Chapter 10

Chapter 10 Synchronization 10.2 RECEIVER SYNCHRONIZATION 10.2.1 Frequency and Phase Synchronization

Normalized input : r (t ) = cos[ ω Normalized VCO output:

0

+θ (t )]

(10.1)

(10.2)

x (t ) = −2 sin[ ω0 t +θˆ(t )]

Phase detector output: e(t ) = x (t )r (t ) = 2 sin[ ω0 t +θˆ(t )] cos[ ω0 t +θˆ(t )] = sin[ θ (t ) −θˆ(t )] + sin[ 2ω t +θ (t ) +θˆ(t )]

(10.3)

d ˆ [θ (t )] = K 0 y (t ) = K 0 e(t ) * f (t ) dt ≈ K [θ (t ) −θˆ(t )] * f (t )

(10.4)

0

∆ω(t ) =

0

Example 10.1 Linearized Loop Equation

Fourier Analysis: ˆ (ω) = K [Θ(ω) − Θ ˆ (ω)] F (ω) (10.5) jωΘ 0

The closed-loop transfer function of the PLL. ˆ (ω) K 0 F (ω) Θ = = H (ω) (10.6) Θ(ω) jω + K 0 F (ω)

Order of PLL: The highest-order term in jω in the denominator of H(ω ). ( One more than the order of loop filter F(ω ).) 10.2.1.1 Steady-State Tracking Characteristics The steady-state error is the residual error after all transients have died away, and thus provides a measure of a loop’s ability to cope with various tpes of changes in the input. Use final value theorem to judge for the tracking: ˆ (ω) = [1 − H (ω)]Θ(ω) = E (ω) = ℑ{e(t )} = Θ(ω) − Θ

lim e(t ) = lim jωE (ω) lim e(t ) = lim

jω→0

t →∞

(10.7)

(10.8)

jω→0

t →∞

jωΘ(ω) jω + K 0 F (ω)

( jω) 2 Θ(ω) jω + K 0 F (ω)

(10.9)

Example 10.2 Response to a Phase Step Θ(ω) = ℑ{∆Φu (t )} =

∆Φ jω

(10.10)

1 for t > 0 u (t ) =  0 for t < 0 t

= ∫ δ (τ) dτ −∞

lim e(t ) = lim

jω→0

t →∞

jω∆Φ =0 jω + K 0 F (ω)

The loop will eventually track out any phase step that appears at the input if the loop filter has a nonzero dc response. Example 10.3 Response to a Frequency Step Θ(ω) =

∆ω ( jω) 2

lim e(t ) = lim

jω→0

t →∞

(10.11) ∆ω ∆ω = jω + K 0 F (ω) K 0 F (0)

(10.12)

The steady-state result depends on more properties of the loop filter than merely a nonzero dc response. If (1) F (ω) =1 (10.13) ap

(2) Flp (ω ) = (3)

ω1 jω + ω1

ω Fll (ω ) =  1  ω2

(10.14)

 jω + ω 2   jω + ω 1

(10.15)

Then the velocity error will exist regardless of the order of the filter, unless the denominator of F(w), contains jω as a factor. Having jω as a factor in denominator is equivalent to having a perfect integrator in the loop filter. lim e(t ) = t →∞

∆ω K0

Example 10.4 Response to a Frequency Ramp Θ(ω) =

∆ω ( jω) 3

(10.16)

lim e(t ) = lim

jω→0

t →∞

∆ω / jω ∆ω = lim jω + K 0 F (ω) jω→0 jωK 0 F (ω)

(10.17)

Need F(ω ) to have (jω )2 as denominator to track a frequency ramp with a constant phase error. 10.2.1.2 Performance in Noise r (t ) = cos( ω t +θ ) + n(t ) (10.18) n(t ) = n (t ) cos ω t + n (t ) sin ω t (10.19) 0

c

0

s

0

e(t ) = x (t ) r (t ) = sin( θ −θˆ ) + n c (t ) cos θˆ + n s (t ) sin θˆ + (terms at twice

(10.20) Let n′(t ) = n

c

ˆ +n (t ) sin θ ˆ (t ) cos θ s

the carrier frequency)

(10.21)

