Effect of skywave interference on coverage of radiobeacon DGPS stations

ADVANCES IN NAVIGATION

Effect of skywave interference on coverage of radiobeacon DGPS stations

J.D. Last, CEng, FlEE D.C. Poppe, PhD

Inde,xing terms: Radiobracon stations, Skywave interjerence, Groundwaves, Global Positioning System

Abstract: Coastal radiobeacon stations, which have long served marine direction-finding receivers, are increasingly being employed to transmit differential satellite navigation system (DSNS) correction messages to marine users. The range of these 300kHz data transmissions extends beyond the nominal ranges of the radiobeacons into regions where skywave effects become significant, especially at night. The paper demonstrates by statistical analysis and experimental measurements that skywave- propagated components interact with the wanted groundwaves, causing deep signal fading that can lead to loss of data. Also, in Europe, skywave signals from other radiobeacons on the same, or adjacent, channels cause severe interference. A coverage and performance prediction model that takes these effects into account, together with atmospheric noise and attenuation due to ground losses, is used to illustrate the reduction of usable range caused by skywave propagation. The paper argues that it is essential to allow for skywave factors when predicting the operating areas of beacons and when allocating frequencies within the radiobeacon DSNS service.

1 Introduction

Differential global satellite navigation systems (DSNS) are based on the principle that the chief sources of position errors are correlated over substantial geographical areas. Differential GPS (DGPS) and differential GLONASS systems employ reference receivers at fixed, known locations. These receivers measure pseudo-range errors and broadcast corrections that allow mobile receivers in the region to adjust their position measurements. This substantially improves accuracy. In addition, the reference receivers detect and signal satellite malfunctions. DSNS systems for

0 IEE, 1997 IEE Pvoceedings online no. 1997 1 177 Paper first received 25th April 1996 and in revised form 12th February 1997 J.D. Last is with the School of Electronic Engineering & Computer Sys- tems, University of Wales, Bangor, Gwynedd LL57 IUT, UK D. Poppe is at 10199 East Celtic Drive, Scottsdale, A 2 85260, USA

maritime use commonly employ existing coastal radiobeacons operating in the 283.5-325 kHz frequency band [ 11. The differential corrections are modulated onto the main, or a subsidiary, carrier using low baud- rate minimum-shift keying (MSK). Radiobeacon DGPS systems are being deployed throughout coastal regions of North America and Europe and are planned for many other areas of the world.

Coastal radiobeacons have nominal ranges of up to about 400km. The coverage of their DSNS transmissions is limited by attenuation of the groundwave signals and by atmospheric noise and interference [2]. When stations are designed and frequencies are allocated, the radiated power, ground conductivity, atmospheric noise and potential interfering transmissions must be considered. All these parameters have been built into a computer coverage- prediction model intended to automate and speed up the design process [3-51.

A factor that has not so far been fully considered, however, is the propagation of transmissions via ionospheric paths. In radio systems operating at low and medium frequencies (LF and MF), such ‘skywave’ components can interfere with the groundwaves causing fading, especially at night. Close to the station, of course, the groundwave is dominant. With increasing range, the groundwave becomes weaker, the skywave becomes stronger, and the usable signal is then a com- bination of the two. Still further away, the groundwave may be attenuated to a negligible level, leaving only the skywave. Where the strengths of the groundwave and skywave are comparable, ‘own-skywave’ fading occurs because of cancellations due to the phase differences between them. The received signal may then fall below the usable minimum, causing data to be lost. This paper presents analyses based on experimental data that demonstrate that such own-skywave fading can significantly reduce DSNS radiobeacon coverage.

In addition, skywave-propagated signals from other beacons on, and adjacent to, the channel can cause unacceptable levels of interference to the DSNS transmission. Both this skywave-propagated interference and own-skywave fading have been built into the coverage prediction model that will be used to demonstrate their effects on DSNS radiobeacon coverage at night.

163 IEE Proc.-Radar, Sonar Navig., Vol. 144, No. 3, June 1997

2

2.1 Strength of the groundwave signal The strength of the groundwave signal at a given site is determined by the effective power radiated along the earth's surface in the direction of the site and the attenuation of the path to it from the transmitter. It can be considered constant in time. Experimentally-derived graphs of attenuation with distance, as a function of ground conductivity, are published in CCIR Report 717-2 [6]. The groundwave-only curves in Fig. 1 show the attenuation with distance of DSNS signals at 300kHz, a typical radiobeacon frequency. Curve a is for a sea-water path (conductivity 5000mS/m), and curve h is for a land path of low Conductivity (0.3mSi in). Where signals follow paths that cross terrain of more than one conductivity, CCIR [7] recommends the use of Millington's method, a semi-empirical technique, for calculating the total attenuation.

