Cell Radius Inaccuracy: A New Measure of Coverage Reliability

Transcription

1 Cell adis Inaccracy: A ew Measre of Coverage eliaility PETE BEADI, MEMBE, IEEE, MEG YEE, MEMBE, IEEE, AD THOMAS ELLIS Preprint of Plication in the ovemer 1998 Isse (Vol. 47, o.4, pp ) of the IEEE Transactions on Vehiclar Technology Astract * -- A rost method for determining the ondaries of cells and the associated reliaility of the F coverage within these ondaries is presented. The procedre accrately determines the effective cell radis sing a linear regression of F signal strength samples. The accracy of this estimate is qantified oth as a radis ncertainty (e.g., ± 100 meters) and as a coverage (i.e., area/edge) reliaility error throgh 1) simlation, ) analysis of real data, and 3) theoretical analysis. It is shown that if the estimate of the cell radis meets the desired accracy, then the corresponding estimates of coverage reliaility (oth area and edge) are more than sfficiently accrate. Throgh a sensitivity analysis, it is discovered that estimating the cell radis is a mch more critical step in determining the qality of F coverage than the more common practice of simply estimating the area reliaility. In addition, a formla for estimating area reliaility is given and shown to e more accrate than can e otained y crrent approaches. The verification method presented here is particlarly sefl in wireless planning since it effectively determines the geographic extent of reliale F coverage. It is recommended that radio srvey analyses select cell radis estimation as the preferred method of coverage verification. I. ITODUCTIO THE TWO MOST COMMOLY sed measres of the reliaility of F coverage are 1) cell edge reliaility and ) cell area reliaility. Cell edge reliaility refers to the proaility that the F signal strength measred on a circlar contor at the cell edge will meet or exceed a desired qality threshold (e.g., -90 dbm). Whereas, cell area reliaility is the proaility that F signal will meet or exceed the qality threshold after integrating the contor proaility over the entire area of the cell (i.e., across all of the contors of the cell, inclding the cell edge). D.O. edink showed that, for a given propagation environment, cell edge reliaility and cell area reliaility are deterministically related [] (see also section IV and the Appendix of this paper). Becase of this relationship, estimating the distance to the cell edge can e shown to e theoretically eqivalent to determining the reliaility of the signal strength within the cell (e.g., see eqation (5) and eqation (a7) in the Appendix). In this stdy we descrie a new measre of F reliaility that has previosly not een reported in other wireless investigations. * Manscript received Agst 1, 1996; revised Ferary 17, 1997 and Ferary 7, P. Bernardin and M. Yee are with orthern Telecom Wireless Engineering Services, ichardson, TX, T. Ellis is with Wireless Software Design & Conslting, Forestway Drive, Dallas, Tx, We call this coverage criterion cell radis inaccracy,. We have fond this criterion to e very sefl in answering the following two (related) qestions: 1) How many signal strength measrements are needed to accrately estimate the spatial extent of reliale coverage? ) How do we est estimate the coverage reliaility of isolated cells with a finite nmer of signal strength measrements? In answering the first qestion, the eqivalent circlar contor (i.e., the effective radis, ) of the cell is estimated, as shown in Figre 1. The relationship etween the inaccracy ( ) of this radis estimate and the amont of lognormal fading, σ, in each cell is empirically derived as a fnction of the nmer of independent signal strength measrements, (see eqation (14)). egarding the second qestion, perhaps the most important finding of this stdy is that it is the accracy of the cell radis estimate (i.e., ), not the accracy of the area reliaility estimate that is the limiting factor in determining the qality of F coverage. The relationship etween cell radis inaccracy and area reliaility is also discssed. Typically, cell radis estimation and area reliaility analyses are not considered together in propagation optimization. It is fond that these two prolems cannot e considered independently, and the conseqence of doing so can lead to inaccrate estimates of F coverage. Many vendors of F predictive tools already se regression to determine the est linear approximation to the median path loss for the prpose of tning F prediction models. In this paper we investigate the advantages of also measring the lognormal fading within each cell to more precisely determine the radis of reliale celllar coverage. These measrements are sed to compte a fade margin for each cell, which is then incorporated in the estimation of the cell s radis. Ths, the cell radis is defined explicitly in terms of the desired qality of coverage. It is recommended that, in addition to area reliaility, ftre wireless verifications also consider cell radis inaccracy. II. OVEALL APPOACH Figre 1(a) shows a high level view of the coverage estimation process for an omni cell. The signal strength is shown as a path loss srface which decreases with the logarithm of the distance from the celllar ase station. After signal strength measrements have een taken niformly over the area of the

