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

The typical experimental record ofreceived signal envelope as a function of distance shown in Figure 5.2 illustrates thispoint.. The e€ect of the di€erential time delays will be tointrod

go into more detail about the propagation mechanism in built-up areas, not onlyqualitatively but also in terms of a mathematical model In that way we canunderstand the full signi®cance of the prediction techniques and indicate the waysforward towards a global model that includes the e€ects of topographic andenvironmental factors.

The major problems in built-up areas occur because the mobile antenna is wellbelow 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 possiblemechanisms by which energy can arrive at a vehicle-borne antenna In practiceenergy arrives via several paths simultaneously and a multipath situation is said toexist in which the various incoming radio waves arrive from di€erent directions withdi€erent time delays They combine vectorially at the receiver antenna to give aresultant signal which can be large or small depending on the distribution of phasesamong the component waves

Moving the receiver by a short distance can change the signal strength by severaltens of decibels because the small movement changes the phase relationship betweenthe incoming component waves Substantial variations therefore occur in the signalamplitude The signal ¯uctuations are known as fading and the short-term

¯uctuation caused by the local multipath is known as fast fading to distinguish itfrom the much longer-term variation in mean signal level, known as slow fading

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 overdistances large enough to produce gross variations in the overall path between thetransmitter and receiver Because the variations are caused by the mobile movinginto the shadow of hills or buildings, slow fading is often called shadowing.Unfortunately there is no complete physical model for the slow fading, butmeasurements indicate that the mean path loss closely ®ts a lognormal distributionwith 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 aspatial phenomenon, but spatial variations are experienced as temporal variations by

a receiver moving through the multipath ®eld The typical experimental record ofreceived signal envelope as a function of distance shown in Figure 5.2 illustrates thispoint The fast fading is observed over distances of about half a wavelength Fadeswith a depth less than 20 dB are frequent, with deeper fades in excess of 30 dB beingless 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 Areceiver moving at 50 kph can pass through several fades in a second, or moreseriously 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, secondarye€ects often disturb the ®eld pattern, easing the problem signi®cantly

Whenever relative motion exists between the transmitter and receiver, there is anapparent shift in the frequency of the received signal due to the Doppler e€ect Wewill return to this later; for now it is sucient to point out that Doppler e€ects are amanifestation in the frequency domain of the envelope fading in the time domain.Although physical reasoning suggests the existence of two di€erent fadingmechanisms, in practice there is no clear-cut division Nevertheless, Figure 5.2Figure 5.1 Radio propagation in urban areas

LOS path

shows how to draw a distinction between the short-term multipath e€ects and thelonger-term variations of the local mean Indeed, it is convenient to go further andsuggest 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 concentrateprincipally on the short-term e€ects for narrowband channels; in other words, weconsider 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 betaken to mean that the spectrum of the transmitted signal is narrow enough to ensurethat 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 energytravels from the transmitter to the receiver If we begin with the case of a stationaryreceiver then we can imagine a static multipath situation in which a narrowbandsignal, e.g an unmodulated carrier, is transmitted and several versions arrivesequentially at the receiver The e€ect of the di€erential time delays will be tointroduce relative phase shifts between the component waves, and superposition ofthe di€erent components then leads to either constructive or destructive addition (at

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 isconstant 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 twoextreme possibilities The resultant signal arising from propagation via paths A and

B will be large because of constructive addition, whereas the resultant signal frompaths 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 theelectrical length of every propagation path and thus the relative phase shifts betweenthem change as a function of spatial location Figure 5.5 shows how the receivedamplitude (envelope) of the signal varies in the simple case when there are twoincoming paths with a relative phase that varies with location At some positions

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 Inpractice 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 inFigure 5.2 The space-selective fading which exists as a result of multipathpropagation is experienced as time-selective fading by a mobile receiver whichtravels through the ®eld

The time variations, or dynamic changes in the propagation path lengths, can berelated directly to the motion of the receiver and indirectly to the Doppler e€ects thatarise The rate of change of phase, due to motion, is apparent as a Doppler frequencyshift in each propagation path and to illustrate this we consider a mobile movingwith velocity v along the path AA' in Figure 5.6 and receiving a wave from ascatterer S The incremental distance d is given by d ˆvDt and the geometry showsthat the incremental change in the path length of the wave is Dl ˆ d cos a, where a isthe spatial angle in Figure 5.6 The phase change is therefore

