Chapter 5 Mobile Radio Propagation: Small-Scale Scale Fading and Multipath

Transcription

1 Chaper 5 Moble Rado Propagaon: Small-Scale Scale Fadng and Mulpah Ymn Zhang, Ph.D. Deparmen of Elecrcal & Compuer Engneerng Vllanova Unversy hp://ymnzhang.com/ece878 Ymn Zhang, Vllanova Unversy

2 Oulnes Impulse Response Model of a mulpah channel Relaonshp beween bandwdh and recever power Parameers of moble mulpah channels -Tme dsperson and coheren bandwdh -Doppler spread and coheren me Fla fadng and frequency-selecve fadng Raylegh and Rcean dsrbuons Clarke s model for fla fadng Level crossng Ymn Zhang, Vllanova Unversy

3 Propagaon Fadng n Wreless Communcaons Ymn Zhang, Vllanova Unversy 3

4 Small-Scale Mulpah Propagaon The hree mos mporan effecs of small-scale mulpah fadng: Rapd changes n sgnal srengh over a small ravel dsance or me nerval Random frequency modulaon due o varyng Doppler shfs on dfferen mulpah sgnals Tme dsperson caused by mulpah propagaon delays Facors nfluencng small-scale fadng Mulpah propagaon: reflecon objecs and scaers Speed of he moble: Doppler shfs Speed of surroundng objecs Transmsson bandwdh of he sgnal Coheren bandwdh: bandwdh of he mulpah channel. The receved sgnal wll be dsored f he ransmsson bandwdh s greaer han he channel coheren bandwdh. Ymn Zhang, Vllanova Unversy 4

5 Propagaon Fadng n Wreless Communcaons Local scaerng Iner-Symbol Inerference ISI Remoe scaerng Ymn Zhang, Vllanova Unversy 5

6 Small-Scale Mulpah Propagaon Doppler Shf A moble moves a a consan velocy v, along a pah segmen havng lengh d beween pons X and Y. Pah lengh dfference l d cosθ v cosθ Phase change φ Doppler shf π l πv λ λ cosθ φ v f d cosθ π λ Ymn Zhang, Vllanova Unversy 6

7 Impulse Response Model of a Mulpah Channel A moble rado channel may be modeled as a lnear fler wh a me varyng mpulse response me varaon s due o recever moon n space flerng s due o mulpah The channel mpulse response can be expressed as hd,. Le x represen he ransmed sgnal, hen he receved sgnal yd, a poson d can be expressed as : convoluon y d, x h d, x h d, d For a causal sysem y d, x h d, d Ymn Zhang, Vllanova Unversy 7

8 Impulse Response Model of a Mulpah Channel The poson of he recever can be expressed as d v We have y v, x h v, d Snce v s a consan, y v, s jus a funcon of. y x h v, d h In general, he channel mpulse response can be expressed as h,, where : me varaon due o moon : channel mulpah delay for a fxed value of. Ymn Zhang, Vllanova Unversy 8

9 Ymn Zhang, Vllanova Unversy 9 ECE 878 Wreless Communcaons : Propagaon Small-Scale Fadng Wh he channel mpulse response, we may have he oupu For bandlmed bandpass channel, may be equvalenly descrbed by a complex baseband mpulse response The equvalen baseband oupu where c and r are he complex envelops of x and y., h,, h x d h x y, h b, or, h c r h c r b b Impulse Response Model of a Mulpah Channel, h

10 Impulse Response Model of a Mulpah Channel x Re r c hb, y Re { c exp jωc } { r exp jω } c Ymn Zhang, Vllanova Unversy

11 Impulse Response Model of a Mulpah Channel The baseband response of a mulpah channel can be expressed as N h, a, exp jπf + jφ, δ b a, : amplude of he h mulpah componen : excess delay of h mulpah componen πf c + φ, : oal phase shf Defne θ, πf φ, c + If he channel mpulse response s assumed o be me nvaran, he channel mpulse response may be smplfed as h b N a exp The mpulse response may be measured by usng a probng pulse p whch approxmaes a dela funcon. p δ c jθ δ Ymn Zhang, Vllanova Unversy

12 Impulse Response Model of a Mulpah Channel Ymn Zhang, Vllanova Unversy

13 Relaonshp Beween Bandwdh and Receved Power Consder a pulsed, ransmed sgnal of he form { p exp jπf } x Re p c real response magnary response T bb T REP The sgnal p s a repeve baseband pulse ran wh very narrow pulse wdh T bb and repeon perod T REP, wh. T REP >> max Now, le p / T max T bb bb Ymn Zhang, Vllanova Unversy 3

