Bernard Sklar Raylegh Fading Channels

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Sklar, B.Rayleigh Fading Channels

Mobile Communications Handbook

Ed. Suthan S. Suthersan

Boca Raton: CRC Press LLC, 1999

c1999 by CRC PressLLC


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Rayleigh Fading Channels1

Bernard SklarCom m uni cations Engineering Services

18.1 Introduction18.2 TheChall engeof a Fading Channel

18.3 Mobi le-Radio Propagati on: Large-Scale Fading andSmall-ScaleFadingLarge-Scale Fading: Path-LossM ean andStandard Deviation Small-Scale Fading: Statisticsand Mechanisms

18.4 Signal Time-Spreading Viewed in the Time-Delay Domain:Figure18.1, Block 7TheMulti path Intensity ProfileDegradati onCategori esduetoSignalTime-SpreadingViewedin theTimeDelay Domain

18.5 Signal Ti me-Spreading Viewed in the Frequency Domain:Figure 18.1, Block 10The Spaced-Frequency CorrelationFunctionDegradati onCategori esduetoSignalTime-SpreadingViewedin theFrequency Domain

18.6 Typical Examplesof Flat Fading and Frequency-Selecti veFading Manifestations

18.7 Time Vari anceViewed in the Time Domain: Figure18.1,Block 13The Spaced-Time Correlation FunctionTheConcept of Dualit yDegradation Categoriesdueto Time

VarianceViewed in theTimeDomain18.8 Time Vari anceViewed in the Doppler-Shift Domain: Figure

18.1, Block 16The Doppler Power Spectrum

18.9 Analogy Between Spectral Broadening in Fading Channelsand Spectral Broadening in Di gital Signal Keying

18.10 Degradati on Categoriesdueto Time Vari ance, Viewed in theDoppler-Shift Domain

18.11 Mi tigation MethodsMi tigation to Combat Frequency-SelectiveDistortion Mit-igation to Combat Fast-Fading Distorti on Mitigation toCombat Lossin SNR

18.12 Summary of the Key Parameters Characteri zing FadingChannelsFast-Fading Distorti on: Example #1 Frequency-SelectiveFading Distorti on: Example#2Fast-Fading andFrequency-

SelectiveFading Distorti on: Example #318.13 TheViterbi Equali zer asApplied to GSM

18.14 The RakeReceiver Applied to Di rect-SequenceSpread-Spectrum (DS/SS) Systems

18.15 ConclusionReferences

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18.1 Introduction

When the mechanisms of fading channels were first modeled in the 1950s and 1960s, the ideas

were primari ly applied to over-the-horizon communications coveri ng a wide range of frequencybands. The330 MH z high-frequency(HF) band is used for ionospheric communications, and the300MH z3 GHzultra-high-frequency( UHF) and 330GHz super-high-frequency( SHF) bandsareused for tropospheric scatter. Although the fading effects in a mobile radio system are somewhatdifferent from thosein ionospheric and tropospheric channels, the early models aresti ll quiteusefulto help characterizefading effectsin mobiledigital communicati on systems. Thischapter addressesRayleigh fading, primari ly in theUHFband, that affectsmobilesystemssuch ascellular and personalcommunicati on systems(PCS). The chapter itemizesthe fundamental fading manifestations, typesof degradati on, and methods to mitigate the degradation. Two part icular mitigation techniquesare examined: the Viterbi equali zer implemented in the Global System for Mobile Communicati on(GSM) , and the Rakereceiver used in CDMA systemsbuilt to meet Interim Standard-95 (I S-95).

18.2 TheChallengeof a FadingChannel

In the study of communicati on systems, the classical (i deal) addit ive-white-Gaussian-noise( AWGN)channel, with statistically independent Gaussian noisesamplescorrupti ngdata samplesfreeof inter-symbol interference (I SI) , is the usual starting point for understanding basic performance relation-ships. The primary source of performance degradati on is thermal noise generated in the receiver.Often, external interferencereceived by the antennais moresignificant than the thermal noise. Thisexternal interferencecan someti mesbecharacterized ashavingabroadband spectrum and quantifi edby a parameter called antenna temperature [ 1]. The thermal noise usually hasa flat power spectraldensit y over the signal band and a zero-mean Gaussian voltage probabilit y densit y function (pdf ).When modeling practi cal systems, the next step is the int roduction of bandlimit ing filters. Thefilterin the tr ansmitter usually serves to sati sfy some regulatory requirement on spectral containment.The filter in the receiver often serves the purpose of a classical matched filter [2] to the signalbandwidth. Due to the bandlimiting and phase-distort ion propert iesof fi lters, special signal designand equalization techniquesmay berequired to miti gate thefilter-induced ISI.

If a radio channels propagati ng characteristicsare not specifi ed, oneusually infers that the signalattenuati on vs. distancebehavesasif propagati on takesplaceover i deal freespace. Themodel of freespacetreatstheregion between thetr ansmit and receiveantennasasbeingfreeof all objectsthat mightabsorb or reflect radio frequency(RF) energy. It also assumesthat, within thisregion, theatmospherebehavesasaperfectly uniform and nonabsorbing medium. Furthermore, theearth istreated asbeinginfi nitely far away from thepropagatingsignal (or, equivalently, ashaving ar eflection coefficient thatis negligible). Basically, in thisidealized free-spacemodel, the attenuati on of RFenergy between thetransmitter and receiver behaves accordi ng to an i nverse-square law. The received power expressedin termsof transmitted power is attenuated byafactor, Ls (d), wherethisfactor is calledpathlossor

1A version of this chapter has appeared as two papers in the IEEE Communications M agazine,September 1997, underthe titlesRayleigh Fading Channels in Mobile Digital Communication Systems, Part I: Characterization and Part II :Mitigation.



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freespaceloss.When the receiving antenna is isotropic, thisfactor is expressed as[1] :

Ls (d) =

4d

2(18.1)

In Eq. (18.1), disthedistancebetween thetr ansmitter and thereceiver, and isthewavelength of thepropagati ng signal. For this caseof ideali zed propagati on, received signal power isvery predictable.

For most practical channels, where signal propagati on takesplacein the atmosphere and near theground, the free-spacepropagati on model is inadequate to describethe channel and predict systemperformance. In a wireless mobile communication system, a signal can tr avel from tr ansmitter toreceiver over multiple reflective paths; this phenomenon is referred to asmultipathpropagation.Theeffect can causefluctuationsi n thereceived signalsamplitude, phase, and angleof arr ival, givingrise to theterminologymultipathfading.Another name, scintillation,having originated in radioastronomy, is used to describe the mult ipath fading caused by physical changes in the propagati ngmedium, such asvariationsin thedensity of ionsin thei onospheric layerst hat reflect highfrequency(HF) radio signals. Both names, fading and scint il lation, refer to a signals random fl uctuations orfading dueto multipath propagation. The main differenceis that scinti llation involvesmechanisms(e.g., ions) that are much smaller than a wavelength. The end-to-end modeli ng and design ofsystemsthat mit igate the effects of fading are usually morechallengingthan thosewhosesolesourceof performancedegradati on is AWGN.

18.3 Mobile-Radio Propagation: Large-Scale Fading and Small-ScaleFading

Figure 18.1represents an overview of fading channel manifestations. It starts with two t ypes offading effects that characterizemobile communicati ons: large-scale fading and small-scale fading.Large-scale fading represents the average signal power attenuation or the path loss due to motionover large areas. In Fig.18.1,the large-scale fading manifestation i s shown in blocks 1, 2, and 3.This phenomenon is affected by prominent terrain contours (e.g., hil ls, forests, billboards, clumpsof bui ldi ngs, etc.) between the tr ansmitter and receiver. The receiver is often represented as beingshadowed by such prominences. The statisti csof large-scale fading provide a way of computingan estimate of path loss as a function of distance. This is described i n terms of a mean-path loss(nth-power law) and a log-normally distr ibuted variation about the mean. Small-scale fading refersto the dramati c changes in signal ampli tude and phase that can be experi enced as a result of smallchanges(assmall asa half-wavelength) in the spati al separation between a receiver and tr ansmitter.As indicated in Fig.18.1,blocks4, 5, and 6, small-scale fading manifests itself in two mechanisms,namely, time-spreading of thesignal (or signal dispersion) and time-variant behavior of thechannel.For mobile-radio applicati ons, the channel is ti me-variant because motion between the tr ansmitterand receiver resultsi n propagation path changes. Ther ateof changeof thesepropagati on conditi onsaccounts for the fading rapidity (rate of change of the fading impairments). Small-scale fading isalso calledRayleighfadingbecause if the multiple reflectivepaths are large in number and there isno line-of-sight signal component, the envelope of the received signal is statisti cally described by a

Rayleighpdf. When thereis adominant nonfading signal component present, such asa li ne-of-sightpropagati on path, the small-scale fading envelope is described by a Rician pdf [3]. A mobileradioroamingover alargeareamust processsignalsthat experi enceboth typesof fading: small-scalefadingsuperi mposed on large-scale fading.

There are three basic mechanisms that impact signal propagation in a mobile communicati onsystem. They are reflecti on, diffraction, and scattering [ 3].



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FIGURE 18.1: Fading channel manifestati ons.

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Reflection occurs when a propagati ng electromagnetic wave impinges upon a smoothsurfacewith very largedimensions compared to the RF signal wavelength ().

Diffracti on occurswhen ther adio path between thetransmitter and receiver isobstructedby a dense body with large dimensions compared to, causing secondary waves to be

formed behind the obstructing body. Diffraction is a phenomenon that accountsfor RFenergy tr avelling from tr ansmitter to receiver without a line-of-sight path between thetwo. It isoften termedshadowingbecausethe diffracted field can reachthe receiver evenwhen shadowed by an impenetrable obstr uction.