R (t1 , t 2 ) = E{n ′(t1 ) n ′(t 2 )} = E{nc (t1 ) nc (t 2 )} cos 2 θˆ + E{n s (t1 ) n s (t 2 )} sin 2 θˆ +[ E{n (t ) n (t )} + E{n (t ) n (t )}] sin θˆ cos θˆ c

1

s

2

s

1

R(τ ) = Rc (τ ) cos 2 θˆ + Rs (τ ) sin 2 θˆ (10.23) G (ω) = ℑ[ R (t )] = G (ω) cos 2 θˆ + G (ω) sin 2 θˆ c

From 10.19,

s

c

(10.22)

2

(10.24)

G s (ω) = Gc (ω) = G n (ω0 − ω) + G n (ω0 + ω)

(10.25) For special case of white noise: Gn(ω )=Gs(ω )=N0/2 and G (ω) = N (10.26) For the small-angle approximations, the spectral density of the VCO phase is elated to the spectral density of the noise process through the loop transfer function. That is, G (ω) = G n (ω0 − ω) + G n (ω0 + ω)

0

Gθˆ (ω) = G (ω) H (ω)

2

(10.27)

Variance of the output phase: σθ

2 ˆ

For white noise: σθ2ˆ

=

N0 2π

∞

∫

−∞

1 2π

= 2

H (ω) dω

The two-sided loop bandwidth: W

L

∞

∫

−∞

2

G (ω) H (ω) dω

(10.29)

= 2 BL =

1 2π

∞

∫

−∞

2

H (ω) dω Hertz

(10.30) σ θ2ˆ = 2 N 0 B L

(10.31)

Phase variance is a measure of the amount of jitter or wobble in the VCO output due to noise at the input. Phase variance is hoped to be small but BL is hoped to be wider to enhance the capability of tracking  left for design ! 10.2.1.3 Nonlinear Loop Analysis

when

sin( θ −θˆ ) ≈θ −θˆ

is no more satisfied.

d ˆ [θ (t )] = K 0 f (t ) * sin[ θ (t ) −θˆ (t )] + K 0 f (t ) * n ′(t ) dt

(10.33)

let the phase variation with constant phase inputθ and time varying VCO output : Φ(t ) =[θ −θˆ(t )] modulo 2π (10.34) d [Φ(t )] = K 0 f (t ) * sin Φ(t ) + K 0 f (t ) * n ′(t ) dt

(10.35)

Viterbi determined that for a 1st-order PLL, the probability density function of φ is of the term: p (Φ) =

exp( ρ cos Φ) for Φ ≤ π 2πI 0 ( ρ)

(10.36)

The mean time to the first cycle slip, Tm, beginning at some arbitrary reference time: Tm = Tm ≈

π 2 ρI 02 ( ρ ) 2 BL

(10.37)

π exp( 2 ρ)

(10.38)

4 BL

P (T ) = 1 − exp( −

T ) Tm

(10.39) The probability that a loop will cycle-slip

within time T, starting from zero phase error. 10.2.1.4 Suppressed Carrier Loops To acquire the carrier by suppressing the modulated signal. Let r(t) be suppressed carrier signals, that is, the average energy at ω o is zero. r (t ) = m(t ) sin( ω t +θ ) + n(t ) (10.40) 0

r 2 (t ) = m 2 (t ) sin 2 (ω0 t + θ ) + n 2 (t ) + 2n(t ) m(t ) sin( ω0 t + θ ) =

1 1 − cos( 2ω0 t + 2θ ) + n 2 (t ) + 2n(t )m(t ) sin( ω0 t + θ ) 2 2

(10.41)

Gardner shows that if the input noise process n(t) is a narrowband Gaussian noise of bandwidth Bi, the squaring loss is upper bounded by: S ≤ 1 + N B (10.42) L

0

i

The SNR in the input filter bandwidth,

ρi =

1 2 N 0 Bi

(10.43)

For large loop SNR, the output phase variance can now be expressed as: σ θ2ˆ = 2 N 0 BL S L = 2 N 0 B L (1 +

1 ) 2 ρi

(10.44)

10.2.1.5 Costas Loops Implementation of suppressed carrier loop without square circuit.

10.2.1.6 High-Order Suppressed-Carrier Loops SL ≤ 1 +

9 6 3 + 2 + ρ i ρ i 2 ρ i3

(10.44)

