represents a ray within the nth cluster.
Figure BDA0001328394340000188
Is the azimuth departure angle phin,m,AoDAnd a vertical departure angle thetan,m,EoDNormalized angle vector of (2) is obtained by the following equation
Figure BDA0001328394340000189
Figure BDA00013283943400001810
In addition to this, the present invention is,
Figure BDA00013283943400001811
respectively, the position vectors of the receive antenna u and the transmit antenna s. Kappan,mIs the cross-polarization power ratio in the linear range. Lambda [ alpha ]0Is the wavelength of the carrier frequency. Doppler frequency component vn,mThe velocity vector that can be determined by the angle of arrival (AoA, EoA) and the UE
Figure BDA00013283943400001812
Obtaining:
Figure BDA0001328394340000191
wherein
Figure BDA0001328394340000192
When an LOS path exists, calculating the channel coefficient of the LOS path:
Figure BDA0001328394340000193
and verifying the validity of the channel model. To validate the proposed parameterized model, the validity of the channel model is represented by a normalized spatial cross-correlation function (CCF):
Figure BDA0001328394340000194
where, subscripts k, k 'and l, l' respectively denote the kth, k 'antenna element and the l, l' antenna element of the BS and the UE, n denotes the nth cluster in the propagation environment, and γ and ∈ are the respective antenna element intervals on the BS and UE sides. Rhokl,k′l′,n(gamma, epsilon; t) denotes the connection of the transmitting antenna k at the instant t
Figure BDA0001328394340000195
And a receiving antenna l
Figure BDA0001328394340000196
Link with transmit antenna k 'connecting the nth cluster'
Figure BDA0001328394340000197
And a receiving antenna l'
Figure BDA0001328394340000198
Is measured by the normalized spatial cross-correlation function CCF between the links.
The application effect of the present invention will be described in detail with reference to the simulation.
Consider link-level simulation verification as the UE moves away from the BS along a linear trajectory during measurement activities. First, the BS distance to the cell boundary is set to 350m, and the UE initial position is placed at a distance BS50 m. Between 50m and 350m, 7 random and independent measurement routes are used to obtain simulation results. In fig. 7, curves | ρ, reflecting the channel spatial cross-correlation characteristics, are plotted according to different normalized antenna spacings γ of the BS side under LOS and NLOS conditions, respectivelyk1,k′1,1(gamma, 0; t) |, the minimum interval of the antenna array elements used by the BS side is 0.5 wavelength and the maximum is 1.5 wavelength in actual test.
In fig. 7, it can be seen that the actual measurement results and the simulation results almost coincide in LOS and NLOS scenes, which illustrates that the proposed 3d mimo channel model can well represent the statistical characteristics of the real propagation environment. It is worth noting that the absolute value of the spatial CCF under the LOS path is significantly higher than that under the NLOS path. This is because in NLOS scenarios, the angular spread of the multipath is larger as well as the cluster spread, resulting in a more diffuse transmission.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.

Claims (1)

1. A3D MIMO statistical channel modeling method based on actual measurement is characterized in that the 3D MIMO statistical channel modeling method based on actual measurement extracts statistics of large-scale parameters through external field measurement to generate a cross-correlation matrix of the large-scale parameters; expressing the dependency relationship between the angle expansion of the vertical domain and the distance by using a linear model; introducing mixed VMF distribution to depict the interdependence of a plurality of clusters of azimuth angles and pitch angles and generating an arrival angle and a departure angle of a 3D space; determining each characterization parameter of the channel model according to the statistical analysis of the external field measurement, and finally generating a 3D MIMO channel coefficient;
the actual measurement-based 3D MIMO statistical channel modeling method comprises the following steps:
step one, carrying out channel measurement activity, obtaining channel impulse response through channel measurement, and extracting channel multipath components according to the channel impulse response;
step two, modeling to generate large-scale parameters LSPs and generating a cross-correlation matrix thereof by using a circular filtering method, wherein the seven large-scale parameters are respectively as follows: delay spread, shadow fading, azimuth departure angle spread, azimuth arrival angle spread, elevation departure angle spread, elevation arrival angle spread, and rice K factor;
seven large-scale parameters are modeled as lognormal distribution, and the generated large-scale parameters (LSPs) are as follows:
Figure FDA0002452667230000011
wherein s is a large-scale parameter vector, and μ and σ are mean and standard deviation vectors of lognormal distributionThe measured data of the external field is subjected to statistical analysis to obtain,
Figure FDA0002452667230000012
is a parameter vector describing the correlation between large-scale parameters,
Figure FDA0002452667230000013
a is a cross-correlation matrix of seven large-scale parameters, ξ is generated by using a circular filtering method by utilizing a correlation distance, and is randomly generated by a Gaussian independent same-distribution variable with a mean value of 0 and a variance of 1 under the condition that the correlation distance is not obtained;
