ARIMAbased time variation model for beneath the chassis UWB channel
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Abstract
Intravehicular wireless sensor network (IVWSN) enables the integration of the wireless sensor network technology into the vehicle architecture through either eliminating the wires between the existing sensors and the corresponding electronic controller units (ECUs) or empowering new sensor technologies that are not currently implemented due to technical limitations. Ultrawideband (UWB) has been determined to be the most appropriate technology for IVWSNs since it provides energy efficiency through the low dutycycle operation and high reliability by exploiting the large bandwidth. In this paper, we propose a time variation model for UWBbased IVWSNbased on the extensive amount of data collected from the transmitter and receiver antennas at various locations and separation distances beneath the chassis of a vehicle moving at different speeds on different types of roads. We adopt the commonly used SalehValenzuela (SV) model to represent the clustering phenomenon in the received power delay profiles (PDPs). The proposed novel time variation model then determines the time evolution of the PDPs by representing the changes in their cluster breakpoints, slopes, and break point amplitudes with the autoregressive integrated moving average (ARIMA) model. ARIMA(5,1,0) has been demonstrated to fit the breakpoint, cluster slope, and breakpoint amplitude sequences collected at different vehicle speeds with different transmitter and receiver locations on asphalt and stone roads by using BoxJenkins procedure. This model is validated with diagnostic checking. The absolute values of the model coefficients are observed to be mostly larger on asphalt road than their counterparts on the stone road while exhibiting no dependence on the vehicle speed nor the location of transmitter and receiver antennas.
Keywords
Channel model Ultrawideband Vehicle Wireless sensor networks Timevariation model SalehValenzula model ARIMA model1 Introduction
Intravehicular wireless sensor network (IVWSN) is a specific type of wireless sensor network between the vehicle sensors and their corresponding ECUs deployed with the purpose of either eliminating the currently existing wires or enabling new sensor technologies that cannot be integrated into the vehicle using wired means. The elimination of the wires provides savings in part, assembly and maintenance cost together with fuel efficiency whereas new sensor technologies enable new vehicle applications. The full adoption of a IVWSN requires providing the same performance and reliability as the wired equivalent that has been tested for a long time with vehicles on the road. The first IVWSN examples are therefore expected to be in the integration of either new sensor technologies such as intelligent tire [1] or some sensor technologies for noncritical vehicle applications such as park sensors and steeringwheel angle sensors. Proving the robustness of these applications on the vehicle will then pave the way for the usage of IVWSN in more critical vehicle applications such as the transmission of automotive speed data from the wheel speed sensors to the ECU in an antilock braking system [2].
Among various modulation techniques investigated for IVWSN, ultrawideband (UWB) has been demonstrated to be the most suitable satisfying the highreliability requirement of vehicle control systems and the strict energyefficiency requirement of the sensor nodes at short distance and low cost [3, 4, 5]. The vast literature on UWB channel measurement campaigns performed in such locations as indoor [6, 7, 8], outdoor [9, 10], around the human body [11, 12], industrial environments [13], vehicletovehicle, and vehicletoroadside [14, 15, 16, 17] cannot be applied to the IVWSNs since the intravehicular environment is very different containing a large number of metal reflectors and operating at extreme temperatures.
Although the locations of the transmitter and receiver within the vehicle do not change over a short time scale in IVWSN, the movement of the vehicle generates small variations in the sensor locations due to vibrations and considerable changes in the environment such as road conditions and nearby vehicles. The development of an efficient UWB communication system for IVWSNs therefore requires introducing these time variations into the channel model. A timevarying channel model allows optimizing system performance and robustness under realistic radio propagation conditions. The time variations of the wireless channel can also be used in the design of channel estimation algorithms and rake receivers. The UWB channel models developed for different parts of the vehicle, including engine compartment [3, 4, 18], beneath the chassis [3, 4, 19, 20], passenger compartment [21, 22, 23, 24, 25, 26], and trunk [27], all aim to represent the average behavior and the variations around the average behavior of the UWB channel without adopting timeseries analysis. Even when these works collect data on the vehicle moving on the road, they still consider the distribution of the power delay profiles (PDPs) around their average value without considering the ordering of the data in time. Such modeling therefore does not allow the regeneration of the successive PDPs. Timeseries analysis on the other hand models the data as an ordered sequence of values. Embedding this ordering into the model allows the representation of the temporal variations in the channel.

