Chapter 9. Carrier Acquisition and Tracking. March 5, 2008

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1 Chapter 9 Carrier Acquisition and Tracking March 5, 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 9.1 Non-Data Aided Carrier Recovery Methods

4 9.1.1 Binary PSK with a rectangular pulse-shape x BPSK (t) = x(t) cos(2πf c t) where x(t) = s[n]p T (t nt b ), n= s[n] {+1, 1} are the transmit symbols (bits).

5 9.1.1 Binary PSK with a rectangular pulse-shape x BPSK (t) = x(t) cos(2πf c t) where x(t) = s[n]p T (t nt b ), n= s[n] {+1, 1} are the transmit symbols (bits). When p T (t nt b ) = Π ( t T b ), x(t) = ±1, and thus x 2 BPSK(t) = cos 2 (2πf c t) = cos(4πf ct).

6 9.1.2 Binary PSK with a band-limited pulse-shape y BPSK (t) = y(t) cos(2πf c t + θ 0 )

7 9.1.2 Binary PSK with a band-limited pulse-shape y BPSK (t) = y(t) cos(2πf c t + θ 0 ) y 2 BPSK(t) = 1 2 (y 2 rms + y ac (t))(1 + cos(4πf c t + 2θ 0 ))

8 9.1.2 Binary PSK with a band-limited pulse-shape (continued) MATLAB Script CRExp1.m Tb=0.0001; L=100; Ts=Tb/L; fc=100000; fs=1/ts; alpha=0.5; N=8*L; sigmav=0; c=1; s=sign(randn(1000,1)); pt=sr cos p(n,l,alpha); xt=conv(expander(s,l),pt); t=[0:length(xt)-1] *Ts; xt=cos(2*pi*fc*t).*xt; xr=conv(c,xt); xr=xr+sigmav*randn(size(xr)); xr2 = xr.ˆ2; [ X, F ]=spec analysis(xr2,fs); figure, axes( position,[ ]), plot(f,x, k ) xlabel( FREQUENCY, Hz ), ylabel( AMPLITUDE )

9 9.1.2 Binary PSK with a band-limited pulse-shape (continued) 6 x AMPLITUDE FREQUENCY, Hz x 10 5

10 9.1.3 Quadrature Amplitude Modulation In the case of QAM, the modulation process results in the complex baseband signal where θ(t) = 2π f c t + θ 0. y(t) = (x R (t) + jx I (t))e jθ(t)

11 9.1.3 Quadrature Amplitude Modulation In the case of QAM, the modulation process results in the complex baseband signal y(t) = (x R (t) + jx I (t))e jθ(t) where θ(t) = 2π f c t + θ 0. Taking the 4th power of y(t), we get or y 4 (t) = ( x 4 R (t) + x 4 I (t) 6x 2 R (t)x 2 I (t) + j2x R (t)x I (t)(x 2 R (t) x 2 I (t))) e j4θ(t) y 4 (t) = m y4 e j4θ(t) + v(t)e j4θ(t) where [ ] m y4 = avg xr(t) 4 + xi 4 (t) 6xR(t)x 2 I 2 (t).

12 Numerical Study of m y4 m y4 = m y4 E [ y(t) 4 ] Table: Numerical values of the normalized mean m y4 for different QAM constellations and three choices of the roll-off factor α. Constellation α = 0.25 α = 0.5 α = 1 4-QAM/QPSK QAM QAM QAM

13 MATLAB Script CRExp2.m Tb=0.0001; L=100; M1=20; Ts=Tb/L; fs=1/ts; fc=100000; Dfc=10; N=8*L; phic=0.5; sigmav=0; alpha=0.5; c=1; Nb=12000; b=sign(randn(nb,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+sigmav*randn(size(xr)); t=[0:length(xr)-1] *Ts; y=2*exp(-i*(2*pi*(fc-dfc)*t-phic)).*xr; pr=pt; y=conv(y,pr); y=y(1:m1:end); fs1=fs/m1; y4=y.ˆ4; [ X, F ]=spec analysis(y4,fs1); figure, axes( position,[ ]), plot(f,x, k ) xlabel( FREQUENCY, Hz ), ylabel( AMPLITUDE )