2 Path Loss Srface ± ADIUS IACCUACY EQUAL POWE COTOU A y x B= y x Frstrm of ight Circlar Cone Cell Edge EQUIVALET CICULA COTOU (a) () Figre 1. (a) Fitting the path loss srface via linear regression. Signal strength as a decreasing fnction of the logarithm of the distance from the ase station for 360 degrees of azimth. The area of the path loss srface is niformly sampled and approximated with a cone via linear regression. The cell edge is approximated as the radis of the ase of the est-fitting cone. () Enlargement of the ase of the cone in (a). The measrement approach comptes the est circlar approximation to the eqal power contor. The effective radis,, of the cell is measred and the accracy qantified in terms of a radis inaccracy ring, ±. The average signal strength on the circlar contor is eqal to the signal strength of the eqal power contor. srface, the path loss is fit with a cone via linear regression. The cell edge is approximated as the radis of the ase of the est-fitting cone. The proposed method estimates the est circlar ondary that matches the cell edge at the desired area reliaility, as illstrated in Figre 1(). It shold e emphasized that this the signal power is aove -90 dbm). It is the radis of this fitted circle that is estimated. Ths, this radis can e considered the effective radis of the cell and is well defined for any cell, circlar or otherwise. The accracy of the cell radis estimate is qantified in terms of a radis inaccracy ring, ±, also shown in Figre 1(), where the dimension of is expressed in nits of distance. The width of this ring depends mostly on the nmer of signal strength samples in the regression, and also pon the amont of lognormal fading in the cell. Two methods for determining area reliaility from drive test data are compared. The first method is the standard approach of estimating the proportion of signal strengths that are aove a desired reliaility threshold [3]. The second techniqe is the preferred method, which is sed throghot this stdy. This method involves determining the propagation parameters of individal cells and sing this information in conjnction with edink s analysis (see eqation (a7) in the Appendix) to estimate the area reliaility. The propagation path is approximated with a two-parameter model similar to Hata [1]. A fade margin ased on the actal signal variation within each cell is calclated to ensre the desired cell edge reliaility. It is shown that this techniqe provides area reliaility estimates that are mch more accrate than those otained from the first method. approach does not in any way reqire that the tre cell edge e circlar. ather, even the most irreglar cell edge can e fitted with a circle sch that the average power along the circmference is eqal to the power of the tre cell edge. This circle encloses the area over which the F signal meets or exceeds the desired area reliaility (e.g., over 90% of the area, The proposed approach can e sed to qickly determine the validity of drive test data. Given enogh measrements, simlations show that this techniqe can e made almost aritrarily exact. It is recommended that this method e inclded as part of the pre-ild verification procedre for any wireless technology (TDMA, AMPS, CDMA, etc.). III. APPOACH FO ESTIMATIG THE CELL ADIUS We seek to characterize the propagation effects of the environment (terrain and cltter), not of the antenna. Ths, the signal strength measrements are first inverse filtered to remove the anisotropic weighting introdced y the horizontal antenna pattern. The proposed approach for estimating the cell radis is graphically smmarized in Figre. The measrement method is ased on a two-parameter propagation model similar to the prediction formlas of Hata Pr = Pt PL = Pt A Blog 10 r (1) where P r is the received power (dbm), P t is the transmitted power (EIP) of the ase station pls the receiver gain (e.g., P t =50 dbm EIP+G r ), P L is the path loss (db), r is the range (km) from the ase station, and A and B are the nknown constants to e estimated from the F data via linear regression [1]. A fade margin ased on the actal signal

3 - } A SL (dbm) P THESH y x y B= x CELL ITEIO MEA PATH LOSS CELL ADIUS } FADE MAGI }+ 1 km log 10 r Figre. The graphical approach to estimating the cell radis to within ±. The received signal strength level (SL) is plotted verss the range from the ase station to each measrement. The mean path loss is compted via linear regression and offset y the fade margin. The cell radis is defined in terms of the desired coverage reliaility as the point where the faded line crosses the reliaility threshold, P THESH. variation within each cell is calclated to ensre the desired cell edge reliaility. Becase of the similarity to Hata s model, it is important to clarify that the method does not incorporate Hata s coefficients. Instead, the salient propagation parameters are estimated from the data since the major goal in this stdy is F verification, not F prediction. The interior of each cell is divided into approximately 5000 ins which are niformly sampled oth in range and azimth (i.e., niform area sampling). The signal strength measrements in each in are averaged to prodce a single (average power) vale per in [4]. The range is then compted from the ase station to the center of all of the ins that contain measrements. Ths, each in represents an average power measrement at a certain range from the ase station. The range axis is then mapped to a logarithmic (common log) scale, the transmit power is comined with the parameter, A, and the two parameters of the following eqivalent model are estimated via linear regression P = A Br () r where rl = log 10 r and A = P t A. L Once the constants A and B have een estimated, the mean trend of the propagation data is stracted from the signal strength measrements and the standard deviation, σ, of the remaining zero-mean process is estimated. The vale of σ represents the composite variation de to two primary factors: lognormal fading and measrement error. Both of these factors tend to introdce ncorrelated variations arond the the mean since the regression is compted for range measrements across all azimth angles which greatly redces most spatial correlation effects. A fade margin, FM σ, that ensres the desired cell edge reliaility, F(z), can then e approximated (see eqation (a4) in Appendix) FM z σ σ = (3) t 1 z where Fz ()= e dt π For example, cell edge reliailities of 75% and 90% correspond to fade margins of aot 0.675σ and 1.8σ, respectively. It is now straightforward to derive the distance to the cell edge,, at any desired signal strength threshold, P THESH, and service reliaility, F(z). From eqations (1), (), (3), and (4) = PTHESH + FM A B 10 ( )/ (4) σ (5) Any additional static (nonfading) margin, sch as ilding penetration losses, can also e easily incorporated into the P THESH term. Ths, A,B and, σ are all that is needed to determine the range from the ase station to the cell edge. Example: Compte the range to the cell edge assming the Hata (Cost- 31) ran model constants for 1900 MHz and a ase station antenna height of 30 meters: A= 140 B= 35. Also assme σ = 8 db P THESH = -95 dbm P t = 50 dbm (EIP) F(z) = 75% (i.e., FM σ = 0.675σ ) From eqation (5), for 75% cell edge reliaility the estimated radis is ( )/ 35. = 10 = km Similarly, the radis for 90% cell edge reliaility is given y ( )/ 35. = 10 = km Ths, exact knowledge of the propagation parameters A, B, and σ is eqivalent to the exact knowledge of. The remainder of this paper deals with the details of how to estimate the parameters A, B, and σ from drive test data that has een corrpted with measrement error and the precision that reslts from doing so. 3