Levels of groundwave and skywave signals

100 IO' 102 i o 3 104 distance, km

Fig. 1 Gi-oundwave-only propagation values shown for sea-water and poor land. Skywave-only curves are median (50%) and 95-percentile Gaussian values. Median skywavc can equal groundwave at ranges from 45 to 550kiii U Groundwavc only, 5OOOmSim h Groundwave only. O.?mS/m c Skywave only. 50% d Skywdve only, 95%

Field strength curva f i r I ItW, 300 IzHz, trirnsniitter

Fig.2 Contours of groundwuve field ,men th (in dB with respect to lpv/mJ of U radiobeacon at Porkkalu. Finlanf Nominal range: lhOltm Note the short ranges over laud paths and much greater ranges over sea-water.

Our computer model for predicting DSNS radiobeacon coverage [3, 41 stores extensive maps of ground conductivity and implements Millington's method [8]. It determines the attenuation from a

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transmitter specified by the user to each point in a computation array spaced at 0.1" of latitude by 0.1" of longitude (approximately 11 x 7km at UK latitudes). Fig. 2 shows contours of the groundwave field strength of the Finnish DGPS radiobeacon at Porkkala calculated by the model; the high rate of signal attenuation due to the low-conductivity land north of the beacon contrasts strongly with the low rate of attenuation over the sea to the east and west.

2.2 Strength of skywave signal The field strength of the skywave component received depends on the effective power radiated in the horizontal direction and the vertical angle of the skywave path. The proportion of the signal refracted back to earth by the ionosphere is a function of the time of day, the sea- son of the year and the point in the 11-year sunspot cycle. A skywave signal is normally the sum of components received via multiple paths, and its intensity can vary substantially over just a few seconds. It is also delayed with respect to the groundwave, having travelled a greater distance.

CCIR reports 575-4 [9] and 435-6 [lo] present methods for calculating skywave intensities at LF and MF. We chose to use the report 435-6 method as it is valid over the widest range of distances and in all regions of the world. This method predicts a median night skywave field strength SkyciR at range d km from a 1 kW transmitter of

S l C y d ~ = -4 - 2010gp - 10-3kp + G, + G, + Ap (1) where A = 106.6-2sinQ. Here @, the 'geomagnetic latitude', is the latitude with respect to the geomagnetic poles

4 = arcsin(sinasin78.5" + cosacos(-69" + p ) ) where a is geographical latitude, and /3 is longitude. The slant-propagation distance y km in eqn. 1 is the path length via the ionosphere, e.g.

p = dd2 + 2002 for a typical ionospheric height of 100km.

absorption is The 'basic loss factor' k due to ionospheric

k = 3.2 + O . l 9 [ f k ~ , ] O - ~ tan2[$ + 31 wherefk, is the frequency in kHz. Factor Ap accounts for transmitter power (OdB at 1 kW). G, represents the gains of the radiobeacon and receiver antennas. These are normally assumed to be vertical monopoles, short compared with the wavelength of 1000m, and omni- directional in the horizontal plane. In report 435-6, the variation of G, with distance is computed: signals transmitted and received vertically have infinite attenuation; low-angle, long-distance skywave signals are unattenuated. Finally, G,y, a sea gain factor or maximum value 1.5dB, is applied when either end of the path lies within 5km or the sea.

All these factors have been built into the coverage prediction model.

2.3 The skywave field strength given by eqn. 1 is the night median, or 50th percentile, value. Report 435-6 states that the level not exceeded 90% of the time is 6.5dB greater, but no statistical distribution is specified. Other authors, working at around 100kHz, have assumed a Rayleigh distribution for skywave [9, 131. The Rayleigh

Variation of skywave strength with time

IEE Proc.-Rndur, Sonar .Vuri,q.. Vol. 144, No, 3, June 1997

distribution, however, does not provide a good fit to the known points in this case, deviating from the CCIR 90% value by 1.3dB. A Gaussian distribution provides a much better fit and so has been adopted. At other probability levels, the discrepancy in skywave intensity between a Rayleigh and a Gaussian assumption can be significant, reaching 3.7dB at the 99th percentile. The Gaussian distribution is a reasonable choice of distribution in situations such as this, where few percentile values are known, and the physics suggests that many independent components sum to form the measured skywave [ 1 1, 121.