Example 10.5 Squaring Loss Bounds

10.2.1.7 Acquisition θ (t ) = ωi t t

θˆ(t ) = ω0 t + ∫ K 0 sin e(t )dt +θˆ(0) 0

(10.46)

e(t ) =θ(t ) −θˆ(t ) t = (ωi −ω0 )t + ∫ K 0 sin e(t )dt +θˆ(0)

(10.47)

0

de = ∆ω − K 0 sin e dt

(10.48)

The necessary but not sufficient condition for phase lock:

de =0 dt

point a: a sable point of the system where phase lock can be obtained and will be maintained. Point b: a point of marginal staility for the loop, if there is any slight offset from b, the sign of the derivative term will be such that the error will be driven away from b. For phase lock, ( spec. for locking range.) ∆ω K0

≤1

(10.50)

For second-order loop with loop transfer function, H (ω ) =

1 − ( jω / ω n ) + 2ζ ( jω / ω n ) + 1 2

(10.51)

The lock range should be ∆ω ≈

1 2 ωn (1 − 2σ θˆ ) 2

(10.52)

where the natural frequency of the PLL is ω

n

=

8ζ BL 4ζ 2 + 1

10.2.1.8 Phase Tracking Errors and Link Performance

Example 10.6 PLL Signal-to-Noise Ratio

2π

PB = ∫ Q( 0

2 Eb cos β exp( ρ cos β ) ) dβ N0 2πI 0 ( ρ )

10.2.1.9 Spectrum Analysis Techniques

10.2.2.1 Open-Loop Symbol Synchronizers  1 for x > 0 sgn x =  - 1 otherwise

For a BPF that effectively average K input symbols (bandwidth 1/KT), the magnitude of the fractional mean time error(delay) is approximated by

ε T

≈

0.33 KE b / N 0

for

Eb > 5, K ≥ 18 N0

(10.54)

At high SNR, the fractional standard deviation of the fractional timing error is given by

σε ≈ T

0.411 KE b / N 0

for

Eb >1 N0

(10.55)

10.2.2.2 Closed-Loop Symbol Synchronizers Disadvantage of open loop symbol synchronization: there is an unavoidable non-zero-mean tracking error which can be made small with large SNR but will never vanish. Closed-loop use comparative measurements on the incoming signal and a locally generated data-clock signal to bring the locally generated signal into synchronism with the incoming data transitions.

10.2.2.3 Symbol Synchronization Errors and Symbol Error Performance

Example 10.7 Effect of Timing Jitter

10.2.3 Synchronization with Continuous-Phase Modulations (CPM) 10.2.3.1 Background s (t ) = exp{ j[ω t +θ +ψ (t −τ , α)]} (10.57) 0

ψ (t , α) = η(t , C k , αk ) + Φk kT ≤ t ≤ (k +1)T

η(t , C k , αk ) = 2πh

(10.58)

k

∑α q(t − iT ) (10.59)

i =k −L +1

i

 0 for t ≤ 0 q (t ) =  1/2 for t ≥ LT

(10.60)

C k = (αk −L +1 ,  , αk −2 , αk −1 ) k −L

Φ k = πh∑α i mod 2π

(10.61)

i =0

 1 /( 2T ) 0 ≤ t ≤ T g (t ) =  t < 0, t > T  0 r (t ) = s (t , γ ) + n(t )

(10.62)

(10.63)

  e − jηl (T −t ,C0 h (t ) ≡    0

(l )

(l )

α0( l ) )

ηl (t , C , α ) = 2πh (l ) 0

(l ) 0

0 ≤t ≤T elsewhere 0

∑α

i =− L +1

τ +( k +1)T

Z k( l ) (C k , αk , τ ) ≡ ∫

τ +kT

(l ) i

(10.64)

q (t − iT )

(10.65)

r (t ) h ( l ) (t −τ − kT )e − jω0t dt

(10.66)

10.2.3.2 Data-Aided Synchronization Likelihood analysis: L0 −1 − j (θˆ +Φ k )  ˆ ˆ Λ ( R | θ , τ ) = exp ]  ∑ Re[ Z k (C k , α k , τˆ)e x  k =0  L0 −1