4 1 Area eliaility vs Edge Coverage verss Proailities Edge eliaility. CELL EDGE ELIABILTY EUDIK S OUTAGE POBABILITY MODEL FO 90% AEA ELIABILITY ESTIMATES OF POPOTIOS OUTAGE POBABILITY MODEL FO 90% AEA ELIABILITY AEA ELIABILITY Decreasing adis, 3.6% 3.3%.4% 0.7% / %.5%.5%.5% σ/B Figre 3. Area reliaility (ordinate) and cell edge reliaility (see parameter associated with each crve) verss10σ/b, where σ is the standard deviation of the F signal and B is the propagation path loss exponent. For a given vale of σ/b, knowledge of the cell edge reliaility directly determines the area reliaility. (The figre is redrawn from Chapter of reference[]). ote, increasing cell edge reliaility is eqivalent to decreasing the radis of coverage. IV. AEA ELIABILITY ESTIMATIO APPOACH The relationship etween the reliaility of coverage over a circlar area and the reliaility of coverage on the perimeter of the circle was first estalished y D.O. edink (circa 1974) []. The main finding of this stdy was that cell area reliaility and cell edge reliaility oey the simple relationship illstrated in Figre 3. As long as the propagation path follows a power law this relationship is completely determined y the ratio of σ/b, where σ is the standard deviation of the lognormal fading within the cell and B is the path loss exponent (e.g., typical vales are σ=8db and B=35.). As shown in Tale 1, given exact knowledge of σ and B, the cell area reliaility (and cell edge reliaility) can e exactly compted (see also eqation (a7) in the Appendix). ote that althogh 75% cell edge reliaility approximately corresponds to 90% cell area reliaility, and 90% cell edge reliaility approximately corresponds to 97% cell area reliaility, their exact vales depend on the propagation parameters of each cell (i.e., σ and B) σ B Edge eliaility 75% 90% 75% 90% 75% 90% Area eliaility 91.65% 96.87% 90.7% 96.57% 87.4% 96.03% Tale 1. The relationship etween area and edge reliaility for varios propagation parameters σ and B. Example 1 Example Edge eliaility 75% 75% σ 8 10 B Area eliaility at 75% edge reliaility 90.7% 87.4% adis at 75% edge reliaility 1 km km adis at 90% area reliaility 1.01 km km Edge reliaility at 90% area reliaility 73.5% 79.5% Tale. The relationship etween cell area reliaility, cell edge reliaility and cell radis for different propagation parameters σ and B and the same transmit power. Figre 4. For eqal area portions of a cell and their corresponding otages for two proaility models: edink s method assmes a linear path loss. The Estimate of Proportions method implicitly assmes no path loss. That is, otages directly nder the ase station and otages at the cell edge are eqally likely. edink s path loss model is clearly a more valid assmption. The relationship in Figre 3 is apparently independent of the asolte cell radis, as well as eing independent of the transmit and receive power, which only serve to scale the radis. This seems to ncople the prolem of determining the coverage reliaility from the prolem estimating the size of a cell. However, the relationship in Figre 3 does not mean that the cell radis has no effect on coverage reliaility. On the contrary, for the same two-way gain and transmit power, making the cell radis larger redces the coverage reliaility and decreasing the cell radis increases the coverage reliaility. The dependency on cell radis is implicit throgh the desired edge reliaility and eqation (5). Tale demonstrates some of the relationships etween cell radis, cell area reliaility, and cell edge reliaility for two cells designed with the same transmit power. In this tale, the cell radis is compted from eqation (5) and the area reliaility is compted from eqation (a7). For example 1, the cell radis at 75% edge reliaility (90.7% area) is 1 km, the radis at 73.5% edge reliaility (90% area) is 1.01 km. The reslts are similar for example. Oserve that changing the cell radis can have a significant effect on the reliaility of F coverage. In edink s original derivation of area reliaility, the explicit dependence of coverage on cell radis was prposely eliminated. Since the cell radis is one of the estimated qantities of interest in this paper, it is reintrodced into edink s expression in the Appendix in eqation (a7). This is the formla (i.e.,! F ) that is sed throghot this paper to estimate the reliaility of F coverage over a circlar area. Ths, the approach for measring area reliaility in this stdy is as follows: 1) measre the propagation parameters A! ', B!, and!σ for each cell via linear regression ) se these parameters to estimate the cell radis! from eqation (5) 4