14 AMPLITUDE FREQUENCY, Hz x 10 4

15 9.2 Non-Data Aided Carrier Acquisition and Tracking Algorithms

16 9.2.1 Coarse carrier acquisition MATLAB Script CRExp2.m (Extension 1) Coarse carrier acquisition and compensation [xmax,imax]=max(x); Dfc est=f(imax)/4; y1=y.*exp(-j*2*pi*dfc est*ts*m1);

17 9.2.2 Fine carrier acquisition and tracking y[n] ( ) 4 PLL 4 Interpolator e j

18 9.3 Costas Loop Costas loop for AM signal x AM [n] = x[n] cos(2πf c nt s + θ) + ν[n]

19 9.3 Costas Loop Costas loop for AM signal x AM [n] = x[n] cos(2πf c nt s + θ) + ν[n] We wish to synthesize and adjust φ to track θ. y[n] = cos(2πf c nt s + φ)

20 9.3 Costas Loop Costas loop for AM signal x AM [n] = x[n] cos(2πf c nt s + θ) + ν[n] We wish to synthesize y[n] = cos(2πf c nt s + φ) and adjust φ to track θ. Cost function: [ ( ) ] 2 ξ = E LP(x AM [n]y[n])

21 9.3 Costas Loop Costas loop for AM signal x AM [n] = x[n] cos(2πf c nt s + θ) + ν[n] We wish to synthesize y[n] = cos(2πf c nt s + φ) and adjust φ to track θ. Cost function: [ ( ) ] 2 ξ = E LP(x AM [n]y[n]) Here, x AM [n]y[n] = x[n] 2 and [ ] cos(θ φ)+cos(4πf c nt s +θ+φ) +ν[n] cos(2πf c nt s +φ) LP(x AM [n]y[n]) = x[n] 2 cos(θ φ) + ψ[n].

22 9.3 Costas Loop Costas loop for AM signal x AM [n] = x[n] cos(2πf c nt s + θ) + ν[n] We wish to synthesize y[n] = cos(2πf c nt s + φ) and adjust φ to track θ. Cost function: [ ( ) ] 2 ξ = E LP(x AM [n]y[n]) Here, x AM [n]y[n] = x[n] 2 and Hence, [ ] cos(θ φ)+cos(4πf c nt s +θ+φ) +ν[n] cos(2πf c nt s +φ) LP(x AM [n]y[n]) = x[n] 2 cos(θ φ) + ψ[n]. ξ = σ2 x 4 cos2 (θ φ) + σ 2 ψ

23 9.3 Costas Loop Costas loop for AM signal x AM [n] = x[n] cos(2πf c nt s + θ) + ν[n] We wish to synthesize y[n] = cos(2πf c nt s + φ) and adjust φ to track θ. Cost function: [ ( ) ] 2 ξ = E LP(x AM [n]y[n]) Here, x AM [n]y[n] = x[n] 2 and Hence, [ ] cos(θ φ)+cos(4πf c nt s +θ+φ) +ν[n] cos(2πf c nt s +φ) LP(x AM [n]y[n]) = x[n] 2 cos(θ φ) + ψ[n]. ξ = σ2 x 4 cos2 (θ φ) + σ 2 ψ Clearly, maximizing ξ leads to the desired tracking, i.e., the PLL.

24 9.3 Costas Loop Derivation of Costas loop for AM signals ˆξ = ( ) 2 LP(x AM [n]y[n]) φ[n + 1] = φ[n] + µ ˆξ φ ˆξ φ = 2LP (x AM[n]y[n]) (LP(x AM[n]y[n])) φ ( ) ˆξ φ = 2LP(x AM[n]y[n])LP x AM [n] y[n] φ φ[n + 1] = φ[n] + 2µLP(x AM [n] cos(2πf c nt s + φ[n])) LP (x AM [n] ( sin(2πf c nt s + φ[n]))).

