Outline. 1 Path-Loss, Large-Scale-Fading, Small-Scale Fading. 2 Simplified Models. 3 Multipath Fading Channel

Transcription

1 Outline Digital Communiations Leture 11 Wireless Channels Pierluigi SALVO OSSI Department of Industrial and Information Engineering Seond University of Naples Via oma 29, Aversa CE), Italy homepage: Path-Loss, Large-Sale-Fading, Small-Sale Fading 2 Simplified Models 3 Multipath Fading Channel 4 WSSUS Channel, Correlations and Power Spetra 5 Underspread and Overspread Channels P. Salvo ossi SUN.DIII) Digital Communiations - Leture 11 1 / 25 P. Salvo ossi SUN.DIII) Digital Communiations - Leture 11 2 / 25 Wireless Channels Main Phenomena 1/2) Path-Loss due to dissipation of the radiated power variations our over very-large distanes 100m-1000m) modeled deterministially Shadowing or Large-Sale Fading due to obstrution of obstales between TX and X variations our over obstale size 1m-100m) modeled stohastially Multipath Fading or Small-Sale Fading due to onstrutive/destrutive ombination of signal replias at the X variations our over signal wavelength 1mm-1m) modeled stohastially P. Salvo ossi SUN.DIII) Digital Communiations - Leture 11 3 / 25 P. Salvo ossi SUN.DIII) Digital Communiations - Leture 11 4 / 25

2 Main Phenomena 2/2) One Path, Stati TX, Stati X TX in the origin transmits one single sinusoid Path-Loss green) Shadowing red) Multipath Fading blue) X at point p = r, θ, ψ) reeives xt) = expj2πft) r exp j2πf t r )) Hf) = yt) = exp j2πf r ) xt) r We will fous on Small-Sale Fading P. Salvo ossi SUN.DIII) Digital Communiations - Leture 11 5 / 25 P. Salvo ossi SUN.DIII) Digital Communiations - Leture 11 6 / 25 Two Paths, Stati TX, Stati X TX in the origin transmits one single sinusoid xt) = expj2πft) X at point p = r, θ, ψ) reeives one diret ray LOS) and one refleted ray overall distane r 1 = r + r) r Hf) = r θ = 2π r λ exp j2πf t r )) + exp j2πf t r 1 r 1 )) exp j2πf r ) + r + r exp j2πf r + r ) Small reeived power for r = nλ/2 and n Z P. Salvo ossi SUN.DIII) Digital Communiations - Leture 11 7 / 25 One Path, Stati TX, Mobile X TX in the origin transmits one single sinusoid xt) = expj2πft) X at point pt) = rt), θ, ψ), with rt) = r 0 + vt, reeives exp j2πf t rt) )) rt) = exp j2πf r ) 0 exp j2πf 1 v ) ) t r 0 + vt yt) = exp xt) r 0 + vt Doppler shift of fv/ j2πf r 0 ) exp j2πf v ) t P. Salvo ossi SUN.DIII) Digital Communiations - Leture 11 8 / 25

3 Multipath Fading Channel 1/2) Assume no AWGN for simpliity Multiple paths from TX to X y BP t) = Nt) α l t)x BP t τ l t)) Nt) is the number of resolvable satterers at time t α l t) is the attenuation introdued by the lth satterer at time t τ l t) is the delay introdued by the lth satterer at time t Baseband representation: s BP t) = {st) expj2πf t)} Multipath Fading Channel 2/2) Baseband I/O relation Nt) Stohasti LTV hannel: α l t) exp j2πf τ l t)) xτ τ }{{} l t)) l t)=α l t) exp jϑ l t)) ht, τ) = Nt) ht, τ)xt τ)dτ l t)δτ τ l t)) Generalization for Continuum of Multipath Components ht, τ) = αt, τ) exp j2πf τ) P. Salvo ossi SUN.DIII) Digital Communiations - Leture 11 9 / 25 P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25 Stohasti Impulse esponse 1/2) Large variations of α l t) are needed to impat the reeived signal Small variations of ϑ l t) do impat the reeived signal For a given t 0, we assume that ht 0, τ) is deterministi summable For a given τ 0, we assume that ht, τ 0 ) is stohasti not summable The time-variant) F-transform of the hannel response is Ht, f) = ht, τ) exp j2πfτ)dτ Stohasti Impulse esponse 2/2) The hannel response is modeled as a omplex-valued Gaussian random proess Central Limit Theorem) Components may add up onstrutively or destrutively ausing flutuations of the reeived signal power, i.e. fading ayleigh fading ht, τ) is a zero-mean omplex-valued Gaussain proess suitable for NLOS hannel iean fading ht, τ) is a nonzero-mean omplex-valued Gaussain proess suitable for LOS hannel Nakagami fading mathing empirial measurements P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25 P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25

