Handout Shankhar Fading Chapter

8/3/2019 Handout Shankhar Fading Chapter

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24 CHAPTER 2 PROPAGATION CHARACTERISTICS OF WIRELESS CHANNELS

FIGURE 2.15 Middle zone. A base stationis localed within the structure to serve theMU in a portion of the structure. Similar basestations are present in other parts of thebuilding.

2.2.4 Small Zone and MicrozoneA bu ilding can have many part it ions, wi th the penetration of the signal depending heavilyon the material propert ies of the walls an d partitions. This requ ires the provision of onebase station for each room in the building. The loss models for the sma ll zone can be obtained using the results for the large zone by incorporating the appropriate value of the lossparameter, v, based on the number and types of obstacles between the transmitter and receiver. If LOS conditions exist, v will be 2; for N-LOS conditions, vwiU be close to 3.5.

The conditions of the building may be such th ai there is heavy traffic withineach room, and this will require the use of several base stations within a single room.Path loss calculations for the microzone can be do ne simi lar ly to those for the smallzone, using slightly smaller va lues of the loss parameter v.

2.3 FADINGAs discussed earlier, the transmission characte ristics are not determined by attenuation alone. The loss or attenuation observed may also fluc tu ate with distance and time,and this can be described in terms of fad ing.

When a signal leaves the transmitting antenna, it gets reflected, scattered, diffracted, or refracted by the various structures in its path (Arre 1973, lake 1974,Gupl 1985, Fleu 1996, Hamm 1998, Stee 1999). We examined Ihe signal loss ar isingfrom the presence of various obs tacles in the cha nnel. We also observed that the transmiss ion loss fluc tu ates around a mean or median value. This aspect of the transmi ssionloss curve, where the received signal loses its determini stic nature and becomes randomin time and space, is described in terms of fading. In o ther words, fading is th e processthat describes the fluctuations in the received signal as the signal travels to the receiv inganlelma. Fading can be described either in terms of the primary cause (multipath or Doppler), the statist ical distribution of the received envelope (Ray leigh, Rician, or lognormal), the durat ion of fad ing (long-term or shan-term), or fast versus slow fading (Tu ri1972, Slei 1987). We will look at these different characterizations of fading to understand the ir orig in and specific consequences. We will also look at the various forms offading 10 es tablish the relat ions hips that exist among them.

2.3.1 Multipath FadingMu ltipath fading, as the name suggests, arises from the existence of multip le path s between the transmitter and receiver. When a signal leaves the lransmitt ing antenna, itcan lake a number of different paths to reach the receiver, as shown in Figure 2.16.


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2.3 FADING 25

Building 1 Building 2 Building 3

Building 4 Building 5 Building 6FIGURE 2.16 The multipath concept.

The mobile unit receives signal components that are scattered, reflected, diffracted,etc. by the buildings or by other artificial or natural structures, creating a number ofdifferent paths (Hodg 1990a, Ikeg 1980, Rumm 1986, Brau 1991, Rapp 1996b).

We assume that the signal components scattered by these different structures arrive at the receiving antenna independently of each other, as shown in Figure 2.17.Under this condition, the signal received at the antenna can be expressed as the vectorsum of the components coming from these structures .

For the time being, we also assume that the receiver is stationary. The receivedsignal e,.(t) can then be expressed as a sum of delayed components:

Ne,.(t) = L aiP(t- t) , (2.31)

i=1where a i is the amplitude of the scattered component, pet) is the transmitted pulseshape, and ti is the time taken by the pulse to reach the receiver. N is the number of different paths taken by the signal to reach the receiver. Note that we also make the assumption that no direct path exists between the transmitting and receiving antennae.Instead of writing the received signal amplitude as the sum of delayed components,we can also use phasor notation to represent the received signal:

Ne,.(t) = L aicos(2'TT'fot + cp), (2.32)

i = !where fo is the canier frequency. The ith signal component has an amplitude of ([ i anda phase of CPi. The difference between eq. (2.31) and eq. (2.32) is the assumption of asingle signal of canier frequency fo in eq. (2.32) versus the transmission of a pulse ineq. (2.31). We will go back to eq . (2.31) later.

Equation (2.32) can be rewritten in terms of in-phase and quadrature notation as

Transmitter

N Ne,.(t) = cos(2'TT'fot)La; cos( 0/) - sin(2'TT'fot) L ([ i sine cp), (2.33)

i= ! i= !

FIGURE 2.17 Conceptual"ray" diagram of themultipath between thetransmitter and receiver.


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where the fi rst sum mation is identified as the ill-phase term and the second summation is ident ified as the quadralllre term.

If the locations of the structures are complete ly random, one can safely assume that the ph ase cPj will be uniformly dist ributed (Papo 1991) in the range(0, 27T). Under conditions of large N, the amplitude of the received signal can thenbe ex pressed as

er(t) = XCOS(27Tfot) - y sin(27Tfot), (2.34,)where

N NX = L fjcos(;), Y = L J sine ;). (2.34b)

;=1 i=1X and Y will be independent, identically distr ibuted Gaussian random variables by virtue of the Central Limit Theorem (Papa 1991). Under these conditions, the envelopeof the received signal, A, given by (X 2 + y 2)1 /2, will be Rayle igh distributed. Theenvelope can be recovered through demodulation, which is discussed in Chapter 3.The probabili ty density function f ia) is given by

fA(a) = ~ e x p ( - ( / )U(O) , (2 .35)2 2 2" "where the parameter a2 is the variance of the random variable X (o r Y) and U(.) is theunit step function. Note that if the envelope of the s igna l is Rayle igh di stributed, thepower, P, will have an exponent ial distribution, given by

(1') p - ) U ( P ) ' p 2u2 2u2 (2.36)

111e Rayleigh and exponential pmbability density functions are shown in Figllle 2. 18.The Rayleigh-distributed envelope is characterized by a mean given by

E(A) = uH (2.37)d . 2 . ban a vanance, U A ,gIven y

u! = 2 [ 2 - ~ l (2.38)Note tha t the Rayleigh distribution is also unique in telms of its ratio of the mean tothe standard dev iat ion:

E(A) 1.9 1. (2.39)"A

Plots of the radio-frequency (rf) s ignal and the corresponding envelope underRay leigh fading are shown in Figure 2. 19 . We see that the received signal power israndom even in the absence of noise introduced by the elec tronic system, namely,the additive white Gaussian noise. This is a consequence of the existence of multiple pa ths and the randomness of the phase (Auli 1979). Multipath fading thus leadsto fluctuations in the received signal when the MU moves from place to place.


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2.3 FADING 27

0 .7 , - - - - - - - - - - - - - - - - - - - - - - ,

0.6

0.5

Ci.;;::: 0.4....o

0.3

0.2

0.1

2 4 6 8 10Envelope A or power P

FIGURE 2.18 The density functions of the Rayleigh-distributed envelope andexponentially distributed power.

co?l:0a.Ql>QiII :

20,----------------------,

- 2 0 ~ - - - ~ - - - - ~ - - - ~ - - - ~

20

0

- 20

-40

o 0.05 0.1Time (ms)

(a)

0.15 0.2

' ~ \ i ~ ~ i ~ ~ \ I W ~ ' \ ~ , I I ~ ~ i \ i 1 W l j l i ' ~ ~ ~ ! I i I ~ l l ' i ' ! ~ ' l l i ~ ~ l K I M ~ W i ~ , l ! ' ~ I I I !

o 0.05

I 'I I0.1Time (ms)(b)

I

0.15 0.2

FIGURE 2.19 Rayleigh-faded rf signal (8) and its power (b). The plots weregenerated from 11 multiple paths. The envelope was obtained by demodulatingthe rf signal.


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28 CHAPTER 2 PROPAGATtON CHARACTERtST tC S OF WIRELESS CHANNELS

Note that we have not taken into accou nt the fact that the MU may be mov ing at acertain speed.To understand the imp lications of this fluctuation in the received power, CO Ilsider the follow ing scenario. I f th e re ceive r is designed to operate at an acceptablelevel only if a certain minimum power, P1hr, is be ing received, the receiver goesinlo olltage whenever the power goes below this threshold value (Jake 1974, Ikeg1980, Pale 1991 ). In Figure 2.1 9, we can clea rl y see that th e sys tem goes into out-age if the threshold is set to -20 dB of relative power (indicated by the line drawnpa rallel to th e freq uency axis in Figure 2.19b) . The Ol/tage probability, Pout> cannow be calculated as

J.. J' ''' 1 (P) ( P)Jout = f(p)dp = -ex p -- clp = I - exp - - ,o 0 Po Po Po (2.40)where Po is the av erage power, given by 2u2 . One of the adverse conse quences offad ing is the ex istence of outage. When o lLtage occurs, the perfonnan ce of the wireless system becomes unacceptable.EXAMPLE 2,4Consider the case of a Rayleigh-fading channel. If the average power being received is100 I.A-W, what is the probability that the received power wi ll be less than 50 I.A-W?Answer Using eq. (2.40), the probability is ( l - exp(- 50/100 ) = 0.3935. EXAMPLE 2.5If the mi nimum required power fo r acceptable perfonnance is 25),1,W, what is the outage probabi lity in a Rayleigh channel with an average received power of 100 ),I,W?Answer Using eq. (2.40), the probability of outage is0.2212, or 22.1 %.