5 SL (dbm) CELL ITEIO 1 AEA ELIABILITY = SL (dbm) CELL ITEIO 1 P THESH P THESH (a) log 10 r () log 10 r CELL ITEIO CELL ITEIO SL (dbm) 1 SL (dbm) 1 =0 P THESH P THESH =0 (c) log 10 r log 10 r Figre 5. Comparison of area reliaility estimators for three different drive test scenarios (a) Graphical approach to estimating the cell radis with the Estimate of Proportions method: the radis is redced ntil the desired area reliaility is reached () Most typical drive test; half of the signal strength measrements are within the cell; half of the signal strength measrements are otside the cell (c) Least likely( ideal) drive test; all of the signal strength measrements are within the cell (d) Worst case drive test; all of the signal strength measrements are otside the cell. The area reliaility estimate ased on edink s method can tilize the measrements otside the cell (d) 3) se the radis and the propagation parameters to estimate the reliaility of coverage,! F, over the cell area sing edink s expression (see eqation (a7)). In the following section, this method of estimating coverage reliaility is shown to e mch more accrate than crrent approaches [3]. V. ESTIMATE OF POPOTIOS COMPAED TO EUDIK S APPOACH In this section, the following two area reliaility estimators are compared: 1) edink s approach with linear regression. We will refer to this techniqe as edink s method, even thogh his original stdy did not address the se of linear regression, or the significance of his reslts to verification []. ) Estimate of Proportions [3]. It is worthwhile to compare the path loss models that nderlie oth of these estimators. To facilitate this comparison, it is assmed that the measrements are independent and niformly distrited throghot the area of the cell, oth in range and in azimth. Figre 4 shows the distrition of otages for edink s method, which assmes a linear path loss. The cell is divided into for eqal area regions and otage proailities for each region are generated, sing eqation (a7), for a typical 90% cell area reliaility design. In edink s techniqe the median path loss is adaptively compted via linear regression within each cell. Oserve that, as the cell edge is approached, the otage proaility increases. In contrast, the nderlying assmption of the Estimate of Proportions techniqe is that otages are eqally distrited throghot each cell, as also shown in this figre. The implicit assmption of the Estimate of Proportions approach is that there is no path loss. Specifically, otages at the cell edge and otages nder the ase station are eqally likely events. edink s path loss model is clearly a more appropriate choice and this is the major reason that edink s area reliaility estimates are always more precise. The asic approach of the Estimate of Proportions method is illstrated in Figre 5(a). The cell radis is determined iteratively y redcing the radis ntil the desired area reliaility (eqation (6)) is reached. The precision of this method is compared with edink s techniqe in the analysis that follows. Consider the following estimate of area reliaility made y calclating the proportion of signal strength vales that are aove a qality threshold!f = = where is the total nmer of signal strength measrements within the cell (6) 5

6 1 is the nmer of signal strength vales in the cell aove P THESH is the nmer of signal strength vales in the cell elow P THESH Provided oth F >5 and (1-F )>5, F! is approximately a ormal random variale with a mean of F and a standard deviation as indicated in the following eqation:! F ~ F, F( 1 F) + 1 (7) where F is the tre area reliaility (this eqation may e fond in reference [3]). The error, F, (one sided 95% confidence interval) of this coverage estimate is F = F( 1 F) + 1 The application of this expression to celllar verification is somewhat anomalos since this eqation is completely independent of the amont of lognormal fading, σ, in the cell. This sggests that coverage estimation needs no more measrements in hilly terrain than for flat terrain, which is inconsistent with reality. The precision of edink s method is empirically determined from the simlation reslts in section VII and expressed in eqation (15). The relative precision of the area reliaility estimates can e directly compared y dividing eqation (8) y eqation (15): F F F( 1 F) ( σ ) F ( 1 F ) F 1 F 3 (0.0695σ ) F ( 1 F ) = + 1 (8) (9) where it is also assmed that edink s method is not sing any signal strength samples otside of the cell (i.e., = 1 +, Figre 5(c)). Given the aove assmptions, Eqation (9) can e shown to e a completely general expression, provided 6 σ 10, F 90% and 100. Eqation (9) is independent of the nmer of signal strength samples,, and dependent only on the area reliaility and the lognormal fading within the cell, σ. The exact vale of σ is nimportant to the point of this comparison; assme that σ=8 db. Both area reliaility estimators can e directly compared for the following two vales of reliaility F =90% and F =97%: F F F 507. and F 87. σ= 8 σ= 8 F = 090. F = 097. By sstitting different vales of σ into eqation (9), the reader can easily verify that edink s area reliaility estimate is always more than for times the precision of the Estimate of Proportions approach. Alternatively, edink s approach reqires fewer measrements to achieve the same area reliaility accracy as the Estimate of Proportions techniqe. For the same assmptions as aove, and sing eqation (9), the Estimate of Proportions method reqires : 1) 6 times (=5.07 ) as many points as edink s method for a 90% area reliaility design. ) 68 times (=8.7 ) as many points as edink s method for a 97% area reliaility design. Hence, edink s method makes mch etter se of a finite set of signal strength measrements. Figres 5(), 5(c), and 5(d) show typical distritions of signal strength measrements within single cells. All three drive test scenarios in this figre are possile. The est scenario (and least likely) is in Figre 5(c) where all of the measrements are within the cell. edink s method can also e sed to predict coverage of sparsely driven areas. Often, dring the drive data collection phase, the intended coverage area is not exactly known. Figres 5() and 5(d) have a significant nmer of measrements otside the cell. edink s method easily exploits this data, and ths can e very sefl in optimizing celllar handoff performance. The est soltion to the scenario in Figre 5(d) is to retest the cell. However, this is not always possile and for these cases a tned prediction of the coverage may e desired. edink s approach is ideal for this application. In smmary, it was shown that edink s area reliaility estimator is more than for times the precision of approaches that are ased on the Estimate of Proportion method. Eqivalently, edink s techniqe reqires an order of magnitde fewer measrements to achieve the same accracy as the Estimate of Proportions method. VI. F POPAGATIO SIMULATIO ESULTS To test the validity of the radis estimation and area reliaility estimation approaches, an F propagation simlation was written, as shown in Figre 6. To redce the comptation, we se a single radial component that is niformly sampled along its length. It is assmed that the simlated measrements reslt from a niform azimthal sampling of the cell. Hence, the samples along this single radial represent the composite path 6

