Repeatability of Large-Scale Signal Variations in Urban Environments

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1 Repeatability of Large-Scale Signal Variations in Urban Environments W. Mark Smith and Donald C. Cox Department of Electrical Engineering Stanford University Stanford, California Abstract This paper quantifies the repeatability of largescale signal variations in an urban propagation environment. We propose a model and use field measurements taken in downtown San Francisco to quantify how similar the received signals appear when a mobile receiver revisits the same general area. The results establish bounds for repeated measurements within a given street that are taken on different days as well as in different lanes. I. INTRODUCTION Researchers have often proposed the use of signal-strength databases to improve the performance of existing cellular infrastructure (e.g., resource allocation [1]) or to propose methods for offering new services (e.g., mobile location [2]). The viability of these schemes, however, is contingent upon the reliability of the underlying signal-strength databases in terms of measurement accuracy and whether or not those measurements change over time. Not only does the repeatability of a signal within a geographic area directly impact the performance of these systems, but the rate at which these patterns change over time would determine the interval at which these databases must be updated in order to optimize system performance. The propagation of high-frequency signals from transmitter to receiver is characterized by three properties: (1) a distance-dependent trend, (2) large-scale variation over tens of wavelengths, and (3) the small-scale fading indicated by rapid fluctuations in the received signal strength over distances less than a wavelength. The large-scale variation is caused by shadowing of large objects, such as buildings and hills, in the propagation environment, and the small-scale fading is caused by the phase combinations of multipath components incident on the receiver. Since the distance-dependent trend and the large-scale variation are governed by the physics of wave propagation and the placement of large objects in the environment that do not move over time, one expects these characteristics to be repeatable at a given location. Furthermore, if one could control the precise locations of the transmitter, receiver, and all the scatterers in the environment from one measurement to the next, then one would expect the small-scale variations to be the same in both cases as well. Since the positions of all these items are not completely controllable, it is usually expected that only the statistics of the small-scale fading will remain the same over repeated measurements of the same area. Received Signal Level [dbm] Feb 2002 May Distance Along Street [ft] Fig. 1. Two Drives Down Sansome Street For example, Figure 1 shows a comparison of two different drives along Sansome Street located in the financial district of downtown San Francisco. Each trace shows the signal after it has been averaged over 10 wavelengths to estimate the largescale variation. While the curves do not lie directly on top of one another, they are very similar. We are interested in quantifying this degree of similarity and determining what factors contribute to the differences. II. EXPERIMENT SETUP We wish to characterize the repeatability of signal propagation in a built-up urban environment. This type of environment forms the so-called urban canyons with streets between very tall buildings. The San Francisco financial district is such an area. Although San Francisco is known for its rolling hills, the financial district itself is located on a piece of flat terrain near the San Francisco Bay. The buildings in this area are typically 10 stories or taller, and there is very little vegetation, so seasonal changes in the propagation environment are not expected. The measurement area is as shown in Figure 2. The measurements are taken in the streets. As suggested in [3], radial streets (those that proceed directly out from the transmitter) exhibit different path-loss characteristics from those of nonradial streets. For this reason, this paper analyzes the sections /03/$ IEEE. 16

2 Montgomery Fig. 2. Sansome Coverage Area Battery Clay Sacramento California Pine transmitter This sensor generates a pulse every 4cm of travel, providing a drive path distance measurement. A video camera is mounted perpendicular to the direction of travel and indicates the receiver s location relative to buildings along the drive route. Using the video record of each drive down a street, multiple runs can be compared by aligning the locations of the buildings along the route for each run. This provides a systematic way to align two traces (such as those shown in Figure 1) without having to adjust the relative distance offsets by hand. The video camera and the samples from the spectrum analyzer and the ABS sensor are synchronized to the laptop s system clock, providing a method to match signal strength to receiver location during off-line processing of the data. III. THE REPEATABILITY MODEL Using the previously described statistical model, the received signal level can be represented as P (x) =D(x)L(x)Y (x) (1) of street that have the best fit to an exponential path-loss characteristic. These measurement locations are indicated by the thick lines in the streets on the map. Since Montgomery Street intersects several streets in which the signals are stronger, there is not a continuous section longer than a