Diversity Combining Techniques

Transcription

1 Diversity Combiig Techiques Whe the required sigal is a combiatio of several waves (i.e, multipath), the total sigal amplitude may experiece deep fades (i.e, Rayleigh fadig), over time or space. The major problem is to combat these deep fades, which result i system outage. SISO x Most popular ad efficiet techique for doig so is to use some form of diversity combiig. Diversity: multiple copies of the required sigal are available, which experiece idepedet fadig (or close to that). Effective way to combat fadig. Basic priciple: create multiple idepedet paths for the sigal, ad combie them i a optimum or ear-optimum way. Lecture -Dec-5 (4)

2 Combiig techiques Selectio combiig (SC). Maximum ratio combiig (MRC). Equal gai combiig (EGC). Hybrid combiig (two differet forms) Micro-diversity ad macro-diversity. Selectio Combiig db time brach brach. output Lecture -Dec-5 (4)

3 Types of Diversity Space diversity: Ateas are separated i space. Frequecy diversity: Multiple copies are trasmitted at differet frequecies. Time diversity: Multiple copies are trasmitted over differet time slots. Polarizatio diversity: Multiple copies have differet field polarizatios. For all types of diversity, multiple sigal copies must be ucorrelated (or weakly correlated, corr. coeff..5 ) for diversity combiig to be most effective. Hece differet types of diversity are ormally efficiet i differet scearios. All the forms have their ow advatages ad disadvatages. I ay particular case, some forms are better tha the other. Example: if the chael is static (or-slow-fadig), time diversity is ot good. If the chael is frequecy-flat, frequecy diversity is ot good. A diversity form is efficiet whe the sigals are idepedet (ucorrelated). Multiple sigals have to be combied i a certai way. Lecture -Dec-5 3(4)

4 Selectio combiig (SC) Key idea: at ay give momet of time, pick up the brach with the largest SR. h h h System model: y = hx + ξ (.) Output: select the brach with the largest SR. Lecture -Dec-5 4(4)

5 Selectio Combiig db time brach brach. output Lecture -Dec-5 5(4)

6 Aalysis of a selectio combier Cosider the SC i a frequecy-flat slow-fadig Rayleigh chael. Assume that some diversity form provides idepedet paths (each is i.i.d. Rayleigh-fadig). Istataeous SR i brach i is, E γ i = h i (.) where h i is the chael (complex) gai, E is symbol eergy ad is the oise spectral power desity (assumed to be the same i all the braches). The PDF of γ i is where ρ γ = (.3) γ ( ) e i γ γ E γ = γ i = h i is the average SR. The CDF (i.e., the probability that γ i < γ ) is γ i i i γ / γ ( ) ( ) Fγ γ = ρ γ dγ = e (.4) Lecture -Dec-5 6(4)

7 This is also outage probability of i-th brach. The output SR is The SC outage probability is P γ out = maxi γ i (.5) { } = Pr{ γ < γ } = Pr γ,.., γ < γ out out ( ) ( ) ( ) γ = F γ = F γ = e γ γ i= Asymptotically, for γ << γ (low outage prob.), P out ( ) ( ) i = F γ = γ γ γ γ (.6) (.7) where is the diversity order. ote sigificat decrease i outage probability compared to = case. The average output SR is Q.: prove it! γ out = γ i (.8) i= The most importat improvemet is i terms of outage probability rather tha the average SR! Example: =, γ γ = ; P out =? γ out / γ =? Lecture -Dec-5 7(4)

8 PDF of a selectio combier: f ( ) df ( γ) γ = dγ (.9) As icreases, the peak of the output PDF moves to the right ad the curve becomes more ad more closer to Gaussia. The Rayleigh chael becomes a Gaussia oe as. Q.: prove it! Lecture -Dec-5 8(4)

9 Selectio Combiig Diversity gai: how much less SR is required to achieve the same Pout Selectio Diversity Outage Probability. =,,3,4, Gd = db 4 3. γ / γ P ( γ ) = P ( γ ) = P out γ G = G = γ γ db d or d γ ( )[ ] Lecture -Dec-5 9(4)

10 Diversity order d : umber of idepedet braches, d P c ~ out d d ( γ ) ( γ ) =, (.) which is also the SR expoet i i.i.d. Rayleigh fadig. Formal defiitio: d = lim γ l P out l γ (.) Diversity gai: the differece i SR required to achieve certai outage probability usig ad braches. P ( γ ) = P ( γ ) = P out γ G G db ( )[ ] d = or d = γ γ γ (.) Effect of brach correlatio: ca sigificatly degrade the performace if too high. Practical cosideratios: atea switch ca be used. The, oly oe R x is required > very simple! Lecture -Dec-5 (4)

11 Maximal Ratio Combiig (MRC) Key idea: use liear coheret combiig of brach sigals so that the output SR is maximized h w h h w w The equivalet basebad model is as follows. The idividual brach sigals are x = A h + ξ (.3) where A - complex evelope (amplitude), h - chael (complex) gai, ( ) ~ C, ξ σ is AWG. ote that h acts as multiplicative oise. Lecture -Dec-5 (4)

