IV - Synchronization∗

1 Introduction

Coherent (or synchronous) detection of digitally modulated signals using a receiver as

the ML receiver of Fig. 2, chapter 3, requires two types of synchronization:

a) carrier (or phase) synchronization: the generation of reference signals ψ0(t),

ψ1(t), . . . , ψN−1(t) requires the knowledge of the carrier frequency and phase.

b) symbol synchronization: integration and sampling of the integrators outputs

require the knowledge of the starting and finishing times of each symbol (symbol clock).

Carrier synchronization is usually obtained from the modulated signal (affected by

noise) through a feedback circuit based on the phase locked-loop (PLL). Obviously, in

a noncoherent system, carrier synchronization is of no concern.

Synchronization can be implemented in one of two fundamentally different ways

[4]:

• Data-aided synchronization

In data-aided synchronization systems, a preamble is transmitted along with the

data-bearing signal in a time-multiplexed manner on a periodic basis. The pream-

ble contains information about the carrier and symbol timing, which is extracted by

appropriate processing of the channel output at the receiver. Such an approach is com-

monly used in digital satellite and wireless communications, where the motivation is to

minimize the time required to synchronize the receiver to the transmitter. Its limita-

tions are two-fold: (1) reduced data-throughput efficiency that results from assigning

a certain portion of each transmitted frame to the preamble; and (2) reduced power

efficiency by allocating a certain fraction of the transmitted power to the transmission

of the preamble.

• Nondata-aided synchronization

In this approach, the use of a preamble is avoided, and the receiver has the task of

establishing synchronization by extracting the necessary information from the modu-

∗FDNunes, IST 2013.

76

2 PHASE-LOCKED LOOP (PLL) 77

lated signal. Both throughput and power efficiency are thereby improved but at the

expense of an increase in the time taken to establish synchronization.

In this chapter we consider only nondata-aided synchronization schemes where we

may identify two approaches for solving the synchronization problem: the classical ap-

proach based on the phase-locked loop (PLL), and the algorithmic (modern) approach

that resorts to the maximum likelihood (ML) estimation.

2 phase-locked loop (PLL)

A PLL consists of three basic components [3]: (1) a phase detector, (2) a low-pass filter,

and (3) a voltage-controlled oscillator (VCO). The VCO is an oscillator that produces a

periodic waveform, whose frequency may be varied about some free-running frequency

(f0) according to the applied voltage v2(t) (see Fig. 1). The free-running frequency

f0 is the frequency of the VCO output when v2(t) = 0. The phase detector produces

an output signal v1(t) that is a function of the phase difference between the incoming

signal vin(t) and the oscillator signal vo(t). The filtered signal v2(t) is the control signal

that is used to change the frequency of the VCO output. The PLL configuration may

be designed so that it acts as a narrowband tracking filter when the low-pass filter

(LPF) is a narrowband filter.

phasedetector (PD)

low-pass filter (LPF) F(f)

VCO

vo(t)

vo(t)

vin(t) v

1(t)

v2(t)

Figure 1: Basic PLL

If the applied signal has an initial frequency of f0, the PLL will acquire lock and the

VCO will track the input signal frequency over some range, provided that the input

frequency changes slowly. The loop will remain locked only over some finite range of

frequency shift (hold-in or lock range). The frequency range over which an applied

input signal will cause the loop to lock is called the pull-in or capture range. If a PLL

2 PHASE-LOCKED LOOP (PLL) 78

is realized using analog circuit techniques, it is said to be an analog PLL (see Fig. 2).

On the other hand, if digital circuits are used, it is said to be a digital PLL.

+ LPFF(f)

VCO

vin(t)

vo(t)

v1(t)=Kvin(t)vo(t) v2(t)

Figure 2: Analog PLL

The PLL may be studied by examining the analog PLL shown in Fig. 2. Assume

that the input signal is

vin(t) = Ai sin[ω0t + θi(t)]

and the VCO output signal is

vo(t) = Ao cos[ω0t + θo(t)]

where

θo(t) = Kv

∫ t

−∞v2(τ) dτ (1)

and Kv is the VCO gain constant (rad/s/volt). The PD (multiplier) output is

v1(t) = KmAiAo sin[ω0t + θi(t)] cos[ω0t + θo(t)]

=KmAiAo

2sin[θi(t)− θo(t)] +

KmAiAo

2sin[2ω0t + θi(t) + θo(t)]

where Km is the gain of the multiplier circuit. The sum frequency term does not pass

through the LPF, so the LPF output is

v2(t) = Kd sin[θe(t)] ∗ f(t) (2)

where

θe(t) ≡ θi(t)− θo(t), Kd =KmAiAo

2(3)

f(t) is the impulse response of the LPF and θe(t) is called the phase error.

The overall equation describing the operation of the PLL may be obtained by taking

the derivative of (1) and (3) and combining the result by use of (2). The resulting

nonlinear equation that describes the PLL becomes

2 PHASE-LOCKED LOOP (PLL) 79

dθe(t)

dt=

dθi(t)

dt−KdKv

∫ ∞

−∞sin[θe(λ)]f(t− λ) dλ

where θe(t) is the unknown and θi(t) is the forcing function.

