Outline. Digital Communications. Lecture 12 Performance over Fading Channels and Diversity Techniques. Flat-Flat Fading. Intuition

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1 Digital Counications Lecture 2 Perforance over Fading Channels and Diversity Techniques Pierluigi SALVO ROSSI Outline Noncoherent/Coherent Detection 2 Channel Estiation Departent of Industrial and Inforation Engineering Second University of Naples Via Roa 29, 83 Aversa CE, Italy 3 Diversity hoepage: eail: pierluigi.salvorossi@unina2.it P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 / 27 P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 2 / 27 Flat-Flat Fading The signal odel is rt Hst + wt where H α jϑ is a coplex-valued r.v. Rayleigh fading is a coon and siple odel: H N C, 2σ 2 h, i.e. RH, IH N, σ2 h α Rayleighσ h ϑ U π, π usually E H 2 Eα 2 2σ 2 h The effect of fading is ainly changing the receive SNR into a r.v. γ α 2 E /η Intuition Consider a BPSK over a channel with average SNR γ db such that half of the tie is in state with γ db half of the tie is in state with γ 3 db The BERs corresponding to the two channel states are Pre state /2 Pre state The average BER is P e 2 Pre state + Pre state.25 2 The BER for BPSK over an AWGN channel with SNR γ γ db is P e 4 6 P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 3 / 27 P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 4 / 27

2 Noncoherent Detection /2 Noncoherent Detection 2/2 f r H,α,θr H : π r αe jθ s + w H N r αejθ s 2 r 2 +α 2 E π N r 2 +α 2 E π N 2 Rαe jθ s H η r o 2α r E cos r s θ f r H,αr 2π r 2 +α 2 E f 2π r H,α,θrdθ π N I 2α r E P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 5 / 27 f r H r The ML decision is f α af r H,arda f α a r 2 +a 2 E π N I û arg ax fr H r 2a r E da There is no dependence on the phase of the transitted signal s Constant-odulus odulations e.g. PSK and also QAM cannot e considered unless perforing channel estiation coherent detection P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 6 / 27 Coherent Detection /2 We assue that the specific realization of the r.v. H α jθ is known at the receiver This assuption is coonly denoted Channel State Inforation at the Receiver CSIR û arg ax fr H,α,θr arg in r αe jθ s 2 arg in α 2 E 2α E r cos r s θ Binary odulation with equal-energy signals û arg ax Rαe jθ s H r,2 arg ax,2 cos r s θ Coherent Detection 2/2 Copute the average BER considering H, as P e P e H The error event is Reeer that r αe jθ s 2 > r αe jθ s 2 2 αe jθ s + w αe jθ s 2 > αe jθ s + w αe jθ s 2 2 w 2 > αe jθ s s 2 + w 2 Re jθ s s 2 H w < α 2 s s 2 2 Re jθ s s 2 H w N, s s 2 2 then if we define γ α 2 E/ P e α Q s s ρ2e ρ α2 E Q ργ P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 7 / 27 P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 8 / 27

3 Average BER The SNR is now a r.v. thus the BER is a r.v. The average BER is an iportant perforance easure Average BER ay e coputed as follows: Copute the BER w.r.t. to a given SNR as done for AWGN P e γ Copute the pdf of the SNR f γ γ Use the total proaility theore P e f γ gp e gdg R SNR distriution Rayleigh fading: SNR is onentially distriuted γ Exp/ f γ γ γ uγ 2σh 2 Γ E Ricean fading: f γ γ + K Nakagai fading: + Kγ K f γ γ I γ uγ 4 + KKγ uγ P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 9 / 27 P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 / 27 Binary Modulations over Rayleigh Fading /3 BPSK BER for a given SNR is P e γ Q 2γ P e 2 Γ + 4 Binary Modulations over Rayleigh Fading 2/3 2 BER 3 BFSK BER for a given SNR is P e γ Q γ P e 2 Γ BPSK over AWGN BFSK over AWGN BPSK over Rayleigh fading BFSK over Rayleigh fading 5 5 SNR db P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 / 27 P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 2 / 27

