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
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.
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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
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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.
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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.
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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
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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
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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
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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.
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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
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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
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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.
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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.
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