[1 - exp(- 25/100)] =2.3 .2 Dispersive Characteristics of the ChannelThe fluctuat ion of the received power is not the on ly effect of fading. Fading mayalso affect the shape of the pulse as it is being transmitted through the chann el(Cox 1972, 1975; Hash 1979, 1989). Cons ider the diagram of multipath fadi ngshown in Figure 2.20, correspond ing to eq. (2.31). Only four different pa th s areshown. Because of the different paths taken, the repl icas of th e pu lse will arrive atthe receiver at four different times. If these pulses are not resolvable, the effect ofthe mullipalh is to produce a broadened pulse, as shown by the envelope of th eoverlapping pulses. In other words, the mu ltipath effect can result in broadening ofthe transmitted pulse, leading to intersymbol interference (lSI) (see Section B.6,Appendix B).The situat ion show n in Figure 2.20 can be simul ated using MATLAB. Theresults are shown in Figure 2.2 1. A Gaussian pu lse of width ud = 14.14ms is beingtransmitted through a wireless channel. Ten different pa th s have been selected torepl icate the scenario described in the previous paragraph. These pu lses have di fferenttim e delays, randomly chosen and ha ving random power as they reach th e receiver. A


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JLTransmitted pulse

'(a)

E.... 0.5OJ:l:00-

0400 500

E.... 0.5J:l:00-

0400 500

Pulses overlap andresult in a broadenedpulse

(b)

f\ C5d= 14.14 ms

E.... 0.5OJ:l:00-

) \ 0600 700 800 400Time (ms)

(a)

C5d= 22.64 msE

0.500-

0600 700 800 400l ime (ms)

(c)

2 .3 FADING 29

FIGURE 2.20 (8) Atransmitted pulse. (b ) Themultiple pulses produced dueto the multipath arriving atdifferent times and withdifferent powers, leading to abroadened envelope of thepulse.

500 600 700 800l ime (ms)

(b)

C5d= 27.38 ms

500 600 700 800l ime (ms)

(d)FIGURE 2.21 The frequency-selective fading channel simulated using MATLAB.

single pulse corresponding to one of the paths is shown in Figure 2.21a; this pulse issimply a delayed version of the transmitted pulse. In the absence of multiple paths,the pulse width remains the same as that of the transmitted pulse. Figures 2.21b, c,and d show the received pulse (the sum of the 10 multipath components) for threedifferent simulations. The power has been normalized to unity. The standard deviationof each pulse is also indicated. It is obvious that the pulses have broadened. This isdemonstrated by the increase in the pulse width of the received pulse compared withthe transmitted pulse (Figure 2.21a). The situation depicted in Figure 2.20b is shownin Figures 2.21b, c, and d.

This dispersive behavior of the channel can be qualitatively described in thefollowing manner. (The reasons for referring to this behavior of the channel asdispersive will be clear later.) Consider the transmission of a very narrow pulse


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30 CHAPTER 2 PROPAGATION CHARACTERIST ICS OF WIRELESS CHANNELS

Rural area Urban area0 0"'

ii i: :- 10 i -100 0 >'i -20 "i - 20.. ..

'" '"- 30 -300 5 10 0 5 10

TImelms) TIme (ms)I_I Ibl

FIGURE 2 .22 Impulse responses of two channels . (a) A typical rural area. (b) An urban area.