Scattering occurs when a radio wave impinges on either a large rough surface or anysurface whose dimensions are on the order of or less, causing the reflected energy tospread out (scatter) in all directions. In an urban environment, typical signal obstr uctionsthat yield scatteri ng are lampposts, str eet signs, and foliage.

Figure18.1 may serveasa table of contents for the sectionsthat follow. We will examine the twomanifestationsof small-scale fading: signal time-spreading (signal dispersion) and the ti me-variantnature of t he channel. These examinations will take place in two domains: time and frequency,as indicated i n Fig.18.1,blocks7, 10, 13, and 16. For signal dispersion, we categorize the fading

degradati on typesasbeing frequency-selectiveor frequency-nonselecti ve( flat) , asli sted in blocks8,9, 11, and 12. For theti me-vari ant manifestation, wecategorizet hefading degradati on typesasfast-fading or slow-fading, asl isted in blocks14, 15, 17, and 18. Thelabels indicating Fourier tr ansformsand duals will beexplained later.

Figure18.2 il lustratesthevari ouscontributi onsthat must beconsidered when estimatingpath lossfor a link budget analysis in a cellular application [4]. Thesecontributionsare:

Mean path lossasa function of distance, dueto large-scale fading

Near-worst- case vari ati ons about the mean path loss (t ypically 610 dB) or large-scalefading margin

Near-worst- caseRayleigh or small-scale fading margin (t ypically 2030 dB)

InFig.18.2, theannotations12% indicateasuggestedarea(probabili ty) under thetail of each

pdf asadesign goal. Hence, theamount of margin indicated isintended to provideadequatereceivedsignal power for approximately 9899% of each type of fading vari ati on (large- and small-scale).

A received signal, is generally described in terms of a tr ansmitted signals(t)convolved with theimpulseresponseof the channelhc(t). Neglecting the degradation dueto noise, wewrite:

r(t) = s(t) hc(t) (18.2)

where denotes convoluti on. In the case of mobile radios,r(t)can be partitioned in terms of twocomponent random variables, asfollows[ 5]:

r(t) = m(t) r0(t) (18.3)

where m(t) is called the large-scale-fading component, and r0(t) is called the small-scale-fading

component. m(t) is sometimes referred to as the local mean orlog-normal fadingbecause themagnitude ofm(t) is described by a log-normal pdf (or, equivalently, the magnitude measuredin decibels has a Gaussian pdf). r0(t) is sometimes referred to as multi path or Rayleigh fading.Figure 18.3illustrates the relati onship between large-scale and small-scale fading. In Fig.18.3(a),received signal powerr(t)vs. antenna displacement ( typically in unitsof wavelength) is plotted forthe case of a mobile radio. Small-scale fading superimposed on large-scale fading can be readily



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FIGURE 18.2: Link- budget considerationsfor a fading channel.

identi fied. The typical antenna displacement between the small-scale signal nulls is approximatelya half wavelength. In Fig.18.3(b), the large-scale fading or local mean,m(t), has been removed i norder to view the small-scale fading, r0(t), about some averageconstant power.

In thesectionsthat follow,weenumeratesomeof thedetailsregardingt hestatisti csand mechanismsof large-scale and small-scale fading.

18.3.1 Large-Scale Fading: Path-Loss Mean and Standard Deviation

For the mobile radio application, Okumura [6] made some of the earl ier comprehensive path-lossmeasurements for a wide range of antenna heights and coverage distances. Hata [ 7] transformedOkumuras data into parametr ic formulas. For the mobile radio application, the mean path loss,Lp(d), as a function of distance, d, between the tr ansmitter and receiver isproportional to annth-power ofdrelativeto a referencedistance d0[ 3].

Lp(d)

d

d0

n(18.4)

Lp(d) is often stated in decibels, asshown below.

Lp(d) (dB) = Ls(d0) (dB) + 10 n log

dd0

(18.5)

Thereferencedistance d0, correspondsto apoint located in thefar field of theantenna. Typically,thevalueofd0 is taken to be1 km for largecells, 100 m for microcells, and 1 m for indoor channels.Lp(d) is the average path loss (over a multit ude of dif ferent sites) for a given value of d. Linear



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FIGURE18.3: Large-scale fading and small-scale fading.

regression for a minimum mean-squared estimate (M MSE) fit ofLp(d) vs. don alog-log scale (fordistancesgreater than d0) yieldsastraight linewit h aslopeequal to 10n dB/decade. Thevalueof theexponentn dependson the frequency, antennaheights, and propagati on environment. In freespace,n = 2, asseen in Eq. (18.1). In thepresence of a very strong guided wavephenomenon (likeurbanstreets), n can belower than2. When obstr uctionsarepresent, n islarger. Thepath lossLs (d0) tothereference point at a distance d0from the transmitter is typically found through field measurementsor is calculated using the free-spacepath l oss given by Eq. (18.1). Figure18.4 showsa scatter plotof path loss vs. distancefor measurements made in several German cities[ 8]. Here, the path losshasbeen measured relativeto thefree-spacereferencemeasurement atd0 = 100m. Also shown arestr aight-l ine fitsto variousexponent values.

The path lossvs. distance expressed in Eq. (18.5)is an average, and therefore not adequate todescribe any part icular setting or signal path. It is necessary to provide for variations about themean sincetheenvironment of different sitesmay bequitedifferent for simi lar transmitter-receiverseparations. Figure18.4 illustr ates that path-loss vari ati ons can be quite large. Measurementshave shown that for any value ofd, the path loss Lp(d) is a random variable having a log-normal

distribution about t he mean distant- dependent value Lp(d) [ 9]. Thus, path loss Lp(d) can be



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FIGURE18.4: Path lossvs. distancemeasured in several German cities.

expressed in termsofLp(d) plusarandom variableX, asfollows[ 3].

Lp(d) (dB) = Ls(d0) (dB) + 10 n log10 d

d0

+ X (dB) (18.6)

where X denotesa zero-mean, Gaussian random variable (i n decibels) wit h standard deviation(also in decibels). X is siteand distancedependent. Thechoiceof a value for X is often based onmeasurements; i t is not unusual for X to take on values as high as 610 dB or greater. Thus, theparametersneeded to statistically describepath lossdueto large-scalefading for an arbitr ary locati onwith aspecific tr ansmitter-receiver separation are:

The referencedi stanced0

Thepath-lossexponentn

Thestandard deviation ofX

There are several good references deali ng wit h t he measurement and esti mati on of propagati onpath lossfor many different applicationsand configurations[ 3], [7][11].



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18.3.2 Small-Scale Fading: Statistics andMechanisms

When thereceived signal is madeup of multiple reflectiver aysplusa significant li ne-of-sight (non-faded) component, theenvelopeamplitudedueto small-scale fading hasa Rician pdf, and isreferredto asRicianfading[3]. Thenonfaded component is called thespecularcomponent. Astheampli-tudeof thespecular component approacheszero, theRicianpdf approachesaRayleighpdf, expressedas:

p(r) =

r

2 exp

r2

22

forr 0

0 otherwise

(18.7)

where r is the envelope amplitude of the received signal, and22 is the predetecti on mean powerof the multipath signal. TheRayleigh faded component i ssometimescalled therandom,scatter,ordiffusecomponent. TheRayleigh pdf resultsfrom having no specular component of thesignal; thusfor asinglelink it representsthepdf associated with theworst caseof fading per mean received signalpower. For theremainder of thischapter, it will beassumed that lossof signal-to-noiseratio (SNR)due to fading followsthe Rayleigh model described. It will also be assumed that the propagating

signal is in the UHFband, encompassingpresent-daycellular and personal communicati onsservi ces(PCS) frequency allocati onsnominally 1 GHz and 2 GHz, respectively.

Asindicated in Fig. 18.1, blocks4, 5, and 6, small-scale fading manifests it self in two mechanisms:

Time-spreading of the underlying digital pulseswithin the signal

A time-variant behavior of thechannel dueto motion (e.g., areceiveantennaon amovingplatform).

Figure18.5illustrates the consequences of both manifestations by showing the response of amultipath channel to anarrowpulsevs. delay, asa function of antennaposition (or time, assumingaconstant velocityof motion). In Fig.18.5, wedistinguish between two different ti mereferencesdelaytime and tr ansmission or observation timet. Delay timerefersto thetime-spreading manifestationwhich results from the fading channels nonoptimum impulse response. The tr ansmission t ime,however, is related to the antennas motion or spati al changes, accounti ng for propagati on pathchanges that are perceived as the channels ti me-variant behavior. Note that, for constant velocit y,as is assumed in Fig.18.5,either antenna posit ion or t ransmission time can be used to i llustratethis time-variant behavior. Figures 18.5(a)(c) show the sequenceof received pulse-power profilesas the antenna moves through a succession of equally spaced posit ions. Here, the interval betweenantenna positions is 0.4 , where is the wavelength of the carr ier frequency. For each of thethree cases shown, the response-pattern differs significantly i n the delay t ime of the largest signalcomponent, the number of signal copies, t heir magnitudes, and the total received power (area) inthe received power profile. Figure 18.6 summari zes these two small-scale fading mechanisms, thetwo domains (time or ti me-delay and frequency or Doppler shift) for viewing each mechanism andthe degradati on categories each mechanism can exhibi t. Note that any mechanism characterizedin the ti me domain can be characterized equally well in the frequency domain. Hence, as outli ned

in Fig.18.6,the time-spreading mechanism will be characterized in the ti me-delay domain as amulti path delay spread and in the frequency domain asa channel coherence bandwidth. Similarly,the time-variant mechanism will becharacteri zed in t he ti me domain as a channel coherence timeand in the Doppler-shift (frequency) domain as a channel fading rate or Doppler spread. Thesemechanisms and their associated degradati on categories wil l be examined i n greater detail in thesectionsthat follow.