city block that has a consistent distance trend, so it is not included in the analysis of this paper. Control channels from active cellular base stations are transmitted continuously at constant power levels, providing beacons that can be characterized throughout the coverage area. AT&T Wireless has provided base station locations and center frequencies for cell sites in this area. The base station at 22 Battery Street has antennas that are mounted on the north face of the building at a height of 178 feet, which is lower than the rooftops of the tall buildings in the vicinity. Since we are interested in the signal variation imposed by the multipath environment itself, independent of any modulating signal, the frequency-shift keyed (FSK) data modulation of the IS-136 Forward Analog Control Channel is removed. A spectrum analyzer operating in zero-span mode acts as a tunable filter. The center frequency is tuned to the control channel and the resolution bandwidth is set to 10kHz, which is narrower than the channel bandwidth of 30kHz to remove the data modulation. The output of the spectrum analyzer is sampled and stored on a laptop computer. Additional baseband filtering is performed to remove any residual modulation components. To verify that the same base station is being monitored throughout the coverage area, the received signal is split off (before the spectrum analyzer) to an FM receiver, which is used to decode and verify the base station s digital color code (DCC). The spectrum analyzer is connected to a roof-mounted antenna and is placed inside a 1992 Chevrolet Cavalier to move through the coverage area. The laptop computer also stores samples of the car s anti-lock braking system (ABS) sensor. where P (x) is the received power at location x, D(x) is the distance-dependent trend, L(x) is the large-scale variation, and Y (x) is the small-scale variation. We can obtain an estimate of the large-scale variation by looking at the spatial average of the received signal and removing the distance trend: ˆL(x) = P (x) W (2) ˆD(x) W denotes the spatial average of the process over x in the averaging window W. Using the transmitter as the origin of the coordinate system, ˆD(x), when expressed in db, is a regression fit of P (x) W to the formula below: ˆD db (x) =K 0 +10nlog 10 ( x ) (3) K 0 is a constant that accounts for transmit power, antenna gains, cable losses, etc., and n is the path-loss exponent. The correlation length of the small-scale fading process is frequency-dependent, and at cellular frequencies, this process decorrelates at a relative distance much shorter than the correlation length of the large-scale process. Because of this, one can estimate the large-scale process by averaging over a distance that is several times longer than the correlation distance of the small-scale fading process. There is, however, a tradeoff in the choice of the averaging interval, W, in estimating the large-scale variation. The objective is to average out the rapid fluctuations caused by the small-scale fading process, and the remaining variations would be interpreted as the large-scale variation. Since averaging is performed over a finite interval, not all of the randomness of the small-scale process will be averaged out. For smaller values of W,the estimate ˆL(x) will be dominated by variations in the smallscale fading process that were not completely averaged out. On the other hand, as W gets larger, the variations in L(x) will be smoothed out as well. Austin and Stüber[4] suggest averaging intervals of 20-40λ. (In this paper, λ will be used to denote units that have been normalized by the signal s wavelength) /03/$ IEEE. 17

3 For purposes of illustrating the comparison of one data set against another (Figure 1), we have used a slightly smaller window (10λ), the reasons for which will become apparent later in the analysis. To determine if the large-scale process is repeatable for two different trips down the same street, we denote the part of L(x) that is repeatable as L r (x), and we introduce an additional process, l(x), that accounts for any differences in the largescale variation on the second run. In order to quantify the repeatability, we can bound the variance of the dissimilarity process l(x). Next, we divide the length of the data set into averaging regions, W i, of equal size (the length of each region W i is W ). We then consider the ratio of the large-scale variation estimates for different window sizes: ˆL 1 (x i ) ˆL 2 (x i ) = P 1(x) Wi P 2 (x) Wi = D(x)L r(x)l(x)y 1 (x) Wi D(x)L r (x)y 2 (x) Wi Here, i is used to index equally-spaced locations along the drive route. If the large-scale process is repeatable (l(x) =1) and there are no small-scale variations (Y 1,2 (x) = 1), this ratio would always be 1. Including these uncertainties (l(x) and Y 1,2 (x)), however, yields a distribution of ratios over the data set. By determining the properties of l(x), we can then quantify how repeatable the two measurements are. For the data sets analyzed in this paper, the properties of l(x) are determined indirectly, by observing the statistics of the ratio in (4) through simulation. IV. SIMULATION FOR COMPARISON For comparison to the measurement data, the simulations must generate the quantities used on the right-hand side of equation (4). The statistics of the ratio of large-scale variation estimates (equation (4)) taken at locations x i along the route are compared for a number of values of the averaging window length W. The repeatable large-scale variation L r (x) is a log-normal sequence that has been filtered