(delta function). The impulses corresponding to the multiple paths arrive atthe receiver at different tim es and with different amounts of power depend ing on thenatu re of the scatteringlreflectionlrefraction/diffraction that is respons ible forthe generation of the particular component. These mult iple arrival times of signalswith different powers can be used to define the impulse response of the channel asshown in Figu re 2.22. For example, in a rural area, Ihese impulses are likely toarrive at almost the same time and wi ll most likely take a shorter time to reach thereceiver. This is due to the fa ct that there are fewer tall structures, and therefore thepaths are close to each other (Figu re 2.22a). This means that the diffe rence bctweenarrival times of any information received will be too small to be observable ormeasurable. On the other hand, for an urba n area (Figure 2.22b), the multiple pathswill be mo re diverse and the received pu lses will be spread out much morc (Bu lt1983, Has h 1993, Akai 1998, Hanz 1994). Under these cond itions, info rmationarriving in the form of fin ite-size pulses will overlap and resu lt in a broadenedpu lse, as shown in Figure 2.21.

We ca n now write an expression for the average time taken by a pu lse to reachthe receiver. A typical impulse response is shown in Figure 2.23. The average delay,(T), experienced by the pulse as it traverses the channel is

N2:>;;

i " l

The quantity Pi represents the power coming along the ith pa th, and Ti is the timetaken by the ith component. The rms (root-mean-squ are) delay spread, lTd' isgiven by(2.42)


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...


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32 CHAPTER 2 PROPAGATION CHARACTERISTICS OF W IRELESS CHANNELS

- - - - ______ ____ r - ' ~ l t - 101

sin Channel

B, fIbl

FIGURE 2.24 The basebandchannel response Itransferfunction) of a " f l a t ~ channelalong with the spectrum of thetransmi tted signal 5(11.

subjected to dispersive be havior, resulting in pulse broadening, and consequentl y Ihecomponents will experience lSI (see Figure 2.25). The channel is thu s classified as afrequency-selective channe l (Be ll 1963a, lake 1974). The impu lse response shown inFigure 2.22h can be viewed as an ex ample of a frequency-selective channel. Note,however, that the dist inction between nat and freq uency-selective chan nels mu st bebased on the relationship between Ihe informal on bandwidth and Ud ' and not on theabsolule val ue of O"d. Thus, a fi al channel becomes a frequency-selective one ifthe information is Iransmilled at a hi gher and hi gher data rale. We can nowunderstand why we refer 10 a freq uency-selective ch annel as a di spersive channel.The frequency-se lective ch annel behaves as if different frequency co mponents travelal di ffe ren l speeds (ph enomenon of dispersion) and arrive at different times at thereceiver, leading 10 pu lse broaden ing.

- - - - - - ~ ' L I ______ ____ ~ r - ~ - - 181

IHlfll r


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2.3 FADING 33

EXAMPLE 2.6A typical impulse response of a wireless channel is given in Figure E2.6. I f the data rate is240 kbps, classify the channel as frequency-selective or flat.

OJ 0L-Q):::0 -10.Q)>.;; -20tJQi0::

-30

0FIGURE E2.6

0.5 1.5 2Time (J.1S)

Answer P(0.5) = O.Ol,P(l) = 0.1,PCl.5) = 0.001, andP(2) = 1, with L P, = 1.111.,< ) = 0.5 x 0.01 + 1 x 0.1 + 1.5 x 0.001 + 2 x 1 = 1 8961" 1.111 . f.L s

< 2 ) = 0.25 x O.01 + 1 x O.1 +2.25 xO.OOl +4 x 1 = 3695 21" 1.111 . f.Lsad = J3.695 - 1.8962 = 0.315 f.LS

Be = 675kHz > 240 kHz. The channel is therefore "flat." We see that the two effects, the randomness of the received signal envelope and

the frequency selectivity of the channel, are separate manifestations of the multi pathpropagation and can exist alone or in combination. However, in most practical cases,the received signal phases are random, and the Rayleigh distribution of the envelopewill be exhibited ilTespective of the frequency-selective nature of the channel.

One of the best ways to describe the frequency-dependent behavior of the channel is to use a two-ray model to represent the fading. In this model (Walk 1966, Clar1968, Hash 1979, Ball 1982, Casa 1990), the impulse response, hc(t), of the channel iswritten as the sum of two Rayleigh fields having random phases and a delay of T:

(2.45)where a l and a2 are independent, identically distributed Rayleigh variables, and l{1,and l{12 are uniformly distributed in the range (0, 27T). I f a2 is zero, we have a flat fad ing channel. By varying T, it is possible to create channels with different bandwidths .Consider a simple case where a, and a2 are scalars and deterministic with b = a21a, .Assuming l{1, and l{12 to be deterministic and equal, eq. (2.45) for the impulse responsecan be rewritten as

hc(t ) = o(t) + bo(t- T), (2.46)


