Subsequent, we deduce a specific vehicular 5.9 GHz NLOS path-loss model--VirtualSource11p--from the measurements and characterize small-scale fading in NLOS areas.

### 3.1 Data quality and system loss

To deduce path-loss and fading from an of-the-shelf radio, it needs to provide reliable reception power values. We got per packet reception power values with 1 dBm resolution via the level value in the *iw* _*statistics*.*iw* _*quality* struct from a *SIOCGIWSTATS* socket call. Measured values equal to the reported values in iwconfig and the Linkbird wlan11p tool. Observed values range between -5 and -92 dBm. This corresponds well to the reception sensitivity of -92 as reported in data sheets such as [20]. Figure 7 shows the good quality of the reported power values over time for two exemplary 20 m street stretches. Differences in small-scale fading between NLOS and LOS are clearly visible. The resulting power histograms (compare Figure twelve in later Section 3.3) show very reasonable results too. More of these detailed plots are available at the website [19] covering this work.

There was only one issue: power histograms revealed that there are no packets reported with -69, -68 and -67 dBm. A figure illustrating this can be seen on the website [19]. The same gap can be seen in [21]. We believe that the chipset changes its sensitivity in this power range, and reports values above -69 dBm by 3 dB too strong. We corrected this by subtracting 3 dB from all reported values >-69 dBm. The reported reception sensitivity was not changed by the correction.

We measured received power, where in dBm space: RxPower = Txpower - SystemLoss - PathLoss. To determine path-loss, we need to know system loss. The cables lead to a combined loss of ≈ 4.8dB and the antennas to a gain of two times some value between 0 (0°) and 5 dB (15°).

To determine the average loss, we took the LOS measurements from most intersections (excluding free space, special case 9, and 1) and determined the deviation between average power curve and the theoretical limit. The fit equation used is

\begin{array}{c}\hfill \mathsf{\text{LogDist}}\left(x\right)={P}_{\mathsf{\text{tx}}}-{L}_{S}-\mathsf{\text{P}}{\mathsf{\text{L}}}_{\mathsf{\text{ref}}}-10*{E}_{L}*{log}_{10}\left(\frac{x}{1}\right)\hfill \\ \hfill \mathsf{\text{P}}{\mathsf{\text{L}}}_{\mathsf{\text{ref}}}:=\mathsf{\text{FSPL}}\left(1\right)=10{log}_{10}\left({\left(\frac{4\pi 1}{\lambda}\right)}^{2}\right)=47.86\phantom{\rule{0.3em}{0ex}}\mathsf{\text{dB}}\hfill \\ \hfill \mathsf{\text{FitDimensions}}:{L}_{S}=\mathsf{\text{SystemLoss}},{E}_{L}=\mathsf{\text{LossExponent}}.\hfill \end{array}

It comprises the common Log Distance model and free space path loss (FSPL) for determination of reference loss. Unfortunately, curve attenuation interferes with slope variation. Therefore, three fits were performed (Variable, SL = 0, LE = 2). Figure 8 visualizes fit input and results. We limited the fit input to 20 < *x* < 150 m, as packet loss occurred at *x* < 150 m and small distances are inaccurate as the transmitter was not exactly positioned in intersection centers.

The fit reveals a loss of 2.75 dB with LE = 2. With SL = 0, it shows a loss exponent of 2.1, being higher as in FSPL. Subsequently, we will assume 1.75 dB system loss, as revealed by fitting both variables. The resulting average gain of 1.5 dB per antenna seems realistic, given its characteristic. Note that such loss determination absorbs the problem that real transmitted power might slightly differ from the configured value.

### 3.2 NLOS path-loss model development

To determine path-loss with respect to variable street-width and suburban/urban differences, we fit the intersection wide average power distance curves of multiple intersections (such as in Figure 6) to a unified path-loss equation. The basis for our fit equation is the cellular model proposed in [14]. The original VirtualSource equation (as given in [14], but indices modified to Figure 9 and as positive path-loss in dB) is

\begin{array}{c}\hfill \mathsf{\text{PathLoss}}=\left\{\begin{array}{cc}\hfill 10\phantom{\rule{0.1em}{0ex}}{log}_{10}\left(\frac{1}{\alpha}{\left(\sqrt{\frac{2\pi}{{x}_{t}{w}_{r}}}\frac{4\pi {d}_{t}{d}_{r}}{\lambda}\right)}^{2}\right),\hfill & \hfill {d}_{r}\le {d}_{b}\hfill \\ \hfill 10\phantom{\rule{0.1em}{0ex}}{log}_{10}\left(\frac{1}{\alpha}{\left(\sqrt{\frac{2\pi}{{x}_{t}{w}_{r}}}\frac{4\pi {d}_{t}{d}_{r}^{2}}{\lambda {d}_{b}}\right)}^{2}\right),\hfill & \hfill {d}_{r}>{d}_{b}\hfill \end{array}\right.\hfill \\ \hfill \begin{array}{cc}\hfill {d}_{b}=\frac{4{h}_{t}{h}_{r}}{\lambda}\left(\begin{array}{c}\hfill \mathsf{\text{BreakEven}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{Dist}}\mathsf{\text{.}}\hfill \end{array}\right)\hfill & \hfill \begin{array}{c}\hfill {h}_{t},{h}_{r}=\mathsf{\text{Tx,RcvHeight}}\hfill \\ \hfill \alpha =\mathsf{\text{StreetParameter}}\hfill \end{array}\hfill \end{array}\hfill \end{array}

