Điện thoại di động vô tuyến điện - Tuyên truyền Channel P5 pps

Chapter 5 Characterisation of Multipath Phenomena 5.1 INTRODUCTION In Chapter 3 we described some methods for predicting path losses, concentrating on those applicable to mobile communication systems. The discussion centred around techniques that deal principally with radio propagation over irregular terrain; methods of predicting signal strength in urban areas or in other environments, e.g. inside buildings, were deliberately left until Chapter 4. These propagation models are extremely important since the vast majority of mobile communication systems operate in and around centres of population. Having introduced them, we can now go into more detail about the propagation mechanism in built-up areas, not only qualitatively but also in terms of a mathematical model. In that way we can understand the full signi®cance of the prediction techniques and indicate the ways forward towards a global model that includes the eects of topographic and environmental factors. The major problems in built-up areas occur because the mobile antenna is well below the surrounding buildings, so there is no line-of-sight path to the transmitter. Propagation is therefore mainly by scattering from the surfaces of the buildings and by diraction over and/or around them. Figure 5.1 illustrates some possible mechanisms by which energy can arrive at a vehicle-borne antenna. In practice energy arrives via several paths simultaneously and a multipath situation is said to exist in which the various incoming radio waves arrive from dierent directions with dierent time delays. They combine vectorially at the receiver antenna to give a resultant signal which can be large or small depending on the distribution of phases among the component waves. Moving the receiver by a short distance can change the signal strength by several tens of decibels because the small movement changes the phase relationship between the incoming component waves. Substantial variations therefore occur in the signal amplitude. The signal ¯uctuations are known as fading and the short-term ¯uctuation caused by the local multipath is known as fast fading to distinguish it from the much longer-term variation in mean signal level, known as slow fading. The Mobile Radio Propagation Channel. Second Edition. J. D. Parsons Copyright & 2000 John Wiley & Sons Ltd Print ISBN 0-471-98857-X Online ISBN 0-470-84152-4 Slow fading was mentioned in Chapter 3 and is caused by movement over distances large enough to produce gross variations in the overall path between the transmitter and receiver. Because the variations are caused by the mobile moving into the shadow of hills or buildings, slow fading is often called shadowing. Unfortunately there is no complete physical model for the slow fading, but measurements indicate that the mean path loss closely ®ts a lognormal distribution with a standard deviation that depends on the frequency and the environment (Chapter 3). For this reason the term lognormal fading is also used. The terms `fast' and `slow' are often used rather loosely. The fading is basically a spatial phenomenon, but spatial variations are experienced as temporal variations by a receiver moving through the multipath ®eld. The typical experimental record of received signal envelope as a function of distance shown in Figure 5.2 illustrates this point. The fast fading is observed over distances of about half a wavelength. Fades with a depth less than 20 dB are frequent, with deeper fades in excess of 30 dB being less frequent but not uncommon. The slow variation in mean signal level, indicated in Figure 5.2 by the dotted line, occurs over much larger distances. A receiver moving at 50 kph can pass through several fades in a second, or more seriously perhaps, it is possible for a mobile to stop with the antenna in a fade. Theoretically, communication then becomes very dicult but, in practice, secondary eects often disturb the ®eld pattern, easing the problem signi®cantly. Whenever relative motion exists between the transmitter and receiver, there is an apparent shift in the frequency of the received signal due to the Doppler eect. We will return to this later; for now it is sucient to point out that Doppler eects are a manifestation in the frequency domain of the envelope fading in the time domain. Although physical reasoning suggests the existence of two dierent fading mechanisms, in practice there is no clear-cut division. Nevertheless, Figure 5.2 Characterisation of Multipath Phenomena 115 Figure 5.1 Radio propagation in urban areas. LOS path shows how to draw a distinction between the short-term multipath eects and the longer-term variations of the local mean. Indeed, it is convenient to go further and suggest that in built-up areas the mobile radio signal consists of a local mean value, which is sensibly constant over a small area but varies slowly as the receiver moves; superimposed on this is the short-term rapid fading. In this chapter we concentrate principally on the short-term eects for narrowband channels; in other words, we consider the signal statistics within one of the small shaded areas in Figure 5.3, assuming the mean value to be constant. In this context, `narrowband' should be taken to mean that the spectrum of the transmitted signal is narrow enough to ensure that all frequency components are aected in a similar way. The fading is said to be ¯at, implying no frequency-selective behaviour. 