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and the con'csponding transfer function, HAf ) , of the frequency-select ive channelwill be give n by

(2.47)The behavior of a typica l channel may be observed by plotting the absolute

va lue of the transfer fun ction:IH,(nl JI+b'+2bcos(2"/T). (2.48)

A plot of IH( /)1 is shown in Figure 2.26.The transfer func tion has "notches" at intervals offT = I . For different values of

T, the band width of the channel measured by the zero crossing wi ll vary, causing thechann el to go from being !lat to being frequency select i;ve. Note thai in this si mple description we have assumed the scaling factors to be dete rministic. In practice. the scaling factors are random (Rayleigh distributed).2 .3.3 Time-Dispersive Behavior of the ChannelSo far we have considered on ly the case of a stationary mobile unit. Consider now thecase of a mobile unit traveling at speed 1', as shown in Figure 2.27. The motion of themobile unit will result in a Doppler shift in the frequency of the s igna l being received.The maximum Doppler shift ,!,/. can be expressed as

(2.49)where c is the veloc ity of the elec tro magnet ic wave in free space. Taking all possibledirections into account, the instantaneous frequency, fin. will be given by

fin = 10 +Id cos( 8;).3 - - - - - - - - - - - - - , - - - - - - - - - - - - ~

2.82.62.42.22.01.81.61.41.2

L - - - - - - - - - - - - - ~ ______ ____a 0.5 1 1.5 2( ,

FtGURE 2.26 The transfer fu nction of a "two-ray " model to describe thefrequencyselec l ive channel.

(2.50)


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will be /'/. I f we make the assumpt ion that the phases, !/Ji' and 0i are uniform in therange (0, 27T), the power spectrum, Sd(f), of the received signal e,(t) can be expressedas (Gans 1972, Jake 1974)

" If s f"rrfdR ilSd(f) ~ (2.52)

0 otherwise.A plot of the power spectrum is shown in Figure 2.29 . It shows that most of

th e energy is concentrated around the maximum Doppler shift , fd' Note that thepower spectrum depends on th e radiation pattern of the antenna and the polarizationused.

Consider now the transmission of a short rf pulse as the vehicle is in motion.The motion of the MU will now in troduce changes in the channel at a rate of fd Hz. Ifthe duration of the pulse is very short, the changes introduced by the motion will beve ry slow and w ill have very litt le or no impact on the transmission and, therefore, onthe reception of the pulse. In o ther words, if the bandwidth of the signal measured interms of the inverse of the pulse duration is much larger than the maximum Dopplershift, the channel will vary very slowly or will be a slow-fading channel. On the otherhand, if the duration of the pulse is large , changes introduced in the channel from themotion of the mobile unit will be "fast" and thus will affect the transm ission. In otherwords, fo r transmission at a very low data rate, a moving vehicle will introduce fastfading if the bandwidth of the signal is no t much larger than the maximum Dopplershifl.

,,,,,,,,,,,,,,,,,,,,,,,"'" '" ------ ---- --- ,/

,,,,,,,,,,,,,,,,,,,,,

- - - - - - - - - ~ - - - - - - - - - - -fd 0 fdFIGURE 2.29 Spectrum of the Doppler-shifted signal.


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2.3 FADING 37

The condition for slow versus fast fading can now be expressed in terms of thecoherence time, Tc ' of the channel (Stei 1987), measured in terms of the inverse of themaximum Doppler shift given by (see Section C.l, Appendix C)

9T "" - - . (2.53)c 167TjdI f the pulse duration is smaller than Tc' the pulses are unlikely to undergo distortion(slow fading), and if the pulse duration is larger than the coherence time, the pulsesundergo fast fading and will be distorted. Fast fading is thus a frequency-dispersiveproperty of the channel brought on by the motion of the mobile unit. The differencebetween distortion and dispersion is explained in Appendix C.EXAMPLE 2.7Consider an antenna transmitting at 900 MHz. The receiver, a MU, is traveling at a speed of30 km/h and is receiving/transmitting data at 200 kbps. Examine whether the channel fading isslow or fast.Answer The Doppler shift is given by

8fd = 9 x 10 x 30 x 1~ O O = 25 Hz.3600 x 3 x 10

The coherence time is9 1Tc = 400 = 7162fLS 3 .

'TT" 200 x 10The channel is therefore a "slow-fading" one.