Df ˆ 2pl Dl ˆ 2plvDtcos aand the apparent change in frequency (the Doppler shift) is

It is clear that in any particular case the change in path length will depend on thespatial angle between the wave and the direction of motion Generally, wavesarriving from ahead of the mobile have a positive Doppler shift, i.e an increase infrequency, 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 tothe maximum rate of change of phase, giving fmˆ v=l

In a practical case the various incoming paths will be such that their individualphases, as experienced by a moving receiver, will change continuously and randomly.The resultant signal envelope and RF phase will therefore be random variables and itremains to devise a mathematical model to describe the relevant statistics Such amodel must be mathematically tractable and lead to results which are in accordance

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 statisticalcharacteristics of the electromagnetic ®elds and the associated signal envelope andphase The earliest of these was due to Ossanna [1], who attempted an explanationbased on the interference of waves incident and re¯ected from the ¯at sides ofrandomly located buildings Although Ossanna's model predicted power spectra thatwere in good agreement with measurements in suburban areas, it assumes theexistence of a direct path between transmitter and receiver and is limited to arestricted range of re¯ection angles It is therefore rather in¯exible and inappropriatefor urban areas where the direct path is almost always blocked by buildings or otherobstacles

A model based on scattering is more appropriate in general, one of the mostwidely quoted being that due to Clarke [2] It was developed from a suggestion byGilbert [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 wavesare vertically polarised with spatial angles of arrival and phase angles which arerandom and statistically independent Furthermore, the phase angles are assumed tohave a uniform probability density function (PDF) in the interval (0, 2p) This isreasonable at VHF and above, where the wavelength is short enough to ensure thatsmall changes in path length result in signi®cant changes in the RF phase The PDFfor the spatial arrival angle of the plane waves was speci®ed a priori by Clarke interms of an omnidirectional scattering model in which all angles are equally likely, sothat pa…a† ˆ 1=2p A model such as this, based on scattered waves, allows theestablishment of several important relationships describing the received signal, e.g.the ®rst- and second-order statistics of the signal envelope and the nature of thefrequency spectrum Several approaches are possible, a particularly elegant onebeing 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 dimensional model In practice, di€raction and scattering from oblique surfacescreate waves that do not travel horizontally It is clear, however, that those waveswhich make a major contribution to the received signal do indeed travel in anapproximately horizontal direction, because the two-dimensional model successfullyexplains almost all the observed properties of the signal envelope and phase.Nevertheless, there are di€erences between what is observed and what is predicted, inparticular the observed envelope spectrum shows di€erences at low frequencies andaround 2 fm

two-An extended model due to Aulin [5] attempts to overcome this diculty bygeneralising Clarke's model so that the vertically polarised waves do not necessarilytravel 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

sometimes produces rather di€erent results The detailed mathematical analysis isavailable in the original references or in textbooks [6,7] In this chapter weconcentrate on indicating the methods of analysis, the physical interpretation of theresults, 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 Atypical component wave is shown in Figure 5.7, which illustrates the frame ofreference The nth incoming wave has an amplitude Cn, a phase fnwith respect to anarbitrary reference, and spatial angles of arrival anand bn The parameters Cn, fn, anand bn are all random and statistically independent The mean square value of theamplitude C is given by

where E0 is a positive constant

The generalisation in this approach occurs through the introduction of theangle bn, which in Clarke's model is always zero The phase angles fnare assumed to

be uniformly distributed in the range (0, 2p) but the probability density functions ofthe spatial angles an and bn are not generally speci®ed At any receiving point(x0, y0, z0) the resulting ®eld can be expressed as

E…t† ˆXN

nˆ1

where, if an unmodulated carrier is transmitted from the base station,

Figure 5.7 Spatial frame of reference: a is in the horizontal plane (XY plane), b is in thevertical plane

