Channel Modelling ETIM10 Lectue no: 3 Fading Statistical desciption of the wieless channel Ghassan Dahman \ Fedik Tufvesson Depatment of Electical and Infomation Technology Lund Univesity, Sweden 013-01-8 Fedik Tufvesson - ETIM10 1
Contents Why statistical desciption Lage scale fading Fading magin Small scale fading without dominant component with dominant component Statistical models Measuement example 013-01-8 Fedik Tufvesson - ETIM10
Why statistical desciption Unknown envionment Complicated envionment Can not descibe eveything in detail Lage fluctuations Need a statistical measue since we can not descibe evey point eveywhee Thee is a x% pobability that the amplitude/powe will be above the level y 013-01-8 Fedik Tufvesson - ETIM10 3
The WSSUS model Assumptions A vey common wide-band channel model is the WSSUS-model. Roughly speaking it means that the statistical popeties emain the same ove the consideed time (o aea) Recalling that the channel is composed of a numbe of diffeent contibutions (incoming waves), the following is assumed: The channel is Wide-Sense Stationay (WSS), meaning that the time coelation of the channel is invaiant ove time. The channel is built up by Uncoelated Scattees (US), meaning that contibutions with diffeent delays ae uncoelated. 013-01-8 Fedik Tufvesson - ETIM10 4
What is lage scale and small scale? 013-01-8 Fedik Tufvesson - ETIM10 5
Lage-scale fading Basic pinciple Received powe D C d B Position A B C C A 013-01-8 Fedik Tufvesson - ETIM10 6
Lage-scale fading Log-nomal distibution A nomal distibution in the db domain. pdf L db Deteministic mean value of path loss, L 0 db db pdf L 1 exp L db L0 db db F db F db 013-01-8 Fedik Tufvesson - ETIM10 7
Lage-scale fading Example What is the pobability that the small scale aveaged amplitude will be 10 db above the mean if the lage scale fading can be descibed as log-nomal with a standad deviation of 5 db? pdf L db P out 10 P 0 db db L L 10 Q Q 0. 0 L 0 db What if the standad deviation is 10 db instead? 013-01-8 Fedik Tufvesson - ETIM10 8
The Q(.)-function Uppe-tail pobabilities x Q(x) x Q(x) x Q(x) 4.65 0.00001 4.107 0.0000 4.013 0.00003 3.944 0.00004 3.891 0.00005 3.846 0.00006 3.808 0.00007 3.775 0.00008 3.746 0.00009 3.719 0.00010 3.540 0.0000 3.43 0.00030 3.353 0.00040 3.91 0.00050 3.39 0.00060 3.195 0.00070 3.156 0.00080 3.11 0.00090 3.090 0.00100.878 0.0000.748 0.00300.65 0.00400.576 0.00500.51 0.00600.457 0.00700.409 0.00800.366 0.00900.36 0.01000.054 0.0000 1.881 0.03000 1.751 0.04000 1.645 0.05000 1.555 0.06000 1.476 0.07000 1.405 0.08000 1.341 0.09000 1.8 0.10000 0.84 0.0000 0.54 0.30000 0.53 0.40000 0.000 0.50000 013-01-8 Fedik Tufvesson - ETIM10 9
Lage-scale fading, why log-nomal? Many diffaction points adding exta attenuation to the pathloss. N 1 Total pathloss: 1 Ltot L d N L L d tot db db 1 db db N db If these ae consideed andom and independent, we should get a nomal distibution in the db domain. 013-01-8 Fedik Tufvesson - ETIM10 10
Two waves Wave 1 E RX1 (t)=a 1 cos(f c t-/*d 1 ) k 0 =/ E=E 1 exp(-jk o d 1 ) E TX (t)=a cos(f c t) E RX (t)=a cos(f c t-/*d ) E E=E exp(-jk o d ) E 1 Wave 013-01-8 Fedik Tufvesson - ETIM10 11
Two waves Wave 1 Wave 1 + Wave Wave 013-01-8 Fedik Tufvesson - ETIM10 1
Dopple shifts v Receiving antenna moves with speed v at an angle elative to the popagation diection of the incoming wave, which has fequency f 0. c Fequency of eceived signal: f f whee the dopple shift is 0 v f0 cos c The maximal Dopple shift is f max 0 v c 013-01-8 Fedik Tufvesson - ETIM10 13
Dopple shifts How lage is the maximum Dopple fequency at pedestian speeds fo 5. GHz WLAN and at highway speeds using GSM 900? v max f0 c f 0 =5. 10 9 Hz, v=5 km/h, (1.4 m/s) 4 Hz f 0 =900 10 6 Hz, v=110 km/h, (30.6 m/s) 9 Hz 013-01-8 Fedik Tufvesson - ETIM10 14
Two waves with Dopple Wave 1 v E E1exp( jk d f cos( ) t) 0 1 c 1 c0 E TX (t)=a cos(f c t) v Wave v E Eexp( jk d f cos( ) t) 0 c c0 E E1 The two components have diffeent Dopple shifts! The Dopple shifts will cause a andom fequency modulation 013-01-8 Fedik Tufvesson - ETIM10 15
Many incoming waves Many incoming waves with independent amplitudes and phases Add them up as phasos 1, 1, 3 3 1 1 3, 3 4, 4, 4 4 exp j exp j exp j exp j exp j 1 1 3 3 4 4 013-01-8 Fedik Tufvesson - ETIM10 16