2.3.4 Level Crossing and Average Fade Duration

One of the important consequences of Doppler fading is that the signal will experience deep fades occasionally as the vehicle is in motion (Lee 1967, Kenn 1969, Bodt1982, Adac 1988b, Pars 1992). The analysis of fading in terms of Rayleigh statisticsdoes not allow a clear understanding of how often deep fades occur or how long theylast; Rayleigh statistics merely provide infOlmation on the overall percentage of timethat the signal goes below a certain level. InfOlmation is needed on the rate at whichdeep fades occur and their duration, so that system designers can choose specific approaches for appropriate data rates, word lengths, and coding schemes to mitigate theeffects of deep fades .

Deep fades can be quantitatively expressed using the parameters level crossing rate, NA , and average fade duration, Tav. To understand the concept of levelcrossing, consider the envelope of a signal received from a moving vehicle asshown in Figure 2.30. The envelope of the signal is seen to fluctuate in time. Thelevel crossing rate is defined as the expected rate at which the envelope crosses aspecified signal level A in the positive direction, as shown in Figure 2.30, and isgiven by

(2.54)


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38 CHAPTER 2 PROPAGATION CHARACTERISTICS OF W IRELESS CHANNELS

1iocw

Slope line

~ r ~ r - 7 " - - - - - ; c - i - - - - - - - - - - - - - L e v e I A 't Fade duration

Time, tFtGURE 2.30 The concept of tevet crossing .where a is the derivative of aU) and peA, a) is the joint probability density functionof the level A and the rate of change of a(r). Based on the results of Rice (Lee 1967,Kenn 1969), the ex pression fo r the level crossing rate, NA, becomes

(2.55)where a is the ratio AI AmI&" (AmlS is the rms va lu e of th e envelope.) The level crossing rate also depends on the maximum D o p ~ l e r sh ift,Jd' and therefore on the speed ofthe mobile unit. By vi rtue of the factor a ra , there will be fewer crossings at low values of the signal level as we ll as at high values of the signal level. Cer tain ly there willbe more level crossings at higher speeds of the mob ile uni!.

Another parameter of interest is the average fade duration, 7"av' which is the average period of time the signal stays below a cert ai n level A. For the case of Rayleighfading, the average fade duration is given by

I7" = -prob (a :s: A)av N ', (2.56)where prob (a :s: A ), the probabi lity that the in stantaneous signal is less th an A, isgiven by ,prob(a :S: A) = l _e - a . (2.57)Using eqs. (2.56) and (2.57), the average fade duration Tav is given as

(2.58)

These two parameters are useful in enabling assessment of the instantaneous bit error rates, since the overa ll per formance is determined not only bythe error probab ilit ies on a long-range basis but al so by how the error rates varyon a shor t-term basis. The prese nce of deep fades as we ll as th e numbe r of suchfades will change the instantaneous signal-la-noise ratio and hence, th e bit errorrates.


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2.3 FADING 39

EXAMPLE 2.8Continuing Example 2.7, calculate the average fade duration if a = 0.1.Answer

Doppler shift = 25 Hz2

e


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2.4 OTHER FADING MODELS2_4.1 Rician FadingRayleigh fading is not the only consequence of the multipath phenomenon (Stei 1964,Hash 1993). In addition to a number of random paths taken by the signal, it is possibleto have a line-of-sight (LOS) propagation from the transmitter to the receive r. ThisLOS signal adds a deterministic component to the mu lt i path signal. This is shown inFigure 2.32 and is compared wilh the case of Rayleigh fading.

The determ inistic component makes the Gaussian random variable (eq. 2.32)one of nonzero mean, and consequen tly the envelope is Rician distributed. The pdf ofthe envelope can be expressed as (Papo 1991, Dave 1958)

fA(a) :'ex{ a ~ : : i H;,o} (2.59)where 10(. ) is the modified Bessel fu nction and Ao is the component arising from theLOS signal. In the literature, it is cus tomary to refer to the contribution of the randomly located scattering centers as the "diffuse" component and lIle contribution ofthe LOS component as the "steady" component. We therefore say th at Rayleigh fad ing is the result of diffuse components and Rician fading is the result of the presenceof a steady component along with the diffuse components.

The Rician probability density func tion is often characterized by the ratio of thepower of the direct component to the power of the diffuse component , K (dB ):

K(d8) IO IOg LO(2A; , )- (2.60)For K = - 0 0 , we have no direct path and the Rician distribution becomes Rayleigh.For higher and higher va lues of K, the Rician distribution becomes almost Gaussian.