5.2 THE NATURE OF MULTIPATH PROPAGATION A multipath propagation medium contains several dierent paths by which energy travels from the transmitter to the receiver. If we begin with the case of a stationary receiver then we can imagine a static multipath situation in which a narrowband signal, e.g. an unmodulated carrier, is transmitted and several versions arrive sequentially at the receiver. The eect of the dierential time delays will be to introduce relative phase shifts between the component waves, and superposition of the dierent components then leads to either constructive or destructive addition (at 116 The Mobile Radio Propagation Channel Figure 5.2 Experimental record of received signal envelope in an urban area. Figure 5.3 Model of mobile radio propagation showing small areas where the mean signal is constant within a larger area over which the mean value varies slowly as the receiver moves. any given location) depending upon the relative phases. Figure 5.4 illustrates the two extreme possibilities. The resultant signal arising from propagation via paths A and B will be large because of constructive addition, whereas the resultant signal from paths A and C will be very small. If we now turn to the case when either the transmitter or the receiver is in motion, we have a dynamic multipath situation in which there is a continuous change in the electrical length of every propagation path and thus the relative phase shifts between them change as a function of spatial location. Figure 5.5 shows how the received amplitude (envelope) of the signal varies in the simple case when there are two incoming paths with a relative phase that varies with location. At some positions Characterisation of Multipath Phenomena 117 Figure 5.4 Constructive and destructive addition of two transmission paths. Figure 5.5 How the envelope fades as two incoming signals combine with dierent phases. there is constructive addition, at others there is almost complete cancellation. In practice there are several dierent paths which combine in dierent ways depending on location, and this leads to the more complicated signal envelope function in Figure 5.2. The space-selective fading which exists as a result of multipath propagation is experienced as time-selective fading by a mobile receiver which travels through the ®eld. The time variations, or dynamic changes in the propagation path lengths, can be related directly to the motion of the receiver and indirectly to the Doppler eects that arise. The rate of change of phase, due to motion, is apparent as a Doppler frequency shift in each propagation path and to illustrate this we consider a mobile moving with velocity v along the path AA' in Figure 5.6 and receiving a wave from a scatterer S. The incremental distance d is given by d  v Dt and the geometry shows that the incremental change in the path length of the wave is Dl  d cos a, where a is the spatial angle in Figure 5.6. The phase change is therefore Df À 2p l Dl À 2pvDt l cos a and the apparent change in frequency (the Doppler shift) is f À 1 2p Df Dt  v l cos a 5:1 It is clear that in any particular case the change in path length will depend on the spatial angle between the wave and the direction of motion. Generally, waves arriving from ahead of the mobile have a positive Doppler shift, i.e. an increase in frequency, whereas the reverse is the case for waves arriving from behind the mobile. Waves arriving from directly ahead of, or directly behind the vehicle are subjected to the maximum rate of change of phase, giving f m Æv=l. In a practical case the various incoming paths will be such that their individual phases, as experienced by a moving receiver, will change continuously and randomly. The resultant signal envelope and RF phase will therefore be random variables and it remains to devise a mathematical model to describe the relevant statistics. Such a model must be mathematically tractable and lead to results which are in accordance 118 The Mobile Radio Propagation Channel Figure 5.6 Doppler shift. with the observed signal properties. For convenience we will only consider the case of a moving receiver. 