ELEG 635 Digital Communication Theory. Lecture 11

Transcription

1 EEG 635 Digital Cmmuiati Thery eture

2 Ageda Carrier Syhrizati Phase k p (P) Deisi Direted ps N-Deisi Direted ps Timig Syhrizati Deisi Direted ps N-Deisi Direted ps frmati Thery Sha's aw iear Blk Cdes

3 Phase k p e(t)=s( π f 1+τ s G(s)= 1+τ s 1 t+ϕ)si ( π f p filter t+ ϕ)= 1 si ( ϕ ϕ)+ 1 si(4 π f t+ϕ+ ϕ) π f t+ ϕ= π f t t+k v(t)dt Assume the phase differee is small s that we a apprximate the P as a liear system si( ϕ ϕ) ϕ ϕ ϕ ϕ H (s)= KG(s)/ s 1+KG (s)/s = 1+(τ 1+ τ +1 /K ) s+(τ / K )s 1 s φ si ( ϕ ϕ) G(s) ( ζ ω H (s)= s ω / K ) s+ω + ζ ω s+ω ω = K / τ 1 ζ=ω (τ +1/ K )/ ϕ VCO

4 First Order P Respse t a abrupt hage i phase G(s)= s 1+s %First rder Phase k p um = [0.01 1]; de = [ ]; [a b d]=tfss(um,de); dt = 0.01; u = es(1,000); x = zers(,001); fr i = 1:000 x(:,i+1) = x(:,i) + dt.*a*x(:,i)+dt*b.*u(i); y(i) = *x(:,i); ed t = [0:dt:0]; plt(t(1:000),y) Phase time

5 P fr Phase Estimati %First rder Phase k p dt = 0.01; f = 10; f = 10; phi1 = *pi/4; phi(1) = *pi/8; u11 = 0; u131 = 0; t = [0:dt:0]; u1 = s(*pi*f*t(1:000)+phi1); fr i = 1:000 % Syig t Carrier if i==1 u3 = si(*pi*f*t(i)+phi(i)); else u3 = si(*pi*f*t(i)+phi(i-1)); ed u13 = u1(i).*u3; u(i) = sum(u13).*dt; if i==1 phi(i)=phi(i)-u(i); else phi(i)=phi(i-1)-u(i); ed ed plt(t(1:000),u//pi)

6 Additive Nise ad Phase Estimati - 1 s(t )= A s( π f (t)= x(t)s(π f (t)= i (t)s( π f t+ϕ(t)) t) y(t )si ( π f t+ϕ(t)) t ) (t)si ( π f q Narrw bad ise t+ϕ(t)) i (t)= x(t )s(ϕ(t ))+ y (t )si (ϕ(t)) q (t)= x(t)si(ϕ(t))+ y(t)s(ϕ(t)) i (t)+ j (t )=[x(t )+ jy(t)]e j ϕ(t) q e(t)= A si Δ ϕ+ i (t)si Δϕ (t )s Δ ϕ=a si Δ ϕ+ q l (t) l (t) ϕ ϕ φ - + A si( ϕ ϕ) + G(s) ϕ K/s VCO

7 Additive Nise ad Phase Estimati - Assume P = A / is muh larger tha the ise pwer (t) (t) i q (t)= si Δ ϕ s Δϕ A A The ise term has a pwer spetral desity f N A σ ϕ N = A H ( f ) df = N A 0 H ( f ) N df = B eq A SNR= N A B eq = 1 σ ϕ

8 Deisi-Direted ps deisi direted parameter estimati we assume that the ifrmati sequee,, is kw r we treat as a radm sequee ad average ver it. Csider the fllwig mdulati type r Λ Λ l (t)=e j ϕ (θ)=r[ 1 N T [ (θ)=r e j ϕ 1 N g (t T )=s r(t )s K 1 =0 * l * l (t )e j ϕ +z (t) (t)e j ϕ dt] =R [ 1 N (+1)T T r l (t) g * ( T r (t)s (t T )dt ] * l (t)dt ) e j ϕ] Λ (θ)=r[ e j ϕ 1 N K 1 =0 * y ] y ( +1 )T = T r l (t )g * (t T )dt Λ ϕ (ϕ)=r[ 1 N = ta M 1( [ K 1 =0 K 1 =0 R[ K 1 =0 * y ] s(ϕ) [ 1 N * y ] * ]) y K 1 =0 * ] y si(ϕ)