Many incoming waves Re and Im components ae sums of many independent equally distibuted components Re( ) N(0, ) Re() and Im() ae independent 3 3 1 1 The phase of has a unifom distibution 4 4 013-01-8 Fedik Tufvesson - ETIM10 17
Rayleigh fading No dominant component (no line-of-sight) TX X RX Tap distibution D Gaussian (zeo mean) Amplitude distibution 0.8 Rayleigh 0.6 Im a Rea a 0.4 0. 0 0 1 3 No line-of-sight component pdf exp 013-01-8 Fedik Tufvesson - ETIM10 18
Rayleigh fading Rayleigh distibution pdf exp min 0 ms Pobability that the amplitude is below some theshold min : P pdf d 1 exp min min min 0 ms 013-01-8 Fedik Tufvesson - ETIM10 19
Rayleigh fading outage pobability What is the pobability that we will eceive an amplitude 0 db below the ms? min P min 1exp 1 exp( 0.01) 0.01 ms What is the pobability that we will eceive an amplitude below ms? min P min 1exp 1 exp( 1) 0.63 ms 013-01-8 Fedik Tufvesson - ETIM10 0
Rayleigh fading fading magin To Ensue that we in most cases eceive enough powe we tansmit exta powe fading magin ms min ms db 10log10 min M M 0 min ms 013-01-8 Fedik Tufvesson - ETIM10 1
Rayleigh fading fading magin How many db fading magin, against Rayleigh fading, do we need to obtain an outage pobability of 1%? Some manipulation gives 10.01 exp min P 1exp min ms min min ms ln 0.99 0.01 ms ln 0.99 013-01-8 Fedik Tufvesson - ETIM10 ms min 1% 0.01 min ms M 1/ 0.01 100 M db 0
Rayleigh fading signal and intefeence Both the desied signal and the intefeence undego fading Fo a single use intefee and Rayleigh fading: pdf cdf SIR SIR () ( ) 1 whee is the mean signal 1 to intefeence adio ( ) ( ) pdf 0.5 0.45 0.4 0.35 0.3 0.5 0. 0.15 0.1 0.05 0-0 -15-10 -5 0 5 10 15 0 SIR(dB) pdf fo 10 db mean signal to intefeence atio 013-01-8 Fedik Tufvesson - ETIM10 3
Rayleigh fading signal and intefeence What is the pobability that the instantaneous SIR will be below 0 db if the mean SIR is 10 db when both the desied signal and the intefee expeience Rayleigh fading? min 10 P min 1 1 0.09 ( (10 1) min ) 013-01-8 Fedik Tufvesson - ETIM10 4
one dominating component In case of Line-of-Sight (LOS) one component dominates. Assume it is aligned with the eal axis Re( ) N( A, ) The ecieved amplitude has now a Ricean distibution instead of a Rayleigh The fluctuations ae smalle Im( ) N(0, ) The phase is dominated by the LOS component In eal cases the mean popagation loss is often smalle due to the LOS The atio between the powe of the LOS component and the diffuse components is called Ricean K-facto k Powe in LOS component A Powe in andom components 013-01-8 Fedik Tufvesson - ETIM10 5
Rice fading Ima Tap distibution D Gaussian (non-zeo mean) A A dominant component (line of sight) Rea Amplitude distibution.5 1.5 1 0.5 Rice k = 30 k = 10 k = 0 TX RX 0 0 1 3 Line-of-sight (LOS) component with amplitude A. exp A A pdf I 0 013-01-8 Fedik Tufvesson - ETIM10 6
Rice fading, phase distibution The distibution of the phase is dependent on the K-facto 013-01-8 Fedik Tufvesson - ETIM10 7
Nakagami distibution In many cases the eceived signal can not be descibed as a pue LOS + diffuse components The Nakagami distibution is often used in such cases with m it is possible to adjust the dominating powe inceasing m {1 3 4 5} 013-01-8 Fedik Tufvesson - ETIM10 8
Both small-scale and lage-scale fading Lage-scale fading - lognomal fading gives a cetain mean Small scale fading Rayleigh distibuted given a cetain mean The two fading pocesses adds up in a db-scale Suzuki distibution: log-nomal mean 4 pdf () e e 0 0 1 ln(10) F 0log( ) log-nomal sdt small-scale std fo complex components F 013-01-8 Fedik Tufvesson - ETIM10 9
Both small-scale and lage-scale fading An altenative is to add fading tems (fading magins) independently Pessimistic appoach, but used quite often The communication link can be OK if thee is a fading dip fo the small scale fading but the log-nomal fading have a athe small attenuation. 013-01-8 Fedik Tufvesson - ETIM10 30
Some special cases Rayleigh fading pdf Rice fading, K=0 exp Rice fading with K=0 becomes Rayleigh Nakagami, m=1 Nakagami with m=1 becomes Rayleigh 013-01-8 Fedik Tufvesson - ETIM10 31
Example, shadowing fom people Two pesons communicating with each othe using PDAs, signal sometimes blocked by pesons moving andomly 013-01-8 Fedik Tufvesson - ETIM10 3
Example, shadowing fom people 10 line-of-sight 0 Received powe [db] -10-0 -30-40 0 4 6 8 10 Time [s] obstucted LOS 013-01-8 Fedik Tufvesson - ETIM10 33