5.3 SHORT-TERM FADING Several multipath models have been suggested to explain the observed statistical characteristics of the electromagnetic ®elds and the associated signal envelope and phase. The earliest of these was due to Ossanna [1], who attempted an explanation based on the interference of waves incident and re¯ected from the ¯at sides of randomly located buildings. Although Ossanna's model predicted power spectra that were in good agreement with measurements in suburban areas, it assumes the existence of a direct path between transmitter and receiver and is limited to a restricted range of re¯ection angles. It is therefore rather in¯exible and inappropriate for urban areas where the direct path is almost always blocked by buildings or other obstacles. A model based on scattering is more appropriate in general, one of the most widely quoted being that due to Clarke [2]. It was developed from a suggestion by Gilbert [3] and assumes that the ®eld incident on the mobile antenna is composed of a number of horizontally travelling plane waves of random phase; these plane waves are vertically polarised with spatial angles of arrival and phase angles which are random and statistically independent. Furthermore, the phase angles are assumed to have a uniform probability density function (PDF) in the interval (0, 2p). This is reasonable at VHF and above, where the wavelength is short enough to ensure that small changes in path length result in signi®cant changes in the RF phase. The PDF for the spatial arrival angle of the plane waves was speci®ed a priori by Clarke in terms of an omnidirectional scattering model in which all angles are equally likely, so that p a a1=2p. A model such as this, based on scattered waves, allows the establishment of several important relationships describing the received signal, e.g. the ®rst- and second-order statistics of the signal envelope and the nature of the frequency spectrum. Several approaches are possible, a particularly elegant one being due to Gans [4]. The principal constraint on the model treated by Clarke and Gans is its restriction to the case when the incoming waves are travelling horizontally, i.e. it is a two- dimensional model. In practice, diraction and scattering from oblique surfaces create waves that do not travel horizontally. It is clear, however, that those waves which make a major contribution to the received signal do indeed travel in an approximately horizontal direction, because the two-dimensional model successfully explains almost all the observed properties of the signal envelope and phase. Nevertheless, there are dierences between what is observed and what is predicted, in particular the observed envelope spectrum shows dierences at low frequencies and around 2 f m . An extended model due to Aulin [5] attempts to overcome this diculty by generalising Clarke's model so that the vertically polarised waves do not necessarily travel horizontally, i.e. it is three-dimensional. This is the generic model we will use in this chapter. It is necessarily more complicated than its predecessors and Characterisation of Multipath Phenomena 119 sometimes produces rather dierent results. The detailed mathematical analysis is available in the original references or in textbooks [6,7]. In this chapter we concentrate on indicating the methods of analysis, the physical interpretation of the results, and ways in which the information can be used by radio system designers. 5.3.1 The scattering model At every receiving point we assume the signal to be the resultant of N plane waves. A typical component wave is shown in Figure 5.7, which illustrates the frame of reference. The nth incoming wave has an amplitude C n , a phase f n with respect to an arbitrary reference, and spatial angles of arrival a n and b n . The parameters C n , f n , a n and b n are all random and statistically independent. The mean square value of the amplitude C is given by EfC 2 n g E 0 N 5:2 where E 0 is a positive constant. The generalisation in this approach occurs through the introduction of the angle b n , which in Clarke's model is always zero. The phase angles f n are assumed to be uniformly distributed in the range (0, 2p) but the probability density functions of the spatial angles a n and b n are not generally speci®ed. At any receiving point (x 0 , y 0 , z 0 ) the resulting ®eld can be expressed as Et X N n1 E n t5:3 where, if an unmodulated carrier is transmitted from the base station, 120 The Mobile Radio Propagation Channel Figure 5.7 Spatial frame of reference: a is in the horizontal plane (XY plane), b is in the vertical plane. E n tC n cos  o 0 t À 2p l x 0 cos a n cos b n  y 0 sin a n cos b n  z 0 sin b n f n  5:4 If we now assume that the receiving point (the mobile) moves with a velocity v in the xy plane in a direction making an angle g to the x-axis then, after unit time, the coordinates of the receiving point can be written ( v cos g, v sin g , z 0 ). The received ®eld can now be expressed as EtIt cos o c t ÀQt sin o c t 5:5 where It and Qt are the in-phase and quadrature components that would be detected by a suitable receiver, i.e. It X N n1 C n coso n t y n  Qt X N