7 F GEEATIO F ESTIMATIO σ Gassian (0, σ) SIMULATED DATA - ZEO MEA AC VAIATIO x σ^ A = P T - A B - x Uncorrelated ormal Fading AC VAIATIO MEA PATH LOSS (MPL) + Y LIEA EGESSIO X ^ A ^ B - x ESTIMATED MEA PATH LOSS (EMPL) - COV x ρ MSE per sample AGE log X 10 Figre 6. Block diagram of F propagation simlation. The inpts to the simlation are A, B, and σ. The corresponding otpts are estimated as shown aove. The mean-sqare-error per sample etween the est fitting line and the tre mean path loss is sed to measre the performance of the estimation process. The correlation coefficient, ρ, of the est fitting line with the data is also compted. loss flctations of radials in all azimth directions. The single radial of this model is ths considered to e a linear sperposition of mltiple radials that are niformly spaced in azimth (i.e., two-dimensional niform area sampling). The simlation cold explicitly calclate this sperposition, t this is comptationally inefficient and wold have no effect on the reslts. For a fixed azimth angle, the fading is correlated along the radial from the ase station. However, we assme that the distance etween the measrements on a radial is large enogh to neglect correlation effects. In addition, the fading etween radial components at eqally spaced azimth angles is nearly ncorrelated. Since the regression is actally evalated across all azimth angles simltaneosly, an ncorrelated Gassian fading model is chosen. The standard deviation, σ, (typically 5-10 db) is inpt into a Gassian random nmer generator which prodces ncorrelated normal random variales with zero mean and variance σ. The mean path loss is compted for each range vale as the prodct of the logarithm of range and the path loss coefficient, B, to which is added the intercept vale A. The variation de to fading is then added to the mean path loss (MPL) also shown in Figre 6. This concldes the F signal generation portion of the simlation. The remainder of the simlation is concerned with estimating A, B, and σ. Both A! ' and B! are compted via linear regression. The estimated mean path loss is then stracted from the simlated signal strength vales and an estimate of the standard deviation,!σ, is made from the reslting zero-mean process. Two major criteria are sed to evalate the performance of the estimation procedre in the simlation: 1) the correlation coefficient, ρ, etween the simlated data and the est fit line (EMPL). The closer ρ is to nity, the more linear the data. This measre is also sed to characterize the reliaility of the field data. ) the mean sqare error (MSE) per sample etween the est fit line (EMPL) and the tre path loss in the simlation (MPL). This measre cannot e sed in the field, since the tre path loss is nknown. eceived Signal Level (dbm) A F Propagation Simlation. SL(dBm) Best Line Fit ange from Base Station (km) Figre 7. Simlated received signal strength verss distance from the ase station and the est fitting linear approximation. Typical reslts from the simlation are shown in Figre 7. The following parameters were the inpts sed to generate the 515 data points in this figre: 7

8 hence A= 140 P t = 50 dbm (EIP) A = Pt A=-90dBm B= 35. σ = 10 db The corresponding otpts were A! ' = !B = σ! =10.63 db ρ = 0.86 (correlation coefficient, where ρ =1 for a line) MSE per sample = 0.39 db The vale of ρ=0.86 is typical of that fond in actal drive test data. The simlation estimated the inpt parameters very well since A - A! ' =-1.65 B- B! =-1.15 σ- σ! =-0.63 MSE =0.39 The aove for vales can e made as close to zero as desired y increasing the nmer of simlated data points,. Since these errors depend on the nmer of data samples sed to compte the regression, a natral qestion is How many data samples are necessary to achieve a given precision? The accracy of the measrement approach is examined in more detail in the next section. VII. MEASUEMET ACCUACY VESUS THE UMBE OF SAMPLES This section deals with determining the measrement error of the overall estimation process. The simlation is sed to determine the proaility densities of the following two random variales: where e F = F F! and e F =! (10) e F is the relative error of the area availaility estimate as compted from eqation (a7) e is the relative error of the cell radis estimate F is the tre area reliaility compted from eqation (a7! F is the estimated area reliaility compted from eqation (a7) is the tre cell radis compted from eqation (5)! is the cell radis estimate compted from eqation (5) The transformations specified y eqation (10) allow a direct comparison of the cell radis estimate with the area reliaility estimate, which wold otherwise e difficlt de to the differences in the dimensions of these two estimators (i.e., is in kilometers and F is a percentage). Typical proaility densities for e F and e are shown in Figre 8. Oserve that the error of the cell radis estimate, e, is comparale in Figres 8(a) and 8() (ote the scale change etween the ordinates of these two figres). However, e F is almost a factor of two smaller for the 90% cell edge reliaility design. The evidence in Figre 8 and all of the histograms processed in this stdy demonstrate that oth e and e F are well modeled as zero-mean ormal random variales, and ths, only their respective variances are needed to characterize the precision of the estimates! and F!. These are determined empirically via Monte Carlo simlation. We are interested in determining the inaccracy,, of the estimate of the cell radis at a 95% confidence level. The inaccracy is measred from empirical histograms y simlating e and determining sch that ( ) P! + = 95% The inaccracy of the radis estimate,, is determined y the following two-sided test t 1 z c c = F( zc) = e dt π z (11) c where the z c variale in eqation (11) is chosen to yield the desired confidence level, c. For example, if c=95%, then z c =1.96. Since e has a mean of zero, the corresponding twosided normalized radis inaccracy, δ, is ± δ = ± = ± 196. VA( e ) (1) where δ is a dimensionless percentage of the cell radis,. Likewise, the inaccracy of the area reliaility estimate, F!, is estimated from histograms of e F and determining F sch that ( ) PF! F + F = 95% Since e F also has a mean of zero, the inaccracy, F, (one sided 95% confidence interval) of the coverage estimate is δ F F = = VA( ef ) (13) F 8

