Chapter 10. Timing Recovery. March 12, 2008

Transcription

1 Chapter 10 Timing Recovery March 12, 2008

2 b[n] coder bit/ symbol transmit filter, pt(t) Modulator Channel, c(t) noise interference form other users LNA/ AGC Demodulator receive/matched filter, p R(t) sampler Equalizer slicer/ decoder ˆb[n] Carrier recovery Timing recovery

3 10.1 Non-Data Aided Timing Recovery Methods

4 Fundamental results In a data modem, the demodulated received signal is given by y(t) = n= s[n]c BB (t nt b )

5 Fundamental results In a data modem, the demodulated received signal is given by y(t) = n= s[n]c BB (t nt b ) The key point: the ensemble mean square ρ(t) = E[ y(t) 2 ] = σ 2 s is periodic with period of T b. n= c BB (t nt b ) 2

6 Fundamental results In a data modem, the demodulated received signal is given by y(t) = n= s[n]c BB (t nt b ) The key point: the ensemble mean square ρ(t) = E[ y(t) 2 ] = σ 2 s n= c BB (t nt b ) 2 is periodic with period of T b. When c BB (t) is a bandlimited pulse within the range 1 T b f 1 T b, one finds that ( ) 2π ρ(t) = ρ ρ 1 cos t + ρ 1 T b where ρ 0 = σ2 s T b C ) BB(f ) 2 df, ρ 1 = σ2 s T b C BB(f )CBB (f 1Tb df.

7 The timing recovery cost function 1.5 ρ(τ) = ρ ρ 1 cos ( ) 2π τ + ρ 1 T b 1 ρ(τ) τ/t b

8 The optimum timing phase When p(t) be a Nyquist pulse, and p s(t) = X n= P s(f ) = 1 T b p(nt b )δ(t nt b ) = δ(t) X n= P f n! = 1 T b

9 The optimum timing phase When p(t) be a Nyquist pulse, and p s(t) = X n= P s(f ) = 1 T b p(nt b )δ(t nt b ) = δ(t) X n= P f n! = 1 T b When P(f ) has an excess bandwidth of less than 100%, we get!! P s(f ) = 1 P(f ) + P f 1 T b T b, for 0 f 1/T b

10 The optimum timing phase When p(t) be a Nyquist pulse, and p s(t) = X n= P s(f ) = 1 T b p(nt b )δ(t nt b ) = δ(t) X n= P f n! = 1 T b When P(f ) has an excess bandwidth of less than 100%, we get!! P s(f ) = 1 P(f ) + P f 1 T b T b, for 0 f 1/T b 1 P s (f) 0.8 AMPLITUDE (1/T b )P(f 1/T b ) (1/T b )P(f) ft b

11 The optimum timing phase (continued) Now, let us introduce a delay in the sampling time:

12 The optimum timing phase (continued) Now, let us introduce a delay in the sampling time: X p s(t, τ) = p(τ + nt b )δ(t τ nt b ) and n= P s(f, τ) = 1 ««P(f ) + P f 1Tb e j2πτ/t b T b

13 The optimum timing phase (continued) Now, let us introduce a delay in the sampling time: X p s(t, τ) = p(τ + nt b )δ(t τ nt b ) and n= P s(f, τ) = 1 ««P(f ) + P f 1Tb e j2πτ/t b T b τ = 0.2T b AMPLITUDE τ = 0.3T b 0.2 τ = 0.5T b ft b

14 Improving the cost function

15 10.2 Non-Data Aided Timing Recovery Algorithms

16 Early-late gate timing recovery ρ(τ) τ τ + δτ ρ (τ + δτ) ρ (τ δτ) τ δτ τ opt T b 2 τ opt τ opt + T b 2 τ τ opt T b 4 τ opt + T b 4 τ[n + 1] = τ[n] + µ (ρ (τ[n] + δτ) ρ (τ[n] δτ))