The model takes the distance of transmitter and receiver to intersection center (*d*_{
t
} and *d*_{
r
} ), receiver street width (*w*_{
r
} ), and distance of transmitter to wall (*x*_{
t
} ) as input. Last two values reflect building position influence. Adaption to differing streets is enabled by a street parameter (*α*). A higher loss is present at high rcv-distances (due to a diffraction, rather than reflection predominance), determined by a break even distance (*d*_{
b
} ).

The geometric input parameters *d*_{
t
} , *d*_{
r
} , *w*_{
r
} , *h*_{
r
} , *h*_{
t
} , and *x*_{
t
} are given by the measured data. Therefore, we first fitted the path-loss exponent and *α*. As both are fitted globally, *α* represents a relative shift of the fitting curve. These first fits showed that especially the influence of street width *w*_{
r
} is not properly reflected by the existing equation. Therefore, we replaced the fixed factor \sqrt{2\pi \u2215.} for *x*_{
t
}*w*_{
r
} by a fittable exponent. As *d*_{
t
} also influences the "view" and therefore energy into a crossing street, it was also made fittable. A suburban loss factor was added to incorporate the observation of less power in suburban scenarios from Section 2.4. The following fit equation (in dBm) was found:

\begin{array}{c}\hfill \mathsf{\text{RxPower}}={P}_{\mathsf{\text{tx}}}-\mathsf{\text{SystemLoss}}-\mathsf{\text{PathLoss}}\hfill \\ \hfill \mathsf{\text{PathLoss}}:=\mathsf{\text{VirtualSource11p}}\left({d}_{r},{d}_{t},{w}_{r},{x}_{t},{i}_{s}\right)=\hfill \\ \hfill C+{i}_{s}{L}_{SU}+10\phantom{\rule{0.1em}{0ex}}{log}_{10}\left({\left(\frac{{d}_{t}^{{E}_{T}}}{{\left({x}_{t}{w}_{r}\right)}^{{E}_{S}}}\frac{4\pi {d}_{r}}{\lambda}\right)}^{{E}_{L}}\right)\hfill \\ \hfill \mathsf{\text{FitDimensions}}:C=\mathsf{\text{CurveShift}},{L}_{SU}=\mathsf{\text{SubUrbanLoss}},\hfill \\ \hfill {E}_{L}=\mathsf{\text{LossExponent}},{E}_{S}=\mathsf{\text{StreetExp}}.,{E}_{T}=\mathsf{\text{TxDistExp}}\mathsf{\text{.}}\hfill \end{array}

Value *i*_{
s
} is specifying suburban (*i*_{
s
} = 1) or urban (*i*_{
s
} = 0). As *d*_{
b
} ≈ 180 m for our setup, which exceeds the highest distance *d*_{
r
} with reception, the *d*_{
r
} > *d*_{
b
} equation is of no use for the fitting.

The final fit to determine the five variable parameters is visualized in Figure 10. We fitted the intersection wide average median reception power per tx-distance curve. Each of these curves abstracts (averages) eight measurements, thus providing a stable input to the fit and keeping the complexity on a moderate level. We showed in [17] that this averaging is viable, as the performance is very similar despite the transmitter being in the different side streets.

The fit input values are form the regular shaped (≈90° and *w*_{
t
} ≈ *w*_{
r
} ) intersections with buildings at each corner: intersection 2, 3, 10, 11, and 20, with *w*_{
r
} being 21, 21, 23, 26 and 30 m, respectively, and intersection 2 and 3 having *i*_{
s
} = 1. The fitted measurements have *d*_{
t
} values of 30 and 60 m (plus 100 m for intersection 11).

Each input value (visualized as a cross in Figure 10) is complemented by the reception rate in the bin and the intersection wide *w*_{
r
} and *i*_{
s
} values as input to the equation and for pre-selection. *w*_{
r
} is set to \frac{{w}_{t}+{w}_{r}}{2} as *w*_{
t
} and *w*_{
r
} were selected similar per intersection and we fit intersection wide average values. System loss is set to 1.75 dB and {x}_{t}=\frac{{w}_{r}}{2} (as this dimension was not tested).