In general, this PLL equation is difficult to solve. However, it may be reduced to

a linear equation if the gain Kd is large, so that the error θe(t) is small. In this case,

sin[θe(t)] ≈ θe(t) and the resulting linear equation is

dθe(t)

dt=

dθi(t)

dt−KdKvθe(t) ∗ f(t) (4)

A block diagram that follows from this linear equation is shown in Fig. 3. It should

be emphasized that in this linear PLL model, the phase of the input signal and the

phase of the VCO output signal are used instead of the actual signals themselves.

+ LPFF (f)=F(f)1

VCO

F (f)=Kv

j2πf2

θ

θ

+

-

θi

o

e(t) (t)

(t)

K d

v (t)2

Figure 3: Linear model of the analog PLL

Consider the Fourier transform of (4). The PLL closed-loop transfer function is

H(f) =Θo(f)

Θi(f)=

KdKvF (f)

j2πf + KdKvF (f)(5)

where Θo(f) = F(θo(t)) and Θi(f) = F(θi(t)).

By reorganizing (5) we can obtain an expression for the Fourier transform of the

phase error

Θe(f) = Θi(f)−Θo(f)

= [1−H(f)]Θi(f) =j2πfΘi(f)

j2πf + KdKvF (f)(6)

The order of the PLL is defined to be the order of the highest-order term in j2πf

in the denominator of H(f). For instance, if F (f) = 1 (all-pass filter) or F (f) =

1/(1+ jf/fc) (passive low-pass filter), the PLL will be first-order. A second-order PLL

has to include an integrator: F (f) = 1/(j2πf).

2 PHASE-LOCKED LOOP (PLL) 80

Equation (6) can be used in conjunction with the final value theorem of Fourier

transform to determine the steady-state error response of a loop to a variety of possible

input characteristics. The steady-state error is the residual error after all transients

have died away, and thus provide a measure of a loop’s ability to cope with various

types of changes in the input. The final value theorem states that [6]

limt→∞ θe(t) = lim

f→0j2πfΘe(f) (7)

Combining equations (6) and (7) yields

limt→∞ θe(t) = lim

f→0

(j2πf)2Θi(f)

j2πf + KdKvF (f)(8)

Let us analyze the PLL response to a (1) phase step and a (2) frequency step.

Assuming that the PLL was originally in phase lock, a phase step will throw the

loop of of lock. Having abruptly changed, however, the input phase again becomes

stable. The Fourier transform of a phase step is

Θi(f) = F{∆φu(t)} =∆φ

j2πf(9)

where ∆φ is the magnitude of the step in radians and u(t) is the unit step function.

Replacing (9) in (8) leads to

limt→∞ θe(t) = lim

f→0

j2πf∆φ

j2πf + KdKvF (f)= 0

assuming that F (0) 6= 0. Thus, a first-order PLL will automatically tend to recover

phase lock if the input is displaced by a constant phase. This is clearly a very desirable

loop characteristic.

Next, consider the PLL steady-state response to a frequency step at the input. A

frequency step (or phase ramp) can approximate the effect of a frequency Doppler shift

in the incoming signal due to the relative motion between the transmitter and the

receiver.The Fourier transform of the phase characteristic will be the transform of the

integral of the frequency characteristic. Thus

Θi(f) =∆ω

(j2πf)2(10)

where ∆ω is the magnitude of the frequency step in rad/s. Replacing (10) in (8) gives

limt→∞ θe(t) = lim

f→0

∆ω

j2πf + KdKvF (f)=

∆ω

KdKvF (0)(11)

Equation (11) indicates that the loop will track the input phase ramp with a con-

stant steady-state error whose value will depend on the gain term, KdKv, and the

3 MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS 81

magnitude of the frequency step. One way to achieve a null error in (11) is that the

denominator of F (f) contains j2πf as a factor which is equivalent to having a perfect

integrator in the loop filter (second-order PLL). It is not possible to build a perfect

integrator, but one may be closely approximated either digitally or by using active

integrated circuits. It should be noted that even with a nonzero Doppler velocity, the

frequency is still being tracked. There are applications where tracking a zero phase

error is not important provided that frequency is tracked. Noncoherent signaling, such

as M-FSK modulation is an example [6].

The steady-state analysis above assumed that the input signal was noise free, which

is not true in practical transmission systems. If the noise process is white with power

spectral density Gw(f) = N0/2 and the small-angle approximation holds (that is, if

the loop is successfully tracking the input phase), the output phase variance is given

by [2], [10]

σ2e =

2N0BL

A2i

rad2 (12)

where

BL =1

|H(0)|2∫ ∞

0|H(f)|2 df Hz

is the single-sided equivalent noise bandwidth.