14 Ymn Zhang, Vllanova Unversy 4 ECE 878 Wreless Communcaons : Propagaon Small-Scale Fadng The channel oupu r closely approxmaes he mpulse response and s gven by Insananeous mulpah power delay profle max rec exp exp N bb bb N T T j a p j a r θ θ max max max * max exp Re 4 θ θ d j p p a a d r r r N j N j j j Relaonshp Beween Bandwdh and Receved Power

15 Ymn Zhang, Vllanova Unversy 5 ECE 878 Wreless Communcaons : Propagaon Small-Scale Fadng If all he mulpah componens are resolved by he probe p, hen Then we have The oal recevng power s relaed o he sum of he powers n he ndvdual mulpah componens. bb j T > j max max max rec 4 max max N k k bb k bb N k k N k k k a d T T a d p a r Relaonshp Beween Bandwdh and Receved Power

16 Ymn Zhang, Vllanova Unversy 6 ECE 878 Wreless Communcaons : Propagaon Small-Scale Fadng Assumng ha he receved power from he mulpah componens forms a random process where each componen has a random amplude and phase a any me, he average small-scale receved power s Now, consder a CW sgnal whch s ransmed no he exac same channel, and le he complex envelope be gven by c. Then he receved sgnal can be expressed as The nsananeous power s gven by,, exp ] [ N N a WB a a j a E P E θ θ θ, exp N j a r θ, exp N j a r θ Relaonshp Beween Bandwdh and Receved Power

17 Relaonshp Beween Bandwdh and Receved Power In a local area, a vares lle, bu θ wll vary grealy due o changes n propagaon dsance over space, resulng n large flucuaons of r. The average receved power over a local area s gven by where E a, θ [ P ] CW E N a, θ a N a + r E j N exp jθ, N, j a[ aa j r j ] cos θ θ j The receved power for CW wave has large flucuaons han ha for WB sgnal. Ymn Zhang, Vllanova Unversy 7

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19 Relaonshp Beween Bandwdh and Receved Power Fadng and Bandwdh CDMA example Cumulave probably Raylegh Bandwdh Normalzed power db Ymn Zhang, Vllanova Unversy 9

20 Small-Scale Mulpah Measuremen Mulpah channel measuremen echnques Drec pulse measuremens Spread specrum sldng correlaor measuremens Swep frequency measuremens Ymn Zhang, Vllanova Unversy

21 Drec RF Pulse Sysem Drec RF pulse sysem Transms a repeve pulse of wdh Τ bb, and uses a recever wh a wdeband fler wh bandwdh BW/Τ bb. Envelope deecor o deec he amplude response. Mnmum resolvable delay Τ bb. No phase nformaon can be measured. Ymn Zhang, Vllanova Unversy

22 Spread Specrum Sldng Correlaor Channel Soundng Sysem descrpon A carrer s spread over a large bandwdh by usng a pseudo-nose sequence havng chp duraon T c and a chp rae R c. Despread usng a PN sequence dencal o ha used a he ransmer. The probng sgnal s wdeband. Use a narrowband recever preceded by a wdeband mxer. The ransmer chp clock s run a a slghly faser rae han he recever chp clock sldng correlaor. Ymn Zhang, Vllanova Unversy

23 Spread Specrum Sldng Correlaor Channel Soundng PN sequence provdes low and consan auocorrelaon for non-zero lags. N auocorrelaon - N Ymn Zhang, Vllanova Unversy 3

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25 Spread Specrum Sldng Correlaor Channel Soundng The me resoluon of mulpah componens usng a spread specrum sysem wh sldng correlaon s Tc R The me beween maxmum correlaon can be calculaed γ l T Tcγ l Rc T c : chp perod s γ : sldng facor R c : chp rae Hz l : sequence lengh chps The sldng facor can be expressed as α γ α β α: ransmer chp clock rae β: recever chp clock rae c Ymn Zhang, Vllanova Unversy 5

26 Spread Specrum Sldng Correlaor Channel Soundng The ncomng sgnal s mxed wh a PN sequence ha s slower han he ransmer sequence. The sgnal s down convered o a low-frequency narrow band sgnal. The observed me scale on he osclloscope usng a sldng correlaor s relaed o he acual propagaon me scale by Acual Propagaon Tme Observed Tme γ Ymn Zhang, Vllanova Unversy 6