En…t† ˆ Cn cos



o0t 2pl …x0 cos an cos bn‡ y0 sin an cos bn‡ z0 sin bn† ‡ fn

…5:4†

If we now assume that the receiving point (the mobile) moves with a velocityvin the

xy plane in a direction making an angle g to the x-axis then, after unit time, thecoordinates of the receiving point can be written (v cos g,vsin g, z0) The received

®eld can now be expressed as

E…t† ˆ I…t† cos oct Q…t† sin oct …5:5†where I…t† and Q…t† are the in-phase and quadrature components that would bedetected by a suitable receiver, i.e

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† areindependent Gaussian processes which are completely characterised by their meanvalue and autocorrelation function Because the mean values of I…t† and Q…t† areboth zero, it follows that EfE…t†g is also zero Further, I…t† and Q(t) have equalvariance s2equal to the mean square value (the mean power) Thus the PDF of I and

5.4 ANGLE OF ARRIVAL AND SIGNAL SPECTRA

If either the transmitter or receiver is in motion, the components of the receivedsignal will experience a Doppler shift, the frequency change being related to thespatial angles of arrival anand bn, and the direction and speed of motion In terms ofthe frame of reference shown in Figure 5.7, the nth component wave has a frequencychange given by eqn (5.7) as

fnˆo2pnˆvl cos…g an† cos bn …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 toassume they will all be a€ected in the same way We can therefore take the carriercomponent as an example and determine the spread in frequency caused by theDoppler shift on component waves that arrive from di€erent spatial directions Thereceiver must have a bandwidth sucient to accommodate the total Dopplerspectrum

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 asEfE…t†E…t ‡ t†g ˆ EfI…t†I…t ‡ t†g cos oct EfI…t†Q…t ‡ t†g sin oct

The correlation properties are therefore expressed by a…t† and c…t†, which Aulin [5]has shown to be

In general, the power spectrum is given by the Fourier transform of eqn (5.13); forthe particular case of Clarke's two-dimensional model pb…b† ˆ d…b† and in this caseeqn (5.13) becomes

Taking the Fourier transform, the power spectrum of I…t† and Q(t) is given by

A0… f † ˆ F ‰a0…t†Š ˆ

E04pfm

fm

p

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 havethe 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 arealistic PDF for b is one that has a mean value of 08, is heavily biased towards smallangles, does not extend to in®nity and has no discontinuities The PDF shown inFigure 5.8(b) meets all these requirements and can be represented by

pb…b† ˆ 4jbpmj cos

p2

b

bm

jbj4jbmj4p

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 A2… f †using standard numerical techniques Figure 5.9 shows the form of the powerspectrum obtained using eqns (5.13) and (5.18), together with the spectrum A1… f †given by eqn (5.17) and A0… f † given by eqn (5.15) All the spectra are strictly

Figure 5.8 Probability density functions for b, the arrival angle in the vertical plane: (top)proposed by Aulin, (bottom) as expressed by equation (5.18) In each case the values of bmare(a) 108, (b) 158, (c) 308, (d) 458.

band-limited to j f j < fm but in addition, the power spectral density in the ®rst twocases is always ®nite The spectrum given by eqn (5.17) is actually constant for

fm cos bm< j f j < fmbut the spectrum obtained from eqn (5.18) does not have thisunrealistic ¯atness In contrast, A0… f † is in®nite at j f j ˆ fm There is a muchincreased low-frequency content even when bm is small

We conclude therefore that the RF signal spectrum is strictly band-limited to arange  fm around the carrier frequency However, within those limits the powerspectral density depends on the PDFs associated with the spatial angles of arrival aand b The limits of the Doppler spectrum can be quite high; for example, in a vehiclemoving at 30 m/s (  70 mph) receiving a signal at 900 MHz the maximum Dopplershift is 90 Hz Frequency shifts of this magnitude can cause interference with themessage information Hand-portable transceivers carried by pedestrians experiencenegligible Doppler shift

5.5 THE RECEIVED SIGNAL ENVELOPE

Practical radio receivers do not normally have the ability to detect the components