n1 C n sino n t y n  5:6 and o n  2pv l cosg Àa n  cos b n y n  2pz 0 l sin b n  f n 5:7 In these equations, o n  2pf n  represents the Doppler shift experienced by the nth component wave. Equations (5.3) to (5.7) reduce to the two-dimensional Clarke model if all waves are con®ned to the xy plane (i.e. if b is always zero). If N is suciently large (theoretically in®nite but in practice greater than 6 [8]) then by the central limit theorem the quadrature components It and Qt are independent Gaussian processes which are completely characterised by their mean value and autocorrelation function. Because the mean values of I t and Qt are both zero, it follows that EfEtg is also zero. Further, It and Q(t) have equal variance s 2 equal to the mean square value (the mean power). Thus the PDF of I and Q can be written as p x x 1 s  2p p exp  À x 2 2s 2  5:8 where x  I t or Q(t) and s 2  EfC 2 n gE 0 =N. We will return later to the signi®cance of the autocorrelation function. Characterisation of Multipath Phenomena 121 5.4 ANGLE OF ARRIVAL AND SIGNAL SPECTRA If either the transmitter or receiver is in motion, the components of the received signal will experience a Doppler shift, the frequency change being related to the spatial angles of arrival a n and b n , and the direction and speed of motion. In terms of the frame of reference shown in Figure 5.7, the nth component wave has a frequency change given by eqn. (5.7) as f n  o n 2p  v l cosg Àa n  cos b n 5:9 It is apparent that all frequency components in a transmitted signal will be subjected to this Doppler shift. However, if the signal bandwidth is fairly narrow it is safe to assume they will all be aected in the same way. We can therefore take the carrier component as an example and determine the spread in frequency caused by the Doppler shift on component waves that arrive from dierent spatial directions. The receiver must have a bandwidth sucient to accommodate the total Doppler spectrum. The RF spectrum of the received signal can be obtained as the Fourier transform of the temporal autocorrelation function expressed in terms of a time delay t as EfEtEt tg  E fI tI t  tg cos o c t ÀEfItQt  tgsin o c t  at cos o c t Àct sin o c t 5:10 The correlation properties are therefore expressed by at  and ct, which Aulin [5] has shown to be at E 0 2 Efcos ot g ct E 0 2 Efsin ot g 5:11 To proceed further we need to make some assumptions about the PDFs of a and b. Aulin followed Clarke in assuming that waves arrive from all angles in the azimuth (xy) plane with equal probability, i.e. p a a 1 2p 5:12 With this assumption, at is given by at E 0 2  p Àp J 0 2pf m tcosbp b bdb 5:13 where J 0 : is the zero-order Bessel function of the ®rst kind and ct0. In general, the power spectrum is given by the Fourier transform of eqn. (5.13); for the particular case of Clarke's two-dimensional model p b bdb and in this case eqn. (5.13) becomes a 0 t E 0 2 J 0 2pf m t5:14 122 The Mobile Radio Propagation Channel Taking the Fourier transform, the power spectrum of It and Q(t) is given by A 0 f F a 0 t  E 0 4pf m  1  1 Àf=f m  2 q  jf j 4 f m 0 elsewhere 8 > < > : 5:15 This spectrum is strictly band-limited within the maximum Doppler shift f m Æv=l but the power spectral density becomes in®nite at ( f c Æ f m ). Returning to eqn. (5.13), in order to ®nd a solution in the more general case we must assume a PDF for b. Aulin wrote pb cos b 2 sin b m jbj4jb m j4 p 2 0 elsewhere 8 < : 5:16 This is plotted in Figure 5.8(a) and was claimed to be realistic for small b m . There are sharp discontinuities at Æb m , however, and although it has the advantage of providing analytic solutions, it does not seem to be realistic, except at very small values of b m (a few degrees). Nevertheless, Aulin used this equation to obtain the RF spectrum as A 1 f F at  0 jf j > f m E 0 4 sin b m  1 f m  f m cos b m 4jf j4 f m 1 f m  p 2 À arcsin 2 cos 2 b m À 1 Àf=f m  2 1 Àf=f m  2  jf j < f m cos b m 5:17 8 > > > > > > > < > > > > > > > : Although Aulin's point that all incoming waves do not travel horizontally is valid, it is equally true that Clarke's two-dimensional model predicts power spectra that have the same general shape as the observed spectra. It is therefore clear that the majority of incoming waves do indeed travel in a nearly horizontal direction and therefore a realistic PDF for b is one that has a mean value of 08, is heavily biased towards small angles, does not extend to in®nity and has no discontinuities. The PDF shown in Figure 5.8(b) meets all these requirements and can be represented by p b b p 4jb m j cos  p 2 b b m  jbj4jb m j4 p 2 0 elsewhere 8 < : 5:18 This PDF is limited to Æb m , which depends on the local surroundings. It was originally intended to be relevant for land mobile paths, but with suitable parameters it could also be useful in the satellite mobile scenario. Using (5.18) in eqn. (5.13) allows us to evaluate the RF power spectrum A 2 f  using standard numerical techniques. Figure 5.9 shows the form of the power spectrum obtained using eqns (5.13) and (5.18), together with the spectrum A 1 f  given by eqn. (5.17) and A 0 f  given by eqn. (5.15). All the spectra are strictly Characterisation of Multipath Phenomena 123 [...]