12 The output of the combier is x = w x = A w h + w ξ out = sigal oise (.4) where w are the combiig weights. The sigal ad oise compoets are give by st ad d term correspodigly. The sigal ad oise power at the output are: s = = P A w h A w h P = w ξ = w σ ξ (.5) where σ = ξ is the brach oise power. Output SR is SR out = Ps P = ξ A w h w σ We ca maximize it usig the Lagrage multipliers or better, Cauchy-Schwarz iequality. (.6) Lecture -Dec-5 (4)

13 Cuachy-Schwartz iequality: * ab a b (.7) With the equality achieved whe a * = cb Usig (.7), the best combiig weights are w * h / ad the output SR of the best combier is = σ (.8) where γ = γ out (.9) is -th brach SR. γ = A h σ (.) The resultig combier is called MRC. It is the best combier i terms of the SR. Q.: prove (.8)! Lecture -Dec-5 3(4)

14 * ote: w = h / σ is true for ay chael, ot ecessarily Rayleigh. Whe all the brach oise powers are equal, * h w, out σ = σ = γ = γ h (.) The brach gai is proportioal to the chael gai +coheret combiig. ote -fold icrease i average SR (but this is ot the mai gai!). Q.: explai it! Lecture -Dec-5 4(4)

15 Outage Probability Cosider a Rayleigh chael ad assume there is o correlatio betwee braches (i.i.d. chael gais). The outage probability ca be expressed as P out = e k ( γ γ ) ( k ) γ γ (.)! k= Asymptotically, for γ γ (small P out ), ( γ γ ) γ out = ~ out, ~! γ ( γ ) P c P (.3) Diversity order = (prove it!) Q.: Diversity gai? Lecture -Dec-5 5(4)

16 MRC Outage Probability MRC.. =,,3,4,6 d =? γ / γ Q.: compare to SC ad EGC! Lecture -Dec-5 6(4)

17 MRC PDF P.M. Shakar, Itroductio to Wireless Systems, Wiley,. Lecture -Dec-5 7(4)

18 Implemetatio Issues Combier complexity varies sigificatly from type to type ( as well as performace). Overall, there exists performace/complexity trade-off. Highperformace combier (MRC) is difficult to implemet; simpler combier (SC) does ot perform that well as MRC, EGC is i betwee. Selectio diversity: a brach with the strogest sigal is used; i practice - with the largest sigal + oise power sice it is difficult to measure SR aloe. This must be doe o cotiuous basis may be difficult. Ca be implemeted with some time costat or as scaig (switched) diversity. SC is the simplest techique. Maximum ratio combiig: coheret techique, i.e., sigal s phase has to be estimated. All the sigals are co-phased ad weighted accordig to their sigal voltage to oise power ratios. It requires for a lot of circuitry ( idividual receivers) Provides the best performace. DSP makes it practical. The output may be acceptable eve whe all the iputs are ot. Equal gai combiig: compromise. Less complexity tha MRC (all the weights are equal), but co-phasig is till required. Ca still produce acceptable output from uacceptable iputs. If the idividual brach sigals are ot idepedet (correlated), the performace degrades. Lecture -Dec-5 8(4)

19 Average BER Improvemet Usig Diversity Combiig Istataeous SR is a R.V. Hece, BER is a RV as well. Itroduce average BER (frequecy flat, slow fadig), P = P γ ρ γ dγ e e ( ) ( ) (.4) where ρ( γ ) is a pdf of γ at the output of a diversity combier. Recall that the BER ca be expressed i may cases as ( ) P ( ) e γ = Q αγ (.5) where α = for BPSK ad QPSK, α = for BFSK (all detected coheretly) ad sub-optimal DPSK. Other modulatio formats have similar BER expressios (or bouds). Hece, we evaluate average BER based o this expressios. Lecture -Dec-5 9(4)

20 Cosider -brach MRC (the best oe), ( ) /! ( γ) γ/ γ γ F ( γ ) = e i γ f i= df γ γ = = dγ! γ ( ) i e γ/ γ (.6) The average BER of a coheret demodulatio, P i i e = ( µ ) C + i ( ) + µ (.7) i= µ = αγ + αγ, C k =!/ ( k!( k)!). where /( ) Asymptotically, γ, (.8) αγ Pe C αγ Diversity order =. Lecture -Dec-5 (4)

21 The average BER for MRC ad the coheret BPSK receiver.. Perror SR, db = = =3 P ( ) =4 e γ P out γ. =5 <- explai this! Lecture -Dec-5 (4)

22 ELG479: Wireless Commuicatio Fudametals S.Loyka Average BER for various modulatio formats J. Proakis, Digital Commuicatios, McGraw Hill, Bosto,. Lecture -Dec-5 (4)

23 How to reduce P, P i a fadig chael? e out Diversity combiig techiques Strog LOS (large K-factor) via proper locatio of ateas BER i Ricea fadig chael.. Perror SR, db = = =3 =4 = K= K= K= K= K= AWG 3 4 average SR [db] Lecture -Dec-5 3(4)

24 Summary Diversity combiig techiques. Forms of diversity ad types of combiig Performace improvemet due to diversity combiig Average BER ad outage probability Readig: Rappaport, sec. 6., 7. Stuber, Ch. 6 Ay other text that covers the topics above. ote: Do ot forget to do ed-of-chapter problems. Remember the learig efficiecy pyramid! Lecture -Dec-5 4(4)