to give rise to an exponential autocorrelation as described in [5]. We define the decorrelation length to be the spatial offset required to cause the normalized autocorrelation function to be 0.5. The decorrelation length and the standard deviation of L r (x) are estimated from the data for each street in which the comparison is made. The process l(x) is generated in the same manner as, but independent of, L r (x). By construction, it has a decorrelation length that is less than or equal to that of the corresponding L r (x). The unknown quantities, which are the standard deviation (σ l ) and the decorrelation length (D l )of the dissimilarity process l(x), are determined through trial and error to make the simulated comparisons match the behavior of the measured-data comparisons. Since D(x) varies much more slowly than either the smallscale fading or the large-scale variations in the signal, it is (4) St. Dev. of Ratio of Averages [db] σ l increasing, ρ =0 ρ increasing, σ l =0 Measurement Simulation Averaging Window [λ] Fig. 3. Battery Street (Left Lane) Comparison approximately constant over the decorrelation length of any of the simulated random processes. For this reason, it is considered to be constant (=1) in all the simulations. The small-scale fading processes Y 1,2 (x) are generated from independent complex Gaussian sequences. The sequences are filtered to have the Doppler spectrum that is characteristic of two-dimensional isotropic scattering as described in [6, Ch. 1]. In some cases these two processes are correlated with each other, and in others, they are uncorrelated. To generate the appropriate cross-correlation, the unfiltered Gaussians are generated as follows: y 1,i = αx 1,i +(1 α)x 2,i y 2,i = αx 1,i +(1 α)x 3,i (5) x 1, x 2, and x 3 are independent, complex Gaussian sequences; α is a fractional weighting constant; and y 1 and y 2 are the sequences that are subsequently filtered to generate Y 1 (x) and Y 2 (x). Throughout the rest of this paper, the correlation coefficient ρ = COV(Y 1,Y 2 ) σ Y1 σ Y2 (6) is used to describe the normalized cross-correlation of Y 1 (x) and Y 2 (x) where σ Y1 and σ Y2 are the standard deviations of Y 1 (x) and Y 2 (x), respectively. V. COMPARING DATA SETS The standard deviation of the ratio of the large-scale variation estimates (equation (4)) when the averaging window length W is varied provides the basis for comparing two different sets of measurements along the same street. The ratio is converted to decibels before the statistics are computed. The comparison plots for drives along Battery Street are shown in Figure 3. The lower curve represents a comparison between two measurement drives that were taken on the same night. In this case, the standard deviation of l(x) is zero, meaning that the large-scale variation is completely repeatable /03/$ IEEE. 18

4 TABLE I REPEATABILITY OVER TIME Street (lane) Same Night Same Day 10 Days Apart 15 Months Apart ρ σ l [db] D l [λ] ρ σ l [db] D l [λ] ρ σ l [db] D l [λ] ρ σ l [db] D l [λ] Battery (center) , California (left) Sansome TABLE II LONG-TERM REPEATABILITY Street (lane) Same Night 15 Months Apart ρ σ l [db] D l [λ] ρ σ l [db] D l [λ] Battery (left) Battery (center) , California (left) California (right) Clay Sacramento Sansome Pine (left) Pine (right) Furthermore, the correlation coefficient ρ between the smallscale variation processes is 0.45, meaning that the smallscale fading processes are not independent between the two runs. 100% cross-correlation is not expected since the receiver location is the same only to within several inches between measurement runs, and there is no control over the positions of other scatterers in the environment. The upper curve in Figure 3 represents a comparison between two drives that were made 15 months apart. In this case, ρ =0, so the small-scale processes are uncorrelated and σ l =3.0dB, meaning that the large-scale variations are not exactly the same, and the value of σ l quantifies the degree to which they are dissimilar. The dashed line in Figure 3 corresponds to both ρ = 0 and σ l = 0dB. All comparisons in which the curve falls below the dashed line have repeatable large-scale variations (σ l = 0dB) and small-scale variations that are correlated with each other (ρ 0). These curves are characterized by the single parameter ρ. Asρincreases, the comparison curve moves downward and flattens out. Comparisons in which the curve falls above the dashed line have small-scale fading processes that are uncorrelated (ρ =0), and the degree of dissimilarity between large-scale processes is measured by σ l and the decorrelation length (D l ) of the process l(x). These curves are characterized by two parameters (σ l, D l ). Generally, the value of σ l shifts the curve up or down, and the value of D l determines the slope of the asymptote as the averaging window increases. For smaller D l, the asymptote becomes steeper. The results for the entire coverage area are summarized in Tables I and II. The first column of Table I shows comparisons made on May 6, 2003, when all measurements were made between midnight and 4:00 A.M., and there was little or no traffic. The second column shows comparisons made on February 15, 2002, when measurements were taken between 6:00 and 7:30 A.M. under moderate traffic conditions. The comparisons in the third column were made 10 days apart (Feb , 2002). The times of day were 6:30/9:30 A.M. and 6:00/8:20 A.M., respectively. The fourth column compares the data between the 2002 and 2003 measurement campaigns. In these cases, the comparisons were made between