We employ autoregressive integrated moving average (ARIMA) model to determine the time evolution of the parameters of the modified SalehValenzuela (SV) model derived for the beneath the chassis UWB channel [20]. ARIMA model is a flexible tool that provides a good understanding of trends, correlation, and forecasting in time series. The modified SV model aims to generate the clustering phenomenon in the PDP by statistically determining its parameters including cluster arrival times, cluster amplitude, and ray amplitude decay rate. The ARIMA model is then used to determine the time evolution of the PDPs by representing the changes in the time series of these parameters. This is the first work to propose a timevarying UWB channel model based on the time series analysis employing ARIMA model on the SV model parameters, allowing the regeneration of successive PDPs.

We validate the proposed ARIMA model on the SV model parameters as a timevarying UWB channel model beneath the chassis of the vehicle based on the analysis of the residuals between generated and observed values and the sensitivity of the model parameters to different vehicle speeds, road types, and distances and locations of transmitter and receiver antennas. This is the first work to analyze the validity of a time variation model across a wide range of scenarios.
The rest of the paper is organized as follows: Section 2 provides the experimental setup. Section 3 presents the data processing required to derive the coefficients of the ARIMA model. This includes deriving the PDP corresponding to each received pulse, determining the parameters of the modified SV model based on the separation of the PDP into clusters, and generating the ARIMA model corresponding to the time series of the parameters of the SV model. Section 4 provides and analyzes the coefficients of the ARIMA model at different vehicle speeds, road types, locations, and separation distances of transmitter and receiver antennas. Finally, Section 5 concludes the paper and gives the future work.
2 Experiment setup
The receive antenna is connected to the oscilloscope via lowloss coaxial cables. The oscilloscope is used to record the received signal. The segmented memory feature of the digital oscilloscope is used to increase the number of pulses that can be captured with the limited memory: The oscilloscope stores information only during the active periods so that the memory is not used during the inactive periods. We capture 1024 successive pulses by sampling 200 ns long signal at 40 G samples/s for each pulse. The output of the impulse generator is also connected to the trigger input of the oscilloscope via another lowloss cable for synchronization. The distance between the antennas and the road is around 20 cm. The data was collected at low vehicular traffic on asphalt and stone roads within the Koc University campus on mostly sunny or cloudy days. We have chosen Koc University campus to have full control on the speed of the vehicle and road conditions. On the other hand, we have chosen sunny or cloudy days to avoid the damage of the antennas on the wet roads. Moreover, we have collected the data over roads without any impulsive interference, which is ensured as a result of measuring the signals in the data collection environment without any pulses transmitted. We have also checked the collected data to guarantee that there is no impulsive interfering noise other than white noise. The collected data are provided in [37].
3 ARIMAbased time variation model
3.1 Derivation of PDP
3.2 Estimation of SV model parameters
In the PDPs recorded beneath the chassis, the multipath signals arrive in the form of clusters. The arrival time of the PDP clusters is determined by using the automatic clustering algorithm [39] and validated by visual inspection. Based on the assumption that the slope changes at the beginning of each cluster, automatic clustering algorithm identifies the change points of the partial slopes of the PDP. The clustering is exemplified for the PDP corresponding to the pulse transmission from 25cm distance beneath the chassis of the vehicle moving on the stone road at 20 km/h in Fig. 5.
where a _{ l,k } and θ _{ l,k } are the gain and phase of the kth component in the lth cluster, respectively; T _{ l } is the delay of the lth cluster; τ _{ l,k } is the delay of the kth multipath component in the lth cluster relative to the lth cluster arrival time T _{ l }; K is the number of the multipath components within a cluster; and L is the number of clusters. The phases θ _{ l,k } are uniformly distributed in the range [ 0,2π].
List of SV parameters used in timeseries analysis
Abbreviation  SV parameter 

B _{1}  First break point (ns) 
B _{2}  Second break point (ns) 
S _{1}  First cluster slope (dB/ns) 
S _{2}  Second cluster slope (dB/ns) 
S _{3}  Third cluster slope (dB/ns) 
X _{1}  First break point amplitude (dB) 
X _{2}  Second break point amplitude (dB) 
3.3 Estimation of ARIMA model coefficients for SV parameter sequences
3.3.1 Description of ARIMA model
ARIMA models are the most general class of models for understanding and forecasting a possibly nonstationary time series.
where ϕ _{ i } and θ _{ i } represent the ith AR and MA coefficients, respectively; and a _{ t } is zero mean white Gaussian error term with standard deviation σ _{ a }.
3.3.2 Determination of ARIMA model coefficients
3.3.3 Convert time series data to a stationary process
The stationarity of the process is determined by using DickeyFuller test. In this test, the existence of a unit root in the AR model corresponds to the nonstationarity of the model. The test provides a p value, which is used whether to reject or accept the null hypothesis of the existence of the unit root (the p value is the probability of obtaining a test statistic result at least as extreme or as close to the one that was actually observed, assuming that the null hypothesis is true. A researcher will often reject the null hypothesis when the p value turns out to be less than a predetermined significance level, often 0.05 or 0.01).