We did not fit intersection 1, 9, and 21 due to differing reasons: intersection 1 was the very first tested intersection. Here, we measured with alternating transmission power (20 dB, 10 dB) and rate (3 Mbps, 6 Mbps) in each second. In consequence, there are spatial gaps in the data for each of the four configurations, leading to empty bins at the 5 m bin width in the fit. Anyhow, the performance is very close to intersection 2 and 3, as shown in [17]. Intersection 9 has missing reflection facades. This dimension was not incorporated in the fit, as it would have complicated the fit by another dimension. Furthermore, we only tested one intersection of such type (as it is rare), leading to insufficient data to provide a reliable fit in this additional dimension. Intersection 21 was excluded due to two reasons: First, one of the street legs has a non 90° angle. Second, the inter-building distance in the two streets differs a lot (55 m against 30 m). It is questionable whether the averaging over the four side street simplification is applicable for this particular intersection. Despite the exclusion of these three intersections, the fit covers 11 data rows from five intersections, stemming from 88 test-runs.

We fitted the median reception power curve, as it is more stable at lower reception rates. The average reception power curve suffers (bin values are too high) from incomplete data as soon as the reception rate sinks below 1.0. The median is technically accurate as long as reception rate is greater than 0.5. However, due to small-scale fading leading to variations and potential measurement inaccuracies around the reception threshold of the radio, median values also turn out to be slightly too high at reception rates close to 0.5. This is visible in plots. To prevent a negative influence on the fit, an exclusion criterion of reception-rate >0.65 was selected.

We also excluded small distances to center, as they are in LOS. The root mean square (RMS) error of the fit showed that *x* > 10 m is a good exclusion criterion: RMS error decreases from 2.4 with *x* > 0 to 0.8 with *x* > 10, but not much further with higher *x*. The very low RMS error of 0.8 corresponds to the good fit quality (compare input values to resulting model curves in Figure 10). The resulting VirtualSource11p path-loss equation, as determined by the fit, is

\begin{array}{c}\mathsf{\text{VirtualSource11p}}\left({d}_{r},{d}_{t},{w}_{r},{x}_{t},{i}_{s}\right)=3.75+{i}_{s}2.94\\ +\left\{\begin{array}{c}\hfill 10\phantom{\rule{0.1em}{0ex}}{log}_{10}\left({\left(\frac{{d}_{t}^{0.957}}{{\left({x}_{t}{w}_{r}\right)}^{0.81}}\frac{4\pi {d}_{r}}{\lambda}\right)}^{2.69}\right),\phantom{\rule{1em}{0ex}}{d}_{r}\le {d}_{b}\hfill \\ \hfill 10\phantom{\rule{0.1em}{0ex}}{log}_{10}\left({\left(\frac{{d}_{t}^{0.957}}{{\left({x}_{t}{w}_{r}\right)}^{0.81}}\frac{4\pi {d}_{r}^{2}}{\lambda {d}_{b}}\right)}^{2.69}\right),\phantom{\rule{1em}{0ex}}{d}_{r}>{d}_{b}\hfill \end{array}\right.\end{array}

Despite the no available measurement data for high *d*_{
r
} distances, an increased loss at high distances (*d*_{
r
} > *d*_{
b
} ) due to diffraction rather than reflection being dominant is incorporated (as in [11, 12, 14, 15]).

Note that close to intersection center, loss is really low (similar to FSPL having a heavy slope close to 0). Figure 11 depicts a representative example for intersection 10. In consequence, the VirtualSource11p path-loss equation only applies to NLOS conditions and not to the complete crossing street. At LOS on the crossing street, either the normal LOS path-loss should be used with distance as *d*_{
t
} + *d*_{
r
} or a percental value between LOS at intersection center and NLOS value at the first point of NLOS. The latter is potentially more accurate.

### 3.3 NLOS small-scale fading classification

Small-scale fading leads to a distribution of power values around an average value. Figure 12 shows the different power probability distributions of received packets for different distances to the center in suburban intersection 3. It reveals e.g. a high variation in the 10-20 m bin as it includes LOS and NLOS conditions, or a variance limitation due to failed receptions (measurement limitation) for larger distances.

To determine fading in NLOS conditions, we centered the power probability distribution curves to their average and compared curves from different intersections for a certain street stretch in NLOS. Figure 13 shows the curves for *d*_{
t
} = 30 m and bin 40-50 m. This stretch exhibits NLOS conditions for all intersections and is in most cases (except probably intersection 2) not influenced by the radio reception limit. The curves from different intersections show a very similar shape. While they fit well to both, Nakagami-m and the Normal Distribution, the RMS error is slightly smaller for the Normal Distribution. Using visual judgment, they clearly match the Normal Distribution better. Therefore, we propose to model fading in NLOS as a normal distribution with *σ* = 4.1 dB.

For LOS conditions, we were able to verify that a 5.9 GHz vehicular channel is properly modeled by the often assumed Nakagami-m = 1 distribution. The corresponding fit is given in Figure 14. It reveals a good visible match to the Nakagami curve with fitted *m* = 1.05.