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FIGURE18.5: Responseof amultipath channel to anarrow pulsevs. delay, asafunction of antennaposition.

18.4 Signal Time-SpreadingViewed in theTime-Delay

Domain: Figure18.1, Block 7The MultipathIntensity Profile

A simplewayto model thefading phenomenon wasintroduced byBello [13]in 1963; heproposed thenotion of wide-sensestationary uncorrelated scattering(WSSUS). Themodel treatssignal variationsarr ivingwith different delaysasuncorrelated. It can beshown[4, 13]that such achannel iseffectivelyWSS in both the ti me and frequency domains. With such a model of a fading channel, Bello wasable to define functions that apply for all time and all frequencies. For themobile channel, Fig. 18.7contains four functionsthat make up this model [4], [ 13][16] . We will examine these functions,starti ngwith Fig. 18.7(a) and proceeding counter-clockwisetoward Fig. 18.7(d).

In Fig.18.7(a), amultipath-intensityprofile, S() vs. time delay is plotted. Knowledge ofS()helps answer the questi on, For a transmitted impulse, how does the average received power

vary as a function of time delay, ? The term time delay is used to refer to the excess delay.It represents the signals propagation delay that exceeds the delay of the first signal arr ival at thereceiver. For a typi cal wireless radio channel, the received signal usually consists of several discretemulti path components, someti mesreferred to asfingers. For somechannels,suchasthetropospheri cscatter channel, received signals are often seen asa continuum of mult ipath components [14,16].For making measurements of the multipath intensit y profile, wideband signals (impulsesor spread



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FIGURE 18.6: Small-scale fading: mechanisms, degradati on categories, and effects.

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FIGURE 18.7: Relationships amongthe channel correlation functions and power densit y functions.

spectrum) need to beused [16]. For asingletr ansmitted impulse, thetime, Tm, between thefirst andlast received component represents themaximumexcessdelay,during which the multipath signal

power fallsto somethreshold level below that of thestrongest component. Thethreshold level mightbe chosen at 10 dB or 20 dB below the level of the strongest component. Note, that for an idealsystem (zero excessdelay), the functi on S()would consist of an ideal i mpulsewith weight equal tothe total averagereceived signal power.



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18.4.1 Degradation Categories due to Signal Time-Spreading Viewed in theTime-Delay Domain

In a fading channel, the relati onship between maximum excess delay time, Tm, and symbol time,Ts , can be viewed in terms of two different degradati on categories, frequency-selectivefadingandfrequencynonselectiveorflatfading, as indicated in Fig.18.1,blocks 8 and 9, and Fig.18.6. Achannel is said to exhibit frequency-selective fading ifTm > Ts . This conditi on occurs wheneverthe received multipath components of a symbol extend beyond the symbols ti me duration. Suchmultipath dispersion of thesignal yieldsthesamekind of ISI distorti on that iscaused byanelectronicfilter. In fact, another namefor this category of fading degradation ischannel-inducedISI. In thecaseof frequency-selectivefading, mit igatingthedi stort ion ispossiblebecausemany of themultipathcomponentsare resolvable by the receiver. Later, several such miti gati on techniquesare described.

A channel is said to exhibit frequency nonselectiveor flat fading ifTm < Ts . In this case, all ofthe received multipath components of a symbol arri vewithin the symbol timeduration; hence, thecomponentsarenot resolvable. Here, therei sno channel-induced ISI distort ion, sincethesignal timespreading does not result in significant overlap among neighboring received symbols. There is sti llperformancedegradati on sincetheunresolvablephasor componentscan add up destr uctivelyto yield

asubstantial reducti on in SNR.Also, signalsthat areclassified asexhibit ing flat fading can sometimesexperi encefrequency-selectivedistort ion. Thi swill be explained later when viewing degradati on inthe frequency domain, where the phenomenon is moreeasily described. For lossin SNR dueto flatfading, the mit igation technique called for is to improve the received SNR (or reduce the requiredSNR). For digital systems, introducing some form of signal diversity and using error-correctioncoding is the most efficient way to accomplish this.

18.5 Signal Time-SpreadingViewed in theFrequencyDomain: Figure18.1, Block 10TheSpaced-Frequency Correlation Function

A completely analogouscharacterizati on of signal dispersion can begin in the frequency domain. In

Fig. 18.7(b), thefunction |R(f )| isseen, designated aspaced-frequencycorrelationfunction;it istheFourier transform ofS(). R(f )representsthe correlation between the channels response totwo signals asafunction of thefrequencydifferencebetween the two signals. It can bet hought of asthechannelsfrequencyt ransfer function. Therefore, theti me-spreadingmanifestation can beviewedasif it were the result of a filteri ng process. KnowledgeofR(f )helps answer the questi on, Whatisthe correlation between received signalsthat are spaced in frequency f= f1 f2? R(f ) canbemeasured by tr ansmitting apair of sinusoidsseparated in frequencyby f, cross-correlating thetwo separately received signals, and repeati ngtheprocessmanyti meswith ever-larger separation f.Therefore, themeasurement of R(f ) can bemadewith asinusoid that isswept in frequencyacrossthe band of interest (a wideband signal). Thecoherencebandwidth, f0, is a statistical measure ofthe range of frequenciesover which the channel passesall spectral components with approximatelyequal gain and li near phase. Thus, the coherencebandwidth representsa frequencyrangeover whichfrequency components havea strong potential for amplitude correlation. That is, a signals spectr al

components in that rangeare affected by the channel in asimi lar manner, asfor example, exhibit ingfading or no fading. Note thatf0and Tmare reciprocally related (within a multiplicativeconstant).Asan approximation, it is possible to say that

f0 1

Tm(18.8)



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The maximum excess delay,Tm, i s not necessari ly the best indicator of how any given system willperform on achannel becausedifferent channels with the samevalue ofTmcan exhibit very differentprofilesof signal intensity over the delay span. A more useful measurement of delay spread is mostoften characterized in terms of the root mean squared (rms) delay spread, , where

=

2 ()2 (18.9)

isthe mean excessdelay, ( )2 isthe mean squared, 2 is the second moment, and isthesquareroot of the second central moment ofS()[ 3].

An exact relati onship between coherencebandwidth and delay spread doesnot exist, and must bederived from signal analysis (usually using Fouri er techniques) of actual signal dispersion measure-ments in parti cular channels. Several approximate relati onships havebeen described. If coherencebandwidth is defined asthe frequencyinterval over which the channels complex frequency transferfunction hasacorrelati on of at least 0.9, the coherencebandwidth is approximately [17]

f0 1

50

(18.10)

For the case of a mobile radio, an array of radially uniformly spaced scatterers, all with equal-magnitudereflection coefficients but independent, randomly occurr ing reflection phaseangles[18,19] is generally accepted asa useful model for urban surroundings. This model isreferred to asthedense-scattererchannelmodel.With the use of such a model, coherence bandwidth has similarlybeen defined [ 18] for a bandwidth interval over which the channels complex frequency t ransferfunction hasa correlation of at least 0.5to be

f0 =0.276

(18.11)

Theionospheric-effects community employsthefollowing defini tion

f0 = 12

(18.12)

A morepopular approximation off0corresponding to a bandwidth interval having acorrelation ofat least 0.5 is [ 3]

f0 1

5(18.13)

18.5.1 Degradation Categories due to Signal Time-Spreading Viewed in theFrequency Domain

A channel is referred to as frequency-selective iff0 < 1/Ts W, where the symbol rate 1/Ts isnominally taken to be equal to the signal bandwidthW. In practice,Wmay differ from1/Ts due

to system fi lteri ng or data modulation type (quaternary phase shift keying, QPSK, mi nimum shiftkeying, MSK, etc.) [21]. Frequency-selecti ve fading distort ion occurs whenever a signals spectralcomponents are not all affected equally by the channel. Some of the signals spectr al components,falling outside the coherence bandwidth, will be affected differently (i ndependently) compared tothosecomponents contained withi n the coherence bandwidth. Thisoccurs wheneverf0 < Wandis illustrated in Fig. 18.8(a).



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FIGURE 18.8: Relationships between the channel frequency-transfer function and a signal wit hbandwidth W.



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Frequency-nonselective or flat fading degradati on occurs whenever f0 > W. Hence, all of thesignals spectral components will be affected by the channel in a similar manner (e.g., fading orno fading); t his is il lustrated in Fig.18.8(b). Flat-fading does not introduce channel-induced ISIdistort ion, but performance degradation can sti ll be expected due to the loss in SNR whenever the

signal is fading. In order to avoid channel-i nduced ISI distorti on, the channel is required to exhibitflat fading by insuring that

f0 > W1

Ts(18.14)

Hence, the channel coherencebandwidth f0sets an upper limit on the transmission ratethat canbeused without incorporating an equalizer in the receiver.