In order to tabulate the noise bandwidth for various cases, define the closed-loop

transfer function H(s) of a n-th order PLL by

H(s) =C0 + C1s + C2s

2 + . . . + Cn−1sn−1

d0 + d1s + d2s2 + . . . + dnsn

The loop noise bandwidths are [10]

n = 1 : BL =C2

0

4d0d1

n = 2 : BL =C2

0d2 + C21d0

4d0d1d2

The phase variance σ2e is a measure of the amount of jitter in the VCO output due

to noise at the input. Equations (6) and (12) highlight one of the many trade-offs

in communication theory. Clearly, one would wish σ2e to be small, which implies a

small loop bandwidth, BL, that requires a narrow H(f). However, it can be inferred

from (6) that the narrower the effective bandwidth of H(f) is, the poorer will be the

loop’s ability to track incoming signal phase changes, Θi(f). Thus, a loop design must

balance noise response with desired input phase response [6].

3 Maximum likelihood estimation of parameters

Consider the received complex signal

3 MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS 82

v(t) = c(t, γ) + n(t), −T0/2 ≤ t ≤ T0/2

where γ is an unknown vector parameter to be estimated and n(t) is complex white

Gaussian noise. Consider the Fourier series expansions of v(t) and c(t, γ) restricted to

the interval [−T0/2, T0/2], according to

v(t) =∞∑

i=−∞vi exp(j2πif0t), f0 = 1/T0

c(t, γ) =∞∑

i=−∞ci exp(j2πif0t)

where the Fourier coeffients of v(t) on the interval [−T0/2, T0/2] are

vi =1

T0

∫ T0/2

−T0/2v(t) exp(−j2πif0t) dt = ci + ni

and

ci =1

T0

∫ T0/2

−T0/2c(t, γ) exp(−j2πif0t) dt

ni =1

T0

∫ T0/2

−T0/2n(t) exp(−j2πif0t) dt

Thus, the coefficients {. . . , v−1, v0, v1, . . .} are a set of Gaussian random variables.

Because the noise n(t) is white, the complex random variables vi are independent, with

means ci and equal variances σ2. The probability density function is [9]

p(v−k, . . . , vk|c(t, γ)) =1

(πσ2)2k+1exp

− 1

σ2

k∑

i=−k

|vi − ci|2

From the receiver point of view, {v−k, . . . , vk} is a set of observations (known quan-

tities) and p(v−k, . . . , vk|c(t, γ)) is a function of the unknown vector parameter γ. This

probability density function is known as a likelihood function. The logarithmic form of

the likelihood function (log-likelihood function) is

Λ(γ) = log p(v−k, . . . , vk|c(t, γ))

This function is preferred because it replaces products of likelihood functions by

sums of log-likelihoods. Thus

Λ(γ) = C − 1

σ2

k∑

i=−k

|vi − ci|2

3 MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS 83

where C does not depend on γ. Now let k → ∞ and use the Parseval’s theorem for

the Fourier series

1

T0

∫ T0/2

−T0/2|v(t)|2 dt =

∞∑

i=−∞|vi|2

it yields

limk→∞

Λ(γ) = C − 1

T0σ2

∫ T0/2

−T0/2|v(t)− c(t, γ)|2 dt

The maximum likelihood estimate of the parameter vector γ is the one that max-

imizes Λ(γ). Thus, constants C and T0σ2 are irrelevant and can be omitted. The

maximum likelihood estimate of parameter vector γ is the one that maximizes [7]

Λ(γ) = −∫ T0/2

−T0/2|v(t)− c(t, γ)|2 dt

3.1 ML estimation of the phase

Suppose now that the received signal at complex baseband is

v(t) = s(t)ejθ + nR(t) + jnI(t)

where θ is the phase to be estimated, nR(t) and nI(t) are independent, Gaussian, white

noise processes and s(t) is a finite energy pulse, possibly complex. The log-likelihood

statistic is

Λ(θ) = −∫ ∞

−∞|v(t)− s(t)ejθ|2 dt

= −∫ ∞

−∞[v2

R(t) + v2I (t) + s2

R(t) + s2I(t)] dt

+ cos θ∫ ∞

−∞[2vR(t)sR(t) + 2vI(t)sI(t)] dt

− sin θ∫ ∞

−∞[2vR(t)sI(t)− 2VI(t)sR(t)] dt

Now, doing

∂Λ(θ)

∂θ= 0

leads to the maximum likelihood estimate of the phase [7]

3 MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS 84

θ̂ML = arctan

∫∞−∞[vI(t)sR(t)− vR(t)sI(t)] dt∫∞−∞[vR(t)sR(t) + vI(t)sI(t)] dt

(13)

3.2 ML estimation of the delay

Suppose that a received signal at complex baseband is

v(t) = s(t− α) + nR(t) + jnI(t)

where α is the delay parameter to be estimated, nR(t) and nI(t) are independent,

Gaussian, identically distributed, white noise processes and s(t) is a differentiable finite

energy pulse, possibly complex. The log-likelihood function is

Λ(α) = −∫ ∞

−∞|v(λ)− s(λ− α)|2 dλ

= −Ev − Es + 2∫ ∞

−∞vR(t)sR(λ− α) + vI(t)sI(λ− α) dλ

Doing now

∂Λ(α)