The skywave-only curves of Fig. I show the variation of the CCIR median (50'Yo) and Gaussian 95th percentile intensities with range. Comparing these with the groundwave signals, we see that close to the beacon the groundwave is dominant. Beyond IOOOkm the skywave is dominant. In between lies the region where groundwave and skywave are comparable in strength: the fading zone.

3 Fading zone

3.7 Delay of skywave signal Fig. 1 shows that the median skywave signal can equal the groundwave at ranges from about S50km over sea- water down to 65km over low-conductivity ground; this range can be as short as 45 km for the very lowest conductivity recognised by CCIR, which is not included in the Figure. The corresponding delay of the skywave component with respect to the groundwave (due to its having travelled via the ionosphere) lies in the range 30 -240~ . However, even the maximum delay is short compared with the bit time (Sms) of the data signal at the highest data rate employed in the radiobeacon service, 200 bits per second. We conclude that skywave delay alone cannot lead to significant inter- symbol interference.

There will, however, be signal fading due to cancella- tion between the groundwave and skywave components of the carrier. The skywave delay corresponds to many cycles at 300kHz. Thus the total signal received is the vector sum of a constant groundwave component and a skywave component, whose amplitude is log-normally distributed (Gaussian in dB), and whose phase can be considered random with respect to that of the groundwave. To estimate the performance of the radiobeacon, we need to calculate the probability that its signal strength at any point exceeds the minimum required to support DSNS operation.

3.2 Previous analyses of own-skywave fading CCIR report 266-5 [14] states that the sum of the groundwave and skywave signal components has a Nakagami-Rice amplitude distribution. At ranges where the skywave and groundwave components are equal, the median total signal is 3dB stronger than either component alone, as would be expected. Of more significance, however, is the fading: the signal level that can be guaranteed 90% of the time is SdB weaker than the groundwave. In fact, 1% of the time, the signal will fade by at least 1SdB. These are very significant reduc- tions in field strength.

Where the skywave greatly exceeds the groundwave, the CCIR analysis unfortunately fails: the total signal it predicts does not asymptote to the skywave-only values. The reason is that the Nakagami-Rice analysis

IEE Proc-Radar, Sonav Navig.. Vol. 144, hio. 3, June 1997

incorrectly assumes the amplitude distribution of the skywave signal to be Rayleigh, not Gaussian.

Enge and Olsen [15] have correctly assumed a Gaussian distribution of skywave amplitude. However, their analysis is restricted to situations in which the groundwave is the dominant component. Extrapolating their approach to the skywave-dominant region results in the improbable result that the deepest fading occurs where the skywave exceeds the groundwave, not where they are equal. Thus the method cannot be used throughout the entire fading zone or out to the edge of radiobeacon coverage. A new analysis is required that accurately describes the low-probability, deep fades most likely to cause loss of DSNS data.

3.3 New analysis We have analysed the sum of a constant groundwave and a skywave with Gaussian amplitude and uniform phase distribution, placing no restriction on the relative amplitudes of the two components. Normalising all signal amplitudes to that of the groundwave, the skywave is represented by a vector of amplitude sgv and phase 8. The amplitude of the sum is then F. where

F = 1 + sgr ~3~ (2) The probability of the amplitude of the received signal F being less than or equal to A the depth of fading with respect to the groundwave, is given by

P F ( f ) = I/ 7i PS,,(Q)cJQ (3) U 0

The derivation of this equation is presented in the Appendix, which also discusses the range of integration Dt?

25 m 1

Fig. 3 FI a.\ U ,fuizction of sky~~ave-to-groundi,l'uvc. ratio iSGR) (.a/(.i4-

Maximum fading occurs with SGR = OdB, when field strength that can be guaranteed 90% of the time is 5dB wcakcr than groundwave, that guaranteed 99'% of time is 15dB weaker a 5O%, skywavc only h 9VX, skywavc only c' 99%, skywave only

1ati.d by ' autI7ol.s

Fig. 3 shows our calculated 50th, 90th and 99th percentile values of IF1 as functions of the skywave-to- groundwave ratio (SGN) in dB. Where the groundwave is dominant (SGR large and negative), total signal and groundwave are equal, and IF1 = OdB; there is no fdd- ing. Where the median skywave equals the groundwave (SGR = OdB), the median (50%) total signal exceeds the groundwave by 3dB. The signal exceeded 90% of the time is 5dB below the groundwave; 1% of the time, the signal fades by at least 15dB. These results agree with those of the CClR Nakagami-Rice analysis. In contrast to CCIR's. however, our total signal values in the skywave-dominant region (where SGR is large and

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positive) asymptote to those of the skywave-only component. All these results are physically reasonable.