4 Representations for the Q-function Moent Generating Function The classical definition of the Q-function is Qx t2 dt 2π 2 and corresponds to Pr N, > x x The prole is that the arguent x is in the integration liit and not in the integrand function An alternative representation is Qx π π/2 x2 2 sin 2 dθ θ Consider a r.v. X distriuted according to f X x with f X x x < The Moent Generating Function MGF of X is defined as the Laplace transfor of its pdf with reversed arguent sign Φ X s L f X x s E sx f X x sx dx The nth oent of X is otained as E X n x n f X xdx n s n Φ Xs s P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 3 / 27 P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 4 / 27 Moent Generating Function of the SNR Rayleigh fading Ricean fading Φ γ s Nakagai fading Φ γ s s + K + K s K s + K s Φ γ s Γ s P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 5 / 27 Alternative Method for Coputing the Average BER Assue the following BER for a given SNR P e γ α βγ then it is straightforward to get P e αφ γ β Assue the following BER for a given SNR P e γ c2 then it is straightforward to get P e α c c2 α βxγ dx Φ γ βxdx c P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 6 / 27

5 Binary Modulations over Rayleigh Fading 3/3 BPSK α /π, βθ / sin 2 θ, c, and c 2 π/2 BFSK P e π π/2 + sin 2 θ α /π, βθ /2 sin 2 θ, c, and c 2 π/2 P e π π/2 + 2 sin 2 θ dθ dθ Outage Proaility eaningful when T T c, i.e. a lock of transitted inforation undergoes any different channel realizations which fades alost independently Outage Proaility OP is eaningful when T T c, i.e. a lock of transitted inforation undergoes the sae channel realization and fading causes error ursts OP denotes the proaility that the SNR falls elow a given threshold P out Prγ < γ γ For Rayleigh fading the outage proaility is P out γ f γ gdg P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 7 / 27 P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 8 / 27 Channel Estiation /2 In order to eploy coherent detection we need CSIR One possiility is to transit pilot syols known at the receiver Pilot syols ust e orthogonal in tie or frequency to the data Pilot syols ust e less than coherence tie or coherence andwidth away fro the data Denote u p t and v p t the pth pilot syol and the corresponding received signal v p t Hu p t + w p t, A possile estiator could e Ĥ p,..., P p v ptu ptdt p p u H + w ptu ptdt pt 2 dt P p u pt 2 dt Channel Estiation 2/2 The estiation error is defined as ε Ĥ H p w ptu ptdt p u pt 2 dt It has the following interesting properties Eε ε 2 rs Eε 2 2 P p E p p E p p q SNR p dt dτ u ptu q τew p twqτ P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 9 / 27 P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 2 / 27

6 Diversity Provide the receiver with several replicas of the sae inforation each undergoing an ideally independent channel realization denote p the proaility that the signal fades elow a given threshold the proaility that L independent replicas of the sae inforation all fade elow the sae threshold is p L Most popular fors of diversity are: Tie diversity Frequency diversity Space diversity Transit diversity Receive diversity P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 2 / 27 Coherent Detection with Diversity /3 The signal odel for L independent diversity ranches is H : r l α l e jθ l s l + w l, l,..., L H l ML decision is û arg ax fr,...,r L H,α,...,α L,θ,...,θ L r,..., r L L arg in r l α l e jθ l s l 2 L arg in α 2 l E l 2Rα l e jθ l s H l r l Binary odulation with equal-energy signals L û arg ax Rα l e jθ l s H l r l,2 P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 22 / 27 Coherent Detection with Diversity 2/3 Copute the average BER considering H, i.e. r l α l e jθ ls l + w l, as P e P e H The error event is r l α l e jθ l s l 2 > r l α l e jθ l s l 2 2 w l 2 > α l e jθ l s l s l 2 + w l 2 Rα l e jθ l s l s l 2 H w l < 2 Reeer that αl 2 s l s l 2 2 Rα l e jθ s s 2 H w N, α 2 l s s 2 2 s s ρ2e Coherent Detection with Diversity 3/3 Then L Rα l e jθ s s 2 H w N, 2 ρ2e If we define then P e αl Q α 2 L α 2 l γ α 2 LE/ ρ α2 L E Q ργ α 2 l P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 23 / 27 P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 24 / 27

7 Erlang Distriution Binary Modulations over Rayleigh Fading with Diversity For Rayleigh i.i.d. channels α l Rayleigh/ 2, i.e. αl 2 Exp It is easy to show that αl 2 L α2 l ErlangL,, i.e. f α 2 ξ L L! ξl ξuξ pdf f α 2 L /L ξ L L L! ξl Lξuξ L L2 L3 L5 L pdf.5.5 L L2 L3 L5 L Define then and L f γ γ P e γ E Eγ E α 2 L L Γ αl 2 L L γ L L! L γ uγ f γ γq ρlγ dγ ! L ! 2 /L L P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 25 / 27 P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 26 / 27 BPSK over Rayleigh Fading with Diversity 2 AWGN L L2 L3 4 BER SNR db P. Salvo Rossi SUN.DIII Digital Counications - Lecture 2 27 / 27