17 Early-late gate timing recovery ρ(τ) τ τ + δτ ρ (τ + δτ) ρ (τ δτ) τ δτ τ opt T b 2 τ opt τ opt + T b 2 τ τ opt T b 4 τ opt + T b 4 τ[n + 1] = τ[n] + µ (ρ (τ[n] + δτ) ρ (τ[n] δτ)) LMS-like version of this is τ[n + 1] = τ[n] + µ y (τ[n] + δτ + nt b ) 2 y (τ[n] δτ + nt b ) 2

18 MATLAB Script TR ELG.m: Early-late gate timing recovery Tb=0.0001; L=100; M1=20; Ts=Tb/L; fs=1/ts; fc=100000; delta c=0; N=8*L; phi c=0.5; sigma v=0; alpha=0.5; c=1; b=sign(randn(10000,1)); M=input( QAM size (4, 16, 64, 256) = ); if M==4 s=b(1:2:end)+i*b(2:2:end); elseif M==16 s=2*b(1:4:end)+b(2:4:end)+i*(2*b(3:4:end)+b(4:4:end)); elseif M==64 s=4*b(1:6:end)+2*b(2:6:end)+b(3:6:end)+... j*(4*b(4:6:end)+2*b(5:6:end)+b(6:6:end)); elseif M==256 s=8*b(1:8:end)+4*b(2:8:end)+2*b(3:8:end)+b(4:8:end)+... j*(8*b(5:8:end)+4*b(6:8:end)+2*b(7:8:end)+b(8:8:end)); else print( Error! M should be 4, 16, 64 or 256 ); end pt=sr cos p(n,l,alpha); xbbt=conv(expander(s,l),pt); t=[0:length(xbbt)-1] *Ts; xt=real(exp(i*2*pi*fc*t).*xbbt); xr=conv(c,xt); xr=xr+sigma v*randn(size(xr)); t=[0:length(xr)-1] *Ts; y=exp(-i*(2*pi*(fc-delta c)*t-phi c)).*xr; pr=pt; y=conv(y,pr); %%%%%%%%%%%%%%%%%%%%%%%% % TIMING RECOVER: Early-late Gating % %%%%%%%%%%%%%%%%%%%%%%%% beta=0; mu0=0.01; dtau=12; mu=mu0*(l/4)/dtau; Ly=length(y); tau=0.3*ones(1,round(ly/l)); kk=1; yp=0; ym=0; start=5*l+1 for k=start:l:length(tau)*l tautb=round(tau(kk)*l); yp=sqrt(1-betaˆ2)*y(k+tautb+dtau)-beta*yp; ym=sqrt(1-betaˆ2)*y(k+tautb-dtau)-beta*ym; tau(kk+1)=tau(kk)+mu*(abs(yp)ˆ2-abs(ym)ˆ2); kk=kk+1; end figure, plot(tau(1:kk), k ) xlabel( Iteration Number, n ), ylabel( tau[n] )

19 Early-late gate timing recovery (continued) β = δτ = 25 δτ = 18 δτ = 12 δτ = τ[n] Number of iterations, n

20 Early-late gate timing recovery (continued) β = τ[n] Number of iterations, n

21 Gradient-based algorithm τ[n + 1] = τ[n] + µ ρ(τ) τ

22 Gradient-based algorithm τ[n + 1] = τ[n] + µ ρ(τ) τ ρ(τ) τ y (τ[n] + δτ + nt b) 2 y (τ[n] δτ + nt b ) 2 2δτ

23 Gradient-based algorithm τ[n + 1] = τ[n] + µ ρ(τ) τ ρ(τ) τ y (τ[n] + δτ + nt b) 2 y (τ[n] δτ + nt b ) 2 2δτ When δτ = T b /2 (a case of interest), the following alternative form is obtained τ[n + 1] = τ[n] + µr{y 0 (τ[n] + nt b )y 1 (τ[n] + nt b + T b /2)}

24 Gradient-based algorithm τ[n + 1] = τ[n] + µ ρ(τ) τ ρ(τ) τ y (τ[n] + δτ + nt b) 2 y (τ[n] δτ + nt b ) 2 2δτ When δτ = T b /2 (a case of interest), the following alternative form is obtained τ[n + 1] = τ[n] + µr{y 0 (τ[n] + nt b )y 1 (τ[n] + nt b + T b /2)} Modified timing recovery loop for the realization of the recursion (10.35). for k=start:l:length(tau)*l tautb=round(tau(kk)*l); y0=sqrt(1-betaˆ2)*y(k+tautb)-beta*y0; y1=sqrt(1-betaˆ2)*y(k+tautb+l/2)-beta*y1; tau(kk+1)=tau(kk)+mu*real(y0*y1 ); kk=kk+1; end

25 Gradient-based algorithm (continued) τ[n] Number of iterations, n

26 Tone extraction algorithm y[n] 2 H(z) v[n] z Sample at the peaks of v[n]

27 Tone extraction algorithm (continued) Magnitude Normalized Frequency, f/f s

28 Tone extraction algorithm (continued) Magnitude Normalized Frequency, f/f s

29 Tone extraction algorithm (continued) z L/4

30 10.3 Data Aided Timing Recovery Methods

31 Mueller and Muller s method c BB (τ) c BB (τ T b ) c BB (τ + T b ) Seeks for the value of τ that minimizes the cost function η(τ) = R{c BB (τ + T b ) c BB (τ T b )}

32 Mueller and Muller s method c BB (τ) c BB (τ T b ) c BB (τ + T b ) Seeks for the value of τ that minimizes the cost function η(τ) = R{c BB (τ + T b ) c BB (τ T b )} The following update equation can be used to achieve this: τ[n+1] = τ[n]+µr {y(nt b + τ[n])ŝ [n 1] y((n 1)T b + τ[n])ŝ [n]}

33 Mueller and Muller s method (continued) Muller and Muller s timing recovery method. mu=0.01; Ly=length(y); kk=1; yp=0; ym=0; start=5*l+1; tau=0.3*ones(1,floor((ly-start)/l)); for k=start:l:length(tau)*l-l tautb=round(tau(kk)*l); sk=slicer(y(k+tautb),m); skm1=slicer(y(k+tautb-l),m); tau(kk+1)=tau(kk)+mu*real(y(k+tautb)*skm1 -y(k+tautb-l)*sk ); kk=kk+1; end figure, plot(tau(1:kk-1)) xlabel( Iteration Number, n ), ylabel( tau[n] )

34 Mueller and Muller s method (continued) τ[n] Number of iterations, n

35 Decision directed method τ is adjusted by minimizing where e[n] = s[n] y(nt b + τ). ξ = E[ e[n] 2 ]

36 Decision directed method τ is adjusted by minimizing where e[n] = s[n] y(nt b + τ). ξ = E[ e[n] 2 ] ˆξ τ = e[n] e [n] τ = 2R + e [n] e[n] τ { e [n] y(nt b + τ) τ { = 2R }. e [n] e[n] τ }

37 Decision directed method τ is adjusted by minimizing where e[n] = s[n] y(nt b + τ). ˆξ τ = e[n] e [n] τ = 2R ξ = E[ e[n] 2 ] + e [n] e[n] τ { e [n] y(nt b + τ) τ Moreover, we use the approximation to obtain y[n] τ { = 2R }. e [n] e[n] τ = y(nt b + τ + δτ) y(nt b + τ δτ). 2δτ τ[n + 1] = τ[n] µ ˆξ τ = τ[n] + µr {e [n](y(nt b + τ[n] + δτ) y(nt b + τ[n] δτ))}. }

38 Decision directed method (continued) Decision directed timing recovery method. mu=0.05; Ly=length(y); kk=1; start=5*l+1; tau=0.3*ones(1,floor((ly-start)/l)); for k=start:l:length(tau)*l-l tautb=round(tau(kk)*l); sk=slicer(y(k+tautb),m); tau(kk+1)=tau(kk)+mu*real((sk-y(k+tautb))... *(y(k+tautb+dtau)-y(k+tautb-dtau)) ); kk=kk+1; end figure, plot(tau(1:kk-1)) xlabel( Iteration Number, n ), ylabel( tau[n] )

39 Decision directed method (continued) τ[n] Iteration Number, n