For the flat-fading case, where f0 > W (or Tm < Ts ), Fig.18.8(b) shows the usual flat-fadingpictorial representation. However, as a mobile radio changes it s posit ion, there will be times whenthe received signal experiences frequency-selective distor tion even thoughf0 > W. Thisisseen inFig.18.8(c), where the null of the channels frequency tr ansfer function occurs at the center of thesignal band. Whenever thi soccurs, the baseband pulsewill beespecially muti lated by deprivati on ofitsDC component. Oneconsequenceof thelossof DC (zero mean value) is the absenceof a reliablepulse peak on which to establish the timing synchronizati on, or from which to sample the carrierphase carried by the pulse [18] . Thus, even t hough a channel is categorized asflat fading (based onrms relationships), it can sti ll manifest frequency-selectivefading on occasions. It is fair to say that amobileradio channel, classified ashavingflat-fadingdegradation, cannot exhibit flat fadingall of thetime. As f0becomesmuch larger thanW (orTmbecomesmuch smaller thanTs ), lesst imewill bespent in conditi onsapproximating Fig. 18.8(c). By comparison, it should beclear that in Fig. 18.8(a)the fading is independent of t he posit ion of t he signal band, and frequency-selective fading occursall the time, not just occasionally.

18.6 Typical Examples of Flat FadingandFrequency-Selective FadingManifestations

Figure18.9 showssomeexamplesof flat fading and frequency-selecti vefading for a direct-sequencespread-spectr um (DS/SS) system [20,22]. In Fig.18.9,there are three plots of the output of apseudonoise( PN) codecorrelator vs. delay asa function of time( transmission or observation time).Each amplitude vs. delay plot i sakin to S()vs. shown in Fig. 18.7(a). Thekey differenceis thatthe amplitudes shown in Fig. 18.9 represent the output of a correlator; hence, the waveshapesare afunction not only of thei mpulseresponseof thechannel, but also of thei mpulseresponseof thecor-relator. Thedelay timeisexpressed in unitsof chip durations(chips), wheret hechip isdefined asthespread-spectrum mini mal-durati on keying element. For each plot, theobservation ti meisshown onan axisperpendicular to theamplitudevs. time-delay plane. Figure18.9 isdrawn from asatelli te-to-ground communicati onslink exhibi ti ngscintillati on becauseof atmospheri cdisturbances. However,Fig. 18.9 isstil l auseful illustration of threedifferent channel conditionsthat might apply to amobileradio situation. A mobile radio that moves along the observation-time axis is affected by changingmultipath profilesalongtheroute, asseen in thefigure. Thescale alongtheobservation-time axisis

also in unitsof chips. In Fig.18.9(a), the signal dispersion (one finger of return) is on the orderof a chip time duration,Tch. In a typical DS/SS system, the spread-spectrum signal bandwidth isapproximately equal to 1/Tch; hence, the normalized coherencebandwidth f0Tchof approximatelyunity in Fig.18.9(a) impliesthat t he coherence bandwidth is about equal to the spread-spectrumbandwidth. Thisdescribesa channel t hat can becalled frequency-nonselectiveor sli ghtly frequency-selective. In Fig.18.9(b), where f0Tch = 0.25, the signal dispersion is more pronounced. There is



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FIGURE18.9: DS/SSMatched-filter output time-histor y examplesfor threelevels of channel condi-

ti ons, where Tchis thetimeduration of a chip.

definite interchip interference, and the coherencebandwidth i s approximately equal to 25% of thespread-spectrum bandwidt h. In Fig. 18.9(c), where f0Tch = 0.1, the signal dispersion is even more

pronounced, with greater interchip-interference effects, and the coherence bandwidth is approxi-mately equal to 10% of the spread-spectrum bandwidth. The channels of Figs.18.9(b) and (c) canbe categorized asmoderately and highly frequency-selective, respectively, with respect to the basicsignall ing element, the chip. Later, weshow that aDS/SSsystem operating over a frequency-selectivechannel at the chip level doesnot necessari ly experi encefrequency-selectivedi storti on at the symbollevel.



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18.7 Time VarianceViewed in the TimeDomain:Figure18.1, Block 13TheSpaced-TimeCorrelationFunction

Until now, wehavedescribed signal dispersion and coherence bandwidth, parametersthat describethechannelst ime-spreading propert iesin alocal area. However, they do not offer information aboutthe time-varyingnature of the channel caused byrelativemotion between atransmitter and receiver,or by movement of objects within t he channel. For mobile-radio applications, the channel i s timevari ant because moti on between the transmitter and receiver results in propagati on-path changes.Thus, for a transmitted continuouswave(CW) signal, asa result of such motion, the radio receiversees vari ations in the signals amplitude and phase. Assuming that all scatterers making up thechannel arestationary, then whenever motion ceases, the ampli tude and phaseof thereceived signalremain constant; t hat is, the channel appears to beti me invariant. Whenever motion begins again,the channel appears ti me variant. Sincethe channel characteristicsaredependent on the posit ionsof thetr ansmitter and receiver, ti mevariancein thiscaseis equivalent to spati al vari ance.

Figure18.7(c) showsthefunctionR(t), designated thespaced-timecorrelationfunction;it is

theautocorrelation function of thechannelsresponseto asinusoid. Thisfunction specifi estheextentto which there is correlation between the channels response to a sinusoid sent at time t1 and theresponseto asimilar sinusoid sent at ti met2, wheret= t2t1. Thecoherencetime,T0, isameasureof the expected timeduration over which the channels response is essentially invariant. Earlier, wemademeasurements of signal dispersion and coherencebandwidth by usingwideband signals. Now,to measure the time-variant nature of thechannel, weusea narrowband signal. To measure R(t)we can t ransmit a single sinusoid (f = 0) and determine the autocorrelation function of thereceived signal. The function R(t)and the parameterT0 provide us with knowledge about t hefading rapidity of the channel. Note that for an idealtime-invariantchannel (e.g., a mobile radioexhibit ing no motion at all), the channels response would behighly correlated for all values oft,and R(t)would bea constant function. When using the dense-scatterer channel model describedearlier, with constant velocity of motion, and an unmodulated CW signal, the normalized R(t)isdescribed as

R(t) = J0(kVt) (18.15)

whereJ0() isthezero-order Bessel function of thefi rst kind, V isvelocity, V tisdistancetraversed,and k = 2/ isthefree-spacephaseconstant (transformingdi stanceto radiansofphase). Coherenceti mecan bemeasured in terms of either ti meor distancetr aversed (assuming somefi xed velocity ofmoti on). Amoroso described such a measurement using a CW signal and a dense-scatterer channelmodel [18]. He measured thestatisti cal correlation between thecombination of received magnitudeand phasesampled at apart icular antennalocati on x0, and thecorrespondingcombinati on sampledat some displaced locati onx0 + , with displacement measured in units of wavelength . For adisplacement of0.38 between two antennalocations, thecombined magnitudesandphasesof thereceived CW are statistically uncorrelated. In other words, the stateof the signal at x0saysnothi ngabout the state of the signal atx0 + . For a given velocity of motion, this displacement is readilytransformed into units of t ime (coherencetime).

18.7.1 TheConcept of Duality

Two operators (functions, elements, or systems) are dual when the behavior of onewi th referencetoatime-related domain (time or time-delay) is identi cal to the behavior of the other with referencetothe corresponding frequency-related domain (frequencyor Doppler shift).



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In Fig.18.7,we can identi fy functions that exhibit simi lar behavior across domains. For under-standing the fading channel model, it is useful to refer to such functions as duals. For example,R(f )in Fig.18.7(b), characterizing signal dispersion in the frequency domain, yieldsknowledgeabout the rangeof frequency over which two spectral components of areceived signal havea strong

potential for amplitude and phasecorrelation. R(t)in Fig. 18.7(c), characterizing fading rapidityin thetimedomain, yieldsknowledgeabout thespan of timeover which two received signals haveastrongpotenti al for amplitudeandphasecorrelation. Wehavelabeled thesetwocorrelation functionsas duals. This is also noted in Fig. 18.1 as the duali ty between blocks10 and 13, and in Fig. 18.6 asthe duali ty between the time-spreading mechanism in the frequency domain and the time-variantmechanism in thetimedomain.

18.7.2 Degradation Categories dueto Time Variance Viewed in theTimeDomain

Thetime-variant natureof the channel or fading rapidity mechanism can beviewed in terms of twodegradati on categori es as li sted in Fig.18.6: fast fadingandslowfading. The terminology fast

fading is used for descri bing channelsin which T0 < Ts , whereT0is the channel coherencet imeandTs is the time duration of atr ansmission symbol. Fast fading descri bes a conditi on where the timeduration in which thechannel behavesin acorrelated manner isshort compared to thetimedurationof asymbol. Therefore, it can beexpected that thefading character of thechannel will changeseveraltimes during the time that a symbol is propagating, leading to distorti on of the baseband pulseshape. Analogous to the distort ion previously described as channel-induced ISI, here distort iontakes place because the received signalscomponents are not all highly correlated throughout time.Hence, fast fading can causethe baseband pulseto bedistorted, resulting in a lossof SNR that oftenyields an i rreducible error rate. Such distorted pulses cause synchronization problems (failure ofphase-locked-loop receivers), in addition to difficultiesin adequately defining a matched filter.

A channel isgenerally referredto asint roducingslowfadingifT0 > Ts . Here, thetimedurationthatthe channel behavesin acorrelated manner islong compared to the timeduration of atransmissionsymbol. Thus, one can expect the channel state to virtually remain unchanged during the time

in which a symbol i s transmitted. The propagating symbols will likely not suffer from the pulsedistort ion described above. The primary degradati on in a slow-fading channel, as with fl at fading,is lossin SNR.