for different large-scale parameters on the same link, the correlation coefficients of the two different large-scale parameters are as follows:
Figure FDA0002452667230000014
where ρ isxyThe correlation coefficient, C, of the large scale parameter x, yxyIs the covariance of the large-scale parameter x, y, Cxx，CyyThe variances of the large-scale parameters x and y are respectively;
thirdly, statistically modeling the pitch angle expansion of the 3D MIMO channel by using a linear model, and showing the dependence relationship between the vertical domain angle expansion and the distance in the large-scale parameters;
step four, introducing mixed Von Mises Fisher distribution modeling to generate an arrival angle and a departure angle of a 3D space; in the 3D MIMO statistical channel model, the number of angle parameters is four, namely the azimuth angle and the pitch angle of the receiving end and the transmitting end;
determining various characteristic parameters of the model according to the statistical analysis of the channel measurement, wherein the characteristic parameters comprise a cross-correlation matrix of LSPs (least squares), the slope and the intercept of a linear model, the cluster number, cluster expansion, azimuth angle and pitch angle in mixed VMF (virtual vehicle model) distribution, and other model parameters can be represented by main flow models such as WINNER series or QuaDRiGa;
step six, generating channel coefficients;
(1) setting an initial random phase for the nth clusterSetting random initial phase of mth minor diameter under four polarization modes (vv, vh, hv, hh)
Figure FDA0002452667230000021
The initial phase is uniformly distributed within (-pi, pi);
(2) determining a steering vector and a Doppler frequency of the antenna array; the channel response matrix of the nth cluster from the s-th transmitting antenna to the u-th receiving antenna can be obtained according to the following formula:
Figure FDA0002452667230000022
wherein Hu,s,n(t) is the channel coefficient matrix of the nth cluster, and each cluster has M rays, PnIs the power of the nth cluster, F is the field pattern of the transmit or receive antenna in either horizontal or vertical polarization,
Figure FDA0002452667230000023
is the azimuth angle of arrival phin,m,AoAAnd perpendicular angle of arrival thetan,m,EoAThe normalized angle vector of (a) is,
Figure FDA0002452667230000024
is the azimuth departure angle phin,m,AoDAnd a vertical departure angle thetan,m,EoDNormalized angle vector of (a);
Figure FDA0002452667230000025
and
Figure FDA0002452667230000026
position vectors for the receiving antenna u and the transmitting antenna s, respectively; kappan,mIs the cross-polarization power ratio in the linear range; lambda [ alpha ]0Is the wavelength of the carrier frequency; doppler frequency component vn,mVelocity vector by angle of arrival (AoA, EoA) and UE
Figure FDA0002452667230000027
Obtaining;
when an LOS path exists, calculating the channel coefficient of the LOS path:
Figure FDA0002452667230000031
the concrete process of modeling and generating large-scale parameter LSPs and generating the cross-correlation matrix by using a circular filtering method in the second step is as follows:
(1) generating preliminary transform domain large scale parameters (TLSPs) in a transform domain;
Figure FDA0002452667230000032
obey Gaussian distribution, in
Figure FDA0002452667230000033
Mapping to obtain siBefore the start of the operation of the device,
Figure FDA0002452667230000034
to be mixed with
Figure FDA0002452667230000035
The association is carried out in such a way that,
Figure FDA0002452667230000036
is a transform domain large-scale parameter corresponding to other LSPs or other links; the generation method of TLSPs is different for different network layouts, considering two communication networks as follows:
a) the link is from one BS to multiple UEs
UE coordinate is (x)1,y1),…(xk,yk) Generating a system grid, generating seven Gaussian random variables for each node, respectively corresponding to 7 TLSPs, and finding out the positions loc1 … lock of k users in grid points; generate autocorrelation filter responses for the seven TLSPs:
Figure FDA0002452667230000037
wherein λ ismIs the autocorrelation distance of each LSP; d is coordinate extension value in the system grid; filtering seven Gaussian random variables in each node by using a filter, and recording 7 groups of data with the filtered positions being loc1 … lock as TLSPs of the k links;
b) the link is one BS to one UE
Directly generating seven Gaussian random variables as TLSPs of the link;
(2) adding cross-correlation between TLSPs
Obtaining 7 transform domain large-scale parameters ξ of each link, where the cross-correlation matrix is a (7 × 7), and the final TLSPs is:
Figure FDA0002452667230000041
(3) conversion from TLSPs to LSPs
Figure FDA0002452667230000042
In the third step, the pitch angle expansion of the 3D MIMO channel is statistically modeled by using a linear model, and the dependency relationship between the vertical domain angle expansion and the distance in the large-scale parameter is shown:
the angular characteristics of the vertical domain depend on the distance between the BS and the UE, and the corresponding angular spread is modeled as a lognormal random distribution:
Figure FDA0002452667230000043
the dependence μ ═ λ d + η of ESD and ESA on the relevant distance is represented by a linear model;
where λ and η are linear function coefficients, d is the distance between the BS and the UE in meters;