100 IO’ 102 103 104 distance, km

Fig. 4 Groundwave dominates close to station. In the fading zone the field strength Falls below gi-oundwave or skywave values. Beyond this region skywave is dominant. a Total, 5000mSim b Total, 0.3mSiin c Skywave only, 50% d Skywave only, 95% e Groundwave only, 5000mSim f Groundwave only, 0.3mSim

Total field strength available 95% ofthe time (1 k W, 300kHz)

Now that we know from the above analysis the values of skywave and groundwave strength at points around a beacon, the model can compute the expected level of fading. Fig. 4 shows the total field strength that can be guaranteed 95% of the time, given a 1kW transmitter. Over sea-water (curve a), fading with respect to the groundwave only begins at about 100km. From there to 1000km, the total signal is weaker than either the groundwave or the 95th percentile skywave alone. Beyond 1000km, the groundwave becomes very weak, and the total signal asymptotes to the skywave- only curve. Over low-conductivity land (curve b), fading starts at only 40km and is greatest at 75km, beyond which skywave propagation again takes over. Interestingly, between 600 and 1300km, the 95%0- availability total signal over poor land is actually stronger than over sea: the reason is that the groundwave signal arriving over land is too weak to cause fading of the received skywave, although a stronger groundwave arriving via a sea path can cause fading.

ZL 2 5 - m - 2 0 - 5

1 5 -

- 10-

U

+4

VI

.& 5 - ul

, , I night I 1200 2100 1200 2LOO 1200

time of day ( l o c a l 1 Fig.5 Field strength of signal received neur Bangor, Wule.s, )om beacon at Flumborough Head, Yorkshire By day only groundwave signal is significant. At night maximum signal exceeds groundwave by 9dB and deepest fade is 20dB.

3.4 Experimental results Fig. 5 shows the variation of field strength received at a site near Bangor, Wales, over the 280km path from a DGPS radiobeacon at Flamborough Head, Yorkshire.

166

The Flamborough beacon is exceptional in not sharing its frequency (302.5 kHz) with other stations, so allow- ing the depths of its fades to be measured reliably. The coverage model predicts a groundwave field strength of 22dBpVlm, and a median skywave 4.5dB below this. The measured daytime, groundwave-dominated field strength is seen to be 20dBpV/m, slightly below that predicted. As expected, it is steady, with maximum excursions of only +ldB. In marked contrast, at night, the field strengths of the signals reach a level 9dB greater than the groundwave, and the deepest fade is 20dB.

Fig. 6 compares the distribution of the field strength values measured at night with the predicted values of the analysis above. The agreement between theory and observation is very satisfactory. Neither the predicted groundwave strength nor the statistics of the predicted skywave were varied to fit the experimental data. Not adjusting the groundwave results in the approximately 1dB horizontal shift between the observed and predicted values. In particular, the shape of the curve, which is controlled by the assumed skywave statistics, shows very good agreement with the observed values.

- a J 0 n 0 = 20 B a, LO E

60

6 80

100

a +

m ad

v

-5 0 5 10 15 20 25 30 35 calibrated signal strength, dBp

Fig. 6 Distribution oj jield strengths of signals received from Flambor- ough Head beacon ut night -~ predicted vdlucs

obscrved

3.5 Effect of fading on coverage In Fig. 7 , a is the field strength contour at 20dBbV/m, the minimum level specified for DSNS radiobeacon reception [16]. At night, this level can be guaranteed 95% of the time only within the reduced contour marked b. It is clear that own-skywave fading has a marked effect on the area of reliable reception.

5

20 25 30 15

Fig. 7 At night, the daytime level, marked ‘day’ can be guaranteed 95% of the time only within the reduced contour marked ‘night’

20 dBuV/m minimum field strength contours

IEE Proc.-Radau, Sonar Navig , Vol. 144, No. 3, June 1997

4 Skywave propagation of interfering signals

In the European Maritime Area of ITU Region 1 , some 400 marine, aeronautical and DSNS radiobeacons occupy 64 channels, spaced 500 Hz apart, between 283.5 and 315.0kHz. The nominal ranges of these beacons vary from 9 to 370km [17]. The minimum performance standards for DSNS radiobeacon receivers specify protection ratios that range from 15dB for co-channel interference to -55dB at a2kHz from the channel [16]. Thus, typically, some 40 beacons need to be considered as potential interferers to any wanted beacon. In allocating frequencies, coiiventionally only groundwave propagation of interference has been considered; clearly, from the analysis above, skywave components must also be taken into account.

