Chapter 9

Carrier Acquisition and Tracking

March 5, 2008

Demodulator

coderbit/

symbolModulator

⊕ ⊕

Carrier

recovery

transmit

filter, pT(t)

slicer/

decoder

interference form

other usersnoise

sampler

b[n]

b[n]

Timing

recovery

filter, pR(t)

receive/matchedEqualizer

Channel, c(t)

LNA/AGC

9.1 Non-Data Aided Carrier Recovery Methods

9.1.1 Binary PSK with a rectangular pulse-shape

xBPSK(t) = x(t) cos(2πfc t)

where

x(t) =

∞∑

n=−∞

s[n]pT(t − nTb),

s[n] ∈ {+1,−1} are the transmit symbols (bits).

9.1.1 Binary PSK with a rectangular pulse-shape

xBPSK(t) = x(t) cos(2πfc t)

where

x(t) =

∞∑

n=−∞

s[n]pT(t − nTb),

s[n] ∈ {+1,−1} are the transmit symbols (bits).

When pT(t − nTb) = Π(

tTb

)

, x(t) = ±1, and thus

x2BPSK(t) = cos2(2πfc t) =

12

+12

cos(4πfc t).

9.1.2 Binary PSK with a band-limited pulse-shape

yBPSK(t) = y(t) cos(2πfc t + θ0)

9.1.2 Binary PSK with a band-limited pulse-shape

yBPSK(t) = y(t) cos(2πfc t + θ0)

y2BPSK(t) =

12(y2

rms + yac(t))(1 + cos(4πfc t + 2θ0))

9.1.2 Binary PSK with a band-limited pulse-shape(continued)

MATLAB Script CRExp1.mTb=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’,[0.1 0.25 0.8 0.5]), plot(F,X,’k’)xlabel(’FREQUENCY, Hz’), ylabel(’AMPLITUDE’)

9.1.2 Binary PSK with a band-limited pulse-shape(continued)

−5 −4 −3 −2 −1 0 1 2 3 4 5

x 105

0

1

2

3

4

5

6x 10−4

FREQUENCY, Hz

AM

PLI

TUD

E

9.1.3 Quadrature Amplitude ModulationIn the case of QAM, the modulation process results in thecomplex baseband signal

y(t) = (xR(t) + jxI(t))ejθ(t)

where θ(t) = 2π∆fct + θ0.

9.1.3 Quadrature Amplitude ModulationIn the case of QAM, the modulation process results in thecomplex baseband signal

y(t) = (xR(t) + jxI(t))ejθ(t)

where θ(t) = 2π∆fct + θ0.Taking the 4th power of y(t), we get

y4(t) =(

x4R(t) + x4

I (t) − 6x2R(t)x2

I (t)

+ j2xR(t)xI(t)(x2R(t) − x2

I (t)))

ej4θ(t)

or

y4(t) = my4ej4θ(t) + v(t)ej4θ(t)

wheremy4 = avg

[

x4R(t) + x4

I (t) − 6x2R(t)x2

I (t)]

.

Numerical Study of my4

my4 =my4

E [|y(t)|4]

Table: Numerical values of the normalized mean my4 for differentQAM constellations and three choices of the roll-off factor α.

Constellation α = 0.25 α = 0.5 α = 14-QAM/QPSK −0.65 −0.79 −0.95

16-QAM −0.37 −0.43 −0.4964-QAM −0.33 −0.37 −0.43256-QAM −0.32 −0.36 −0.42

MATLAB Script CRExp2.mTb=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’); endpT=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’,[0.1 0.25 0.8 0.5]), plot(F,X,’k’)xlabel(’FREQUENCY, Hz’), ylabel(’AMPLITUDE’)

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

x 104

0

2

4

6

8

10

12

FREQUENCY, Hz

AM

PLI

TUD

E

9.2 Non-Data Aided Carrier Acquisition and TrackingAlgorithms

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);

9.2.2 Fine carrier acquisition and tracking

⊗

(·)4y[n]

e−j·

Interpolator

PLL ÷4

9.3 Costas LoopCostas loop for AM signal xAM[n] = x [n] cos(2πfcnTs + θ) + ν[n]

9.3 Costas LoopCostas loop for AM signal xAM[n] = x [n] cos(2πfcnTs + θ) + ν[n]We wish to synthesize

y [n] = cos(2πfcnTs + φ)

and adjust φ to track θ.

9.3 Costas LoopCostas loop for AM signal xAM[n] = x [n] cos(2πfcnTs + θ) + ν[n]We wish to synthesize

y [n] = cos(2πfcnTs + φ)

and adjust φ to track θ.Cost function:

ξ = E

[

(

LP(xAM[n]y [n])

)2]

9.3 Costas LoopCostas loop for AM signal xAM[n] = x [n] cos(2πfcnTs + θ) + ν[n]We wish to synthesize

y [n] = cos(2πfcnTs + φ)

and adjust φ to track θ.Cost function:

ξ = E

[

(

LP(xAM[n]y [n])

)2]

Here,

xAM[n]y [n] =x [n]

2

[

cos(θ−φ)+cos(4πfcnTs+θ+φ)

]

+ν[n] cos(2πfcnTs+φ)

andLP(xAM[n]y [n]) =

x [n]

2cos(θ − φ) + ψ[n].