Transmitter Receiver(a) Rayleigh fading (no direct path)

Transmitter - - - - - - - - - - Receiver(b) Aidan fading (includes a direct path)

FIGURE 2.32 Comparisonof the conditions that existfor (8) Rayleigh fading and(b) Rician fading.


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Multiple reflectionsor scattering

yMultiple paths

Transmitter ReceiverFIGURE 2.34 Geometry fo r lognormal fading, or shadowing . Note theex istence o f multiple reflections.

where the strengths of the multipath component s are expressed asM

a" = II ill,, :111=1

(2.62)

The quantilies bill" represent the scattering strengths of the multiply reflected components.Considering the fact that the reg ion where the scattering takes place has a number of individual scatlerers, the bill" can be easily modeled as Rayleigh-distributed random variables.The impactofmu ltiple scattering is to introduce furlher fluctuations in the received power,which will be man ifested in the mean va lue of the received power, itselfbecoming random.

The mean value of the received power will be propor tional to the variance of theRayleigh-distributed envelope. However, the variance of th e Rayleigh envelope willdepend on (b"1II)2 . The average signal power, P LT, can then be expressed as

P LT IIp,,,, (2.63),.,=1

where(2.64)

Applying the Central Lim it Theorem for the product of random variables (Papa1991), the de nsity function j{PdB) of P B the logarithm of P LT, g iven by

P dB = IOlog lO(P LT) = I lOloglO (P ,, ) ,

will be Gaussian and g ive n by

The parameters appeari ng in this equation areP ay = average power (dBm)U dB = standard deviation (dB)

,,=1(2.65)

(2.66)


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2.4 OTHER FADING MODELS 4 3

The pdf of the signal power PLT under lognormal fading can now be expressed as

! ( hT) = 1 exp[ - ~ 1 n 2 ( P L T ) 1 , J2 2 2 2a Po'ITa PLT

(2.67)

where Po is the average power in milliwatts andadBln(lO)

a=- - - -10 (2.68)Equation (2.67) is the lognormal pdf, and the multiple scattering leads to a

lognormal distribution for the power received. The logarithm with respect to thebase e is represented by In. The lognormal pdf is shown in Figure 2.35. Lognormalfading is also refelTed to as shadowing, due to the fact that the shadowing seen inimages can be modeled using an exponential transformation similar to the one givenin eq . (2.63).EXAMPLE 2 .1 0I f the power received at the MU is lognormal with a standard deviation of 8 dB, calculate theoutage probability. Assume that the average power being received is -95 dBm and the threshold power is -98 dBm. (Hint: Use the elf function . See Section B.7, Appendix B.)Answer

_f-98 1 [x - ( -95) l , _ [- 98+95] _Pout - J2i, exp - 2 (h - 0.5 + 0.5 x el f J2 - 0.3538-0 0 27T x 8 2x8 8 2 0.70.6

Pay =- 110 dBm, O'dB =5 dB0.5

-S 0.43..... 0.3

0.20.1

0 0 2 3Power, PLT

FIGURE 2.35 The lognormal probability density function.

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Fluc tuation of the received power from lognormal fadin g creates additional problemsfor designers of til e wireless link. Including lognormal fading, the expression for the received power in any general area can be expressed by modifying eq. (2.9) as fol1ows:

(2.69)where Yg is a zero-mean random variable with a standard deviation of O"dB. In otherwords, the received power in dBm wil1 be Gaussian distributed. with the median valueof the rece ived power (dBm) being the average and O"dB be ing the standard deviation.A typical curve for the case" = 3 and standard deviation of 10gnOimai fading of6 dB is shown in Figure 2.36. It is possi ble to sec that at any given location, the rece ived power could be less than the median value of the power es timated from theHala model or any other model. This necessitates the establishment of a power margin to account fo r fadi ng at least as much as the standard dev iation of fadi ng, whichreduces the max imum transmission distance as indicated in the figure.