The algorithms for calculating groundwavc and median skywave iiitensities in the coverage prediction model allow us to take potential interfering beacons one at a time and compute the strengths of their groundwave and skywave components throughout the computation array. The strongest component received at any location can then be compared with the wanted signal to determine whether the specified protection ratio is met; if so, the location is deemed free of interference. By day, only groundwave components of the wanted and interfering signals need be considered. At night, however, not only does skywave-propagated interference appear, but the wanted signal is reduced by its own-skywave fading.

r , , ;

25 30 20 15

Fig. 8 Erecl of interfirenee on coveruge of Porlzlcala beacon Curve ‘20’ is the 20dBvV/m minimum groundwave field strength contour. Groundwave interference by day reduces coverage in south-wesl to the ‘D’ boundary, At night, skywave interference and fading reduce it much further, to the ‘N’ boundary

In Fig. 8, the outer line is the groundwave 20dBpV/m minimum field strength contour. By day, groundwave interference from a co-channel aeronautical beacon at Barth, Germany, can reduce coverage in the south-west to the area within boundary a. At night, additional skywave interference from both [Note 11 Barth and a beacon at Ordzonikidzegr, Russia, together with own- skywave fading of the Porkkala signal, cuts coverage to boundary b. We conclude that skywave propagation can dramatically reduce the operational areas of

Note 1: ITU has allocated the rreyuency to these beacons, but they do not appear to be operational currently

beacons that suffer from interference. Indeed, powerful co-channel interfering stations located as far as 2000km from the service area of a DSNS radiobeacon need to be considered.

5 Design of radiobeacon DSNS systems

Traditionally, when European radiobeacon systems for direction-finding (DF) have been designed, certain simplifying assumptions have been applied: a minimum field strength of 34dBkV/m (37.5dBpV/m below latitude 45 N), a minimum signal-to-noise ratio (SNR) of 15dB, all propagation paths lying over sea water and interference received via groundwave propagation alone [17]. DSNS receivers, however, operate down to much lower field strengths (20dBpV/m) and lower SNR values (7dB). Thus DSNS ranges are much greater than DF ranges. In addition, many administrations are now installing new beacons intended for DSNS use alone, rather than modifying existing DF beacons. They are employing small numbers of exceptionally long-range stations to minimise costs, especially if the beacons are in inaccessible terrain [IS]. Such DSNS systems are very different from traditional DF configurations that consist of large numbers of short- range beacons designed to provide the intersecting bearings needed for DF position measurements. It is clear that the traditional design assumptions are invalid. Specifically, DSNS propagation paths often lie over land, the lower minimum required field strength compensating for the additional attenuation; thus, at night, own-skywave fading and skywave-propagated interference can be very significant.

It is essential to take these new factors into account when planning systems. The use of a full coverage- prediction program must replace the practice of representing service areas by circles centred on the beacons. As this paper has demonstrated, system designers must consider, not only land-path attenuation of groundwave signals, but also skywave effects. In addition, as the intensities of atmospheric noise and skywave signals vary statistically, published specifications of minimum field strength, SNR and system availability for DSNS systems should now state the percentages of time for which they apply.

6 Conclusions

The paper has demonstrated that skywave-propagated signals play an important role in determining the performance and coverage of radiobeacon DSNS services. On the basis of CCIR data, own-skywave fading has been shown to be a significant factor that can reduce the field strength of the wanted signal at night at only 4Skm from the station. Where the strength of the median skywave equals that of the groundwave, there is a 10% chance of fades exceeding 5dB and a 1% chance of them exceeding 15dB. These analytical results are supported by experimental evidence. In addition, skywave propagation of the signals from other beacons can cause severe interference. The paper demonstrates that these two factors can severely reduce radiobeacon coverage at night. It argues that, as radiobeacons are expected to provide much greater ranges in DSNS than in DF service, a new approach to system design i? required.

I67 IEE Proc.-Radar, Sonar Navig.. Vol. 144, No. 3, June 1997

7 Acknowledgments

The authors gratefully acknowledge financial support for this research from the Corporation of Trinity House and Scorpio Marine Electronics Ltd., and the valuable advice of colleagues within those organisa- tions.