25 9.3 Costas Loop Costas loop for AM signals

26 9.3 Costas Loop Linear model of Costas loop for AM signals φ ν [n] θ[n] ɛ[n] L(z) c[n] µz 1 φ[n] 1 z 1

27 9.3 Costas Loop Costas loop for QAM signals

28 9.3 Pilot Aided Carrier Acquisition Method We begin with y(t) = e j2π fct x(t) + ν(t), where X x(t) = s[n]h 0 (t nt b ) n=

29 9.3 Pilot Aided Carrier Acquisition Method We begin with y(t) = e j2π fct x(t) + ν(t), where X x(t) = s[n]h 0 (t nt b ) n= In discrete-time, when the samples are at the spacing T b, y[n] = x[n]e j2π fcnt b + ν[n]

30 9.3 Pilot Aided Carrier Acquisition Method We begin with y(t) = e j2π fct x(t) + ν(t), where X x(t) = s[n]h 0 (t nt b ) n= In discrete-time, when the samples are at the spacing T b, y[n] = x[n]e j2π fcnt b + ν[n] Assuming that the transmit symbols are periodic and have a period of N symbols, after a transient interval, x[n] = x[n + N]

31 9.3 Pilot Aided Carrier Acquisition Method We begin with y(t) = e j2π fct x(t) + ν(t), where X x(t) = s[n]h 0 (t nt b ) n= In discrete-time, when the samples are at the spacing T b, y[n] = x[n]e j2π fcnt b + ν[n] Assuming that the transmit symbols are periodic and have a period of N symbols, after a transient interval, x[n] = x[n + N] Next, we form the summation J = N X 2 J e j2π fcnt b n=n 1 y[n + N]y [n] and note that N X 2 n=n 1 x[n] 2

32 9.3 Pilot Aided Carrier Acquisition Method We begin with y(t) = e j2π fct x(t) + ν(t), where X x(t) = s[n]h 0 (t nt b ) n= In discrete-time, when the samples are at the spacing T b, y[n] = x[n]e j2π fcnt b + ν[n] Assuming that the transmit symbols are periodic and have a period of N symbols, after a transient interval, x[n] = x[n + N] Next, we form the summation J = Solving this for f c, we get N X 2 J e j2π fcnt b f c n=n 1 y[n + N]y [n] and note that N X 2 n=n 1 x[n] 2 1 2πNT b (J)

33 9.3 Pilot Aided Carrier Acquisition Method (continued) Lock Range: Correct operation of the above procedure requires that π < (J) < π

34 9.3 Pilot Aided Carrier Acquisition Method (continued) Lock Range: Correct operation of the above procedure requires that π < (J) < π This leads to the lock range f b 2N < f c < f b 2N

35 9.4 Data Aided Carrier Tracking Method received signal demodulation/ carrier and timing recovery equalizer s[n] PLL slicer ŝ[n] detected data e j phase detector loop filter ɛ[n] = ( s[n]ŝ [n]) I{ s[n]ŝ [n]} R{ s[n]ŝ [n]}

36 9.4 Data Aided Carrier Tracking Method (continued) MATLAB Script DDCR.m Tb=0.0001; L=100; Ts=Tb/L; fs=1/ts; fc=100000; Dfc=0; N=8*L; phic=pi/8; sigmav=0.05; alpha=0.5; c=1; b=sign(randn(2000,1)); s=b(1:2:end)+i*b(2:2:end); pt=sr cosp(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+sigmav*randn(size(xr)); t=[0:length(xr)-1] *Ts; y=2*exp(-i*(2*pi*(fc-dfc)*t-phic)).*xr; pr=pt; y=conv(y,pr); y=y(1:l:end);y=y(9:end-8); % Extract the received signal samples % at the correct timing phase phi=zeros(size(y)); s1=zeros(size(y)); mu=0.01; for n=1:length(y)-1 s1(n)=y(n)*exp(-j*phi(n)); s2=sign(real(s1(n)))+j*sign(imag(s1(n))); % Slicer s12=s1(n)*s2 ; e=imag(s12)/real(s12); phi(n+1)=phi(n)+mu*e; end figure(1), plot(phi) figure(2), plot(y, * ), axis( square), hold on, plot(s1(1:end-1), *r ), plot([-2 2],[-2 2],[-2 2],[2-2]), hold off

37 9.4 Data Aided Carrier Tracking Method (continued) φ[n] n

38 9.4 Data Aided Carrier Tracking Method (continued) Imaginary part Real part

Outline. EECS 3213 Fall Sebastian Magierowski York University. Review Passband Modulation. Constellations ASK, FSK, PSK.

Outline. EECS 3213 Fall Sebastian Magierowski York University. Review Passband Modulation. Constellations ASK, FSK, PSK. EECS 3213 Fall 2014 L12: Modulation Sebastian Magierowski York University 1 Outline Review Passband Modulation ASK, FSK, PSK Constellations 2 1 Underlying Idea Attempting to send a sequence of digits through