1 Path-loss Model for Broadcasting Applications and Outdoor Communication Systems in the VHF and UHF Bands Constantino Pérez-Vega, Member IEEE, and José M. Zamanillo Communications Engineering Department - University of Cantabria. Santander, Spain. e-mail: constantino.perezv@unican.es Abstract - A simple propagation model for the VHF and UHF bands is presented. The model is a computational form of the data provided by the FCC F(50,50) propagation curves, and it is aimed to be used by practicing engineers. It allows the estimation of median path loss, received power, or electrical field strength which usually is sufficient in many practical applications. The model is independent of frequency and is applicable to outdoor environments in a range of distances from about 0.5 mi (800 m) up to 40 mi (64.36 km) and transmitting antenna heights from 100 ft (30.48 m) up to 2000 ft (609.6 m), and is based on a receiving antenna height of 30 ft (9 m). Index terms - Propagation models. Path loss. Outdoor propagation. I. THEORETICAL BASIS The model studied here is intended for use in coverage prediction of broadcasting services in the VHF and UHF bands. Since the model is not frequency-dependent [6], it can also be used in other types of communication systems in these bands. The model is of easy application and does not require particular computational resources. Actually, the model is no other thing than a mathematical characterization of the FCC F(50,50) propagation curves [1] and allows the estimation of the median value of path loss as a function of frequency, distance and transmitting antenna height. In the model, path loss is characterized by an attenuation factor, in this case, the exponent of distance; i.e. it is assumed that path loss does not follow an inverse square law of distance as in free space propagation, but a law of the inverse of distance raised to an exponent, in general greater than 2, even when in particular environments, such as aisles in buildings or tunnels, it can take values smaller than 2. In this work, the model is intended for outdoor propagation where the values of the exponent are, in general, greater than 2. In this model the received isotropic power at a distance d from the transmitter is given by: 2 Prad λ P iso = n d 4 π watt (1) where P rad is the effective isotropic radiated power in the direction of the receiver, d is the distance between transmitting and receiving antennas in meters and λ is the wavelength in meters. The value of n intrinsically embeds the effects of all propagation mechanisms: attenuation, diffraction, reflection, etc. No attempt is made to establish a relationship between n and the physical aspects of the channel. However, it seems clear that the higher the effects of scattering mechanisms, such as attenuation, diffraction, diffuse reflections, etc., the smaller the received power, and the higher the value of n. Path loss in db is usually expressed as L = 10 log (P rad /P iso ), therefore, from (1), it can be calculated as: L = 10nlog + L db (2) 10 ( d) Where d is in meters. If, in the above expression d is given in miles, path loss can be calculated as: 0
2 where L 0 is the attenuation at 1 m in free space: [ log ( d ) + 3.2066] 0 L = 10n mi + L db (3) 4 π L 0 = 20log10 (4) λ Model (2) has been used for indoor as well as outdoor communications, and several versions have been treated previously in the literature [2]-[5]. It also must be stated that no simple relationship can be established between n and the various and, in general, complex physical processes that affect path loss. In general, it is not possible to quantify the individual effects of the scatterers and, therefore, the predictions that can be obtained with any propagation model, including this one, are only an approximation. In any case, the reasonably accurate estimation of path loss is fundamental in the power budget of any communication system. In the above expressions, no clear dependence of n with frequency, distance or antenna height appears. In order to establish the dependence of n with respect to frequency, a campaign of measurements was performed at various frequencies in the VHF and UHF bands, from 100 MHz to 800 MHz, and the results are reported in [6]. It appears that n is independent of frequency, a result that is in concordance with the findings reported in [6] and with the very nature of the F(50,50) propagation curves. On the other hand, the dependence of n respect to distance and transmitting antenna height is not evident, and a vast amount of measurements would be required in order to establish such dependence experimentally. Instead of that, we attempted an indirect approach using the data obtained from the FCC F(50,50) propagation curves for television broadcasting in the VHF and UHF bands. An empirical model was developed, that is applicable to outdoor propagation environments at distances between around 0.5 and 40 miles (0.8 to 64 km) It is well known that the FCC F(50,50) propagation curves have been in use successfully since many years by practicing engineers, and provide adequate estimations of field strength, and indirectly, of received power for a vast amount of practical cases. Such curves were developed using measured values, taken in different geographical areas over different periods of time, and provide the median values of field strength for service at 50% of locations during 50% of the time. Therefore, such curves reflect reliable experimental data not derived from theoretical models [7]. It must be stated however, that the model proposed here does not pretend to be better than others in use. It is a well known fact that different propagation models often yield conflicting results and it depends on the experience and good judgment of the engineer the final decision on what particular model reflects better the practical results. The model presented here offers an alternative of easy application by practicing engineers which are always faced with the frequently difficult choice between predictions or calculations, and measurements [8] and, in some way, combines both in a simple way. Furthermore, the model is independent of the type of communication system; i.e. if it is analog or digital, but id does not provide any information about angles of arrival, delay spread, etc., neither it provides any other information about channel dynamics; only the mean (or median) value of path loss, which is sufficient for most practical applications. II. DEPENDENCE OF THE MODEL WITH DISTANCE AND TRANSMITTING ANTENNA HEIGHT To establish the dependence of n with distance and transmitting antenna height, field strength values were obtained from a set of F(50,50) curves, for distances from 1 to 40 miles (1.6 to 64.36 km), and for transmitting antenna heights from 100 up to 2000 ft. The curves assume a constant receiving
3 antenna height of 30 ft (9 m) and corrections must be made for other heights [8]. For distances up to about 30 miles (48.3 km) the F(50,50) curves for frequencies above 470 MHz are not based on measured data, and theory would indicate that the field strength should decrease more rapidly with distance beyond the horizon for frequencies in the VHF band [1]. Several sets of field strength values were taken for constant height at different distances, and analyzed afterwards. In this analysis British units were used, so appropriate care must be taken if SI units are used, since the coefficients of the model are not the same in the two systems of units. In Appendix 1, the adequate coefficients in SI units are also presented. Even when field strength values from the curves are referred to an EIRP (P rad in (1)) of 1 kw, calculation of the isotropic received power is straightforward for any other vale of P rad and the value of n can be easily obtained from (1) as: Prad ( dbw) Piso ( dbw) L0 n = (5) 10log ( d) 10 where L 0 is defined by (4) and the equivalent isotropic radiated power (P rad ), as well as the isotropic received power (P iso ) are in the same logarithmic units (dbw or dbm). The latter can be easily obtained from the field strength as: 2 1 E λ P iso = watt (6) 480 π Where E is the field strength in V/m obtained from the F(50,50) curves, and it must be stressed that in the above equations, n is the median value of the exponent of distance. Actually, the mean and median values of the exponent differ only in the order of 1% and, for our purposes, are considered as equivalent here. The procedure followed to obtain the value of n from the F(50,50) curves was, first, to read in the curves the values of E for different transmitting antenna heights and distances, then using (6), obtain P iso and, finally, obtain n using (5). From (5) it seems apparent that n depends on distance; however, the relation is not straightforward since P iso also varies with distance. If the dimensions of the coverage area are small, such as those found in indoor or cellular environments of small radius, the mean value of n is fairly constant [2] and no clear dependence of n with d appears. To establish such dependence, either measurements in a considerable range of distances are required, or alternatively, as done in this work, use already existing data such as provided by the F(50,50) curves. As can be appreciated from figures 1 and 2, the exponent of distance is a function of two variables, distance and transmitting antenna height. It is also a function of the receiving antenna height, however, for our purposes here, a constant receiving antenna height of 30 ft (9 m) was assumed, since the F(50,50) curves are based on this assumption. Actually, three models can be derived from figures 1 and 2. One in which n varies with distance and the transmitting antenna height is kept constant, which is the usual practical case. Other in which n varies with transmitting antenna height keeping the distance constant. The previous two models are unidimensional, and a more complete bidimensional model where n = n (h, d) can also be developed and is the one we present in this paper. III. THE MODEL Following the procedure described in the previous section, a set of values of n for different distances and transmitting antenna heights was obtained, and several attempts to fit the values of n to