9 75% Cell Edge eliaility. 55 (a) 90% Cell Edge eliaility. 697 () Freqency of Occrrence e F e Freqency of Occrrence e F e 0-4% -3% -% -1% 0% 1% % 3% 4% elative Errors of Estimates. 0-4% -3% -% -1% 0% 1% % 3% 4% elative Errors of Estimates. Figre 8. Histograms showing the simlated proaility densities of the relative error e F of the area availaility estimate F! and the relative error e of the cell radis estimate! for (a) a 75% cell edge reliaility design () a 90% cell edge reliaility design. The nmer of samples in the regression is 1000 and the standard deviation of the lognormal fading is σ = 8 db. adis ange Inaccracy Inaccracy for 75% for Cell 75% Edge Cell Edge ( 90% Cell Area) adis ange Inaccracy Inaccracy for 90% for Cell 90% Edge Cell Edge ( 97% Cell Area) δ % 50% 40% 30% 0% 10% σ=10 σ=6 σ=8 0% mer of Samples (a) 5000 δ % 45% 40% 35% 30% 5% 0% 15% 10% 5% 0% σ=10 σ=6 σ=8 () mer of Samples Coverage Coverage Inaccracy Inaccracy for 75% for Cell 75% Edge Cell Edge ( 90% Cell Area) Coverage Coverage Inaccracy Inaccracy for 90% for Cell 90% Edge Cell ( Edge 97% Cell Area) δ F % 1.4% 1.% 1.0% 0.8% 0.6% 0.4% 0.% 0.0% σ=10 σ=6 σ=8 (c) mer of Samples δ F % 0.50% 0.45% 0.40% 0.35% 0.30% 0.5% 0.0% 0.15% 0.10% 0.05% 0.00% σ=10 σ=6 σ=8 (d) mer of Samples Figre 9. Simlated inaccracy (95% confidence) of measrement techniqes verss the nmer of samples in the regression: (a) cell radis estimate! of a 75% cell edge reliaility design, () cell radis estimate! of a 90% cell edge reliaility design, (c) area reliaility estimate! F of a 75% cell edge reliaility design (d) area reliaility estimate! F of a 90% cell edge reliaility design.

10 where δ F is a dimensionless percentage of the area reliaility, F. Each point in the plots in Figre 9 represents the precision (at 95% confidence) that is otained after simlating and processing five million signal strength vales. Close inspection of Figre 9 reveals that for a given nmer of signal strength samples,, the area reliaility is mch more precise (y one to two orders of magnitde) than the estimate of the cell radis (compare Figre 9(a) with Figre 9(c)). For example, 1000 samples in the regression are needed for aot a ±3% inaccracy in the cell radis estimate. However, even with 500 samples in the regression, the area availaility estimate is very precise. The inaccracy of the area availaility estimate is less than 0.5% for cells designed with 75% cell edge reliaility and within 0.% for cells designed with 90% cell edge reliaility. The inaccracies of oth of the estimates! and! F can e approximated y the following expressions which were determined empirically (via least-sqares) from the data in Figre 9 δ δ F 381. σ! = (14) F ( σ! ) F ( 1 F! ) = F (15) where is the nmer of independent samples in the regression! σ is the estimated standard deviation of the lognormal fading in the cell!f is the estimated area reliaility compted from eqation (a7) From eqation (10), it is easy to show that eqations (14) and (15) are completely general expressions, provided 6 σ 10, F 90% and 100. Oserve that the area reliaility inaccracy in eqation (15), F, is inversely proportional to. It is interesting to compare the magnitdes of the inaccracies of the aove area reliaility measrements with those compted y estimating a proportion of signal strength measrements that are aove a desired threshold. From eqation (8), for 500 samples and 90% cell edge reliaility, the inaccracy is aot.%. Ths, the area availaility estimate, F!, shown in Figre 9(d) is aot ten times the precision of estimates that are ased on proportions of signal strengths (i.e., eqation (6)). The radis inaccracy in eqation (14), δ, is inversely proportional to the nmer of samples in the regression,, and directly proportional to the amont of lognormal fading, σ, in the cell. Interestingly, radio srvey engineers have long recognized the negative effects that widely varying terrain and cltter environments have on F coverage tests. They sally compensate for these effects y taking many more measrements in these areas. Eqation (14) is simply the mathematical expression of this practice, specifying the relationship etween the desired coverage inaccracy, δ, the nmer of independent signal strength measrements,, and the terrain fading factor within the cell, σ. It shold e noted that for real data, the radis inaccracy will actally e less than specified y eqation (14), since signal strength samples of adjacent ins are not completely independent. The fact that adjacent samples are correlated actally redces the error of the cell radis estimate making eqation (14) an pper ond. The most important finding of this analysis is that it is the precision of the estimate of the cell radis (i.e., eqation (14)) that is the limiting factor in determining the qality of F coverage, not the precision of the area reliaility estimate. VIII. DISCUSSIO Given the andance of other qality metrics sch as adjacent channel interference, cochannel interference, dropped calls, hand-off failres, call access failres, it error rate, frame error rate, etc., the experienced celllar engineer might qestion or focs on coverage. We jstify or approach y arging that, at least from a design perspective, degradations in these other performance metrics are simply a reslt of either not enogh carrier power or too mch interference power. Frthermore, we smit that all extraneos same-system interference is simply ncontrolled other-cell coverage which wold not exist if the geographic extent of reliale coverage in each cell was properly designed in the first place. Hence, coverage estimation fndamentally remains the most critical step of the design of any network. An accrate method of determining the radis (! ) of individal cells was presented. This led to an even more precise techniqe for estimating the reliaility of coverage over the area of the cell (! F ). For the same precision in area reliaility, the Estimate of Proportions techniqe reqires more than twenty times as many measrements than edink s method (! F ). Ths, edink s area reliaility estimator has some comptational advantages in verification postprocessing. It was shown that it is possile to otain an excellent estimate of the area reliaility even if the nmer of samples is insfficient for estimating the cell radis. This raises an interesting qestion concerning the determination of service reliaility: What is the est metric to se in classifying the qality of F coverage? This stdy indicates that area reliaility alone is insfficient. A major finding of this stdy is that the area reliaility (F, eqation (a7)) and the cell radis (, eqation (5)) are an eqation pair. In F verification, it is not possile to determine an area reliaility withot simltaneosly compting a cell radis, and vice versa.