I…t† and Q…t†, they respond to the envelope and/or phase of the complexsignal E…t†.The envelope r(t) of the complexsignal E…t† is given by

r…t† ˆ ‰I2…t† ‡ Q2…t†Š1=2

Figure 5.9 Form of the RF power spectrum using di€erent scattering models and bmˆ 458:(Ð) Clarke's model, A0… f †; (± ± ±) Aulin's model, A1… f †; (- - - -) equation (5.18), A2… f †

and it is well known [9] that the PDF of r(t) is given by

pr…r† ˆsr2 exp

 r22s2



…5:19†

in which s2, which is the same as a…0†, is the mean power and r2=2 is the short-termsignal power This is the Rayleigh density function, and the probability that theenvelope does not exceed a speci®ed value R is given by the cumulative distributionfunction

2s2



ˆ 0:5hence

rMˆp2s2 ln 2ˆ 1:1774s …5:24†Figure 5.10 shows the PDF of the Rayleigh function with these points identi®ed

It is often convenient to express eqns (5.19) and (5.20) in terms of the mean, meansquare or median rather than in terms of s This is because it is useful to have ameasure of the envelope behaviour relative to these parameters To avoid

cumbersome nomenclature we write Efrg ˆ r and Efr2g ˆ r2, and in these terms,simple manipulation yields the following results In terms of the mean square value,

5.6 THE RECEIVED SIGNAL PHASE

The received signal phase y…t† is given is terms of I…t† and Q…t† by

py…y† ˆ2p1 …5:29†This result is also expected intuitively; in a signal composed of a number ofcomponents of random phase it would be surprising if there were any bias in thephase of the resultant It is random and takes on all values in the range (0, 2p) withequal probability.

The mean value of the phase is

We will return later to a consideration of changes in the signal phase

5.7 BASEBAND POWER SPECTRUM

In Section 5.4 we used the autocorrelation function of the received signal in order toobtain the RF spectrum We saw that the spectrum was strictly band-limited to

fc> fm but that the shape of the spectrum within those limits was determined byother factors, in particular the assumed PDFs for the spatial angles a and b

We can now consider the autocorrelation function of the envelope r…t† and use it toobtain the baseband power spectrum The mean of the envelope is given by eqn.(5.21) as

Efr…t†g ˆ s



p2

approximated by neglecting terms beyond the second order The approximation thenbecomes

rr…t† ˆp

2a…0†



1 ‡14

we are principally interested in the continuous spectral content of the envelope, not

in the carrier component, we can use the autocovariance function (in which the meanvalue is removed), thus

rr…t† ˆ Efr…t†r…t ‡ t†g Efr…t†gEfr…t ‡ t†g …5:36†For a stationary process, Efr…t†g ˆ Efr…t ‡ t†g, so

2a…0†

It is shown in AppendixA that in noisy fading channels the carrier-to-noise ratio(CNR) is proportional to r2, so the autocovariance of the squared envelope is also ofinterest It has been shown [5] that

E‰r2…t†r2…t ‡ t†Š ˆ 4‰a2…0† ‡ a2…t†Š

and we know, from eqn (5.22) that Efr2…t†g ˆ 2a…0†, thus

rr2…t† ˆ 4‰a2…0† a2…t†Š 4a2…0† ˆ 4a2…t† …5:38†The power spectrum of r…t† and r2…t† can therefore be written as

as appropriate; see equations (5.37) and (5.38)

The convolution represented by eqn (5.39) can be evaluated exactly for the RFspectrum represented by eqn (5.15), in which case

S0… f † ˆ CA0… f †*A0… f †

ˆ C



E04p

21

fmK



1

f2fm

21=2

…5:40†where K…:† is the complete elliptic integral of the ®rst kind; as f ! 0, S0… f † ! 1

Again, in the more general case, eqn (5.39) can only be evaluated if pb…b† isknown The expressions for pb…b† given by eqns (5.16) and (5.18) allow numericalevaluation of baseband spectra S1… f † and S2… f † (in the former case, via A1… f † asgiven by eqn (5.17)) A comparison between S0… f †, S1… f † and S2… f † is presented inFigure 5.11, which uses a logarithmic scale Although S0… f † ! 1 at f ˆ 0, S1… f †and S2… f † are always ®nite.