4 mathematical functions were made, namely: logarithmic, exponential and polynomial. The best fit was obtained with a polynomial model of fourth degree with the form 4 4 i j n = a h d (7) i = 0 j = 0 Fitting was performed with Stanford Graphics software to obtain the coefficients a ij when h is in feet and d in miles. Such coefficients are given in Table 1, and in the Appendix, the corresponding coefficients for SI units, i.e. for h in meters and d in km. Table 1. Coefficients of the model ij a 00 a 01 a 02 a 03 a 04 2.6191 0.0318005-9.50112 10-4 1.46844 10-5 -8.30291 10-8 a 10 a 11 a 12 a 13 a 14-3.63991 10-3 2.4824 10-4 -1.15328 10-5 1.98061 10-7 -1.06459 10-9 a 20 a 21 a 22 a 23 a 24 7.20911 10-6 -6.73582 10-7 3.19009 10-8 -5.48948 10-10 2.96093 10-12 a 30 a 31 a 32 a 33 a 34-5.75331 10-9 5.86862 10-10 -2.80726 10-11 4.83956 10-13 -2.61299 10-15 a 40 a 41 a 42 a 43 a 44 1.48675 10-12 -1.5699 10-13 7.53695 10-15 -1.29985 10-16 7.01903 10-19 It must be noted that even when some of the coefficients appear to be very small and, apparently, they could be neglected, for example, if a 44 is made zero, the results deviate considerably from the correct ones as h or d increase. Therefore, all coefficients with the decimal places shown, must be used in implementing the model. Simpler, unidimensional models with less coefficients can also be derived for n as a function of distance or transmitting antenna height alone. The model presented here is the more general bidimensional model, applicable to both cases. IV. PROCEDURE TO OBTAIN THE RECEIVED POWER AND THE ELECTRIC FIELD STRENGTH FROM THE MODEL Model (7) provides the value of the exponent of distance in terms of transmitting antenna height and distance between transmitter and receiver. In order to obtain the received power, or the electric field strength, several calculations are required, as indicated below. a) The value of n being known, path loss in db is calculated using (2) or (3).. b) Isotropic receiver power is now calculated in logarithmic units as: P iso ( dbw) = P L (8) where P rad is the equivalent isotropic radiated power (EIRP) in the direction of the receiver, and L is the path loss in db. c) Finally, the field strength can be calculated from (6) as: rad
5 π E = 480 P iso V/m (9) λ V. RESULTS The model fits very well with the values read from the F(50,50) curves as can be appreciated from figures 1 to 4. In figures 1 and 2, the variation of the exponent of distance is plotted with respect to antenna height and distance, respectively. Figures 3 and 4 show the field strength for both cases. In the figures, broken lines with circles indicate values from FCC curves and continuous lines, the values calculated with the model The maximum deviation between the values of n obtained from the curves, and those calculated with the model is of 1.23%. The maximum deviation between the field strengths obtained from the curves and with the model is of 0.48 db, with maximum errors of about 0.65 db. Fig. 1. Exponent of distance vs antenna height for constant distance
6. Fig. 2. Exponent of distance vs distance for constant antenna height. Fig. 3. Field strength vs antenna height for constant distance.
7 Fig. 4. Field strength vs distance for constant antenna height. VI. CONCLUSIONS A simple computational path loss model has been derived from the FCC F(50,50) curves. Such curves were developed using measured values, taken in different geographical areas over different periods of time, and provide the median values of field strength for service at 50% of locations during 50% of the time. Therefore, such curves reflect reliable experimental data not derived from theoretical models. In the model, the exponent of distance is characterized as a function of distance and transmitting antenna height. From this parameter, mean or median path loss, receiving power and field strength are easily obtained. Fitting of the model with F(50,50) curves is very good for distances between 0.5 mi and 40 mi and antenna heights up to 2000 ft. Values are based on a receiving antenna height of 30 ft, therefore corrections are necessary for other heights. The model does not require particular computational effort and is of simple application for practicing engineers who do not require a deeper knowledge of channel dynamics. The model is also independent of frequency and may be used for broadcasting applications as well as for outdoor communication systems.
8 APPENDIX. MODEL COEFFICIENTS FOR SI UNITS When transmitting antenna height is given in meters, and distance in kilometers, the following coefficients must be used in model (7). a 00 a 01 a 02 a 03 a 04 2.70414 0.00691419 1.64202 10-4 -4.30076 10-6 2.38233 10-8 a 10 a 11 a 12 a 13 a 14-0.0123957 5.24056 10-4 -1.75643 10-5 2.4282 10-7 -1.11177 10-9 a 20 a 21 a 22 a 23 a 24 7.60572 10-5 -3.91766 10-6 1.34 10-7 -1.85925 10-9 8.54657 10-12 a 30 a 31 a 32 a 33 a 34-2.20208 10-7 1.23702 10-8 -4.1595 10-10 5.67899 10-12 -2.58477 10-14 a 40 a 41 a 42 a 43 a 44 2.03856 10-10 -1.18905 10-11 3.9371 10-13 -5.31031 10-15 2.39849 10-17 REFERENCES [1] Code of Federal Regulations, Title 47. Chapter 1. Federal Communications Commission. Part 73. Radio Broadcast Services. Secs. 73.683, 73.684 and 73.699. [2] Perez-Vega, C. and Garcia, J. L. A simple approach to a statistical path-loss model for indoor communications. 27 th European Microwave Conf. Proc. Jerusalem, 1997. [3] Cox, D. C. et al. 800-MHz attenuation measured in and around suburban houses. BTSJ, Vol. 63, Nº 6, pp. 921-955. Aug. 1984. [4] Bach Andersen, J. Rappaport, T. and Yoshida, S. Propagation measurements and models for wireless communication channels. IEEE Comm. Mag. Jan. 1995, pp. 42-49. [5] Perez-Vega, C. and Zamanillo, J. M. Indoor propagation at 2.45 GHz for TV applications. Proc. Microwave Symposium 2000 (MS 2000). Tetuan, Morocco, May, 2000. [6] Perez-Vega C. and Garcia, J. L. Frequency behavior of a power-law path loss model. Proc. 10 th Microcoll. Budapest, March 1999. [7] Barringer, M. H. and Springer, K. D. Radio Wave Propagation. Chap. 2.8 of NAB Engineering Handbook, 8 th Ed. NAB, Washington, DC, 1992. [8] Stielper, J. W. The measurement of FM and TV field strengths (54 MHz 806 MHz). Chap. 2.11 of NAB Engineering Handbook, 8 th Ed. NAB, Washington, DC, 1992.