Novel 3D Geometry-Based Stochastic Models for Non-Isotropic MIMO Vehicle-to-Vehicle Channels

I. INTRODUCTION

• Existing 3D RS-GBSMs treated azimuth and elevation independently and did not account for VTD's impact on channel statistics.The cited measurement campaigns found that VTD significantly affects V2V channel statistics.Source: §I · p. 2
• The model uses the VMF distribution to jointly consider azimuth and elevation angles and can study their and VTD's impacts on channel statistics.The paper derives and investigates amplitude and phase PDFs, space-time correlation, Doppler PSD, envelope LCR, and AFD, and compares the impacts of VTD and elevation with a corresponding 2D model.Source: §I · p. 2
• The corresponding finite-scatterer SoS simulation model uses the MEV method to jointly calculate azimuth and elevation angles.The paper presents MEV as the first method for jointly computing these angles for 3D MIMO channel models.Source: §I · p. 2
• The SoS simulation model is reported to be an excellent approximation of the reference model according to their statistical properties.The paper verifies the simulation model by comparing it with the reference model and simulated results.Source: §I · p. 2

A. Description of the 3D MIMO V2V theoretical RS-GBSM

• The proposed model combines LoS components, a single- and double-bounced two-sphere model, and a single-bounced elliptic-cylinder model.The two spheres represent moving vehicles around the Tx and Rx, while the elliptic cylinder represents stationary roadside environments; the model distinguishes these environments to study VTD.Source: A. Description of the 3D MIMO V2V… · pp. 2–3

B. Statistical properties of the 3D MIMO V2V RS-GBSM

• The section derives and investigates amplitude and phase probability density functions for the proposed 3D MIMO V2V theoretical RS-GBSM.These are among the statistical properties examined in the section.Source: B. Statistical properties of the 3D… · p. 5
• The section derives and investigates the space-time correlation function and Doppler power spectral density.Both properties are included in the section's analysis of the proposed model.Source: B. Statistical properties of the 3D… · p. 5
• The section derives and investigates envelope level crossing rate and average fade duration.These envelope statistics are included in the section's analysis.Source: B. Statistical properties of the 3D… · p. 5

• The amplitude PDF is derived for the proposed 3D V2V reference model.Source: 1) Amplitude and phase PDFs · p. 5
• The expression for the amplitude PDF uses I0, the zeroth-order modified Bessel function of the first kind.Source: 1) Amplitude and phase PDFs · p. 5
• The phase PDF is also derived for the proposed 3D V2V reference model.Source: 1) Amplitude and phase PDFs · p. 5

• The normalized ST CF is defined between any two complex fading envelopes under the wide-sense stationary condition.Source: 2) ST CF · p. 5
• The component expressions are obtained by substituting the channel model into the correlation definition and applying the corresponding VMF distribution.Source: 2) ST CF · p. 5
• The single-bounced components arise from the Tx sphere, Rx sphere, and elliptic cylinder, respectively.Source: 2) ST CF · p. 5
• The double-bounced component considered here results from the Tx and Rx spheres.Source: 2) ST CF · p. 6
• The total normalized theoretical ST CF is the sum of the LoS, single-bounced, and double-bounced component functions.Source: 2) ST CF · pp. 5–6

3) Doppler PSD:

• The Doppler PSD is obtained by applying the Fourier transform to the ST CF.Source: 3) Doppler PSD · p. 6

4) Envelope LCR and AFD:

• The LCR at level r is the rate at which the envelope crosses that level in either the positive or negative direction.Source: 4) Envelope LCR and AFD · p. 6
• The AFD at level r is the average time the signal envelope remains below that level.Source: 4) Envelope LCR and AFD · p. 6

III. THE 3D SOS SIMULATION MODEL FOR MIMO V2V CHANNELS

• The SoS model uses finite numbers of scatterers or sinusoids to simulate the theoretical 3D channel.Source: §III · p. 7
• The model parameters to determine are the discrete departure and arrival angles, while the remaining parameters match the theoretical model.Source: §III · p. 7
• For single-bounce paths, departure and arrival angles are correlated.Source: §III · p. 7
• MEV extends MMEA to jointly calculate azimuth and elevation angles for 3D channel models, whereas the listed existing methods are valid only for 2D horizontal models.Source: §III · p. 7

A. MEV for parameterization of the proposed SoS simulation model

• The VMF CDF is a double integral over azimuth and elevation, and its volume is used in the MEV angle-selection procedure.Source: A. MEV for parameterization of the… · p. 7