9.3 Costas LoopCostas loop for AM signal xAM[n] = x [n] cos(2πfcnTs + θ) + ν[n]We wish to synthesize

y [n] = cos(2πfcnTs + φ)

and adjust φ to track θ.Cost function:

ξ = E

[

(

LP(xAM[n]y [n])

)2]

Here,

xAM[n]y [n] =x [n]

2

[

cos(θ−φ)+cos(4πfcnTs+θ+φ)

]

+ν[n] cos(2πfcnTs+φ)

andLP(xAM[n]y [n]) =

x [n]

2cos(θ − φ) + ψ[n].

Hence,

ξ =σ2

x

4cos2(θ − φ) + σ2

ψ

9.3 Costas LoopCostas loop for AM signal xAM[n] = x [n] cos(2πfcnTs + θ) + ν[n]We wish to synthesize

y [n] = cos(2πfcnTs + φ)

and adjust φ to track θ.Cost function:

ξ = E

[

(

LP(xAM[n]y [n])

)2]

Here,

xAM[n]y [n] =x [n]

2

[

cos(θ−φ)+cos(4πfcnTs+θ+φ)

]

+ν[n] cos(2πfcnTs+φ)

andLP(xAM[n]y [n]) =

x [n]

2cos(θ − φ) + ψ[n].

Hence,

ξ =σ2

x

4cos2(θ − φ) + σ2

ψ

Clearly, maximizing ξ leads to the desired tracking, i.e., the PLL.

9.3 Costas Loop

Derivation of Costas loop for AM signals

ξ =

(

LP(xAM[n]y [n])

)2

φ[n + 1] = φ[n] + µ∂ξ

∂φ

∂ξ

∂φ= 2LP (xAM[n]y [n])

∂ (LP(xAM[n]y [n]))

∂φ

∂ξ

∂φ= 2LP(xAM[n]y [n])LP

(

xAM[n]∂y [n]

∂φ

)

φ[n + 1] = φ[n] + 2µLP(xAM[n] cos(2πfcnTs + φ[n]))

×LP (xAM[n] × (− sin(2πfcnTs + φ[n]))) .

9.3 Costas Loop

Costas loop for AM signals

9.3 Costas Loop

Linear model of Costas loop for AM signals

µz−1

1 − z−1

φ[n]θ[n] c[n]ε[n]⊕ L(z )

φν [n]

9.3 Costas Loop

Costas loop for QAM signals

9.3 Pilot Aided Carrier Acquisition MethodWe begin with y(t) = ej2π∆fc tx(t) + ν(t), where

x(t) =

∞X

n=−∞

s[n]h0(t − nTb)

9.3 Pilot Aided Carrier Acquisition MethodWe begin with y(t) = ej2π∆fc tx(t) + ν(t), where

x(t) =

∞X

n=−∞

s[n]h0(t − nTb)

In discrete-time, when the samples are at the spacing Tb,

y [n] = x [n]ej2π∆fcnTb + ν[n]

9.3 Pilot Aided Carrier Acquisition MethodWe begin with y(t) = ej2π∆fc tx(t) + ν(t), where

x(t) =

∞X

n=−∞

s[n]h0(t − nTb)

In discrete-time, when the samples are at the spacing Tb,

y [n] = x [n]ej2π∆fcnTb + ν[n]

Assuming that the transmit symbols are periodic and have a period of Nsymbols, after a transient interval,

x [n] = x [n + N]

9.3 Pilot Aided Carrier Acquisition MethodWe begin with y(t) = ej2π∆fc tx(t) + ν(t), where

x(t) =

∞X

n=−∞

s[n]h0(t − nTb)

In discrete-time, when the samples are at the spacing Tb,

y [n] = x [n]ej2π∆fcnTb + ν[n]

Assuming that the transmit symbols are periodic and have a period of Nsymbols, after a transient interval,

x [n] = x [n + N]

Next, we form the summation J =

N2X

n=N1

y [n + N]y∗[n] and note that

J ≈ ej2π∆fcNTb

N2X

n=N1

|x [n]|2

9.3 Pilot Aided Carrier Acquisition MethodWe begin with y(t) = ej2π∆fc tx(t) + ν(t), where

x(t) =

∞X

n=−∞

s[n]h0(t − nTb)

In discrete-time, when the samples are at the spacing Tb,

y [n] = x [n]ej2π∆fcnTb + ν[n]

Assuming that the transmit symbols are periodic and have a period of Nsymbols, after a transient interval,

x [n] = x [n + N]

Next, we form the summation J =

N2X

n=N1

y [n + N]y∗[n] and note that

J ≈ ej2π∆fcNTb

N2X

n=N1

|x [n]|2

Solving this for ∆fc, we get

∆fc ≈1

2πNTb∠(J)

9.3 Pilot Aided Carrier Acquisition Method (continued)

Lock Range:Correct operation of the above procedure requires that

−π < ∠(J) < π

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

−fb

2N< ∆fc <

fb2N

9.4 Data Aided Carrier Tracking Method

e−j·

data

detected

slicer

equalizer

received

signal

phase

detector

loop

filter

demodulation/carrier and timing

recovery

PLL

⊗s[n] s[n]

ε[n] = ∠ (s[n]s∗[n]) ≈={s[n]s∗[n]}

<{s[n]s∗[n]}

9.4 Data Aided Carrier Tracking Method (continued)MATLAB Script DDCR.mTb=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 phasephi=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))); % Slicers12=s1(n)*s2’; e=imag(s12)/real(s12); phi(n+1)=phi(n)+mu*e;

endfigure(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

9.4 Data Aided Carrier Tracking Method (continued)

0 100 200 300 400 5000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

n

φ[n]

9.4 Data Aided Carrier Tracking Method (continued)

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Real part

Imag

inar

y pa

rt