SER/BER in a Fading Channl Major points for a fading channl: * SNR is a R.V. or R.P. * SER(BER) dpnds on th SNR conditional SER(BER). * Two prformanc masurs: outag probability and avrag SER(BER). * Ovrall, 3 ffcts in a mobil channl: fading, dlay sprad and Dopplr sprad affct th BER/SER. Concntrat on st on. Exampl: Rayligh Fading Channl SISO x 3 3 4 5 Rcivd powr (SNR) norm. to th avrag [db] vs distanc (location, tim) Lctur 3-Nov-6 (4)
Channl modl: Considr first slow (quasi-static) flat-fading channl, s( t ) h ξ ( t) r ( t) whr ( t) ξ is AWGN, jϕ ( ) ( ) ( ) r t = α s t + ξ t (.) h j ϕ = α is th complx channl gain (an RV for a fading channl), α and ϕ ar its magnitud and phas. Aftr th matchd filtr and samplr, th modl is discrt-tim, r = h s + ξ (.) i i i Slow block-fading: h stays th sam for many symbol intrvals and thn changs to a nw random valu. Introduc conditional SER/BER P ( ) γ = α E / N is instantanous SNR. b Introduc th avrag (ovr fading) SNR: Eb / N γ (i.., for a fixd α), and γ = γ = α (.3) Lctur 3-Nov-6 (4)
For Rayligh fading, α is a Rayligh RV, and xponntial), its pdf is Whn (rfrnc point). ( ) α or γ is χ (or γ ρ γ = γ (.4) γ α = γ = E / N is th AWGN channl SNR b Th pdf of α is ( ) α Avrag SER/BER is ρ x = x. ( ) ( ) (.5) P = P γ ρ γ dγ Th xprssion is gnral, can b usd for any flat-fading channl. Using th SER/BER xprssions of various modulation formats abov, avrag SER/BER in Rayligh channl can b found. Not: avrag SER/BER is not th only prformanc masur! Outag probability is important as wll. Slow / fast fading: P out vs. P Lctur 3-Nov-6 3(4)
Outag probability: a probability that th instantanous SER xcds a givn thrshold ε, whr th P { P } Pr{ } = Pr > ε = γ < γ (.6) out th P γ = ε. γ = thrshold SNR, such that ( ) P out can b xprssd via th CDF F γ of instantanous SNR γ, γ ( ) th Pout = ρ γ dγ = Fγ ( γth) (.7) th Lctur 3-Nov-6 4(4)
Avrag BER Exprssions in Rayligh Fading: Cohrnt BPSK: P γ = + γ 4γ (.8) Cohrnt BFSK: DPSK: P Non-cohrnt orthogonal BFSK: γ = + γ γ P = ( ) + γ γ P = + γ γ (.9) (.) (.) Approximat xprssions hold for larg SNR ( γ ). Q.: Compar with th AWGN channl BER! Dp fad vnts dominat th avrag rror rat at high SNR, P ~ Pout ~ γ S [Ts, Ch. 3] for a dtaild discussion. (.) Lctur 3-Nov-6 5(4)
BER in a Fading Channl ~ SNR Fad margin (Fm) Q SNR ( SNR ) ~ T.S. Rappaport, Wirlss Communications, Prntic Hall, Th diffrnc is dramatic: th avrag BER is invrs in γ rathr than xponntial, as for AWGN channl. Fad margin: additional incras in SNR to kp th sam avrag BER as in a fixd AWGR channl. Larg incras in BER is du to dp fads. Som tchniqus ar rquird to mitigat th ffct of fading. Lctur 3-Nov-6 6(4)
Q-function and its approximation.4 Q( x).3 Qa( x).. 3 4 5 x. Avrag BER of BPSK xact approx.. BER 3 4 5 3 4 SNR [db] Lctur 3-Nov-6 7(4)
BER in a Fast Fading Channl Fast Fading: fdt s > or Ts > Tc Error floor: BER cannot b dcrasd blow a crtain lvl. Avrag BER of diffrntial BPSK * is P [ J ( fdts )] ( + γ ) + γ π = (.3) f d > f d = P.M. Shankar, Introduction to Wirlss Systms, Wily,. * Q: xplain why cohrnt BPSK cannot b usd. Lctur 3-Nov-6 8(4)
Th rror floor is obtaind by γ For small f T <<, d s P [ ( )] P = J π f T / (.4) f d s 5( f T ) or f T P / 5 (.5) f d s d s f P.M. Shankar, Introduction to Wirlss Systms, Wily,. Lctur 3-Nov-6 9(4)
For co-channl intrfrnc (I) and slow fading ( fdts ), P = + γ γi γ γ I I I ( γ γ + γ + γ ) ( γ π) whr γ I =SIR (signal-to-intrfrnc ratio). (.6) Intrprtation: total nois undr Gaussian assumption, Th rror floor: PtN = PN + PI ~ + γ γ Pf as γ γ I, or I (.7) γi (.8) Co-channl intrfrnc has a profound impact on th avrag BER. Ths conclusions ar tru for othr modulations as wll, i.. rror floor du to fast fading and I, frquncy slctiv fading. Th diffrnc is in whn th floor is achivd. For dtails, s W.C. Jaks, Jr.: Microwav Mobil Communications, John Wily and Sons, Nw York, 974. P f Lctur 3-Nov-6 (4)
P.M. Shankar, Introduction to Wirlss Systms, Wily,. Lctur 3-Nov-6 (4)
BER in a Frquncy Slctiv Fading Channl Rcall: th channl is frquncy slctiv whn fc < fs or τ >. T s Thr is an rror floor vn if τ < Ts! = rror floor d = τ / Ts T.S. Rappaport, Wirlss Communications, Prntic Hall, Lctur 3-Nov-6 (4)
For th DPSK, ELG479: Wirlss Communication Fundamntals S.Loyka P BER in a Rican Fading (LOS) + K K γ xp = ( + γ + K) + γ + K whr K A / spcular-to-scattrd powr ration. (.9) = σ is th ky-factor of Rican distribution, i.. BER in Rican fading channl.. 3 4 5 6 K= K= K= K= K= AWGN =? 3 4 avrag SNR [db] Lctur 3-Nov-6 3(4)
Summary BER in a slow flat fading channl. Dramatic consqunc of th fading. BER in fast-fading and frquncy-slctiv channl. Impact of intrfrnc. Error floors. Impact of LOS (Rican fading). Rading: Rappaport, Ch. 6. Any othr txt that covrs th topics abov. Not: Do not forgt to do nd-of-chaptr problms. Rmmbr th larning fficincy pyramid! Fundamntals of digital communications and information thory: strongly rcommndd rfrncs. J.M. Wozncraft, I.M. Jacobs, Principls of communication nginring, Wily: Nw York, 965.. D. Ts, P. Viswanath, Fundamntals of Wirlss Communications, Cambridg, 5. Chaptrs 3, 5, Appndics A and B. Othr books (s th rfrnc list). Not: Do not forgt to do nd-of-chaptr problms. Rmmbr th larning fficincy pyramid! Lctur 3-Nov-6 4(4)