18.8 Time VarianceViewed in the Doppler-Shift Domain:Figure18.1, Block 16TheDoppler Power Spectrum

A completely analogouscharacterizati on of the ti me-variant nature of the channel can begin in theDoppler-shift ( frequency) domain. Figure 18.7(d) showsaDopplerpowerspectral density,S(v),plotted as a function of Doppler-frequency shift, v . For the case of the dense-scatterer model, avert ical receive antenna with constant azimuthal gain, a uniform distr ibution of signals arr iving atall arr ival anglesthroughout therange(0, 2), and an unmodulated CW signal, thesignal spectrumat the antennaterminalsis[19]

S(v) =1

fd

1

vfc

fd

2 (18.16)



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Theequali ty holdsfor frequencyshifts ofv that arein the rangefdabout thecarrier frequency fcand would bezero outsidethat range. Theshapeof theRFDoppler spectr um described byEq.( 18.16)isclassically bowl-shaped, asseen in Fig.18.7(d). Notethat thespectral shapeisa result of thedense-scatterer channel model. Equati on (18.16)has been shown to match experi mental data gathered

for mobile radio channels [23]; however, dif ferent applicati ons yield different spectral shapes. Forexample, the dense-scatterer model does not hold for t he indoor radio channel; the channel modelfor an indoor areaassumesS(v)to bea flat spectrum [24].

In Fig.18.7(d), the sharpness and steepness of the boundaries of the Doppler spectrum are dueto the sharp upper limit on the Doppler shift produced by a vehicular antennatraveling among thestationary scatterers of the dense scatterer model. The largest magnitude (i nfinite) ofS(v)occurswhen thescatterer isdirectly ahead of themovingantennaplatform or directly behind it. In that casethe magnitude of t hefrequency shift is given by

fd=V

(18.17)

where V is relativevelocity and is the signal wavelength. fd is posit ive when the tr ansmitter and

receiver move toward each other and negative when moving away from each other. For scatterersdirectly broadsideof themovingplatform, themagnitudeof thefrequency shift iszero. Thefact thatDoppler components arr iving at exactly 0 and 180 havean infinite power spectral densit y is nota problem, sincetheangle of arri val is continuously distr ibuted and the probability of componentsarr iving at exactly theseanglesiszero [3, 19].

S(v) istheFourier transform ofR(t). Weknowthat theFourier transform of theautocorrelationfunction of atimeseriesisthemagnitudesquared of theFourier transform of theoriginal timeseries.Therefore, measurements can be made by simply transmitti ng a sinusoid (narrowband signal) andusing Fourier analysis to generatethe power spectr um of the received amplitude [16]. ThisDopplerpower spectrum of the channel yields knowledge about the spectral spreading of a transmittedsinusoid (impulsein frequency) in theDoppler-shift domain. Asi ndicated in Fig. 18.7, S(v)can beregarded asthe dual of the multipath intensity profile, S(), sincethe latter yields knowledgeaboutthetimespreading of atransmitted impulsei n thetime-delay domain. Thisisalso noted in Fig. 18.1

as the duali ty between blocks7 and 16, and in Fig.18.6 as the duali ty between the time-spreadingmechanism in thetime-delay domain and thet ime-variant mechanism in theDoppler-shift domain.

KnowledgeofS(v)allowsus to glean how much spectral broadening is imposed on the signal asa function of the rate of change in the channel state. The width of the Doppler power spectrumis referred to asthespectralbroadeningorDopplerspread,denoted by fd, and sometimes calledthefadingbandwidthof the channel. Equati on (18.16) describesthe Doppler frequency shift. In atypical multipath environment, the received signal arr ivesfrom several reflected pathswith differentpath distancesand different anglesof arrival, and the Doppler shift of each arr iving path is generallydifferent from that of another path. The effect on the received signal i sseen asa Doppler spreadingor spectral broadening of thetransmitted signal frequency, rather than ashift. Note that theDopplerspread,fd, and the coherence ti me,T0, are reciprocally related (within a multiplicativeconstant).Therefore, weshow the approximaterelationship between the two parameters as

T0 1fd

(18.18)

Hence, the Doppler spread fdor 1/T0isregarded asthet ypical fadingrateof thechannel. Earlier,T0 was described as the expected t ime duration over which the channels response to a sinusoidis essentially invari ant. When T0 is defined more precisely as the time duration over which the



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channels response to a sinusoid hasa correlati on of at least 0.5, the relationship betweenT0and fdis approximately [ 4]

T0 9

16fd(18.19)

A popular ruleof thumb isto define T0as the geometr ic mean of Eqs. (18.18) and (18.19). Thisyields

T0 =

9

16f2d=

0.423

fd(18.20)

For thecase of a900 MHz mobileradio, Fig. 18.10 il lustratesthe typical effect of Rayleigh fadingon a signals envelope amplitude vs. time [ 3]. The figure shows that the distance traveled by the

FIGURE 18.10: A typical Rayleigh fading envelopeat 900 MH z.

mobile in the time interval corresponding to two adjacent null s (small-scale fades) is on the orderof a half-wavelength (/2)[ 3]. Thus, from Fig. 18.10 and Eq. (18.17), thet ime(approximately, thecoherenceti me) required to tr aversea distance/2when tr aveli ng at a constant velocit y, V, is:

T0 /2V

= 0.5fd

(18.21)

Thus, when the interval between fadesis taken to be/2, asin Fig. 18.10, the result ing expressionfor T0in Eq. (18.21) isquitecloseto therule-of-thumb shown in Eq.(18.20). Using Eq. (18.21),withthe parametersshown in Fig. 18.10 (velocity= 120 km/hr, and carr ier frequency= 900 MHz), it is



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str aightforward to computethat the coherence time is approximately 5 msand theDoppler spread(channel fading rate) is approximately 100 Hz. Therefore, i f this example represents a voice-gradechannel with a typical tr ansmission rate of 104 symbols/s, the fading rate is considerably less thanthe symbol rate. Under such conditi ons, the channel would manifest slow-fading effects. Note that

if the abscissaof Fig. 18.10 werelabeled in units of wavelength instead of time, thefigurewould lookthe samefor any radio frequencyand any antenna speed.

18.9 Analogy Between Spectral Broadeningin FadingChannels andSpectral Broadeningin Digital SignalKeying

Help is often needed in understanding why spectr al broadening of the signal isa function of fadingrate of the channel. Figure 18.11 uses the keying of a digital signal (such asamplitude-shift-keyingor frequency-shift-keying) to illustr ate an analogouscase. Figure18.11(a) shows that a single tone,cos2fct ( < t < )that exists for all t ime is characterized in the frequency domain in termsof i mpulses (atfc). This frequency domain representation is ideal (i .e., zero bandwidth), sincethe tone is pure and neverending. In practical applications, digital signalli ng involves switching(keying) signalson and off at arequired rate. Thekeyingoperation can beviewed asmultiplying theinfinite-duration tonei n Fig. 18.11(a) by an ideal rectangular (switching) function in Fig. 18.11(b).The frequency-domain descripti on of the ideal rectangular function is of the form (sin f )/f. InFig. 18.11(c), theresult of the multipli cation yields a tone, cos 2fct, that istime-duration limitedin the interval T /2 < t < T /2. The resulting spectr um is obtained by convolving the spectralimpulsesin part (a) with the(sin f )/f function in part (b), yielding thebroadened spectrum in part(c). It is further seen that, if the signalling occurs at a faster rate characterized by the rectangle ofshorter duration in part ( d), the resulting spectrum of thesignal in part ( e) exhibitsgreater spectr albroadening. Thechanging stateof a fading channel i ssomewhat analogousto the keying on and offof digital signals. The channel behaves like a switch, turning the signal on and off. The greaterthe rapidity of the change in the channel state, the greater the spectr al broadening of the received

signals. The analogy is not exact because the on and off switching of signals may result in phasedisconti nuiti es, but thetypi cal multi path- scatterer environment inducesphase-continuouseffects.

18.10 Degradation Categories due to Time Variance, Viewed intheDoppler-Shift Domain

A channel isreferred to asfast fading if thesymbol rate, 1/Ts(approximately equal to the signallingrate or bandwidth W) is less than the fading rate, 1/T0 (approximately equal to fd); that is, fastfading is characterized by

W < fd (18.22a)

or

Ts > T0 (18.22b)

Conversely, a channel is referred to asslow fading if the signalling rate is greater than the fading



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FIGURE18.11: Analogy between spectr al broadening in fading and spectral broadening in keying

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rate. Thus, in order to avoid signal distort ion caused by fast fading, the channel must be made toexhibit slow fading by insuri ng that thesignalling ratemust exceed the channel fading rate. That is

W > fd (18.23a)

or

Ts < T0 (18.23b)

In Eq. (18.14), it was shown that due to signal dispersion, the coherence bandwidth, f0, sets anupper limit on thesignalli ngratewhich can beused without sufferingfrequency-selectivedistorti on.Simil arly, Eq. (18.23a18.23b)showsthat dueto Doppler spreading, the channel fading rate, fd, setsa lower l imit on thesignalli ngrate that can beused without suffering fast-fading distorti on. For HFcommunicati ngsystems, when teletypeor Morse-coded messagesweretr ansmitted at al owdata rate,the channels were often fast fading. However, most present- day terrestri al mobile-radio channelscan generally becharacterized asslow fading.