11 The reslts of this paper indicate that estimating the effective radis of a cell is the limiting factor in determining the F coverage reliaility. Specifically, it takes aot fifty times as many signal strength samples to estimate the cell radis,, than to estimate the area reliaility, F. For a given propagation environment, compting the distance to the cell edge is deterministic (i.e., apply eqation (5)). For real drive test data, the tre cell radis is nknown and mst e statistically estimated. It was shown that as long as the radis estimate is sfficiently precise, so is the area reliaility estimate (! F ). If cell edge reliaility is the desired coverage criterion, an accrate estimate of the cell radis is all that is needed since the cell edge reliaility is ensred y the fade margin sed to measre the radis. However, if area reliaility is the desired coverage criterion, then a minor adjstment to the cell edge reliaility (and cell radis) mst e made in each cell to compensate for the variation in the specific vales of the propagation constants A, B, and σ. This is easily done y first compting the propagation constants via linear regression and then compting the area reliaility from eqation (a7) at fine range increments (e.g., steps of /). The desired radis can then e fond y inverse interpolation. The proposed techniqe is est sited for macrocells. However, it can e modified to work eqally well in microcells y eliminating measrements that have line-of-sight to the ase station. In many cells, the path loss is etter descried y a composite of two line segments that intersect at some reakpoint distance near the ase station. For these cases, this distance is approximated and measrements efore this point are eliminated since virtally no otages occr over this region. The est linear fit to the path loss in the oter regions of the cell is extrapolated all the way to the ase station. For most propagation scenarios, the error in the area reliaility estimate de to this approximation is less than 0.5% (see Figre 4) and less than 1.5% for the radis estimate. Ths, the proposed method needs no additional parameters or modifications to accommodate dal-power law propagation environments. However, if these errors are not tolerale, they can easily e eliminated y incorporating reakpoint distance into edink s expression (eqation (a7)). Althogh the precision in this stdy was determined via simlation, we have processed signal strength measrements from hndreds of cell sites and have fond that the reslts are completely consistent with those of or simlation. We have also fond cell radis inaccracy to e very sefl in determining the sampling reqirements of celllar drive tests [5][6][7]. This verification approach is particlarly sefl to anyone involved with cell planning since this eqates the prolem of determining the reliaility of F coverage with that of determining the effective size of the cell. The latter concept is clearly more sefl to the celllar network planner. IX. COCLUSIOS Cell radis inaccracy has een proposed as a new method for measring F celllar coverage. The techniqe measres the average distance from the ase station to the cell edge (eqation (5)) and qantifies the precision y also specifying the ncertainty of the radis estimate. In addition, the approach provides an estimate of the area reliaility (eqation(a7)) which was shown to e mch more accrate than crrent methods that estimate the coverage from a proportion of signal strength measrements [3]. Empirical formlas are given that approximate the precision of oth of these estimates (eqations (14) and (15)). The reslts of this paper show that area reliaility is more sefl in specifying a desired qality of F coverage than in verifying that this qality is actally achieved. The recommended verification techniqe, cell radis inaccracy, ses linear regression to estimate the minimm mean sqare path loss within each cell, and is ths very tolerant to estimation errors de to terrain flctations (e.g., lognormal fading). The approach provides the est circlar approximation to any eqal power contor, at any desired reliaility. Ths, this method is ideal for cell site planning with any wireless technology. For example, sing standard CW drive test measrements, this techniqe can help verify that the F design meets the proper amont of overlap in coverage needed to spport the soft handoff regions of CDMA. The conclsion of this stdy is that cell radis estimation and area reliaility estimation shold not e treated separately, and that cell radis inaccracy is the more critical verification measre. ACKOWLEDGMETS This paper is dedicated to the memory Dr. Charles W. Bernardin and Elizaeth J. Bernardin. The athors wold like to thank the management of OTEL Wireless Engineering Services for providing the fnding and the environment necessary for this research. This stdy originated after several sefl discssions aot coverage estimation with Dr. Ahmad Jalali and Martin Kendal. We wold like to acknowledge Dr. ichard Tang for originally sggesting the regression approach. Also, we wold like to thank Professor Vengopal Veeravalli for his invalale sggestions concerning the error performance of the area reliaility estimation approach inclded here. We are especially gratefl to Professor Veeravalli for performing the rigoros analysis presented in the Appendix that follows. We wold also like to thank Dr. Sdheer Grandhi, Plin Patel, Dr. eid Chang, Mheiddin aji, Mazin Shalash, and Mark Prasse for taking the time to critically evalate this manscript. 11