∂α= 0

leads to the maximum likelihood estimate of the delay [7]

Re

[∫ ∞

−∞ds(λ− α)

dαv∗(λ) dλ

]= 0 (14)

Let u = λ− α. We obtain for the integral in (14)

∫ ∞

−∞ds(λ− α)

dαv∗(λ) dλ =

∫ ∞

−∞ds(u)

du

du

dαv∗(u + α)

dλ

dudu

= −∫ ∞

−∞ds(u)

duv∗(u + α) du

Consider now t = u + α. We obtain

−∫ ∞

−∞ds(u)

duv∗(u + α) du = −

∫ ∞

−∞ds(t− α)

dtv∗(t) dt

which leads finally to

Re

[∫ ∞

−∞ds(t− α)

dtv∗(t) dt

]= 0 (15)

4 CARRIER SYNCHRONIZATION 85

A practical implementation of a circuit that estimates a slowly time-varying delay

α(t) is shown in Fig. 4 and is called delay-locked loop (DLL). This structure is a

discrete-time version of the PLL. The purpose of the DLL is to adjust the time reference

of a voltage-controlled clock (VCC). This is a version of the voltage-controlled oscillator

(VCO) whose output is a sequence of timing impulses rather than a sinusoidal signal.

Whereas a PLL changes the phase of a VCO continuously based on the phase error

signal, a DLL increments the reference time of a VCC based on the timing error signal.

The DLL of Fig. 4 obtains its error signal by sampling the output of the matched

filter at each side of the presumed peak, called the early-gate sample and the late-

gate sample, and computing the difference. This difference is an approximation to the

derivative in (14). When the difference is null the DLL is considered to be synchronized,

which corresponds to the null derivative [7].

sample at time nT-ε

sample at time nT+ε

voltage-controlled clock +

+

-

matched filter

s (-t)*v(t)

Figure 4: A basic delay-locked loop (DLL)

4 carrier synchronization

The PLL is able to generate a signal vo(t) synchronous with vin(t), that is, exhibiting the

same frequency and a reduced phase error. Signal vo(t) is practically free of noise, thus

the PLL can be used as a synchronization circuit in digital and analog communications.

The degree of robustness to noise depends, in a large extent, from the characteristics of

the PLL low-pass filter. The smaller is its bandwidth the larger will be its robustness

to noise but, on the other hand, the process of synchronism acquisition will be slower.

The PLL cannot be used directly to synchronize the carrier in digitally phase mod-

ulated signals because the received signal does not contain any non-modulated compo-

nent at the carrier frequency.

In the case of M-PSK signaling it is possible, through a nonlinear operation per-

formed on the modulated signals, to generate a non-modulated signal with multiple

frequency of that of the carrier (nf0 with n = 2, 4, . . .). The use of a PLL and a

frequency divider allow obtaining a signal of frequency f0.

4 CARRIER SYNCHRONIZATION 86

Let

si(t) = A cos(ω0t +

2πi

M

), 0 ≤ t ≤ Ts, i = 0, 1, . . . , M − 1

be the M-PSK modulated signal. Taking into account that

cosM x =1

2M

2

M/2−1∑

k=0

(M

k

)cos[(M − 2k)x] +

(M

M/2

)

yields

sMi (t) =

(A

2

)M2

M/2−1∑

k=0

(M

k

)cos

[(M − 2k)

(ω0t +

2πi

M

)]+

(M

M/2

)

Therefore, the component at frequency Mf0 (corresponding to k = 0) is free from

digital modulation, that is, it does not depend on the phase 2πi/M of the modulated

signal.

Let us consider, for example, the binary PSK modulation. The received signals are

given by

r(t) = ±A cos(ω0t) + n(t),

where n(t) is channel noise. One solution to eliminate the digital modulation consists

of squaring the signal according to

r2(t) = A2cos2(ω0t)︸ ︷︷ ︸x(t)

+ n2(t)± 2An(t) cos(ω0t)︸ ︷︷ ︸z(t)

,

where

x(t) =A2

2[1 + cos(2ω0t)]

is free from modulation and can used for synchronization purposes. This is carried out

by a PLL with central frequency 2f0 followed by a frequency divider by 2. The signal

component z(t) of r2(t) is noise, behaving as a disturbance to the PLL operation. This

synchronization circuit is called a squaring loop.

We show in Fig. 5 a possible synchronization scheme for the M-PSK signaling (Mth

power loop).

The frequency divider by M introduces a phase ambiguity of 2π/M at the output

of the synchronization circuit. In fact, we can obtain one of the signals cos(ω0t +

2kπ/M), k = 0, 1, . . . , M−1. This drawback is usually solved using differential encoding

of the M-PSK signal.