27 acual channel response expanson by a facor of γ dsplay from osclloscope Ymn Zhang, Vllanova Unversy 7

28 Frequency Doman Channel Soundng Dual relaonshp beween me doman and frequency doman. I s possble o measure he channel mpulse response n he frequency doman. Measure he frequency doman response and hen convered o he me doman usng nverse dscree Fourer ransform IDFT. Ymn Zhang, Vllanova Unversy 8

29 Parameers of Moble Mulpah Channels Power delay profles for dfferen ypes of channels are dfferen Oudoor Indoor Ymn Zhang, Vllanova Unversy 9

30 Ymn Zhang, Vllanova Unversy 3 ECE 878 Wreless Communcaons : Propagaon Small-Scale Fadng Tme Dsperson Parameers Tme dsperson parameers mean excess delay RMS delay spread excess delay spread Mean excess delay RMS delay spread where k k k k k k k k k k P P a a σ k k k k k k k k k k P P a a

31 Tme Dsperson Parameers Depends only on he relave amplude of he mulpah componens. Typcal RMS delay spreads Oudoor: on he order of mcroseconds Indoor: on he order of nanoseconds Maxmum excess delay X db s defned o be he me delay durng whch mulpah energy falls o X db below he maxmum. excess delay X X : maxmum delay a whch a mulpah componen s whn X db : delay for he frs arrvng sgnal Ymn Zhang, Vllanova Unversy 3

32 Tme Dsperson Parameers Example of an ndoor power delay profle; rms delay spread, mean excess delay, maxmum excess delay db, and he hreshold level are shown. Ymn Zhang, Vllanova Unversy 3

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35 Coheren Bandwdh Coheren bandwdh, B c, s a sasc measure of he range of frequences over whch he channel can be consdered o be fla. Two snusods wh frequency separaon greaer han B c are affeced que dfferenly by he channel. If he coheren bandwdh s defned as he bandwdh over whch he frequency correlaon funcon s above.9, hen he coheren bandwdh s approxmaely B c 5σ If he frequency correlaon funcon s above.5 B c 5σ Ymn Zhang, Vllanova Unversy 35

36 Frequency-Fla Fla Fadng When /T s small,.e., s small and B /T s small, ISI s neglgble Impulse response Delay Frequency specra Transm sgnal Receve sgnal Sgnal bandwdh Channel response Frequency Frequency-fla fadng SIGNAL SIGNAL Ymn Zhang, Vllanova Unversy 36

37 Frequency-Selecve Fadng When /T s NOT small,.e., and B /T are NOT small, ISI becomes sgnfcan. Impulse response Frequency specra Transm sgnal Receve sgnal Sgnal bandwdh Channel response Frequency Frequency-selecve fadng SIGNAL SIGNAL Ymn Zhang, Vllanova Unversy 37

38 Doppler Spread and Coheren Tme Doppler spread and coheren me are parameers whch descrbe he me varyng naure of he channel n a small-scale regon. When a pure snusodal one of f c s ransmed, he receved sgnal specrum, called he Doppler specrum, wll have componens n he range f c f d and f c + f d, where f d s he Doppler shf. Channel fc fc f d fc f + c f d Ymn Zhang, Vllanova Unversy 38

39 Doppler Spread and Coheren Tme Doppler shf due o he moon of he recever. For he wave arrvng a an angle α o he movng drecon of he recever, he Doppler shf s gven by f d v :speed of ν cos α λ λ : wavelengh of he moble he lgh f d s a funcon of he relave velocy of he moble, and he angle beween he drecon of moon of he moble and drecon of arrval of he scaered waves. v Ymn Zhang, Vllanova Unversy 39

40 Doppler Spread and Coheren Tme Coheren me T c s he me doman dual of Doppler spread. Coheren me s used o characerze he me varyng naure of he frequency dspersveness of he channel n he me doman. TC fm : maxmum Doppler shf gven by fm v / λ f m Two sgnals arrvng wh a me separaon greaer han T c are affeced dfferenly by he channel A sasc measure of he me duraon over whch he channel mpulse response s essenally nvaran. If he coheren me s defned as he me over whch he me correlaon funcon s above.5, hen T C 9 6πf m Ymn Zhang, Vllanova Unversy 4

41 Types of Small-Scale Fadng Mulpah delay spread leads o me dsperson and frequency-selecve fadng. Doppler spread leads o frequency dsperson and meselecve fadng. Mulpah delay spread and Doppler spread are ndependen of one anoher. Ymn Zhang, Vllanova Unversy 4