If the timeseries data is stationary, we can skip this step. Otherwise, it must be reduced to a stationary process. Generally, there are two types of reasons for the nonstationarity of time series: nonconstant mean and variance. If the mean is not constant, the time series can be reduced to stationary process by applying differencing. The number of differencing operations is denoted by d in the ARIMA model given in Eq. (2). Although theoretically there is no limit on the number of differencing operations, applying it more than twice is not recommended. On the other hand, if the variance is not constant, then power transformation should be applied to the original data to make the time series stationary. If the power transformation is needed, it should be performed before differencing.
3.3.4 Examine the ACF and PACF
Autocorrelation function (ACF) measures the similarity between the observations as a function of the time lag whereas partial autocorrelation function (PACF) is defined as the correlation between Z _{ t } and Z _{ t+k } after the linear dependence of the lags [ t+1,t+k−1] is removed.
ACF and PACF are used in predicting the orders of the ARIMA model. The AR order p and MA order q are predicted by the visual inspection of PACF and ACF, respectively. If PACF or ACF exhibits either exponential or damped sinusoidal behavior, then no order is assigned to p or q, respectively. On the other hand, if either PACF or ACF cuts off at a certain lag, then the cutting lag is assigned as the value of p or q, respectively.
3.3.5 Estimate the coefficients of the ARIMA model
The coefficients of the ARIMA model are determined by using maximum likelihood estimation.
The maximization of the exact likelihood function should be determined numerically. Here, we use BroydenFletcherGoldfarbShanno (BFGS) algorithm to iteratively solve the resulting unconstrained nonlinear optimization problem [18, 42]. The accuracy of the calculated coefficients is checked by using their p values. If a p value is larger than the significance level of 0.05, then this coefficient is omitted.
3.3.6 Diagnostic check
Diagnostic check aims to determine the suitability of the selected ARIMA model by checking whether the distribution of the residuals between the generated and observed data is white Gaussian. Whiteness of the residuals is measured by using LjungBox test on the ACF whereas their normality is tested by using chisquare test. The LjungBox test is a type of statistical test of whether the ACF of a time series except the zero lag is different from zero. Instead of testing randomness at each distinct lag, it tests the overall randomness based on a number of lags. The null hypothesis in this test is the whiteness of the residuals. Chisquare test is a statistical hypothesis test used to determine whether there is a significant difference between two distributions. The null hypothesis used in this test is the normality of the residuals. A researcher will often accept the null hypothesis when the p value turns out to be greater than a predetermined significance level, often 0.05 or 0.01.
4 ARIMA model analysis results
The ARIMA model is determined for the time series data of B _{1}, B _{2}, S _{1}, S _{2}, S _{3}, X1, X _{2} at vehicle speeds of 20, 40, and 60 km/h on asphalt and stone roads for the transmit and receive antennas at 25, 50, and 100 cm distance in different locations beneath the chassis by following the procedure given in Section 3. The main results are summarized as follows:
 ARIMA(5,1,0) consistently fits the data collected over different types of roads, speeds, antenna locations, and antenna separation distances. The suitability of the selected ARIMA model is tested by checking whether the distribution of the residuals between the generated and observed data is white and Gaussian based on LjungBox and chisquare tests, respectively, as explained in detail in Section 3.3. Since all the p values of LjungBox and chisquare tests are greater than 0.1, the residuals are white Gaussian with 90 % confidence. This confirms ARIMA(5,1,0) model. Table 2 shows the ARIMA model coefficients for the parameters of the PDPs collected on asphalt road for the transmit and receive antennas at 25cm distance as an example.Table 2
ARIMA model coefficients for the parameters of the PDPs collected on asphalt road for the transmit and receive antennas at 25 cm distance
ϕ _{1}
ϕ _{2}
ϕ _{3}
ϕ _{4}
ϕ _{5}
σ _{ a }
B _{1}
20k m/h
−0.8104
−0.6506
−0.4808
−0.3280
−0.1566
0.650994
40k m/h
−0.8381
−0.6912
−0.4898
−0.3181
−0.1630
0.636613
60k m/h
−0.8642
−0.6856
−0.5320
−0.3255
−0.1779
0.642207
B _{2}
20k m/h
−0.8436
−0.7004
−0.5297
−0.3380
−0.1837
0.899330
40k m/h
−0.8661
−0.6628
−0.5457
−0.3553
−0.1911
0.895680
60k m/h
−0.8048
−0.6866
−0.5021
−0.3288
−0.1591
0.911310
S _{1}
20k m/h
