This document summarizes techniques for carrier phase estimation, decision-directed loops, and timing estimation in digital communication systems. It discusses maximum likelihood estimation of phase and timing parameters based on maximizing the likelihood function. Carrier phase can be estimated using phase-locked loops or by cross-correlating the received signal with in-phase and quadrature reference carriers. Decision-directed loops allow phase estimation when the transmitted symbols are unknown by assuming symbol decisions. Timing can be estimated for PAM signals by correlating the received signal with pulse shapes centered at different time offsets and choosing the offset that maximizes the likelihood function. Diagrams show example receiver structures for carrier phase tracking and timing estimation.

1. TELE4653 Digital Modulation &
Coding
Synchronization
Wei Zhang
w.zhang@unsw.edu.au
School of Electrical Engineering and Telecommunications
The University of New South Wales

2. Outline
Carrier Phase Estimation
Decision-Directed Loops
Timing Estimation
TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.1/2

3. Signal Model
The received signal may be expressed as
r(t) = [sl (t − τ )ejφ + z(t)]ej2πfc t (1)
where the carrier phase φ, due to the propagation delay τ is
φ = −2πfc τ .
r(t) = s(t; φ, τ ) + n(t) = s(t; θ) + n(t) (2)
where θ denotes the parameter vector {φ, τ }.
By performing an orthonormal expansion of r(t) using N
orthonormal functions {φn (t)}, we may represent r(t) by the
vector of coefficients (r1 r2 · · · rN ) r.
TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.2/2

4. ML Estimation
Since the noise n(t) is white and zero-mean Gaussian, the joint
PDF p(r|θ) may be expressed as
N N
1 [rn − sn (θ)]2
p(r|θ) = √ exp − (3)
2πσ 2σ 2
n=1
where rn = T0 r(t)φn (t)dt and sn (θ) = T0 s(t; θ)φn (t)dt, where
T0 is the integration interval. The maximization of p(r|θ) is
equivalent to the maximization of the likelihood function
1
Λ(θ) = exp − [r(t) − s(t; θ)]2 dt (4)
N0 T0
TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.3/2

5. Receiver Structure
Figure 5.1-1 shows a block diagram of a binary PSK receiver.
Figure 5.1-2 shows a block diagram of an M -ary PSK receiver.
Figure 5.1-3 shows a block diagram of an M -ary PAM receiver.
Figure 5.1-4 shows a block diagram of a QAM receiver.
TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.4/2

6. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

7. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

8. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

9. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

10. Carrier Phase Estimation
Suppose we have an AM signal of the form
s(t) = A(t) cos(2πfc t + φ) (5)
If we demodulate the signal by multiplying s(t) with the carrier
reference
ˆ
c(t) = cos(2πfc t + φ) (6)
and pass c(t)s(t) through a LP filter, we obtain
1 ˆ
y(t) = A(t) cos(φ − φ). (7)
2
A phase error of 30o results in a power loss of 1.25 dB.
TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.9/2

11. Carrier Phase Estimation
The effect of carrier phase errors in QAM and M -ary PSK is
much more severe. The QAM and M -PSK signals may be
expressed as
s(t) = A(t) cos(2πfc t + φ) − B(t) sin(2πfc t + φ). (8)
The signal is demodulated by two quadrature carriers
ˆ ˆ
ci (t) = cos(2πfc t + φ) and cq (t) = − sin(2πfc t + φ). Multiplication
of s(t) with ci (t) and cq (t) followed by LP filtering, respectively,
yields
1 ˆ − 1 B(t) sin(φ − φ)ˆ
yI (t) = A(t) cos(φ − φ) (9)
2 2
1 ˆ + 1 A(t) sin(φ − φ).
ˆ
yQ (t) = B(t) cos(φ − φ) (10)
2 2
TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.10/2

12. ML Phase Estimation
Assume τ = 0. The likelihood function Eq. (4) becomes
1
Λ(φ) = exp − [r(t) − s(t; φ)]2 dt (11)
N0 T0
1 2 2
= exp − r (t)dt + r(t)s(t; φ)dt
N0 T0 N0 T0
1
− s2 (t; φ)dt (12)
N0 T0
The log-likelihood function is
2
ΛL (φ) = r(t)s(t; φ)dt (13)
N0 T0
TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.11/2

13. An Example
Consider the received signal as r(t) = A cos(2πfc t + φ) + n(t),
where φ is the unknown phase and can be estimated by
maximizing
2A
ΛL (φ) = r(t) cos(2πfc t + φ)dt (14)
N 0 T0
L (φ)
A necessary condition for a maximum is that dΛdφ = 0, which
yields
ˆ
r(t) sin(2πfc t + φML )dt = 0 (15)
T0
or, equivalently,
ˆ
φM L = − tan−1 r(t) sin(2πfc t)dt/ r(t) cos(2πfc t)dt (16)
T0 T0
TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.12/2

14. PLL
Eq. (15) implies the use of a loop (PLL) to extract the estimate
as illustrated in Fig. 5.2-1.
Eq. (16) implies an implementation that uses quadrature carriers
to cross-correlated with r(t), as shown in Fig. 5.2-2.
Please refer to TELE3113 lecture notes for details of PLL.
TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.13/2

15. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

16. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

17. Decision-Directed Loops
A problem may arise in maximizing log-likelihood function when
the signal s(t; φ) carries the information sequence {I n }. In
decision-directed parameter estimation, we assume that {I n }
has been estimated.
Consider linear modulation for which the received equivalent LP
signal may be expressed as
rl (t) = e−jφ In g(t − nT ) + z(t) = sl (t)e−jφ + z(t) (17)
n
where sl (t) is a known signal if {In } is assumed known. The
log-likelihood function is
1
ΛL (φ) = rl (t)s∗ (t)dt ejφ
l (18)
N 0 T0
TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.16/2

18. Decision-Directed Loops
If we substitute sl (t) = n In g(t − nT ) into (18) and assume that
the observation interval T0 = KT , where K is a positive integer,
we obtain
K−1
jφ 1 ∗
ΛL (φ) = e In y n (19)
N0 n=0
(n+1)T
where, by definition, yn = nT rl (t)g ∗ (t − nT )dt. The ML
estimate of φ is easily found (by differentiating the log-likelihood)
as
K−1 K−1
ˆ
φM L = − tan−1 In y n
∗
/ In y n
∗
(20)
n=0 n=0
TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.17/2

19. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

20. ML Timing Estimation
If the signal is a basedband PAM, represented as
r(t) = s(t; τ ) + n(t) (21)
where
s(t; τ ) = In g(t − nT − τ ). (22)
n
The log-likelihood function is
ΛL (τ ) = CL r(t)s(t; τ )dt (23)
T0
= CL In yn (τ ) (24)
n
where yn (τ ) = T0 r(t)g(t − nT − τ )dt.
TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.19/2

21. ML Timing Estimation
To get the estimate of τ , we take the differentiation of Λ L (τ ) and
obtain
dΛL (τ ) d
= In [yn (τ )] = 0. (25)
dτ n
dτ
The implementation of the ML estimation of timing for baseband
PAM is illustrated in Fig. 5.3-1.
TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.20/2

22. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi