Tele4653 l5

TELE4653 Digital Modulation &Coding

Synchronization

Wei Zhang

[email protected]

School of Electrical Engineering and Telecommunications

The University of New South Wales

Outline

Carrier Phase Estimation

Decision-Directed Loops

Timing Estimation

TELE4653 - Digital Modulation & Coding - Lecture 5. March 29, 2010. – p.1/21

Signal Model

The received signal may be expressed as

r(t) = <{

[sl(t − τ)ejφ + z(t)]ej2πfct}

(1)

where the carrier phase φ, due to the propagation delay τ is

φ = −2πfcτ .

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 (r1r2 · · · rN ) , r.


ML Estimation

Since the noise n(t) is white and zero-mean Gaussian, the joint

PDF p(r|θ) may be expressed as

p(r|θ) =

(

1√2πσ

)N

exp

(

−N∑

n=1

[rn − sn(θ)]2

2σ2

)

(3)

where rn =∫

T0r(t)φn(t)dt and sn(θ) =

∫

T0s(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

Λ(θ) = exp

{

− 1

N0

∫

T0

[r(t) − s(t; θ)]2dt

}

(4)


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.


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





Suppose we have an AM signal of the form

s(t) = A(t) cos(2πfct + φ) (5)

If we demodulate the signal by multiplying s(t) with the carrier

reference

c(t) = cos(2πfct + φ̂) (6)

and pass c(t)s(t) through a LP filter, we obtain

y(t) =1

2A(t) cos(φ − φ̂). (7)

A phase error of 30o results in a power loss of 1.25 dB.



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(2πfct + φ) − B(t) sin(2πfct + φ). (8)

The signal is demodulated by two quadrature carriers

ci(t) = cos(2πfct + φ̂) and cq(t) = − sin(2πfct + φ̂). Multiplication

of s(t) with ci(t) and cq(t) followed by LP filtering, respectively,

yieldsyI(t) =

1

2A(t) cos(φ − φ̂) − 1

2B(t) sin(φ − φ̂) (9)

yQ(t) =1

2B(t) cos(φ − φ̂) +

1

2A(t) sin(φ − φ̂). (10)


ML Phase Estimation

Assume τ = 0. The likelihood function Eq. (4) becomes

Λ(φ) = exp

{

− 1

N0

∫

T0

[r(t) − s(t;φ)]2dt

}

(11)

= exp

{

− 1

N0

∫

T0

r2(t)dt +2

N0

∫

T0

r(t)s(t;φ)dt

− 1

N0

∫

T0

s2(t;φ)dt

}

(12)

The log-likelihood function is

ΛL(φ) =2

N0

∫

T0

r(t)s(t;φ)dt (13)


An Example

Consider the received signal as r(t) = A cos(2πfct + φ) + n(t),

where φ is the unknown phase and can be estimated by

maximizing

ΛL(φ) =2A

N0

∫

T0

r(t) cos(2πfct + φ)dt (14)

A necessary condition for a maximum is that dΛL(φ)dφ

= 0, which

yields ∫

T0

r(t) sin(2πfct + φ̂ML)dt = 0 (15)

or, equivalently,

φ̂ML = − tan−1

[∫

T0

r(t) sin(2πfct)dt/

∫

T0

r(t) cos(2πfct)dt

]

(16)


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.





A problem may arise in maximizing log-likelihood function when

the signal s(t;φ) carries the information sequence {In}. In

decision-directed parameter estimation, we assume that {In}has been estimated.

Consider linear modulation for which the received equivalent LP

signal may be expressed as

rl(t) = e−jφ∑

n

Ing(t − nT ) + z(t) = sl(t)e−jφ + z(t) (17)

where sl(t) is a known signal if {In} is assumed known. The

log-likelihood function is

ΛL(φ) = <{[

1

N0

∫

T0

rl(t)s∗

l (t)dt

]

ejφ

}

(18)



If we substitute sl(t) =∑

n Ing(t− nT ) into (18) and assume that

the observation interval T0 = KT , where K is a positive integer,

we obtain

ΛL(φ) = <{

ejφ 1

N0

K−1∑

n=0

I∗nyn

}

(19)

where, by definition, yn =∫ (n+1)TnT

rl(t)g∗(t − nT )dt. The ML

estimate of φ is easily found (by differentiating the log-likelihood)

as

φ̂ML = − tan−1

[

=(

K−1∑

n=0

I∗nyn

)

/<(

K−1∑

n=0

I∗nyn

)]

(20)



ML Timing Estimation

If the signal is a basedband PAM, represented as

r(t) = s(t; τ) + n(t) (21)

wheres(t; τ) =

∑

n

Ing(t − nT − τ). (22)

The log-likelihood function is

ΛL(τ) = CL

∫

T0

r(t)s(t; τ)dt (23)

= CL

∑

n

Inyn(τ) (24)

where yn(τ) =∫

T0r(t)g(t − nT − τ)dt.


ML Timing Estimation

To get the estimate of τ , we take the differentiation of ΛL(τ) and

obtain

dΛL(τ)

dτ=∑

n

Ind

dτ[yn(τ)] = 0. (25)

The implementation of the ML estimation of timing for baseband

PAM is illustrated in Fig. 5.3-1.



Tele4653 l5

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