In differential encoding the information is carried in the phase shifts relative the

the phase of the previously transmitted symbol. For instance, in binary PSK bit 1 may

4 CARRIER SYNCHRONIZATION 87

(.) M

+ F(f)

:

VCO

M

y(t)

0 , f ∆

PLL

r(t) r M (t)

B, Mf

M-PSK signal

+noise

output to the

demodulator

frequency

divider

Figure 5: Mth power loop for M-PSK signaling

be transmitted by shifting the phase by 180o relative to the previous symbol, while bit

0 is transmitted by performing a null phase shift regarding the previous symbol. In

this scheme, readily applicable to M-PSK signaling with M > 2, the phase ambiguities

2π/M at the output of the synchronization circuit are irrelevant.

Coherent PSK demodulation with differential encoding leads to larger bit error

probabilities than those achieved with absolute phase modulation (previously described)

since a symbol error leads usually to the incorrect detection of the next symbol.

An important form of a suppressed carrier loop is the Costas loop, shown in Fig.

6. This loop is important because it eliminates the square-law device of Fig. 5 for

2-PSK demodulation. This device can be difficult to implement at carrier frequencies,

and is replaced with a multiplier and relatively simple low-pass filters. Although the

appearance of the circuits in Figs. 5 and 6 are quite different, for M = 2, their

theoretical performance can be shown to be the same [6].

In some cases coherent detection entails the design of rather complicated synchro-

nization circuits as, for instance, in M-FSK with M À 1, where M phase synchronizers

(one for each transmitted carrier) are required. In other cases, synchronous detection

is impossible to carry out, as for instance, in transmissions through fading channels,

due to the random changes of amplitude and phase in the transmitted signals. In

those cases one resorts to non-coherent detection techniques which do not require the

knowledge of carrier phase and frequency to perform detection.

Fig. 7 shows two sub-optimal receivers for differentially encoded 2-PSK signaling

and for binary FSK with carriers of frequencies f0 and f1. In both cases, the carrier

phase and frequency are not determined. These receivers present the advantage of

being simpler than the coherent receivers, although the performances are worse (that

is, they produce larger error rates than their coherent counterparts).

5 SYMBOL SYNCHRONIZATION 88

+

+

90o

VCO F(f) +

demodulatedbit stream

r(t)

Figure 6: Costas loop

5 symbol synchronization

After the signal containing digital information has been heterodyned to baseband using

a locally-generated coherent carrier, it is necessary to establish the symbol clock, that

is, the instants of beginning and end of each symbol; Fig. 8 shows one of those solutions

applied to 2-PSK signals.

The symbol (bit) synchronizer consists of a low-pass filter, a nonlinearity (squarer)

and a PLL or bandpass tuned filter at frequency fb = 1/Tb, where Tb is the bit duration.

The effect of the squaring circuit is to generate a signal, y(t), with a spectral risk at

frequency fb, to allow the PLL synchronization. The synchronizer output (bit clock)

is a periodic sequence of synchronization pulses with frequency fb.

The problem of bit synchronization in the circuit of Fig. 8 consists of symbol timing

recovery in a baseband PAM signal to extract the data values from

x(t) =∞∑

m=−∞amg(t−mTb − ε), am = ±1

where ε is the unknown bit delay (with |ε| < Tb/2). Assume that the data have zero

mean, E{am} = 0, and are independent, E{akaj} = 0, for k 6= j, with E{a2k} = σ2

a.

Thus, E{x(t)} has no periodic components.

Consider now

E{x2(t)} =∞∑

m=−∞

∞∑

n=−∞E{aman}g(t−mTb − ε)g(t− nTb − ε)

= σ2a

∞∑

m=−∞g2(t−mTb − ε) (16)

5 SYMBOL SYNCHRONIZATION 89

++ +

+

r(t)

cos( 0t)ω

ω0t)

0

0

Tb

Tb

(.)dt

(.)dt

delay

Tb

delay

Tb

+ +-

sin(

π/2

binary

decision

t=T b

t=T b

z

z

0

1

{z , z } 0 1

f

f

0

2-FSK signal

+noise

central

frequency=

central

frequency=

envelope

detector

envelope

detector

chooses

the largest detected

bit

1

Figure 7: Non-coherent receiving schemes for 2-PSK and 2-FSK

The Poisson sum formula states that

∞∑

k=−∞s(t− kT ) =

1

T

∞∑

l=−∞S

(l

T

)exp(j2πlt/T ) (17)

with S(f) = F{s(t)}. Thus, using (17) in (16) gives

E{x2(t)} =σ2

a

Tb

∞∑

l=−∞U

(l

Tb

)exp(j2πl(t− ε)/Tb)

where

U(f) = F{g2(t)} =∫ ∞

−∞G(λ)G(f − λ) dλ

6 BLOCK SYNCHRONIZATION 90

+

(.) 2 PLL orbandpass filter

+-

S&H

sampler

g(-t)

x (t)2x(t)

tTb

detected bit

cos(ω0t)

2-PSK signal+noise

bit synchronizer

Figure 8: 2-PSK receiver with bit synchronizer

Thus

E{x2(t)} =σ2

a

Tb

∞∑

l=−∞

∫ ∞

−∞G(λ)G

(l

Tb

− λ

)dλ · exp(j2πl(t− ε)/Tb) (18)