4 WSSUS Channel Wide Sense Stationary WSS): E{ht 1, τ 1 }h t 2, τ 2 )} = E{h0, τ 1 }h t 2 t 1, τ 2 )} Unorrelated Sattering US): { E{ht 1, τ 1 }h E{ht 1, τ 1 }h t 2, τ 1 )} τ 1 = τ 2 t 2, τ 2 )} = 0 τ 1 τ 2 WSSUS Channel E{ht, τ}h t + t, τ )} = φ h t, τ)δτ τ ) Multipath Intensity Profile φ h 0, τ) is alled Multipath Intensity Profile or Delay Power Spetrum and provides the average power output as a funtion of delay ht, τ)xt τ)dτ K y t, t t) = E{yt)y t t)} = φ h t, τ)xt τ)x t t τ)dτ P y t) = E{ yt) 2 } = K y t, t) = φ h 0, τ) xt τ) 2 dτ P y = P y t) = P x φ h 0, τ)dτ P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25 P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25 Spaed-Time Spaed-Frequeny Correlation Funtion Φ h t, f) is alled spaed-time spaed-frequeny orrelation funtion Φ h t, f) = E{Ht, f)h t t, f f)} = φ h t, τ) exp j2π f τ)dτ = F {φ h t, τ)} τ f) ht, τ)xt τ)dτ = Ht, f)xf) expj2πft)df x n t) = e j2πfnt X n f) = δf f n ) y n t) = Ht, f n )e j2πfnt Multipath Delay Spread and Coherene Bandwidth 1/2) φ h t, τ) as a funtion of τ is a power spetrum The essential) duration T m of the funtion φ h 0, τ) is denoted Multipath Delay Spread Φ h t, f) as a funtion of f is an autoorrelation The essential) duration B of the funtion Φ h 0, f) is denoted Coherene Bandwidth E{y n t)y mt)} = E{Ht, f n )H t, f m )} expj2πf n f m )t) = Φ h 0, f n f m ) expj2πf n f m )t) T m B 1 P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25 P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25

5 Multipath Delay Spread and Coherene Bandwidth 2/2) Two spetral omponents of the transmitted signal) whose separation is larger than B are unorrelated at the reeiver The linear distortion is due to frequeny seletivity if B x B, then the signal undergoes linear distortion if B x B, then the signal does not undergo linear distortion The transmitted signal is spread in time due to the onvolution with the hannel response) and its duration is inreased of T m The linear distortion is due to inter-symbol interferene if T x T m, then the signal undergoes linear distortion if T x T m, then the signal does not undergo linear distortion Doppler Power Spetrum s h λ, 0) is alled Doppler Power Spetrum and desribes the time varying effets in terms of Doppler shifts s h λ, f) = Φ h t, f) exp j2πλ t)d t = F {Φ h t, f)} t λ) x n t) = e j2πfnt X n f) = δf f n ) y n t) = Ht, f n )e j2πfnt E{yt)y t t)} = E{Ht, f n )H t t, f n )} expj2πf n t) = Φ h t, 0) expj2πf n t) P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25 P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25 Coherene Time and Doppler Spread 1/2) Coherene Time and Doppler Spread 2/2) Φ h t, f) as a funtion of t is an autoorrelation The essential) duration T of the funtion Φ h t, 0) is denoted Coherene Time s h λ, f) as a funtion of λ is a power spetrum The essential) duration B d of the funtion s h λ, 0) is denoted Doppler Spread T B d 1 Two delayed versions of the transmitted signal) whose separation is larger than T are unorrelated at the reeiver The non-linear distortion is due to time-varying effets if T x T, then the signal undergoes non-linear distortion if T x T, then the signal does not undergo non-linear distortion The transmitted signal is spread in Doppler domain and its spetrum is inreased of B d The non-linear distortion is due to Doppler spreading if B x B d, then the signal undergoes non-linear distortion if B x B d, then the signal does not undergo non linear distortion P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25 P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25

6 Sattering Funtion S h λ, τ) is alled sattering funtion and is related to the previous funtions as follows Four Cases Digital-modulation sheme with symbol duration T and bandwidth B Wireless hannel with oherene time T and oherene bandwidth B or delay spread T m = 1/B and Doppler spread B d = 1/T ) and S h λ, τ) = F {φ h t, τ)} t λ) = φ h t, τ) exp j2πλ t)d t S h λ, τ) = F 1 {s h λ, f)} f τ) = s h λ, f) exp+j2πτ f)d f P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25 P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25 Frequeny Non-Seletive Channel In general rt) = ht, τ)st τ)dτ = Ht, f)sf) expj2πft)df however, B B means that all spetral omponents of st) undergo the same attenuation and phase shift, i.e. Ht, f) Ht, 0) f [ B, B] or equivalently that the ISI is negligible, and then rt) = αt) exp jϑt)) st) }{{} Ht,0) Slow Time-Varying Channel In general rt) = ht, τ)st τ)dτ = Ht, f)sf) expj2πft)df however, T T means that nonlinear distortion introdued by time-varying effets is negligible, i.e. ht, τ) h0, τ) t [0, T ] or equivalently there is no Doppler shift, and then N ) rt) = l δt τ l ) st) = } {{ } h0,t) N l st τ l ) P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25 P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25

7 Flat-Flat Fading Channel with AWGN) Most signals have T B 1, then if T m B d < 1 or equiv. B T > 1) the hannel is said underspread if T m B d > 1 or equiv. B T < 1) the hannel is said overspread Underspread hannels allow to selet st) suh that T < T B < B i.e. transmitting over a frequeny non-seletive slow time-varying hannel often denoted flat-flat fading) The signal model is rt) = Hst) + wt) P. Salvo ossi SUN.DIII) Digital Communiations - Leture / 25