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43 Fla Fadng If he channel has a consan gan and lnear phase response over a bandwdh whch s greaer han he bandwdh of he ransmed sgnal, he receved sgnal wll undergo fla fadng. The receved sgnal srengh changes wh me due o flucuaons n he gan of he channel caused by mulpah. The receved sgnal vares n gan bu he specrum of he ransmsson s preserved. Ymn Zhang, Vllanova Unversy 43

44 Fla fadng channel s also called amplude varyng channel. Also called narrow band channel: bandwdh of he appled sgnal s narrow as compared o he channel bandwdh. Tme varyng sascs: Raylegh fla fadng mos common amplude dsrbuon. A sgnal undergoes fla fadng f B S << B C and >> σ T S Fla Fadng : recprocal bandwdh symbol perod B S : bandwdh of he ransmed sgnal B C : coheren bandwdh σ : rms delay spread T S Ymn Zhang, Vllanova Unversy 44

45 Frequency-Selecve Fadng If he channel possesses a consan-gan and lnear phase response over a bandwdh ha s smaller han he bandwdh of ransmed sgnal, hen he channel creaes frequency-selecve fadng. sgnal specrums f f channel response B C f receved sgnal specrum f Ymn Zhang, Vllanova Unversy 45

46 Frequency-selecve fadng s due o me dsperson of he ransmed symbols whn he channel. Waveform s dsored by ner-symbol nerference ISI Equalzaon s requred Chaper 7 Frequency-selecve fadng channels are much more dffcul o model han fla fadng channels. For frequency-selecve fadng and Frequency-Selecve Fadng B S > B C T S < σ Ymn Zhang, Vllanova Unversy 46

47 Frequency-Selecve Fadng Frequency selecve fadng channel characersc Ymn Zhang, Vllanova Unversy 47

48 Fadng Effecs Due o Doppler Spread Fas Fadng: The channel mpulse response changes rapdly whn he symbol duraon. The coheren me of he channel s smaller hen he symbol perod of he ransmed sgnal. Cause frequency dsperson due o Doppler spreadng. A sgnal undergoes fas fadng f T S > T C and B < Slow Fadng: The channel mpulse response changes a a rae much slower han he ransmed baseband sgnal s. The Doppler spread of he channel s much less hen he bandwdh of he baseband sgnal. A sgnal undergoes slow fadng f T << and B >> B S T C S S B D D Ymn Zhang, Vllanova Unversy 48

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50 Raylegh and Rcean Dsrbuons Raylegh Fadng Dsrbuon The sum of wo quadraure Gaussan nose sgnals Ymn Zhang, Vllanova Unversy 5

51 Consder a carrer sgnal a frequency ω c and wh an amplude a s a exp jω The receved sgnal s he sum of n waves n sr a exp jω + θ r exp j ω + θ where Defne r exp we have [ ] r exp jθ exp jω x n a Raylegh and Rcean Dsrbuons r exp n jθ n a exp jθ jθ a cos θ + j a sn θ x jy n + cos θ r cos θ and y n a sn θ r sn θ Ymn Zhang, Vllanova Unversy 5

52 Raylegh and Rcean Dsrbuons I can be assumed ha x and y are Gaussan random varables wh mean equal o zero due o he followng reasons n s usually very large. The ndvdual amplude a are random. The phases θ have a unform dsrbuon. Because x and y are ndependen random varables, he jon dsrbuon px,y s p x, y x + y p x p y exp πσ σ Ymn Zhang, Vllanova Unversy 5

53 Raylegh and Rcean Dsrbuons The dsrbuon pr,θ can be wren as a funcon of px,y x / r x / θ cosθ r snθ p r, θ J p x, y J y / r y / θ snθ r cosθ r We have p r, θ r r exp πσ σ The Raylegh dsrbuon has a pdf gven by r r π exp p r p r, θ dθ σ σ σ: rms value of σ : me - average power of he receved sgnal before envelop deecon he receved sgnal before envelop deecon Ymn Zhang, Vllanova Unversy 53 r r <

54 Ymn Zhang, Vllanova Unversy 54

55 Ymn Zhang, Vllanova Unversy 55 ECE 878 Wreless Communcaons : Propagaon Small-Scale Fadng Cumulave dsrbuon funcon CDF The mean value of he Raylegh dsrbuon s gven by The varance of he Raylegh dsrbuon s gven by R R dr r p R r R P exp Pr σ σ π σ 533. ] [ dr r rp r E r mean.49 ] [ ] [ σ π σ π σ σ dr r p r r E r E r Raylegh and Rcean Dsrbuons