Note that if the pulse g(t) has a bandwidth in excess of the minimum Nyquist

bandwidth of 1/(2Tb), then E{x2(t)} has periodically varying components due to the

l 6= 0 terms in (18). For most practical pulses, G(f) and its translate overlap only for

l = −1, 0, 1. Therefore

E{x2(t)} =σ2

a

Tb

[U(0) + U

(− 1

Tb

)e−j2π(t−ε)/Tb + U

(1

Tb

)ej2π(t−ε)/Tb

]

In general, U(−1/Tb) = U(1/Tb) leading to

E{x2(t)} =σ2

a

Tb

[U(0) + 2U

(1

Tb

)cos

(2π

t− ε

Tb

)]

This means that E{x2(t)} contains a sinusoidal component at frequency 1/Tb that

can be used for bit synchronization [7], [8].

In the presence of channel noise, there is a degradation of the synchronizer operation

which consists of random changes of the starting time of each synchronization pulse. As

a consequence, the integration of the baseband signal is carried out with a delay or an

advance relative to the bit start, leading to a growth of the symbol error probabilities

relative to the situation of perfect synchronization.

6 Block synchronization

The task of block synchronization is to establish a reference point in time so that

incoming symbols can be assigned their proper indices within a data block or block

frame. If symbols are misframed, then subsequent operations, such as block decoding,

will be incorrect and the entire communication link will be lost.

6 BLOCK SYNCHRONIZATION 91

A binary marker for block synchronization is a special binary sequence that is

embedded into the data to represent the start of a block, as shown in Fig. 9. The sync

markers should be designed that have minimal sensitivity to the data that occur before

and after the marker. The solution to this problem is called a Barker sequence. We

will assume that the nonmarker symbols are random data. We can define the problem

more precisely with the help of Fig. 10. The correlator attempts to match the marker,

which is m − i bits later, with a length-m part of the incoming sequence. The right

m − i bits of the correlator marker will correlate against data and the left i bits will

correlate against the i right bits of the incoming marker. The design problem consists

of making the i-bit prefixes of the marker as little as possible like the i-bit suffixes.

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

data data data

framesync

framesync

framesync

framesync

time

Figure 9: Embedded sync markers in the data stream

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

marker

xxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxx

i-prefix

i-suffixxxxxxxxxxxxxxxxxxxxx

m-i

xxxxxxxxxxxxxxxxxxxxxx

m

data

correlator sequence position

input stream

Figure 10: Illustration of the Barker’s problem [1]

Let the sequence of bits {0, 1} such that the corresponding symbol levels are

{+1,−1}. The aperiodic cross-correlation function for sequences a and b of length

N is

Cab(n) =

1N

∑N−1−ni=0 aibi+n, 0 ≤ n ≤ N − 1

1N

∑N−1−ni=0 ai−nbi = 1

N

∑N−1k=−n akbk+n n−N ≤ n ≤ 0

0, |n| ≥ N

where N−|n| symbols are used in the correlation.The aperiodic autocorrelation function

for sequences is defined as above with b = a.

7 NONCOHERENT DETECTION 92

The Barker sequence a is characterized by the fact that its aperiodic autocorrelation

satisfies

{Ca(0) = 1,

|Ca(n)| ≤ 1N

, n 6= 0

That is, as the code shifts by one or more symbols (|n| ≥ 1) out-of-sync the magni-

tude of the autocorrelation function never exceeds 1/N in magnitude. The only known

Barker sequences are for lengths 2, 3, 4, 5, 7, 11, 13, as shown in the next table. It

is to be noted that the reverse and the inverse of a Barker sequence is still a Barker

sequence [10].

Sequence length Barker sequences

2 1,1 or 1,-1

3 1,1,-1

4 1,1,1,-1 or 1,1,-1,1

5 1,1,1,-1,1

7 1,1,1,-1,-1,1,-1

11 1,1,1,-1,-1,-1,1,-1,-1,1,-1

13 1,1,1,1,1,-1,-1,1,1,-1,1,-1,1

7 Noncoherent detection

Up to this point we have assumed that the receiver is perfectly synchronized to the

transmitter, and the only channel impairment is noise. In practice, besides the noise

there is also uncertainty due to the randomness of certain signal parameters. Perhaps

the most common signal parameter is the carrier phase. Synchronization with the

phase of the transmitted signal may be too complicated, and the designer may simply

choose to disregard the phase information in the received signal at the expense of some

degradation in noise performance. A digital communication receiver with no provision

made for carrier phase recovery is said to be noncoherent [4].

7.1 detection of differential PSK

The name differential PSK (DPSK) sometimes needs clarification because two separate

aspects of the modulation/demodulation format are being referred to: the encoding

procedure and the detection procedure. The term differential encoding refers to the

procedure of encoding the data different, the presence of a binary 1 or 0 is manifested

by the symbol’s similarity or difference when compared to the preceding symbol. The

term differentially coherent detection of differentially encoded PSK, the usual meaning

7 NONCOHERENT DETECTION 93

of DPSK, refers to a detection scheme classified as noncoherent because it does not

require a reference in phase with the received carrier.