introducing mixed Von Mises Fisher distributed modeling to generate an arrival angle and a departure angle of a 3D space in the fourth step, wherein in the 3D MIMO statistical channel model, the number of angle parameters is four, namely an azimuth angle and a pitch angle at the receiving end and the transmitting end; taking AoD and EoD of the BS side as an example, the generation method is as follows, AoA and EoA of the UE side can be generated by using a similar method; the method comprises the following specific steps:
(1) based on external field measurement, fitting a pitch angle and a fitting azimuth angle by adopting truncated Laplace distribution; calculating inverse gaussians and inverse laplacian functions by using the cluster power and the respective angular spread as input to obtain AoD and EoD;
the azimuth angle power spectrum PAS follows truncated Gaussian distribution and is divided into cluster power PnAnd root mean square angular spread σASDTo generate a random angle AoD:
Figure FDA0002452667230000044
wherein sigmaASDIs the azimuth departure angle spread from step two, the constant C is a scale factor related to the cluster number, which depends on the Rice K factor in the case of LOS, with CLOSInstead of:
CLOS＝C·(1.1035-0.028K-0.002K2+0.0001K3)；
the power spectrum PAS of the pitch angle obeys the truncated Laplace distribution and is divided by the cluster power PnAnd root mean square angular spread σESDGenerating a random angle EoD:
Figure FDA0002452667230000051
wherein sigmaESDIs the angular spread of the pitch from step two, C is a scale factor related to the number of clusters, in the case of LOS depends on the Rice K factor, with CLOSInstead of:
CLOS＝C·(1.3086+0.0339K-0.0077K2+0.0002K3)；
(2) randomly pairing the AoD and the EoD of n clusters generated based on external field measurement to form n groups of AoD and EoD angle sets which serve as the average pitch angle and azimuth angle of rays in the clusters to generate the average wave sending/arrival angle vector of the appointed cluster; the average transmission/arrival angle vector of a cluster is represented by a unit vector Δ, Δ ═ sin θocosφosinθosinφocosθo]TWherein thetaoAnd phioRespectively serving as the average pitch angle and the azimuth angle of rays in the cluster;
(3) modeling 3D angle joint distribution of a single cluster by utilizing VonMISESFisher distribution, describing cluster angle expansion of an azimuth angle and a pitch angle, and obtaining an azimuth departure angle phi of the mth sub-diameter of the nth clustern,mAnd a pitch departure angle thetan,m(ii) a According to the VMF distribution, the correlation between the azimuth angle and the pitch angle can be represented;
the probability density function of the VMF distribution is expressed as: f. ofp(Ω；Δ,κ)＝Cp(κ)exp(κΔTΩ)sinθ；
Figure FDA0002452667230000052
Wherein [ sin θ cos φ sin θ sin φ cos θ]TRepresenting any wave sending/arrival direction on a unit ball, wherein theta is a pitch angle and phi is an azimuth angle; delta is the average wave sending/wave reaching angle vector of the cluster, namely the direction of the cluster center; the convergence parameter k describes the degree of diffusion of cluster wave-sending/arrival angles, the larger k is, the more concentrated the cluster angles are, the anisotropy is obtained, and when k is 0, the isotropic scattering occurs in the cluster angles; i isd(kappa) is a modified Bessel function of the first type, having an order d, and
Figure FDA0002452667230000053
p-3 in a 3D spatial scene;
from (2) the cluster mean wave sending/arrival angle vector Δ, Δ ═ sin θocosφosinθosinφocosθo]TAnd κ is modeled as a lognormal distribution, the Probability Density Function (PDF) of the VMF distribution is rewritten as:
fp(θ,φ|θo,φo,κ)＝Cp(κ)exp{κ[sinθosinθcos(φ-φo)+cosθocosθ]}sinθ；
wherein ΔTThe inner product of sum omega is reduced to scalar form,ΔTΩ＝sinθosinθcos(φ-φo)+cosθocosθ；
the edge probability density function of phi and theta can be calculated through the PDF of the VMF distribution; thus producing azimuth and pitch angles with angular spread within the cluster, resulting in azimuth departure angles phi for the respective sub-diametersn,mAnd a pitch departure angle thetan,m；
Then, the correlation between the azimuth angle and the pitch angle is explained through VMF distribution, and the PDF of the VMF is determined by a rotational symmetry axis delta and a convergence parameter kappa; at thetao,φoThe PDF is re-discussed by taking 0 as an example, where the rotational symmetry axis is the z-axis and Δ is [ 001 ═ in]T，
Figure FDA0002452667230000061
The edge PDF of θ is expressed as
Figure FDA0002452667230000062
The edge PDFs for phi obey a uniform distribution, denoted as
Figure FDA0002452667230000063
In this case, the pitch angle and the azimuth angle are independently distributed; but the average directional vector theta of the vast majority of clusters in a practical propagation environmento,φoNot equal to 0,0, the rotational symmetry axis of Δ points beyond the z-axis; the pitch and azimuth angles are correlated;
it is worth noting that a large number of scatterers exist in a wireless channel, and the channel is multi-clustered, so that a mixed VMF distribution is required to represent actual propagation characteristics; constructing a hybrid VMF distribution involves two problems, which cluster and the optimal number of clusters an angle should belong to; a clustering algorithm is required to determine the angles and the number of clusters, and a soft expectation maximization algorithm can be used to cluster the angles and determine the number of clusters.
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