measurements taken with no traffic (2003) to those made with light to moderate traffic (2002). Because measurements were not made multiple times over the entire coverage area during the 2002 campaign, data were not available for the same comparisons on all streets. The remaining comparisons are summarized in Table II, where the columns correspond to the first and fourth columns of Table I. Figure 4 shows a progression of the repeatability of the measurements over time. One item of interest is that the knees of the curves occur around an averaging window length of 10λ, suggesting it is a good choice to use when directly comparing the signal strength values as shown in Figure 1. Measurements taken at night when there was little or no traffic provided the highest degree of repeatability (ρ = 0.64 and σ l = 0dB). The comparison that was made between two measurements that occurred as the morning commute was beginning shows less cross-correlation between the small-scale processes (ρ =0.36). Proceeding up the chart, the next curve shows even less cross-correlation (ρ =0.13) when compared 10 days apart. Finally, when compared 15 months apart, the dissimilarity process has a standard deviation of σ l =1.3dB. A similar trend holds for Battery Street, but there are /03/$ IEEE. 19

5 St. Dev. of Ratio of Averages [db] months apart 10 days apart Measurement Simulation same day same night Averaging Window [λ] Fig. 4. Sansome Street Comparison indications that the time of day at which a signal was measured will have an impact on its repeatability. Notably, the value of σ l = 2.8dB from the 10-day comparison is the result of measurements made during different traffic conditions (6:30/9:30 A.M.). This may also account for the two values of σ l in the fourth column. In this case, the same 2002 data sets were measured against the 2003 data that were taken when there was very little traffic. The lower value (1.8dB) corresponds to the 6:30 A.M. comparison, which had lighter traffic than did the comparison with the 9:30 A.M. run. Time of day, however, is not the sole contributor, as indicated by the 15-month comparison of Sansome Street. In this case, the equivalent comparisons agreed completely as the two sets of measurement data lie on the top simulation curve in Figure 4. TABLE III ADJACENT LANE COMPARISONS Parameters Street (lanes) σ l [db] D l [λ] Battery (left/center) Battery (center/right) Battery (left/right) California Pine Table III shows the results for comparisons of adjacent lanes in the same street. All of these comparisons come from the 2003 measurement campaign, which took place with little or no traffic in the streets. Each line corresponds to four comparisons made between two runs in each lane. Each comparison agreed well with the other comparisons made using the same lanes and with the simulations using the values shown in the table. The dissimilarity process has a greater magnitude here than in almost all of the same-lane comparisons over 15 months (Clay Street being the exception). Battery Street was the only street in which a comparison was made among three lanes traveling in the same direction. It is interesting to note the magnitude (σ l ) of the dissimilarity process was the same regardless of whether the comparison was made with the very next lane or two lanes over. Pine is a one-way street, and both traffic lanes were used for the comparison. California is a wider, two-way street, and only the two eastbound lanes were used for the comparison. VI. CONCLUSION In this paper, we have introduced a model that quantifies the repeatability of signal strength measurements in the streets of a built-up urban area. This model includes a lognormal dissimilarity process that accounts for the changes (if any) in the large-scale variation between measurement sets. The measurements yield a range in the magnitude of the dissimilarity process over time and over distance between the measurements. The measurements are the most repeatable when the two data sets are taken in the same lane with little time in between and when there is little or no traffic. In this case we find that not only is the large-scale variation repeatable, but the small-scale fading will usually be correlated between the two runs. As traffic increases, the small-scale fading tends to decorrelate between the two runs, and in many cases there are small changes in the large-scale variation. Over long periods of time, the repositioning of scatterers in the environment seems to cause small changes in the large-scale variation. Even in this case, however, measurements in the same lane generally showed better agreement after 15 months than did similar comparisons of adjacent lanes on the same night. REFERENCES [1] R. Narasimhan and D. C. Cox, A handoff algorithm for wireless systems using pattern recognition, in Proc. IEEE PIMRC 98, Boston, Massachussetts, 1998, pp [2] M. Hellebrandt, R. Mathar, and M. Scheibenbogen, Estimating position and velocity of mobiles in a cellular radio network, IEEE Trans. Veh. Technol., vol. 46, no. 1, pp , Feb [3] W. M. Smith and D. C. Cox, Urban cell partitioning for improved statistical propagation modeling, in IEEE AP-S Internat. Symp., vol. 4, Columbus, OH, June 2003, pp [4] M. D. Austin and G. L. Stüber, Velocity adaptive handoff algorithms for microcellular systems, IEEE Trans. Veh. Technol., vol. 43, no. 3, pp , Aug [5] M. Gudmundson, Correlation model for shadow fading in mobile radio systems, Electron. Lett., vol. 27, no. 23, pp , 7 Nov [6] W.C.Jakes,Ed.,Microwave Mobile Communications. IEEE Press, /03/$ IEEE. 20