With noncoherent systems, no attempt is made to determine the actual value of

the phase of the incoming signal. Thus, if the transmitted waveform is

si(t) =

√2Es

Ts

cos[ω0t + θi(t)], 0 ≤ t ≤ Ts, i = 0, . . . , M − 1

where Es is the symbol energy and the received signal can be characterized by

r(t) =

√2E

Tcos[ω0t + θi(t) + α] + w(t), 0 ≤ t ≤ T, i = 0, . . . ,M − 1

where α is an arbitrary constant and is typically assumed to be a random variable

uniformly distributed in [0, 2π[, and w(t) is an AWGN process.

For coherent detection, matched filters (or their equivalents) are used; for nonco-

herent detection, this is not possible because α is not known. However, if we assume

that α varies slowly relative to two period times (2T ), the phase difference between

two successive waveforms, θj(T1) and θk(T2) is independent of α.

The essence of differentially coherent detection of differentially encoded PSK (DPSK)

is that the identity of the data is inferred from the changes in phase from symbol to

symbol. Therefore, since the data are detected by differentially examining the wave-

form, the transmitted waveform would first be encoded in a differential fashion shown

in Table I. The differential encoding starts (third row in the table) with the first bit

of the code bit sequence c(k = 0), chosen arbitrarily (here taken to be 1). Then, the

sequence of encoded bits, c(k), can, in general, be encoded in one of two ways (encoding

rules)[6]

c(k) = c(k − 1)⊕

m(k) (19)

or

c(k) = c(k − 1)⊕

m(k) (20)

where the symbol⊕

represents modulo-2 addition and the overline denotes comple-

ment. In Table I the differentially encoded message was obtained by using (20). The

fourth row translates the coded bit sequence, c(k), into the phase shift sequence, θ(k),

where a 1 is characterized by a 180o phase shift, and a 0 is characterized by a 0o phase

shift. The encoding scheme for any of the two encoding rules is depicted in Fig. 11.

Fig. 12 illustrates an implementation of the optimum binary DPSK receiver. It

can be shown that the bit error probability in this receiver is given by [11]

7 NONCOHERENT DETECTION 94

encoding rule

delay Tb

c(k)m(k)

c(k-1)

Figure 11: Binary PSK differential encoder

Table Isample index, k 0 1 2 3 4 5 6 7 8 9 10

information message, m(k) 1 1 0 1 0 1 1 0 0 1

differentially encoded message, c(k) 1 1 1 0 0 1 1 1 0 1 1

corresponding phase shift, θ(k) π π π 0 0 π π π 0 π π

Pb =1

2exp

(−Eb

N0

)(21)

Besides, 2-PSK, 4-PSK and DPSK all have about the same bit error probability as a

function of Eb/N0 as Eb/N0 grows. The idea of differential detection may be extended

to four-phase and higher PSK transmission although the conventional receiver for 4-

DPSK requires asymptotically 2.3 dB more for the same bit error probability compared

to 4-PSK [1].

fcos(2π t)

+delay T

+

r(t)

∫0

T

+- binary

decision+

0

f sin(2π t)0

∫0

T

delay T

+

+

b

b

b

b

Figure 12: Optimum receiver for binary DPSK signaling

7 NONCOHERENT DETECTION 95

7.2 noncoherent detection over the AWGN channel

Assume that there are M signaling waveforms used in the transmission of information,

namely

sm(t) = Re[um(t)ej2πf0t], m = 0, ..., M − 1

where {um(t)} are the (baseband) complex envelopes. The channel is assumed to

introduce an attenuation factor α and a phase shift φm for each signal. The phases

{φm} are assumed to be mutually statistically independent and uniformly distributed

in the interval [0, 2π[.

Consider that symbol sk(t) was transmitted. The received signal is

r(t) = Re{αejφmuk(t)ej2πf0t + n(t)ej2πf0t}

The complex envelope of the received signal is

x(t) = αe−jφmum(t) + z(t) + nR(t) + jnI(t)

where nR(t) and nI(t) are the inphase and quadrature components of the noise. It was

shown in [2] that the optimum receiver for AWGN channels and equally likely signals

with equal energy computes the M decision variables

|Zm| =∣∣∣∣∣∫ T

0x(t)u∗m(t) dt

∣∣∣∣∣ , m = 0, . . . , M − 1

and selects the signal corresponding to the largest decision variable.

Thus

Zm =∫ T

0[xR(t) + jxI(t)][um,R(t) + um,I(t)] dt

and doing Zm = Zm,R + jZm,I leads to

Zm,R =∫ T

0[xR(t)um,R(t) + xI(t)um,I(t)] dt

Zm,I =∫ T

0[xI(t)um,R(t)− xR(t)um,I(t)] dt

The optimum detector, based on a random, uniformly distributed carrier phase,

computes the M envelopes

|Zm| =√

Z2m,R + Z2

m,I , m = 0, . . . , M − 1 (22)

7 NONCOHERENT DETECTION 96

or equivalently the squared envelopes |Zm|2 and selects the signal with the largest

envelope (or squared envelope).