5.8 LCR AND AFD

Figure 5.2 shows that the signal envelope is subject to rapid fading As the mobilemoves, the fading rate will vary, hence the rate of change of envelope amplitude willalso vary Both the two-dimensional and three-dimensional models lead to theconclusion that the Rayleigh PDF describes the ®rst-order statistics of the envelopeover distances short enough for the mean level to be regarded as constant First-order statistics are those for which time (or distance) is not a factor, and the Rayleighdistribution therefore gives information such as the overall percentage of time, or theoverall percentage of locations, for which the envelope lies below a speci®ed value.There is no indication of how this time is made up

We have already commented, in connection with Figure 5.2, that deep fades occuronly rarely whereas shallow fades are much more frequent System engineers areinterested in a quantitative description of the rate at which fades of any depth occurand the average duration of a fade below any given depth This provides a valuable

Figure 5.11 Form of the baseband (envelope) power spectrum using di€erent scatteringmodels and bmˆ 458: (Ð) Clarke's model, S0… f †; (± ± ±) Aulin's model, S1… f †; (- - - -)equation (5.18), S2… f †

aid in selecting transmission bit rates, word lengths and coding schemes in digitalradio systems and allows an assessment of system performance The requiredinformation is provided in terms of level crossing rate and average fade durationbelow a speci®ed level The manner in which these two parameters are derived isillustrated in Figure 5.12.

The level crossing rate (LCR) at any speci®ed level is de®ned as the expected rate

at which the envelope crosses that level in a positive-going (or negative-going)direction In order to ®nd this expected rate, we need to know the joint probabilitydensity function p…R, _r† at the speci®ed level R and the slope of the curve _r…ˆ dr=dt†

In terms of this joint PDF, and remembering that we are interested only in going crossings, the LCR NRis given by [6, Ch 1]:

NRˆp2pfmr exp… r2† …5:44†where

Equation (5.44) gives the value of NRin terms of the average number of crossings persecond It is therefore a function of the mobile speed, and this is apparent from theappearance of fm in the equation Dividing by fm produces the number of levelcrossings per wavelength and this is plotted in Figure 5.13 There are few crossings athigh and low levels; the maximum rate occurs when R ˆ s, i.e at a level 3 dB belowthe RMS level.

It is sometimes convenient to express the LCR in terms of the median value rM,rather than in terms of the RMS value Using eqns (5.24) and (5.43) the normalisedaverage number of level crossings per wavelength is then

EftRg ˆPNr…R†

Substituting for NRfrom eqn (5.43) gives

Figure 5.13 Normalised level crossing rate for a vertical monopole under conditions ofisotropic scattering

Normalised AFD is plotted in Figure 5.14 as a function of r.

Figure 5.14 Normalised average duration of fades for a vertical monopole under conditions

of isotropic scattering

Table 5.1 gives the AFD and average LCR for various fade depths with respect tothe median level and indicates how often a Rayleigh fading signal needs to besampled in order to ensure that an `average duration' fade below any speci®ed levelwill be detected For example, in order to detect about 50% of the fades 30 dB belowthe median level, the signal must be sampled every 0.01l At 900 MHz this is 0.33 cm.

In practice the median signal level is a very useful measure Sampling of the signal

in order to estimate its parameters will be discussed in Chapter 8 but it isimmediately obvious that if a record of signal strength is obtained by sampling thesignal envelope at regular intervals of distance or time, then the median value is thatexceeded (or not exceeded) by 50% of the samples This is very easily determined.Furthermore, it is a relatively unbiased estimator since it is in¯uenced only by thenumber of samples that lie above or below a given level, and not by the actual value

of those samples We note from AppendixB that the mean and RMS values arerespectively 0.54 and 1.59 dB above the median, so conversion of the values given inTable 5.1 is straightforward