Equation (18.23a18.23b) doesnt go far enough i n describing what wedesire of the channel. Abetter way to state the requirement for miti gating the effects of fast fading would be that we desireW fd (orTs T0). If this condition i s not satisfied, the random frequency modulation (FM)due to varying Doppler shifts will limit the system performance significantly. The Doppler effectyields an irreducible error rate that cannot be overcome by simply increasing Eb/N0 [ 25]. Thisir reducible error rate is most pronounced for any modulation that involves switching the carri erphase. A singlespecular Doppler path, wit hout scatterers, registersan instantaneousfrequencyshift,classicall y calculated asfd= V /. However, a combinati on of specular and multi path componentsyields a rather complex time dependenceof i nstantaneous frequency which can cause much largerfrequency swingsthan V /when detected by an instantaneous frequency detector (a nonlineardevice) [26]. Ideally, coherent demodulators that lock onto and track the information signal shouldsuppressthe effect of this FM noise and thus cancel the impact of Doppler shift. However, for largevaluesoff

d, carr ier recovery becomes a problem because very wideband (relativeto the data rate)

phase-lock loops (PLLs) need to be designed. For voice-grade applications with bit-error rates of103 to 104, alargevalueof Doppler shift is considered to beon theorder of0.01W. Therefore,to avoid fast-fading distorti on and the Doppler-induced irreducible error rate, the signalli ng rateshould exceed the fading rate by a factor of 100to 200[27]. The exact factor depends on the signalmodulation, receiver design, and required error-rate [3], [26][29] . Davarian [29] showed that afrequency-tracking loop can help lower, but not completely remove, the ir reducible error rate in amobile system when using differenti al minimum-shift keyed (DM SK) modulation.

18.11 Mitigation Methods

Figure 18.12,subtitled The Good, The Bad, and The Awful, hi ghlights threemajor performance

categoriesin termsof bit-error probabilit y, PB , vs. Eb/N0. Theleftmost exponentially-shaped curverepresentst heperformancethat can beexpected when using anynominal modulati on typei n AWGN.Observe that with a reasonable amount ofEb/N0, good performance results. The middle curve,referred to astheRayleighlimit, shows the performance degradation resulting from a loss in SNRthat is characteristi c of flat fading or slow fading when there is no line-of-sight signal componentpresent. The curve is a function of the reciprocal of Eb/N0 (an inverse-linear function), so for



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reasonable values of SNR, performance wil l generally be bad. In the case of Rayleigh fading,parameters with overbars are often introduced to indicate that amean is being taken over theupsand downs of thefading experience. Therefore, oneoften seessuch bit- error probabili ty plotswithmean parametersdenoted by PBand Eb/N0. Thecurvethat reachesan ir reduciblelevel, someti mes

called an errorfloor,representsawful performance, where the bit-error probabil it y can approachthe value of 0.5. This showsthe severe distort ing effects of frequency-selectivefading or fast fading.

FIGURE 18.12: Error performance: Thegood, the bad, and the awful.



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If the channel introduces signal distort ion as a result of fading, the system performance canexhibit an irreducible error rate; when larger than the desired error rate, no amount ofEb/N0willhelp achieve the desired level of performance. In such cases, the general approach for improvingperformance is to use someform of mitigation to removeor reduce the distorti on. The miti gation

method depends on whether the distort ion is caused by frequency-selective fading or fast fading.Once the distort ion has been mit igated, the PB vs. Eb/N0 performance should have tr ansiti onedfrom the awful bottomingout curveto themerely bad Rayleigh limit curve. Next, wecan furtherameliorate the effects of fading and str ive to approach AWGN performance by using someform ofdiversit y to providethe receiver with acollection of uncorrelated samplesof the signal, and by usinga powerful error-correction code.

In Fig. 18.13, several miti gation techniquesfor combating the effects of both signal distort ion andlossin SNR areli sted. Just asFigs. 18.1 and 18.6 serveasa guidefor characterizingfading phenomenaand their effects, Fig.18.13 can similarly serve to describe mit igation methods that can be used toameliorate the effects of fading. The mit igation approach to beused should follow two basic steps:first, provide distorti on miti gation; second, provide diversity.

18.11.1 Mitigation to Combat Frequency-Selective Distortion

Equali zati on can compensate for the channel- induced ISI that is seen in frequency-selectivefading. That is, it can help movetheoperating point from theerror-performancecurvethat isawful in Fig. 18.12 to the onethat i s bad. Theprocessof equalizing theISI involvessomemethod of gatheringthedispersed symbol energy back together into itsoriginal timeinterval. In effect, equalizationinvolvesinsertion of afil ter to makethecom-binati on of channel and filter yield a flat responsewith linear phase. Thephaselineari tyis achieved bymaking theequalizer filter thecomplex conjugateof thetimereverseof thedispersed pulse[ 30]. Because in a mobile system the channel response vari eswit h ti me,theequalizer filter must also changeor adapt to thetime-varyingchannel. Such equalizerfil ters are, therefore, called adaptive equali zers. An equali zer accomplishes more thandistort ion miti gation; it also provides diversit y. Since distort ion mitigation is achieved

by gatheri ng the dispersed symbols energy back into the symbols original timeintervalso that it doesnt hamper thedetecti on of other symbols, the equali zer issimult aneouslyproviding each received symbol wit h energy that would otherwisebelost.

Thedecision feedbackequali zer (DFE) hasafeedforward section that isalinear tr ansversalfilter [ 30]whoselength and tapweightsareselected to coherently combinevirtually all ofthe current symbolsenergy. TheDFEalso hasafeedback section which removesenergythat remains from previously detected symbols [14] , [ 30][32]. The basic idea behindthe DFE is that once an information symbol hasbeen detected, the ISI that it induceson future symbols can be esti mated and subtracted before the detecti on of subsequentsymbols.

The maximum-l ikelihood sequenceesti mati on (MLSE) equali zer tests all possible datasequences (r ather than decoding each received symbol by i tself ) and chooses the data

sequence that is the most probable of the candidates. The MLSE equali zer was firstproposed by Forney [33] when heimplemented the equalizer using the Viterbi decodingalgorithm [34]. The MLSE is optimal in the sense that it minimizes the probabili tyof a sequence error. Because the Viterbi decoding algorithm i s the way i n which theMLSE equalizer istypically implemented, the equalizer is often referred to astheViterbiequalizer. Later in thischapter, weil lustr ate the adaptiveequalization performed in the



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FIGURE 18.13: Basic miti gati on types.

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Global System for Mobile Communicati ons(GSM) using the Viterbi equali zer.

Spread-spectrum techniques can be used to miti gate frequency-selecti ve ISI distorti onbecausethehallmark of anyspread-spectrum systemisitscapabil it y to reject interference,and I SI is a type of interference. Consider a direct-sequencespread-spectrum (DS/SS)

binary phaseshift keying (PSK) communicati on channel comprisingonedirect path andone reflected path. Assume that the propagation from t ransmitter to receiver resultsin a multipath wave that is delayed by k compared to the direct wave. If the receiveris synchronized to the waveform arri ving via the direct path, the received signal, r(t),neglecting noise, can beexpressed as

r(t) = Ax(t)g(t) cos (2fct)+ Ax(t k ) g (t k) cos (2fct+ ) (18.24)

wherex(t)isthedatasignal, g(t)is the pseudonoise(PN) spreading code, andk isthedifferenti al timedelaybetween thetwo paths. Theangle isarandom phase, assumed tobeuniformly distributed in ther ange(0, 2), and is theattenuation of themultipathsignal relativeto the direct path signal. Thereceiver multi pliesthe incoming r(t)by thecode g(t). If the receiver is synchronized to the direct path signal, multiplication by the

codesignal yields

Ax(t)g2(t) cos (2fct) + Ax(t k ) g(t)g (t k ) cos (2fct+ ) (18.25)

whereg2(t) = 1, and ifkis greater than the chip duration, then,

g(t)g (t k) dt

g(t)g(t)dt (18.26)

over some appropriate interval of integration ( correlation), where indicatescomplexconjugate, and k is equal to or larger than the PN chip duration. Thus, the spreadspectr um system effectively eliminates the multipath interference by vir tue of it s code-correlation receiver. Even though channel-induced ISI is typically tr ansparent to DS/SSsystems, such systemssuffer from thelossin energy contained in all the multipath com-

ponents not seen by the receiver. Theneed to gather up thislost energy belonging to thereceived chip wast he motivati on for developing the Rake receiver [ 35][37]. TheRakereceiver dedicatesa separate correlator to each multi path component (finger). It is ableto coherently add the energy from each finger by selecti vely delaying them (the earl iestcomponent gets the longest delay) so that they can all becoherently combined.

Earlier, we described a channel that could be classified as flat fading, but occasionallyexhibi ts frequency-selecti vedistort ion when the null of the channels frequency tr ansferfunction occursat thecenter of thesignal band. Theuseof DS/SSisagoodwayto mitigatesuch distort ion becausethe wideband SSsignal would span many lobesof the selectivelyfaded frequency response. Hence, a great deal of pulse energy would then be passed bythe scatterer medium, in contr ast to the nulling effect on a relatively narrowband signal[seeFig. 18.8(c)] [ 18].

Frequency-hoppingspread-spectr um (FH/SS) can beused to mit igate the distort ion dueto frequency-selective fading, provided the hopping rate is at least equal to the symbolrate. Compared to DS/SS, mit igati on takes place through a different mechanism. FHreceiversavoid multipath lossesby rapid changesi n thetr ansmitter frequencyband, thusavoiding the interferenceby changing the receiver band posit ion before the arrival of themultipath signal.



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Orthogonal frequency-division multi plexing (OFDM) can beused in frequency-selectivefading channelsto avoid theuseof an equali zer bylengthening thesymbol duration. Thesignal band is part it ioned into multiple subbands, each one exhibit ing a lower symbolratethan the original band. Thesubbandsare then transmitted on multiple orthogonal

carri ers. The goal is to reduce the symbol rate (signalli ng rate), W 1/Ts , on eachcarr ier to be less than t he channels coherence bandwidth f0. OFDM was originallyreferred to asKineplex. Thetechniquehasbeen implemented in theU.S. in mobiler adiosystems [38] , and hasbeen chosen by the European community under the name CodedOFDM (COFDM) , for high-definition television (HDTV) broadcasting [39].