In the special case of M-ary orthogonal FSK signals, the optimum receiver has

the structure depicted in Fig. 13. There are 2M correlators: two for each possible

transmitted frequency. The minimum frequency separation to maintain orthogonality

is ∆f = 1/T .

+

+

cos(2πfc t)

sin(2πfc t)

∫0

T(.)dt

∫0

T(.)dt

+

+

cos[2π(fc t]

∫0

T(.)dt

∫0

T(.)dt

receivedsignal

...

...+ ∆(M-1) f)

sin[2π(fc t]+ ∆(M-1) f)

envelope or square-lawdetector

outputdecision

t=T

...

u

u

u

u

u

u

0,i

0,q

k,i

k,q

M-1,i

M-1,q

Figure 13: Demodulation of orthogonal M-FSK for noncoherent detection

Let us consider the transmission of orthogonal M-FSK equal energy signals over an

AWGN channel, which are envelope-detected at the receiver. We also assume that the

M signals are equally probable a priori and that the signal s0(t) is transmitted in the

interval 0 ≤ t ≤ T . The M decision metrics at the detector are the M envelopes (22)

where

U0,i =T

2

√2Es

Tcos φ0 + n0,i, U0,q =

T

2

√2Es

Tsin φ0 + n0,q

and

Um,i = nm,i, Um,q = nm,q, m = 1, . . . , M − 1

The additive noise components {nm,i} and {nm,q} are mutually statistically inde-

pendent zero-mean Gaussian variables with equal variance σ2 = N0T/4. The envelope

|Z0| =√

U20,i + U2

0,q

7 NONCOHERENT DETECTION 97

is characterized by a Rice distribution with pdf

p|Z0|(r) =r

σ2exp

(−r2 + s2

2σ2

)I0

(rs

σ2

), r ≥ 0

where the noncentrality parameter is

s2 = (E{U0,i})2 + (E{U0,q})2 =TEs

2

and I0(x) is the zeroth-order modified Bessel function of the first kind.

Besides, the remaining envelopes

|Zk| =√

U2k,i + U2

k,q, k = 1, . . . , M − 1

are characterized by Rayleigh distributions with pdf

p|Zk|(r) =r

σ2exp

(− r2

2σ2

), r ≥ 0

It can be shown that the probability of a symbol error is [2]

Ps =M−1∑

n=1

(−1)n+1

(M − 1

n

)1

n + 1exp

[− nEs

(n + 1)N0

](23)

For binary orthogonal signals (M = 2) equation (23) reduces to the simple form

Pb =1

2exp

(− Eb

2N0

)(24)

For M > 2, we may compute the bit error probability by making use of the rela-

tionship

Pb =M/2

M − 1Ps

This expression has the same asymptotic energy efficiency as Q(√

Eb/N0), the error

probability of coherent orthogonal 2-FSK. A similar conclusion applies with M-ary

orthogonal signaling. Thus, noncoherent detection of orthogonal M-FSK in a good

AWGN channel (Eb/N0 À 1) leads to no real loss in energy efficiency [1].

It can be shown that binary DPSK signals are orthogonal over a two-bit interval

[4], [6]. Thus, equation (21) results from (24) by considering that the energy in the

two-bit interval is 2Eb.

REFERENCES 98

References

[1] John B. Anderson, “Digital Transmission Engineering”, IEEE Press, N. York,

1999.

[2] John G. Proakis, “Digital Comunications”, third edition, McGraw-Hill, N. York,

1995.

[3] Leon W. Couch II, “Digital and Analog Communication Systems”, Macmillan, N.

York, 1990.

[4] Simon Haykin, “Communication Systems”, 4.th edition, Wiley, N. York, 2001.

[5] Sergio Benedetto and Ezio Biglieri, “Principles of Digital Transmission with Wire-

less Applications”, Kluwer, N. York, 1999.

[6] Bernard Sklar, “Digital Communications. Fundamentals and Aplications”,

Prentice-Hall, Englewood Cliffs, N.J., 1988.

[7] Richard E. Blahut, “Digital Transmission of Information”, Addison-Wesley, Read-

ing, MA, 1990.

[8] Kamilo Feher, “Digital Communications: microwave applications”, Prentice-Hall,

Englewood Cliffs, NJ, 1981.

[9] Steven M. Kay, “Fundamentals of Statistical Signal Processing: Estimation The-

ory”, Prentice Hall, Upper Saddle River, NJ, 1993.

[10] Jack K. Holmes, “Spread Spectrum Systems for GNSS and Wireless Communica-

tions”, Artech House, Norwood, MA, 2007.

[11] Marvin K. Simon, Sami M, Hinedi, William C. Linsey, “Digital Communication

Techniques”, Prentice-Hall, Englewood Cliffs, NJ, 1995.