9 Deisi Direted Carrier Phase Estimati fr PAM Reeived Sigal x T ()dt 0 Sampler y Amplitude Detetr Time Sy Phase Geeratr x Sigal Pulse Geeratr ϕ M s( π f t + ϕ ) M Carrier Geeratr

10 Deisi-Direted ps deisi direted parameter estimati we assume that the ifrmati sequee,, is kw r we treat as a radm sequee ad average ver it. Csider the fllwig mdulati type r (t)= A g (t)s( π f t+ϕ) m r (t)s( π f t+ ϕ)= 1 [ A m g (t)+ i (t )]sδ ϕ 1 q (t)si Δ ϕ + duble frequey terms e(t)= 1 A g(t) m ( [ A g(t )+ m i (t)]si Δ ϕ (t)sδ ϕ q ) + duble frequey terms e(t)= 1 A 1 g(t) si Δ ϕ+ m A g(t)[ m i (t )si Δϕ (t )s Δ ϕ] + duble frequey terms q

11 Deisi Direted Carrier Phase Estimati fr PAM x T ()dt 0 Sampler Detetr Reeived Sigal 90 Phase Shift s( π f t + ϕ ) M Time Sy VCO p Filter e(t) x si( π f t+ ϕ ) M x Delay T

12 N - Deisi Direted Carrier Phase Estimati There are several ways t estimate the arrier phase usig -deisi direted tehiques, but the easiest is t use the pwer-law devie shw belw Reeived Sigal Mth Pwer Devie Badpass Filter Mf x p Filter Frequey Divider M si(π Mf t+ M ϕ ) M VCO

13 Symbl Timig Revery Maximum ikelihd (M) sluti fr Pulse Amplitude Mdulati (PAM) r Λ Λ y l d Λ (t)=s(t ; τ)+(t)= (τ)=c (τ)=c (τ)= T d τ (τ) l T r(t)s(t ; τ)dt T r (t) g(t T τ)dt = d d τ T g(t T τ)+(t) r(t) g(t T τ )dt =C r(t) g (t T τ )dt= y (τ) d d τ y (τ)=0

14 Symbl Timig Revery Maximum ikelihd (M) sluti fr Pulse Amplitude Mdulati (PAM) r Λ Λ y l d Λ (t)=s(t ; τ)+(t)= (τ)=c (τ)=c (τ)= T d τ (τ) l T r(t)s(t ; τ)dt T r (t) g(t T τ)dt = d d τ T g(t T τ)+(t) r(t) g(t T τ )dt =C r(t) g (t T τ )dt= y (τ) d d τ y (τ)=0

15 Symbl Timig Syhrizati Deisi Direted d y (τ) d τ Reeived Sigal Mathed Filter g(-t) d d τ (.) Sampler x T + τ M VCC

16 N-Deisi Direted Timig Revery Early ate Gate Maximum ikelihd (M) sluti fr Pulse Amplitude Mdulati (PAM) d Λ (τ) = d τ Λ (τ+δ) Λ δ (τ δ) d Λ d τ (τ) = C 4 δ [ y (τ+δ) y (τ δ)] d Λ d τ (τ) C 4 δ ([ T ]) ] r(t) g(t T τ δ)dt [ r (t) g(t T τ+δ)dt T

17 Deisi Direted Carrier Phase Estimati fr PAM x T ()dt 0 Sampler Square-aw Devie Advae By δ Reeived Sigal Symbl Wavefrm Geeratr Symbl Timig VCC Retard By δ p Filter - + x T () dt Sampler 0 Square-aw Devie

18 Early-ate Gate The perati f a early-late gate is based the fat that i a PAM mmuiati system the utput f the mathed filter is the autrrelati futi f the basi pulse sigal used i the PAM system. The autrrelati futi is maximized at the ptimum samplig time ad is symmetri. This meas that, i the absee f ise, at samplig times T + = T + δ ad T - = T δ, the utput will be equal. y(t + )= y(t - ) ad T = T + +T -