56 Raylegh and Rcean Dsrbuons Rcean Fadng Dsrbuon: When here s a domnan saonary non-fadng sgnal componen presen, such as a lne-of-sgh propagaon pah, he small-scale fadng envelope dsrbuon s Rcean. s r x + r A y Scaered waves Drec wave r exp [ x + A + x + r cosθ r snθ [ j ω + θ ] A + jy]exp jω y + Aexp jω Ymn Zhang, Vllanova Unversy 56

57 By followng smlar seps descrbed n Raylegh dsrbuon, we oban r r + A Ar exp I A, r p r σ σ σ r < where Ar π Ar cosθ I θ σ π exp d σ s he modfed Bessel funcon of he frs knd and zero-order. The Rcean dsrbuon s ofen descrbed n erms of a parameer K whch s defned as he rao beween he deermnsc sgnal power and he varance of he mulpah. I s gven by or Raylegh and Rcean Dsrbuons K A /σ A K db log σ db Ymn Zhang, Vllanova Unversy 57

58 The parameer K s known as he Rcean facor. Raylegh and Rcean Dsrbuons As A, we have K db. The domnan pah decrease n amplude, he Rcean dsrbuon degeneraes o a Raylegh dsrbuon. Ymn Zhang, Vllanova Unversy 58

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60 Clarke s Models for Fla Fadng Clarke developed a model where he sascal characerscs of he elecromagnec felds of he receved sgnal are deduced from scaerng. The model assumes a fxed ransmer wh a vercally polarzed anenna. The receved anenna s assumed o comprse of N azmuhal plane waves wh arbrary carrer phase, arbrary angle of arrval, and each wave havng equal average amplude. Equal amplude assumpon s based on he fac ha n he absence of a drec lne-of-sgh pah, he scaered componens arrvng a a recever wll experence smlar aenuaon over small-scale dsance. Ymn Zhang, Vllanova Unversy 6

61 Clarke s Models for Fla Fadng Doppler shf due o he moon of he recever. Assume no excess delay due o mulpah. Fla fadng assumpon For he nh wave arrvng a an angle α n o he x-axs, he Doppler shf s gven by ν cos α fn λ n α n v Ymn Zhang, Vllanova Unversy 6

62 The vercally polarzed plane waves arrvng a he moble have E- feld componens gven by assume a sngle one s ransmed E C n f c θ n N Cn cosπf c + n n E θ The random arrvng phase s gven by θ πf + φ E : real ampludeof z local average E - feld consan : real random varable represenng he ampludeof nh arrvng wave. :carrer frequency. Clarke s Models for Fla Fadng : random phase of he nh arrvng wave. The amplude of E-feld s normalzed such ha n N n C n n Ymn Zhang, Vllanova Unversy 6

63 E z can be modeled as a Gaussan random process f N s suffcen large. Snce he Doppler shf s very small when compared o he carrer frequency, he hree feld componens may be modeled as narrow band random process. E Clarke s Models for Fla Fadng z T cosπ f + T snπf c c s c where T T s c N Cn cosπf n + n N Cn snπf n + n E φ E φ T c and T s are Gaussan random processes whch are denoed as T c and T s, respecvely. Ymn Zhang, Vllanova Unversy 63

64 Ymn Zhang, Vllanova Unversy 64 ECE 878 Wreless Communcaons : Propagaon Small-Scale Fadng T c and T s are uncorrelaed zero-mean Gaussan random varable wh equal varance gven by The envelope of he receved E-feld s gven by I can be shown ha he random receved sgnal envelope r has a Raylegh dsrbuon gven by / E E T T z c c r T T E s c z + < exp r r r r r p σ σ / where E σ Clarke s Models for Fla Fadng

65 Le Pαdα denoe he funcon of he oal ncomng power whn dα of he angle α, and le A denoe he average receved power wh respec o an soropc anenna. N Clarke s Models for Fla Fadng As, Pαdα approached a connuous dsrbuon. If Gα s he azmuhal gan paern of he moble anenna as a funcon of he angle of arrval, he oal receved power can be expressed as π The nsananeous frequency of he receved sgnal arrvng a an angle α s gven by: where f m s he maxmum Doppler shf. P r AG α p α dα v f α f cos α + fc fm cosα + λ f c Ymn Zhang, Vllanova Unversy 65