... rapid fading As the mobile moves, the fading rate will vary, hence the rate of change of envelope amplitude will also vary Both the two-dimensional and three-dimensional models lead to the conclusion that the Rayleigh PDF describes the ®rst-order statistics of the envelope over distances short enough for the mean level to be regarded as constant Firstorder statistics are those for which time (or distance)... K … : † is the complete elliptic integral of the ®rst kind; as f 3 0, S0 … f † 3 I 130 The Mobile Radio Propagation Channel Figure 5.11 Form of the baseband (envelope) power spectrum using di erent scattering models and bm ˆ 458: (Ð) Clarke's model, S0 … f †; (± ± ±) Aulin's model, S1 … f †; (- - - -) equation (5.18), S2 … f † Again, in the more general case, eqn (5.39) can only be evaluated if pb... A1 … f †; (- - - -) equation (5.18), A2 … f † band-limited to j f j < fm but in addition, the power spectral density in the ®rst two cases is always ®nite The spectrum given by eqn (5.17) is actually constant for fm cos bm < j f j < fm but the spectrum obtained from eqn (5.18) does not have this unrealistic ¯atness In contrast, A0 … f † is in®nite at j f j ˆ fm There is a much increased low-frequency... Dy is uniformly distributed with p…Dy† ˆ 1=2p, as would be expected from the convolution of two independent random variables both uniformly distributed in the interval (0, 2p) Dy is also uniformly distributed at all separations for which J0 …bl † ˆ 0, indicating that at spatial separations for which the envelope is 136 The Mobile Radio Propagation Channel Figure 5.16 The PDF of phase di erence Dy between... determine the phase di erence between the signals at two spatially separated points through the time±distance transformation l ˆ vt, and Figure 5.16 shows curves of p…Dy† for the two-dimensional model for various separation distances Two limiting cases are of interest, namely l 3 0 (coincident points) and l 3 I When l 3 0, p…Dy† is zero everywhere except at Dy ˆ 0, where it is a d-function When l 3... of bm in eqns (5.16) and (5.18) is quite large, there is little to choose between the two- and three-dimensional models as far as the PDF of phase di erence is concerned [5] If we consider the phase di erence between the signals at a given receiving point as a function of time delay t, then the PDF of the phase di erence can be expressed as [6, Ch 1]:  p  1 À x2 ‡ x…p À cosÀ1 x† 1 À r2... (b) cumulative distribution 138 The Mobile Radio Propagation Channel separating the range of integration into di erent parts and using appropriate approximations for the Bessel and logarithmic functions The problem has been studied in some detail by Davis [12] and the power spectrum, plotted on normalised scales, is shown in Figure 5.18 We note that, in contrast to the strictly band-limited power... variety of scenarios because in general the mobile has no line-of-sight path to the transmitter and there is no dominant incoming wave However, there are situations, e.g in microcells or picocells within a cellular radio system, where there may be a line-of-sight path or at least a dominant specular component We may then expect the statistics to di er from those already described The problem is analogous... the various samples will therefore have much less severe fading properties than any individual sample alone Space diversity, in which two or more physically separated antennas are used, has received much attention in the literature [6,7,14] but frequency, polarisation and time diversity are also possibilities Time diversity is attractive in digital communication Characterisation of Multipath Phenomena... E fr…t ‡ t†g For a stationary process, E f r…t†g ˆ Efr…t ‡ t†g, so   2  p 1 a…t† p rr …t† ˆ a…0† 1 À À a…0† 2 4 a…0† 2 p 2 a …t† ˆ 8a…0† …5:36† …5:37† It is shown in Appendix A that in noisy fading channels the carrier-to-noise ratio (CNR) is proportional to r2 , so the autocovariance of the squared envelope is also of interest It has been shown [5] that E ‰r2 …t†r2 …t ‡ t†Š ˆ 4 ‰a2 …0† ‡ a2 …t†Š . fading. The Mobile Radio Propagation Channel. Second Edition. J. D. Parsons Copyright & 2000 John Wiley & Sons Ltd Print ISBN 0-4 7 1-9 8857-X Online ISBN 0-4 7 0-8 415 2-4 Slow fading was mentioned. as fading and the short-term ¯uctuation caused by the local multipath is known as fast fading to distinguish it from the much longer-term variation in mean signal level, known as slow fading. The. radio waves arrive from di erent directions with di erent time delays. They combine vectorially at the receiver antenna to give a resultant signal which can be large or small depending on the distribution