12 APPEDIX AEA ELIABILITY AS A FUCTIO OF CELL ADIUS This derivation is similar to D.O. edink s original analysis, which showed that the relationship etween cell edge reliaility and cell area reliaility was theoretically independent of the asolte cell radis []. However, since cell radis is one of the estimated qantities of interest in this stdy, this dependency is prposely reintrodced. Let the received power, P r, at the edge of a cell,, e given y P ( ) = r A B log + X 10 (a1) where X is a normal zero mean random variale with variance σ. Similarly, the received power at a distance, r, is Pr () r = A Blog r + X 10 (a) where it will e assmed that r<. The otage proaility, P ot (r), at a particlar range, r, from the ase station is given y P ( r) = P( A Blog r + X P ) ot 10 THESH + = F P THESH A B log r 10 σ + 10 = 1 Q P THESH A B log r σ (a3) and the corresponding service reliaility on a circlar contor of radis, r, is 1 ot 10 P ( r) = P( A Blog r + X > P ) + 10 = 1 F P THESH A B log r σ = + Q P A B log r THESH 10 σ where P THESH is the desired threshold and Qx ( ) = 1 Q( x) Qx ( ) = 1 Fx ( ) F( x) = P( ξ x) ξ ( 01, ) THESH (a4) Define 1 F( x) = π d dx Qx ( ) = P a = THESH σ t x e dt x 1 e π A and Blog e = 10 σ Then from eqation (a4), the service reliaility on a circlar contor of radis, r, is 1 P r = Q a+ ln r ot () ( ) (a5) The fraction of sale area, F, (i.e., area reliaility) within the cell can e fond y integrating the contor reliaility across range F 1 = [ 1 Pot () r ] πrdr π 0 = Qa ( + ln r) rdr (a6) 0 ow consider the integral set 0 Qa ( + ln r) t = a+ ln r r = e t a t a e dr = Ths Qa ( + ln r) 0 = 1 = e = dt rdr a+ ln Qt () e t = a+ ln Qt () e a rdr t a t a e ( t a) a+ ln Qt () e t dt dt dt 1

13 ow a + ln Qt () e t dt = t a t + Qt e / + ln a ln () + e π e t dt EFEECES [1] Hata, M., Empirical Formla for Propagation Loss in Land Moile adio Services, IEEE Transactions on Vehiclar Technology, vol. VT-9, o. 3, Agst 1980, pp [] edink, D.O, Microwave Moile Commnications, edited y Jakes, W.C., IEEE Press, reprinted 1993, ISB , Chapter, pp = Qa ( + ln ) e e + π ( a+ ln ) a = Qa ( + ln ) e a+ ln e a ln e + e π a 1 4t 4 t t dt = Qa+ e ( ln ) + e 1 Q a + ln Ths, the area reliaility is F = Qa ( + ln r) 0 a rdr a e = e Q ( a + ln ) And finally, a e + e 1 Q a + ln a + e F = Q( a+ ln ) + Qa + 1 ln (a7) dt [3] Hill, C. and Olson, B., A Statistical Analysis of adio System Coverage Acceptance Testing, IEEE Vehiclar Technology Society ews, Fe. 1994, pp [4] Lee, W.C.Y., Moile Commnications Engineering, McGraw-Hill Book Co., 198, p [5] Bernardin, P., Yee, M., and Ellis, T., Estimating the Cell adis from Signal Strength Measrements, 6th WILAB Workshop, March, 0-1, [6] Bernardin, P., Yee, M., and Ellis, T., Estimating the ange to the Cell Edge from Signal Strength Measrements, 47th IEEE Vehiclar Technology Conference, May 5-7, [7] Bernardin, P., Cell adis: A Better F Validation Criterion Than Area eliaility, 1st Annal UCSD Conference on Wireless Commnications, March 8-10, 1998, La Jolla, CA. Pete Bernardin (M 90) received a B.S. in mathematics from Villanova University in 1973 and a Ph. D. in iomedical engineering from The Johns Hopkins University in His gradate work inclded research in the aditory system, applying mathematics and digital signal processing to the neral processes of hearing. He has had over 0 years experience in solving engineering and mathematical prolems with digital methods. He has worked in the adar and Digital Systems division of Texas Instrments, and the Advanced Technology La of CA. He has worked as an image processing consltant for ASA, as a digital signal processing staff specialist at the Tandy Corporation and as a telecommnications engineer at Bell orthern esearch. Crrently, he is a manager in the Wireless F Engineering Services grop of orthern Telecom, ichardson, Texas. Dr. Bernardin is the athor of two ooks concerning nmerical comptation for engineering and scientific applications. He is also the athor of several research papers, covering a road range of technical areas which inclde image processing, radar, spectral estimation, and commnications. 13

14 Meng F. Yee (M 89) received a B. Sc. (Hons) in Electronic Engineering from University of Birmingham, England in 1975, a Master of Bsiness Administration (MBA) from University of Warwick, England in 199 and a Master of Engineering from the University of Toronto, Canada in He has held nmeros engineering and operations positions for Malaysian Telecom, providing oth wireline and wireless network infrastrctre throghot Malaysia. He has also held positions in Strategic Technology and Planning with ogers Cantel, the largest celllar operator in Canada. In this position he participated in oth the TIA and CTIA microcelllar committees. Mr. Yee was also a memer of the CTIA WBSS (Wideand Spread Spectrm) committee in 199, which was formed to evalate the se of spread spectrm technology for celllar applications. Mr. Yee is crrently a Senior Manager in CDMA F Design and Technology Applications with OTEL (orthern Telecom), ichardson, Texas and works on CDMA network optimization and F performance. Thomas Ellis has thirteen years experience, designing and implementing long hal and wireless systems. Additionally, he has several years experience developing software soltions for nmeros commnications and engineering prolems. From 1985 to 199 he worked for MCI Telecommnications, designing and implementing long hal microwave and fier optic systems, as well as designing new two-way UHF systems for field engineering spport. He joined BellSoth Enterprises in 199 as a Senior Systems Engineer in their Worldwide Wireless International Engineering grop. While at BellSoth, he was one of the initial system engineers that planned, designed and developed the technology that is crrently known as BellSoth DCS. From 1993 to 1995 he was a conslting engineer with Moile Systems International (MSI). In 1995 he left MSI and formed Wireless Software Design & Conslting. Since then he has een an engineering consltant to ortel's CDMA Engineering grop, and was the lead software developer on their CDMA and TDMA F Optimizer tools. 14