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References

ENGE, P.K., and DUANE, M.F.: ‘Marine radiobeacons for the broadcast of differential GPS data’. Record of the 1986 IEEE Position, location and navigation symposium, Las Vegas, NV, November 1986 ENGE, P.K., LEVIN, P., HANSEN, A., and KALAFUS, R.: ‘Coverage of DGPS radiobeacons’, Navigation (USA) , 1992, 39, (4), pp. 363-381 POPPE, D.C., LAST, J.D.. and SEARLE, M.D.: ‘Coverage prediction for DGPS radiobeacons’. Proceedings of the Institute of Navigation 1994 technical meeting, San Diego, CA, USA, Janu- ary 1994, pp. 1375-1382 POPPE, D.C., and LAST, J.D.: ‘DGPS radio-beacon coverage prediction i n the European environment’. Proceedings of Royal Institute or Navigation DSNS94 conference, London, 18-22 April 1994, paper 29, pp. 18-22 The Analytic Sciences Corporation: ‘COAST user’s manual’. Technical Report for US Department or Transport, US Coast- guard, 1993 CCIR: ‘Report 71 7-2-World atlas of ground conductivities’, International Telecommunication Union, Geneva, 1988 MILLINGTON, G.: ‘Ground-wave propagation ovcr an inhomo- geneous smooth earth’, Proc. IEE, 1949, 96, Part 111, pp. 53-64 LAST, J.D., SEARLE, M.D., and FARNWORTH, R.G.: ‘The Northwest European Loran-C system: Coverage and performance predictions’, Navigation (LISA), 1993, 40, (3), pp. 209-226 CCIR: ‘Report 575-4-Methods for predicting sky-wave field strengths at frequencies between 150 and 1705 kHz’. International Telecommunication Union. Geneva. 1990

10 CCIR: ‘Recommendation’ 435-6-Prediction of skywave field strength at frequencies between 150 and 1600 kHz’. International Telecommunication Union, Geneva, 1990

1 1 PAPOULIS, A.: ‘Probability, random variables and stochastic processes’ (McGraw-Hill, 1984)

12 HAYKIN, S.: ‘An introduction to analog and digital conimunica- tions’ (John Wilcy & Sons, 1989)

13 FARNWORTH, R.G.: ‘Loran-C coverage prediction in Western Europe’, PhD thesis, University of Wales, Bangor, 1992

14 CCIR: ‘Report 266-5-Ionospheric propagation characteristics per- tinent to terrestrial radio-cominunicatioiis systems design (fading)’. International Telecommunication Union, Geneva, 1982

15 ENGE, P.K., and OLSEN, K.E.: ‘Medium frequency broadcast of differential GPS data’, IEEE Trans. Aerosp. Electvon. Syst., 1990, 26, (4), pp. 607-617

16 International Telecommunication Union: ‘Technical characteristics of differential transmissions for global navigation satellite systems (GNSS) from maritime radiobeacons in the frequency band 285-325 kHz’. Technical Report: Revision of Recommendation M.823, 1995

17 International Telecommunication Union: ‘Final acts of the regional administrative conference for the planning of the maritime radionavigation service (radiobeacons) in the European Mar- itime Area’, Geneva, 1985

18 BAXTER, M., and LAST, J.D.: ‘Designing a marine differential GPS service for the Arabian Gulf‘. Proceedings of Arab lnstitute of Navigation Satellite positioning systems conference, Alexan- dria, Egypt, December 1995

9 Appendix

We wish to find the region of integration of Do. DF is the region where IF1 5 f ( F being defined as in eqn. 2). Thus,

f 2 sgr2 + 2sgrcosO + 1 Solving this quadratic equation bounds sgr as a function of 8 andf

- cos 0 - d m 5 sgr 5 - cos 19 t zlcos”e-l+f2 Phase 8 is uniformly distributed between 0 and x. Amplitude sgr is a real number. Thus eqn. 4 is real for all 8, and so, for f > 1, Do ranges between 0 and x. However, i f f < 1, sgv is only real when sin-” 5 8. Thus the integration region D , is where

T - sinp’(f) 5 o 5 n, Vf > I Thus, for f 5 1,

(4)

Vf 5 1 o 5 O 5 T ,

-P~~~(2010gi , ( -cos19- Jcos20- 1+f2))de and, forJ’> 1,

F 1 dB ( f 7r

0 In these expressions for PF, we assume a Gaussian-distributed skywave amplitude, PSGR

Here 0 is the standard deviation of the Gaussian distribution (90th percentile divided by 1.29). Thus Q is given by

5

from which eqn. 3 follows.

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Effect of skywave interference on coverage of radiobeacon DGPS stations