Pilot signal is the name given to a signal intended to facil it ate the coherent detection ofwaveforms. Pilot signals can be implemented in the frequency domain as an in- bandtone[40], or in thetimedomain asa pilot sequence, which can also provideinformationabout the channel stateand thus improveperformancein fading [41].

18.11.2 Mitigation to Combat Fast-FadingDistortion

For fast fading distort ion, usearobust modulation (noncoherent or differentially coher-ent) that doesnot require phasetr acking, and reducethe detector integration time[20].

Increase the symbol rate,W 1/Ts , to be greater t han the fading rate,fd 1/T0, byadding signal redundancy.

Error-correction coding and interleaving can providemit igation becauseinstead of pro-viding moresignal energy, a codereducesthe required Eb/N0. For a given Eb/N0,withcoding present, the error fl oor will belowered compared to the uncoded case.

An interesti ng filtering technique can provide mit igation in the event of fast-fading dis-torti on and frequency-selecti ve distort ion occurring simult aneously. The frequency-selective distort ion can be mitigated by the use of an OFDM signal set. Fast fading,however, wil l t ypically degradeconventional OFDM becausethe Doppler spreading cor-rupts the orthogonality of the OFDM subcarri ers. A polyphase filtering technique[ 42]is used to provide ti me-domain shaping and duration extension to reduce the spectr al

sidelobes of the signal set and thus help preserve its orthogonali ty. The process int ro-duces known I SI and adjacent channel interference(ACI) which are then removed by apost-processing equali zer and canceli ng filter [ 43] .

18.11.3 Mitigation to Combat Loss in SNR

After implementingsomeform of mit igationto combat thepossibledistort ion (frequency-selectiveorfast fading), thenext step isto usesomeform of diversity to movetheoperatingpoint from theerror-performance curvelabeled as bad in Fig.18.12 to a curve that approaches AWGN performance.Theterm diversit y isused to denote the vari ousmethodsavailable for providing the receiver withuncorrelated renditions of the signal. Uncorrelated is the important feature here, sinceit would nothelp t he receiver to have additional copies of the signal if the copies were all equally poor. Listedbelow are someof thewaysin which diversity can beimplemented.

Time diversityTransmit the signal onLdifferent timeslots with timeseparation of atleast T0. Interleaving, often used with error-correction coding, isaform of timediversity.

Frequency diversityTransmit the signal on L di fferent carr iers with frequency sep-aration of at least f0. Bandwidth expansion is a form of frequency diversit y. The



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signal bandwidth, W, is expanded to be greater than f0, thus providing the receiverwit h several i ndependently fading signal repli cas. This achieves frequency diversity ofthe order L = W/f0. Whenever W is made larger than f0, there is the potential forfrequency-selectivedistort ion unlesswefur ther provide somemiti gati on such asequal-

izati on. Thus, an expanded bandwidth can improvesystem performance (via diversity)only if the frequency-selectivedistortion the diversit y may haveintroduced is mit igated.

Spread spectrum is a form of bandwidth expansion that excels at rejecting interferi ngsignals. In the caseof direct-sequencespread-spectrum (DS/SS), it wasshown earlier thatmultipath components are rejected if they are delayed by more than onechip durati on.However, in order to approach AWGN performance, it is necessary to compensate forthe lossin energy contained in thoserejected components. TheRakereceiver (describedlater) makes it possible to coherently combine the energy from each of the multipathcomponents arr ivi ng along different paths. Thus, used with a Rake receiver, DS/SSmodulation can be said to achievepath diversity. The Rake receiver is needed in phase-coherent reception, but in differentially coherent bit detection, a simple delay line (onebit long) with complex conjugation will do thet rick [ 44].

Frequency-hopping spread-spectrum (FH/SS) is someti mes used as a diversity mecha-nism. The GSM system usesslow FH (217 hops/s) to compensate for thosecaseswherethemobileuser ismoving very slowly (or not at all) and happensto bein aspectral null.

Spatial diversit y is usually accomplished through t he use of multiple receive antennas,separated by a distance of at least 10 wavelengths for a base station ( much less for amobilestation). Signal processing must beemployed to choosethe best antennaoutputor to coherently combine all the outputs. Systems have also been implemented withmultiplespaced transmitters; an example is the Global Posit ioning System (GPS).

Polari zation diversit y [45]isyet another way to achieveadditional uncorrelated samplesof t hesignal.

Anydiversity schememay beviewed asat rivial form of repetition codingin spaceor time.However, there exist techniques for improving the loss in SNR in a fading channel thatare more efficient and more powerful than repeti tion coding. Error-correcti on coding

representsauniquemiti gati on technique, becauseinstead of providingmoresignal energyit reduces the requiredEb/N0 in order to accompli sh the desired error performance.Error-correction coding coupled with interleaving [ 20], [ 46] [51] is probably themostprevalent of the mit igation schemes used to provide improved performance in a fadingenvironment.

18.12 Summary of theKey ParametersCharacterizingFadingChannels

We summarize the conditi onsthat must be met so that the channel does not introduce frequency-selectivedistort ion andfast-fadingdistort ion. Combiningtheinequali tiesofEqs.(18.14) and(18.23a

18.23b), weobtain

f0 > W > fd (18.27a)

or



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Tm < Ts < T0 (18.27b)

In other words, wewant thechannel coherencebandwidth to exceed our signalling rate, which in

turn should exceed the fading rateof the channel. Recall t hat without distort ion mit igation, f0setsan upper li mit on signalli ng rate, and fdsets a lower li mit on it.

18.12.1 Fast-FadingDistortion: Example #1

If the inequalit ies of Eq. (18.27a18.27b)are not met and distorti on mitigation is not provided,distort ion will result. Consider thefast-fading casewhere the signalling rateis lessthan the channelfading rate, that is,

f0 > W < fd (18.28)

Miti gation consists of using oneor moreof the following methods. (SeeFig. 18.13).

Choose a modulation/demodulation technique that i s most robust under fast-fadingconditi ons. That means, for example, avoiding carrier recovery with PLLssincethe fast

fading could keep a PLL from achieving lock conditions.

Incorpor atesuffi cient redundancyso that the tr ansmission symbol rateexceedsthechan-nel fading rate. Aslong as the tr ansmission symbol rate does not exceed the coherencebandwidt h, the channel can be classified asflat fading. However, even flat-fading chan-nelswill experiencefrequency-selectivedistort ion whenever achannel null appearsat theband center.

Since this happens only occasionally, miti gati on might be accomplished by adequate error-correction coding and interleaving.

Theabovetwo mitigation approachesshould result in the demodulator operating at theRayleigh limit [ 20] (see Fig.18.12). However, there may be an i rreducible floor i n theerror-performancevs. Eb/N0curve dueto the FM noise that results from the random

Doppler spreading. The use of an i n-band pi lot tone and a frequency-control loop canlower thisir reducible performancelevel.

Toavoid thiserror floor caused byrandom Doppler spreading, increasethesignalling rateabovethe fading ratestil l further (100200 fading rate) [ 27]. Thisisonearchitecturalmotive behind time-division multipleaccess(TDMA) mobile systems.

Incorporate error-correction coding and interleaving to l ower the floor and approachAWGN performance.

18.12.2 Frequency-Selective Fading Distortion: Example #2

Consider the frequency-selective case where the coherence bandwidt h is less than the symbol rate;that is,

f0

< W > fd

(18.29)

Mi tigation consistsof using oneor moreof the following methods. (SeeFig. 18.13).

Since the t ransmission symbol rate exceeds the channelfading rate, there is no fast-fading distort ion. Mitigation of frequency-selectiveeffects is necessary. Oneor more ofthe following techniquesmay beconsidered:



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Adaptiveequalizati on, spread spectrum (DSor FH), OFDM, pilot signal. TheEuropeanGSM system uses a midamble tr aining sequence in each transmission ti me slot so thatthereceiver can learn theimpulseresponseof thechannel. It then usesa Viterbi equalizer(explained later) for mit igating the frequency-selectivedistort ion.

Once the distort ion effects have been reduced, int roduce some form of diversit y anderror-correction coding and interleaving in order to approach AWGN performance. Fordirect- sequencespread-spectrum (DS/SS) signalli ng, theuseof aRakereceiver (explainedlater) maybeused for providingdiversit ybycoherentlycombiningmultipathcomponentsthat would otherwisebel ost.

18.12.3 Fast-FadingandFrequency-Selective FadingDistortion:Example #3

Consider thecasewherethecoherencebandwidth is lessthan thesignalling rate, which in turn is lessthan the fading rate. The channel exhibits both fast-fading and frequency-selective fading which isexpressed as

f0 < W < fd (18.30a)

or

f0 < fd (18.30b)

Recalli ng from Eq. (18.27a18.27b)thatf0sets an upper limit on signalli ng rate and fdsets alowerlimit on i t, this is a difficult design problem because, unless distorti on mit igation is provided, themaximum allowable signalling rate is (in the strict terms of the above discussion) less than theminimum allowablesignalli ngrate. Mitigation in this casei ssimil ar to theinitial approach outlinedin example #1.

Choose a modulation/demodulation technique that is most robust under fast-fadingconditions.

Usetr ansmission redundancyin order to increasethe transmitted symbol r ate.

Providesomeform of frequency-selectivemitigation in amanner similar to that outlinedin example #2.

Once the distort ion effects have been reduced, int roduce some form of diversit y anderror-correction coding and interleaving in order to approach AWGN performance.