66 If Sf s he power specrum of he receved sgnal, he dfferenal varaon of receved power wh frequency s gven by Dfferenaon df f dα [ p α G α + p α G α ] dα S f df A f f m cosα + f c On he oher hand, we have Clarke s Models for Fla Fadng snα df dα snα m f m α cos f f m f c Ths mples snα f f m f c Ymn Zhang, Vllanova Unversy 66

67 Clarke s Models for Fla Fadng Fnally, we have S f A [ p α G α + p α G α ] f m f f m f c where S f, f f c > f m The specrum s cenered on he carrer frequency and s zero ousde he lms f ±. c f m Each of he arrvng waves has s own carrer frequency due o s drecon of arrval whch s slghly offse from he cener frequency. Ymn Zhang, Vllanova Unversy 67

68 Vercal λ / 4 anenna Gα.5. Example Unform dsrbuon p α /π over o π. The oupu specrum S f πf m.5 f f m f c Ymn Zhang, Vllanova Unversy 68

69 Smulaon of Clarke Fadng Model Produce a smulaed sgnal wh specral and emporal characerscs very close o measured daa. Two ndependen Gaussan low pass nose are used o produce he n-phase and quadraure fadng branches. Use a specral fler o sharp he random sgnal n he frequency doman by usng fas Fourer ransform FFT. Tme doman waveforms of Doppler fadng can be obaned by usng an nverse fas Fourer ransform IFFT. Ymn Zhang, Vllanova Unversy 69

70 RF Doppler fler Baseband Doppler fler Ymn Zhang, Vllanova Unversy 7

71 Smulaon of Clarke Fadng Model Smh smulaor usng N carrers o generae fadng sgnal. Specfy he number of frequency doman pons N used o represen S f and he maxmum Doppler frequency shf f m.. Compue he frequency spacng beween adjacen specral lnes as f fm / N. Ths defnes he me duraon of a fadng waveform, T / f.. Generae complex Gaussan random varables for each of he N/ posve frequency componens of he nose source. 3. Consruc he negave frequency componens of he nose source by conjugang posve frequency and assgnng hese a negave frequency values. 4. Mulply he n-phase and quadraure nose sources by he fadng specrum S f. 5. Perform an IFFT on he resulng frequency doman sgnal from he nphase and quadraure arms, and compue he sum of he squares of each sgnal. 6. Take he square roo of he sum. Ymn Zhang, Vllanova Unversy 7

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73 Frequency-selecon fadng model Ymn Zhang, Vllanova Unversy 73

74 Level Crossng and Fadng Sascs The level crossng rae LCR s defned as he expeced rae a whch he Raylegh fadng envelope crosses a specfed level n a posve-gong drecon. Useful for desgnng error conrol codes and dversy. Relae he me rae of change of he receved sgnal o he sgnal level and velocy of he moble. The number of level crossng per second o he level R s gven by N R rp R, r dr π f ρe r : me dervaon of r slope p R, r : jon densy funcon of r and r a r R. f m : maxmum Doppler frequency m ρ R / R rms : value of he specfed level R, normalzed o he rms amplude of he fadng envelope. ρ A Ymn Zhang, Vllanova Unversy 74

75 Level Crossng and Fadng Sascs Average fadng duraon s defned as he average perod of me for whch he receved sgnal s below a specfed level R. For a Raylegh fadng sgnal, hs s gven by Pr[ r < N R wh Pr[ r < R] T where s he duraon of he fade and T s he observaon nerval. For Raylegh dsrbuon Pr[ r < R] p r dr R] exp ρ Average fadng duraon, usng A, B, C R ρ e ρ f m π B C Ymn Zhang, Vllanova Unversy 75

76 Level Crossng and Fadng Sascs The average duraon of a sgnal fadng helps deermne he mos lkely number of sgnalng bs ha may be los durng a fade. Average fade duraon prmarly depends upon he speed of he moble, and decreases as he maxmum Doppler frequency f m becomes large. Ymn Zhang, Vllanova Unversy 76

77 Oulnes Impulse Response Model of a mulpah channel Relaonshp beween bandwdh and recever power Parameers of moble mulpah channels -Tme dsperson and coheren bandwdh -Doppler spread and coheren me Fla fadng and frequency-selecve fadng Raylegh and Rcean dsrbuons Clarke s model for fla fadng Level crossng Ymn Zhang, Vllanova Unversy 77