In practice [11] the measured average fade rates and durations are closelypredicted by eqns (5.44) and (5.47) Often, however, it is of interest to know thedistribution about this average level and for fade duration this has been measuredusing a Rayleigh fading simulator The results are shown in Figure 5.15 For fadedepths 10 dB or more below the median, all the distributions have identical shapesand for long durations the distributions quickly reach an asymptotic slope of (fadeduration) 3 In general, fades of twice the average duration occur once in every tenand fades of sixor seven times the average duration occur once in every thousand.Very deep fades are short and infrequent Only 0.2 fades per wavelength have adepth exceeding 20 dB and these fades have a mean duration of 0.03l Only 1% ofsuch fades have a duration exceeding 0.1l

5.9 THE PDF OF PHASE DIFFERENCE

It is not very meaningful to consider the absolute phase of the signal at any point; inany case it is only the phase relative to another signal, or a reference, that can bemeasured It is possible, however, to think in terms of the relative phase between thesignals at a given receiving point at two di€erent times, or between the signals at twospatially separated locations at the same time Both these quantities are meaningful

in a study of radio systems

Table 5.1 Average fade length and crossing rate for fades measured with

respect to median value

Fade depth Average fade length

(wavelengths) Average crossing rate(wavelengths 1)

Unless the value of bm in eqns (5.16) and (5.18) is quite large, there is little tochoose between the two- and three-dimensional models as far as the PDF of phasedi€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 phasedi€erence can be expressed as [6, Ch 1]:

r…t† ˆa…t†a…0† and x ˆ r…t† cos DyAssuming that pa…a† ˆ 1=2p, we can determine the phase di€erence between thesignals at two spatially separated points through the time±distance transformation

l ˆvt, and Figure 5.16 shows curves of p…Dy† for the two-dimensional model forvarious separation distances

Two limiting cases are of interest, namely l ! 0 (coincident points) and l ! 1.When l ! 0, p…Dy† is zero everywhere except at Dy ˆ 0, where it is a d-function.When l ! 1, Dy is uniformly distributed with p…Dy† ˆ 1=2p, as would be expectedfrom the convolution of two independent random variables both uniformlydistributed in the interval (0, 2p) Dy is also uniformly distributed at all separationsfor which J0…bl † ˆ 0, indicating that at spatial separations for which the envelope isFigure 5.15 Measured fade duration distribution The data was obtained from a simulatorwith a Rayleigh amplitude distribution and a parabolic Doppler spectrum

uncorrelated then the phase di€erence is also uncorrelated This is to be expectedsince at these separations the electric ®eld signals are uncorrelated.

5.10 RANDOM FM

Since the phase y varies with location, movement of the mobile will produce arandom change of y with time, equivalent to a random phase modulation This isusually called random FM because the time derivative of y causes frequencymodulation which is detected by any phase-sensitive detector, e.g FM discriminator,and appears as noise to the receiver In simple mathematical terms,

_y ˆ dydtˆ d

dt

tan 1Q…t†

The cumulative distribution function is given by

P… _Y† ˆ

…_Y

1 p…_y† d_y

ˆ1 2

1

Figure 5.17 Probability functions for the random FM _y of the received electric ®eld: (a)probability density function, (b) cumulative distribution

separating the range of integration into di€erent parts and using appropriateapproximations for the Bessel and logarithmic functions The problem has beenstudied in some detail by Davis [12] and the power spectrum, plotted on normalisedscales, is shown in Figure 5.18 We note that, in contrast to the strictly band-limitedpower spectrum of the signal envelope (the Doppler spectrum), there is a ®niteprobability of ®nding the frequency of the random FM at any value Nevertheless,the energy is largely con®ned to 2fm, from where it falls o€ as 1=f and is insigni®cantbeyond 5fm The majority of energy is therefore con®ned to the audio band; thelarger excursions, being associated with the deep fades, occur only rarely.

The PDF of the di€erence in random FM between two spatially separated points

is of interest in the context of diversity systems, but is not easily obtained It involvescomplicated integrals and a computer simulation has been used to produce someresults The PDF can be evaluated, however, when there is either zero or in®niteseparation between the points For the case of zero separation a d-function of unityarea at Dy ˆ 0 is obtained For in®nite separation the two values of random FM areindependent and the convolution of two equal distributions p…y† gives the probabilitydensity function [6, Ch 6] as