18.13 TheViterbi Equalizer as Applied to GSM

Figure 18.14shows the GSM time-division multiple access (TDM A) frame, having a duration of

4.615 ms and comprising 8 slots, one assigned to each active mobile user. A normal tr ansmissionburst occupying oneslot of ti mecontains 57messagebitson each side of a26-bit midamblecalled atrainingor soundingsequence. Theslot-time duration is 0.577ms (or theslot ratei s1733 slots/s).Thepurposeof themi dambleisto assist thereceiver in esti matingt heimpulseresponseof thechannelin an adaptiveway (during thetimeduration of each 0.577 msslot). I n order for thetechniqueto beeffective, the fading behavior of thechannel should not changeappreciably during the timeinterval



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FIGURE 18.14: TheGSM TDMA frameand time-slot containing a normal burst.

of oneslot. In other words, thereshould not beany fast-fading degradation during aslot timewhenthe receiver is using knowledgefrom the midamble to compensate for the channelsfading behavior.Consider the example of a GSM receiver used aboard a high-speed train, tr aveli ng at a constantvelocity of 200 km/hr (55.56 m/s). Assumethecarri er frequency to be900 MHz, (the wavelength is = 0.33 m). From Eq.(18.21), wecan calculatethat ahalf-wavelength istr aversed in approximatelythetime(coherencetime)

T0 /2

V 3ms (18.31)

Therefore, thechannel coherencetimeisover 5ti mesgreater thantheslot timeof 0.577ms. Thetimeneeded for asignificant changei n fading behavior isrelatively long compared to theti meduration ofoneslot. Note, that thechoicesmadein thedesign of theGSM TDMA slot timeand midamblewereundoubtedly influenced by the need to preclude fast fading with respect to a slot-t ime duration, asin this example.

The GSM symbol rate (or bit rate, since the modulation is binary) is 271 ki losymbols/s andthe bandwidth is W = 200 kHz. If we consider that the typical rms delay spread i n an urbanenvironment is on the order of= 2s, then using Eq. (18.13)the result ing coherencebandwidthis f0 100kHz. It should therefore beapparent that sincef0 < W, the GSM receiver must uti lizesome form of mitigation to combat frequency-selective distort ion. To accomplish this goal, theViterbi equalizer is typically implemented.

Figure18.15 il lustrates the basic functional blocks used in a GSM receiver for esti mati ng thechannel impulseresponse, whichisthenused to providethedetector wit h channel-corrected reference

waveforms[ 52]. In thefi nal step, theViterbi algorithm isused to computet heMLSEof themessage.Asstated in Eq. (18.2), areceived signal can bedescri bed in termsof thetransmitted signal convolvedwith theimpulseresponseof thechannel, hc(t). Weshowthisbelow,using thenotation of areceivedtraining sequence, rtr (t), and the tr ansmitted tr aini ng sequence, str (t), asfollows:

rtr (t) = str (t) hc(t) (18.32)



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FIGURE 18.15: TheViterbi equali zer asapplied to GSM.

where denotesconvoluti on. At the receiver,rtr (t)is extracted from the normal burst and sent toa filter having impulse response,hmf(t), that is matched to str (t). This matched filter yields at i tsoutput an esti mate ofhc(t), denoted he(t), developed from Eq. (18.32) asfollows.

he(t) = rtr (t) hmf(t)

= str (t) hc(t) hmf(t) (18.33)

= Rs (t) hc(t)

where Rs (t) is the autocorrelation function of str (t). If Rs (t) is a highly peaked (impulse-like)function, then he(t) hc(t).

Next, using a windowing function, w(t), wetruncate he(t) to form a computationally affordablefunction,hw(t). The window length must be large enough to compensate for the effect of t ypical

channel-induced I SI. The required observation interval L0for the window can beexpressed asthesum of twocontributions. Theinterval of length LCISIisduetothecontrolled ISI caused byGaussianfiltering of the baseband pulses, which are then MSK modulated. Theinterval of length LCisduetothe channel-induced ISI caused by multipath propagation; therefore, L0can bewritten as

L0 = LCISI+ LC (18.34)

TheGSM system is required to provide mitigation for distort ion due to signal dispersions of ap-proximately 1520s. Thebit duration is 3.69 s. Thus, the Viterbi equali zer used in GSM hasa memory of 46 bit intervals. For eachL0-bit interval in the message, the function of t he Viterbiequalizer is to fi nd the most likely L0-bit sequence out of the 2

L0 possible sequences that mighthavebeen transmitted. Determining the most l ikely L0-bit sequencerequiresthat2

L0 meaningfulreferencewaveformsbecreated bymodifying (or disturbing) the2L0 ideal waveformsin thesameway

that the channel hasdisturbed the tr ansmitted message. Therefore, the2L0

referencewaveformsareconvolved with thewindowed estimate of the channel impulseresponse, hw(t) in order to derivethedisturbed or channel-corrected reference wavefor ms. Next, the channel-corrected referencewave-formsarecomparedagainst thereceived datawaveformstoyield metr iccalculations. However, beforethecompari son takesplace, thereceived datawaveformsareconvolved with theknown windowed au-tocorrelation function w(t)Rs (t), transforming them in amanner comparableto that applied to the



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reference waveforms. Thisfiltered messagesignal is compared to all possible2L0 channel-correctedreference signals, and metr ics are computed as required by the Viterbi decoding algori thm (VDA).TheVDA yieldsthemaximum likelihood estimate of thetransmitted sequence[ 34].

18.14 TheRake Receiver Applied to Direct-SequenceSpread-Spectrum (DS/SS)Systems

Interim Specificati on 95(I S-95) describesaDS/SScellular system that usesa Rakereceiver [35][37]to providepath diversity. In Fig.18.16, fivei nstancesof chip tr ansmissionscorrespondingto thecodesequence1 0 1 1 1 are shown, with thetransmission or observation timeslabeled t4for theearliesttransmission andt0 for t he latest. Each abscissa shows threefingers of a signal that arr ive at thereceiver with delay times1, 2, and 3. Assumethat the intervals between the ti tr ansmission ti mesand the intervals between the i delay ti mes are each one chip long. From this, one can concludethat the finger arriving at the receiver at time t4, with delay 3, is time coincident wit h two otherfingers, namely the fingers arr iving at timest3and t2with delays2and 1, respectively. Since, inthisexample, thedelayed componentsareseparated byexactly onechip time, they arejustresolvable.

At the receiver, there must be a sounding device that is dedicated to estimating the i delay times.Note that for aterrestr ial mobile radio system, the fading ratei srelatively slow (mill iseconds) or thechannel coherencetimelargecompared to the chip time(T0 > Tch). Hence, the changesin ioccurslowly enough so that the receiver can readily adapt to them.

Once the i delays are esti mated, a separate correlator is dedicated to processing each finger. Inthi sexample, therewould bethreesuch dedicated correlators, each oneprocessing adelayed versionof thesamechip sequence1 0 1 1 1. In Fig. 18.16, each correlator receiveschipswith power profilesrepresented bythesequenceof fingersshown along adi agonal li ne. Eachcorrelator attemptsto matchthesearriving chipswith thesamePN code, similarly delayed in time. At theend of asymbol interval(typically theremaybehundredsor thousandsof chipsper symbol), theoutputsof thecorrelatorsarecoherently combined, and asymbol detecti on is made. At thechip level, the Rakereceiver resemblesan equalizer, but its real function is to providediversit y.

The interference-suppression nature of DS/SSsystemsstemsfrom the fact that a code sequencearriving at thereceiver merely onechip timelate, will beapproximately orthogonal to theparticularPN codewith which the sequenceis correlated. Therefore, any codechipsthat are delayed by oneormorechip timeswill besuppressed bythecorrelator. Thedelayed chipsonly contribute to raising thenoisefloor (correlation sidelobes). Themi tigation provided by theRakereceiver can betermed pathdiversit y, sinceit allowstheenergyof achip that arrivesviamultiplepathsto becombined coherently.Without the Rakereceiver, thisenergy would betr ansparent and therefore lost to the DS/SSsystem.In Fig.18.16,looking vertically abovepoint 3, it isclear that thereisinterchip interferenceduetodifferent fingers arr ivi ng simultaneously. The spread-spectrum processing gain allowsthesystem toenduresuch interferenceat the chip level. No other equalization is deemed necessary in IS-95.

18.15 Conclusion

In thischapter, the major elements that contribute to fading in acommunicati on channel havebeencharacterized. Figure18.1waspresentedasaguidefor thecharacterizati on of fadingphenomena. Twotypesof fading, large-scale and small-scale, weredescribed. Twomanifestationsof small-scale fading(signal dispersion and fading rapidity) were examined, and the examinati on involved two views,ti me and frequency. Two degradati on categories were defined for dispersion: frequency-selecti ve



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FIGURE18.16: Example of received chipsseen by a3-finger rakereceiver.

c1999byCR

CPressLLC


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fading and flat-fading. Two degradati on categori es were defined for fading rapidity: fast and slow.Thesmall-scale fading degradati on categorieswere summari zed in Fig. 18.6.A mathemati cal modelusing correlation and power densit y functions was presented in Fig.18.7. This model yields a nicesymmetr y, a kind of poetr y to help us view the Fourier t ransform and duali ty relationships that

describe the fading phenomena. Further, mitigati on techniquesfor ameliorati ng the effects of eachdegradati on category were treated, and these techniques were summari zed in Fig.18.13. Finally,miti gati on methods that have been implemented in two system types, GSM and CDMA systemsmeeti ng IS-95, were described.

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Bernard Sklar Raylegh Fading Channels

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