19 Example f Early-ate Gate Raised Csie Pulse Shape % Early_ate Gate. alpha=0.4; T=1/4800; t=[-3*t:1.001*t/100:3*t]; x=si(t./t).*(s(pi*alpha*t./t)./(1-4*alpha^*t.^/t^)); y=xrr(x); ty=[t-3*t,t(:legth(t))+3*t]; d=60; ee=0.01; e=1; =500; kk = 0; % Early ad late advae ad delay % Preisi % Step size % The irret samplig time while abs(abs(y(+d))-abs(y(-d)))>=ee kk=kk+1; (kk) = ; if abs(y(+d))-abs(y(-d))>0 =+e; elseif abs(y(+d))-abs(y(-d))<0 =-e; ed ed plt(ty,y) hld plt(ty(),y(),'r*') hld ff

20 frmati Thery - 1 Chael Capaity Biary Symmetri Chael (BSC) Sha's Thery 1-ε ε ε 1-ε C=max p (x) ( X ;Y ) (X ;Y )= x X y Y p( x) p( y x)lg p (x, y) p( x) p ( y)

21 frmati Thery - Chael Capaity Biary Symmetri Chael (BSC) C=1 H b (ϵ)=1 ( ϵ lg Bad limited AWGN hael ϵ (1 ϵ)lg (1 ϵ)) C=W lg ( 1+ P ) N W

22 frmati Thery - 3 BPSK with AWGN Chael C=1 H b(q( E N b )) %Chael Capaity SNR_dB = -0:0; sr = 10.^(SNR_dB/10); p = Qfu((*sr).^0.5); C = 1 -(-p.*lg(p) - (1-p).*lg(1-p)); plt(snr_db, C) grid xlabel('snr') ylabel('capaity')

23 frmati Thery - 4 BPSK with AWGN Chael p ( y)= 1 e ( y+ A) /σ + π σ 1 e ( y A) /σ π σ (X ;Y )= 1 p( y X =+A)lg p( y X =+A) p ( y)dy+ 1 p( y x= A)lg p( y x= A) ( y)dy p (X ;Y )= 1 f ( A σ)+ 1 f ( A σ ) f (a )= 1 π e ( y a) / lg 1+e ay dy

24 frmati Thery - 5 BPSK with AWGN Chael %Chael Capaity SNR_dB = -0:0; sr = 10.^(SNR_dB/10); p = Qfu((*sr).^0.5); C_BPSK = 1 -(-p.*lg(p) - (1-p).*lg(1-p)); x = -6:.01:6; p_f = Qfu(sr); C_BPSK_f = 1 -(-p_f.*lg(p_f) - (1-p_f).*lg(1-p_f)); fr kk = 1:legth(sr) C_f_p(kk) = sum(f_bpsk(x,sr(kk)))*.01; C_f_m(kk) = sum(f_bpsk(x,-sr(kk)))*.01; C_f(kk) = 1 + (.5*C_f_p(kk) +.5*C_f_m(kk)); ed Sft Hard plt(snr_db, C_BPSK_f, SNR_dB, C_f) grid xlabel('a/sigma (db)') ylabel('capaity') futi y = f_bpsk(x,a) y = 1/sqrt(*pi) * exp(-(x - a).^/).*lg(1./(1+exp(-*a.*x)));

25 Chael Cdig Chael dig a be desribed as the lever use f reduday Chael dig a be divided it tw lasses Blk des Biary sure utput sequees f legth k are mapped it biary hael iput sequees f legth Rate k/ bits per trasmissi (,k) blk des Cvlutial des Sure utputs f legth k are mapped it hael iputs, but the hael iputs deped t ly the mst reet k sure utputs, but als the last (-1)k iputs f the eder

26 Simple Example Assume a simple majrity rule blk de where a bit is set times where is a dd umber A errr urs if (+1)/ bits ut f the bits are i errr p e = ( k) ϵ k k=(+1)/ k (1 ϵ) Example: assume = 5 ad ε = p e 5 = ( 5 k) k 5 k 9 ( ) =10 k=3 The hael rate is ly 0% f the raw data rate, but the errr rate is e milli times better