B. Statistical properties of the proposed SoS simulation model

• The SoS model’s statistical properties are derived by applying its discrete parameters to the corresponding expressions.The derivations follow those for the theoretical model, with similar steps briefly given here.Source: B. Statistical properties of the… · p. 7

1) Amplitude and Phase PDFs:

• The amplitude PDF of the SoS simulation model is derived from its amplitude process.The passage introduces the process and refers to a similar procedure in.Source: 1) Amplitude and Phase PDFs · p. 8
• The phase PDF of the SoS simulation model is derived for its phase process.Source: 1) Amplitude and Phase PDFs · p. 8

2) ST CF:

• The ST CF is expressed separately for the three single-bounced components from the Tx sphere, Rx sphere, and elliptic cylinder.Source: 2) ST CF · p. 8
• The LoS ST CF of the SoS simulation model is identical to that of the 3D theoretical RS-GBSM.Source: 2) ST CF · p. 8
• The model also gives a separate ST CF expression for the double-bounced component from the Tx and Rx spheres.Source: 2) ST CF · p. 8

3) Doppler PSD:

• The SoS model’s Doppler PSD is expressed as a statistical property of the simulation model.Source: 3) Doppler PSD · p. 8

4) Envelope LCR and AFD:

• The SoS model’s envelope LCR and AFD are derived using the corresponding theoretical-model expressions.Source: 4) Envelope LCR and AFD · p. 8

IV. SIMULATION RESULTS AND ANALYSIS

• The simulations use distinct low- and high-VTD parameter sets, with Ricean K values of 3.786 and 0.156, respectively.The setup uses 40 discrete scatterers for each component as a tradeoff between realization complexity and accuracy.Source: §IV · p. 9
• Setting β(n1)R = 0° reduces the proposed 3D model to a 2D two-ring and elliptic model for assessing elevation-angle effects.Source: §IV · p. 9

A. Amplitude and phase PDFs

• With N1 = N2 = N3 = 40, the simulation model’s amplitude and phase PDFs agree excellently with the reference model in both low- and high-VTD scenarios.These PDFs depend on scatterer counts, component gains, and LoS amplitude, while other model parameters do not influence them.Source: A. Amplitude and phase PDFs · p. 9, Figure 4 · p. 9, Figure 5 · p. 9

B. Temporal autocorrelation function

• The temporal ACF is obtained from the ST CF by setting the spatial separation to zero.Source: B. Temporal autocorrelation function · p. 9
• The 3D simulation model and simulation results closely match the 3D reference temporal ACF, while the 2D model overestimates temporal correlation.The temporal ACF is higher in the low-VTD scenario than in the high-VTD scenario.Source: B. Temporal autocorrelation function · p. 10, Figure 6 · p. 10

C. Spatial cross-correlation function

• The spatial CCF is obtained from the ST CF by setting the time lag τ to zero.Source: C. Spatial cross-correlation function · p. 10
• The 2D simulation model overestimates spatial correlations and underestimates spatial diversity gain because it misses vertical-plane diversity.The 3D simulation results also match the 3D simulation model closely.Source: C. Spatial cross-correlation function · p. 10, Figure 7 · p. 10

D. Doppler PSD

• Higher VTD makes the Doppler PSD more evenly distributed; the 2D model underestimates it in both low- and high-VTD scenarios.The paper attributes the high-VTD distribution to power reflected from moving vehicles arriving from all directions, versus specific roadside-scatterer and LoS directions at low VTD.Source: D. Doppler PSD · pp. 10–11

E. Envelope LCR and AFD

• Lower VTD produces smaller envelope LCRs and larger AFDs; the supplied figures compare normalized LCRs and AFDs for the 3D and 2D models.The VTD significantly affects both measures.Source: Figure 10 · p. 11, E. Envelope LCR and AFD · p. 11
• At increased elevation parameters of 60°, the 2D model has higher LCR and smaller AFD than the 3D model, although elevation has only a minor overall impact on these measures.With the earlier elevation parameters, the LCR and AFD differences are barely discernible.Source: E. Envelope LCR and AFD · p. 11, Figure 10 · p. 11

V. CONCLUSIONS

• VTD has a great impact on all channel statistical properties, while elevation angle significantly affects only space-time correlation and Doppler PSD.The paper concludes that low VTD shows better channel performance than high VTD.Source: §V · p. 11