−0.8192
−0.6428
−0.4871
−0.2890
−0.1454
0.016715
40k m/h
−0.8605
−0.6842
−0.5309
−0.3393
−0.1492
0.014427
60k m/h
−0.8391
−0.6696
−0.4850
−0.2903
−0.1323
0.014853
S _{2}
20k m/h
−0.8078
−0.6075
−0.4413
−0.3359
−0.1624
0.017013
40k m/h
−0.8354
−0.6547
−0.4905
−0.2903
−0.2102
0.014947
60k m/h
−0.7861
−0.6470
−0.4743
−0.3162
−0.2086
0.016607
S _{3}
20k m/h
−0.8750
−0.6981
−0.5230
−0.3539
−0.1496
0.003219
40k m/h
−0.8655
−0.7276
−0.5553
−0.3257
−0.1873
0.003046
60k m/h
−0.8224
−0.6789
−0.5569
−0.3797
−0.1903
0.003065
X _{1}
20k m/h
−0.7905
−0.6445
−0.4843
−0.3139
−0.2069
0.205739
40k m/h
−0.8106
−0.6493
−0.4490
−0.2330
−0.1196
0.193656
60k m/h
−0.8518
−0.7431
−0.5953
−0.3325
−0.1785
0.187573
X _{2}
20k m/h
−0.8426
−0.6957
−0.4771
−0.3073
−0.1255
0.120691
40k m/h
−0.8578
−0.6502
−0.5158
−0.3853
−0.1886
0.112629
60k m/h
−0.8168
−0.6746
−0.4728
−0.3047
−0.1573
0.108991
 The absolute values of the ARIMA coefficients are mostly larger on asphalt road than their counterparts on the stone road whereas the standard deviation of the innovation is larger on stone road than the asphalt road. Although this trend is not seen very clearly on all of the results, considering that this holds for the majority of the data, we conclude that the absolute value of the ARIMA coefficients decreases as the roughness of the road increases. Smaller absolute value of the coefficients means lower partial correlation between delayed elements of the time series. If the experiment was conducted on a perfectly smooth road, the consecutive PDPs would be very similar to each other resulting in the highest correlation between the delayed elements of the time series. As the roughness of the road increases, the shape of the PDPs are distorted in a random manner, decreasing their correlation to the previously received PDPs with increasing power of the innovation component. The general trend of the model coefficients as a function of the road roughness is exemplified for the ARIMA model coefficient ϕ _{1} on different types of roads under various scenarios in Fig. 8.
 There is no trend in the variation of the absolute values of the ARIMA model coefficients with vehicle speed, antenna locations, and antenna separation distances, as exemplified for the ARIMA model coefficient ϕ _{1} under various scenarios in Figs. 9, 10, 11 and 12. The main reason for this behavior is the independence of the ARIMA model coefficients from the values of the SV model parameters. As the distance between the transmitter and receiver antennas and the location of these antennas vary, the shape of the PDP hence the corresponding SV model parameters change. However, the time variation of these parameters only depends on the variation of the environment around the transmitter and receiver therefore exhibits similar behavior independent of the distance and location. Likewise, as the vehicle speed increases, the distance the vehicle travels between the PDPs increases. Unless the vehicle stays at the same location, the variation in the environment depends on the random changes in the road, independent of the speed.
5 Conclusions
We build a time variation model for the beneath the chassis UWB channel. The clustering phenomenon in the PDPs collected beneath the chassis of the vehicle has been previously represented by using modified SV model. We propose a novel ARIMAbased time variation model for the time series data corresponding to the parameters of the SV model including cluster arrival times, cluster amplitudes, and ray amplitude decay rates. ARIMA(5,1,0) has been demonstrated to fit the time series of all the SV parameters corresponding to the data collected at different vehicle speeds and locations of transmitter and receiver antennas at different separation distances on asphalt and stone roads. The absolute values of the ARIMA model coefficients are demonstrated to be mostly larger on asphalt road than their counterparts on the stone road. On the other hand, these coefficients do not exhibit any dependency on vehicle speed, antenna locations and antenna separation distances. We are planning to investigate the usage of smaller channel bandwidth on the time variation of the channel as future work.
Notes
Acknowledgements
We would like to thank Prof. Ibrahim Akduman and his group for providing the UWB antennas, and Seyhan Ucar for collecting the experimental data used to derive the antenna patterns.
This work is supported by Marie Curie Reintegration Grant on Intra Vehicular Wireless Sensor Networks, PIRG06GA2009256441. Sinem Coleri Ergen also acknowledges support from Bilim Akademisi  The Science Academy, Turkey under the BAGEP program, and the financial support by the Turkish Academy of Sciences (TUBA) within the Young Scientist Award Program (GEBIP).
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