∑Im[ Z k =0

ˆ

k

(C k , α k , τˆ)e − j (θ +Φk ) ] = 0

(10.68)

(10.66)

L0 −1

∑Re[ Y

k

k =0

ˆ

(C k , α k , τˆ)e − j (θ +Φk ) ] = 0

(10.69)

(10.70) τˆ = τˆ + γ e (k −1) (10.71) 10.2.3.3 Non-data-Aided Synchronization θˆk +1 = θˆk + γ P e P (k −1) k +1

k

T

T

L0 −1 ˆ , αˆ ,θˆ, τˆ) = exp  ˆ , αˆ ,τˆ)e − j (θˆ +Φ k ) ] Λ( R | C  ∑ Re[ Z k (C  k k k k  k =0 

p ( r (t ) = R (t ) | γ ) = ∫

all β

1 Λ′( R | θˆ, τˆ) = ML

p[ r (t ) = R (t ) | γ , β] p ( β) dβ

∑Λ( R | Cˆ

all ( Cˆ k ,αˆ k )

e jψ ( t ,α ) ≈ ∑a 0 ,i h0 (t − iT ) i

i

a 0,i = exp( jπh∑α l ) l =0

k

, αˆ k ,θˆ, τˆ)

(10.72)

(10.73)

(10.74)

(10.75)

(10.76)

πt   sin( ) 0 ≤ t ≤ 2T h0 (t ) =  2T  0 elsewhere 

(10.77)

s(t ) ≈ e j (ω0t +θ ) ∑ a 0,i h0 (t − iT − τ ) i

  k s (t ) = exp  j ω 0 t + θ + 2π 1 k2  

(10.78) 

k

∑α q(t − iT ) (10.79)

i = k − L +1

i



k    [ s (t )] k 2 = exp  j k 2 (ω 0 t + θ ) + 2πk1 ∑α i q (t − iT )  i = k − L +1   

(10.80)

10.2.4 Frame Synchronization (Figure 10.17)

N −k

C k = ∑X j X j =1

j +k

(10.81)

Pm =

N  j  p (1 − p ) N − j  j =k +1  j  N

∑ 

N    j  = ∑ N  j =0 2 k

PFA

(10.83)

(10.82)

10.3 NETWORK SYNCHRONIZATION 10.3.1 Open-Loop Transmitter Synchronization T A = Ti +

d c

Te =

re + ∆t c

ωe =

Ve ω 0 + ∆ω c

δ=

∆ω

ω0

 

ω ≈ 1 −

(10.84)

V c

 ω0 

(10.85)

(10.86) (10.87) (10.88)

hertz/hert z/day T

(10.89)

∆ω(T ) = ω0 ∫ δdt + ∆t (0) = ω0δT + ∆ω(0) hertz 0

T

∆ω(t )

0

ω0

∆t (T ) = ∫

dt + ∆t (0)

T

T

∆ω(0)

0

0

ω0

= ∫ δtdt + ∫ = 12 δT 2 +

∆ω(0)T

ω0

dt + ∆t (0)

(10.90)

+ ∆t (0)

10.3.2 Closed-Loop Transmitter Synchronization  sin[(ω 0 + ω s + ∆ω )t + θ ] r (t ) =   sin[(ω 0 + ∆ω )t + θ ] x=

1 T

∫

y=

1 T

∫

T

0 T

0

0 ≤ t ≤ ∆t ∆t < t ≤ T

r (t ) cos ω0 tdt

(10.92)

r (t ) sin ω0 tdt

(10.93)

(10.91)

z 2 = x2 + y2 2

2

 sin[( ωs + ∆ω)∆t / 2]  sin[ ∆ω(T − ∆t ) / 2]   +  =   (ωs + ∆ω)T ∆ωT     cos( ∆ω∆t ) + cos[ ∆ωT − (ωs + ∆ω) ∆t ] − cos( ∆ωT ) − cos( ωs ∆t ) + 2∆ω(ωs + ∆ω)T 2  sin( ∆ωT / 2)  z2 =   ∆ωT   2

2

(10.95)

 sin(ω s ∆t / 2)   T − ∆t  z2 =   +  ω sT  2T   

2

(10.96

(10.94)