We will reexamine the effects of logno rmal fadi ng on the coverage area inChapter 4.2.4.3 Nakagami DistributionThe Rayleigh and Rician models of fad ing assume that the amplitudes of the sca lleredcomponents from the different paths are equal. The Nakagami model is very generalan d allows for the possibility of different strengths for the sca ttered components(Hash 1993, Brau 1991, Hoff 1960, Beck 1962). It can also work under conditions

- 90- 95-100

E - 105'"-- 110&. - 115- - 120 - 125"' - 130

\ Transmitted power", 100 dBm

" I"\ K'I Lognormal fading (OdS) = 6" / AV"\:---.1 \V r--\.. /'\' - ' 'I- 135 V

- 140 o 10 20 30 40Distance (km)FtGURE 2.36 The tognormallong-term behavior (d iscontinuous line) versuspure attenuation. Compare this with Figure 2.5.

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2.4 OTHER FADING MODELS 45

where the possibility of partial conelation exists between scattering elements. Thedensity function of the envelope can be expressed as2 II I 2111-1 ( 2)amafA (a) = exp -t=\ U(a),r (m)n ll1 H

(2.70)

where r(.) is the gamma function,(2.71)

and(A2) 2

n l=- - - - -A2 _fi) 2) (2.72)There is an important restriction on the Nakagami parameter, that m;::: 112.

For a value of In = 1/2, the Nakagami distribution becomes a single-sided Gaussiandisttibution. The Nakagami distribution becomes Rayleigh for In = 1, and for valuesof In > 1 the Nakagami distribution becomes Rician. The Nakagami probabilitydensity function is thus general enough to encompass both the Rayleigh and Riciandistributions. The Nakagami probability density function is shown in Figure 2.37.

2.4.4 Suzuki DistributionRayleigh and lognormal fading have been considered to be two separate effects.However, the phenomena responsible for short-term fading (Rayleigh) and longterm fading (lognormal) occur concurrently (Suzu 1977, Fren 1979). The mean

2 . 5 r - - - - - - - ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,

2

1.5

0.5

\ ~ - - - - - - - - - - m = 1.8 (Rician)'\ .-+- - \ ------ m = 1 (Rayleigh)' ~ T - t - - - - - m = 0.75\,- + - - - \ - - - - \ - \ - - - - - - - - m = 0.5

Q=0.2

O ~ - - - - - - - L - - - - - - ~ ~ ~ ~ - - ~ - - - - - - - - ~ o 0.5 1.5Envelope a

FIGURE 2.37 The Nakagami probability density function for a number ofdifferent values of m.

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46 CHAPTER 2 PRO PAGATION CHA RACTER ISTICS OF W IRELESS CHANNELS

value of the power received under Rayleigh fading conditions typically has a lognormal distri but ion. In other words, the Rayleigh distribution essent ially is not amarginal distribution, bu t a condit ional one:

/(al o") = (/2exp(- a22)u(a),

CF 2CF(2.73)

where 0' has the lognonnal probability density function. The density function of theenvelope can then be obtained as

fA(a) f : f(aW)f(CF)dCF. (2.74)resulti ng in the Suzuki distribution for the envelope of the rece ived signal, givenby

f" ( ') [ ,]a I (lnO' - p.)/A(a) = 2 exp --2 exp - 2 dO' .00 ' 20' &O ' a 2a (2.75)Once again, even though the Suzuki distribution is a more complete model. thefact that the pdf is not ava ilable in analytical form makes it a little di fficult towork with.

2.4.5 Summary of FadingThe various fadi ng mechan isms and the attenuation described can be summarized in adiagram as shown in Figure 2.38.

Note Ihat Rician and Rayleigh fading arise out of multipat h effects, and Nakagami fad ing can represent them both. This is not shown in the figure. For mostcases , analyses based on Rayleigh or Rician fading are suffic ient for understand ingthe na tu re of the mobile channel. A number of recent pub licat ions have suggestedthe use of Nakagami fading models to prov ide a general ized view of fadi ng inwireless systems.

2.5 lESTING OF FADING MODELSWe have stated that the probabili ty densi ty functio ns ofenvelopes under various fading sccnarios can be derived assuming certa in fundamenta l cond it ions such as theexistence of a mu lti path. the avai lability of a direct path. or the existence of multiplereflec tions. It is poss ible to condu ct statistical tests 10 veri fy that the probability density funct ion of the envelope of the faded signa l fo llows a Rayleigh, Rician, or Nakagami distri bution. One such test is thc chi-square (X2) test (Papo 1991). The X2 testis a nonparametric (Le., res ult s are not dependen t on the speci fic shape or parameters of the distributio n) means of testing hypotheses (Papo 1991). Compa ri sons aremade between theoretical popu lations based on assumed models and the actual data.The parameters of the ex pected theoretical probability density fu nctions can be ob

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