Digital communication receivers P5

phase Independent feedfotward symbol sync A kT* phase Independent feedback symbol sync Phase 8 timing dependent feedfoward carrier sync carrier dependent feedback symbol sync I-=- {

Chapter 5 Synthesis of

In this chapter we derive maximum-likelihood (ML) synchronization algo-

rithms for time and phase Frequency estimation and synchronization will be

treated in Chapter 8

The algorithms are obtained as the solution to a mathematical optimization

problem The performance criterion we choose is the ML criterion In analogy to

filter design we speak of synthesis of synchronization algorithms to emphasize that

we use mathematics to find algorithms - as opposed to analyzing their performance

5.1 Derivation of ML Synchronization Algorithms

Conceptually, the systematic derivation of ML synchronizers is straightfor-

ward The likelihood function must be averaged over the unwanted parameters

With the exception of a few isolated cases it is not possible to perform these

averaging operations in closed form, and one has to resort to approximation

techniques Systematically deriving synchronization algorithms may therefore be

272 Synthesis of Synchronization Algorithms

understood as the task of finding suitable approximations The various algorithms are then the result of applying these techniques which can be systematic or ad hoc The synchronizers can be classified into two main categories:

1 Class DD/DA: Decision-directed (DD) or data-aided (DA)

2 Class NDA: Non-data-aided (NDA)

The classification emerges from the way the data dependency is eliminated When the data sequence is known, for example a preamble a0 during acquisition, we speak of data-aided synchronization algorithms Since the sequence a0 is known, only one term of the sum of eq (5-l) remains The joint (0, e) estimation rule thus reduces to maximizing the likelihood function p(rf 1 a = a0 ,8, E) :

( > b DA =arg max p(rf Ia=aO,8,E) (5-2)

e,e When the detected sequence & is used as if it were the true sequence one speaks of decision-directed synchronization algorithms When the probability is high that & is the true sequence ao, then only one term contributes to the sum

Example: NDA for BPSK with i.i.d symbols

n=O

(5-5)

An analogous classification can be made with respect to the synchronization parameters to be eliminated For example,

(DD&De) : data- and timing directed:

p(rf le) = p(rf la = ii, 8, e = it)

phase Independent feedfotward symbol sync

A kT*

phase Independent feedback symbol sync

Phase

8 timing dependent feedfoward carrier sync

carrier dependent feedback symbol sync

I-=- { zF:i } dependent

data detector

tlmlng dependent feedback carrier sync

Figure 5-l Feedforward (FF) and Feedback (FB) Synchronization Algorithms

The other category derives an error signal &I = 8 - 00 and & = t - ~0, respectively These algorithms are called feedback (FB) because they derive an estimate of the error and feed a corrective signal back to the interpolator or phase rotator, respectively Feedback structures inherently have the ability to automatically track slowly varying parameter changes They are therefore also called error-feedback synchronizers

For illustration, in Figure 5-l a typical block diagram of a digital receiver

is sketched together with the various signals required for FF or FEl algorithms Notice that the position of the various blocks may be interchanged, depending on the application For example, interpolator and rotator may be interchanged

When deriving a synchronization algorithm from the ML criterion, one as- sumes an idealized channel model (to be discussed shortly) and constant parame- ters, at least for quasi-static channels In principle, a more realistic channel model and time-variable parameters could be taken into account, but it turns out that this approach is mathematically far too complicated In view of the often crude approx- imations made to arrive at a synchronizer algorithm, it makes no sense to consider accurate channel models Instead, one derives synchronization algorithms under

274 Synthesis of Synchronization Algorithms

idealized conditions and later on analyzes the performances of these algorithms when used in conjunction with realistic channels

We assume Nyquist pulses and a prefilter ]P’(u)]~ being symmetric about 1/2T, In this case the likelihood function [see eqs (4-83), (4-157)] assumes the simplified form

Nc lho,0121a,12 - 2uiz,(&)e-je

n=O

(5-8)

with the short-hand notation Z,(E) = z(nT + ET)

We said earlier that synchronization algorithms can be systematically derived

by finding suitable approximations to remove the “unwanted” parameters in the

ML function The result of these approximations is a function L(O), where 8 denotes the set of parameters to be estimated The estimate 8 is defined as the argument for which L(8) assumes an extremum Depending on the definition of L(8), the extremum can be either a minimum or a maximum:

4 = arg extr L(e)

Strictly speaking, 6 is an ML estimate only if the objective function L(B) is the

ML function P(rf I 6) However, for convenience we frequently speak of ML estimation also in the case that L( 0) is only an approximation to p(rf 1 0)

A first approximation of the likelihood function (5-8) is obtained for large values of N

SfHSf = C l~0,0121ha12

We have shown in Chapter 4 that the inner product

is independent of the synchronization parameters For

a sufficieitly large N the sum

YN =yl%12 n=O

(5-10)

is closely approximated by its expected value This is a consequence of the law

of large numbers We can therefore discard the term C Ian I2

const from maximization to obtain the objective funcion

+ ~~[lha12] =

Several important conclusions can be drawn from this objective function In most digital receivers timing recovery is done prior to phase recovery The reason becomes clear by inspection of (5-11) Provided timing is known, one sample per symbol of the matched filter output is sufficient for carrier phase estimation

and symbol detection To minimize the computational load in the receiver, carrier phase estimation and correction must be made at the lowest sampling rate possible, which is the symbol rate l/T All digital algorithms for phase estimation to be derived later on will therefore be of the DE type running at symbol rate l/T They will be either DD (DA) or NDA

While the number of different carrier phase estimation algorithms is small, there exists a much larger variety of digital algorithms for timing recovery This

is due to the fact that the number of degrees of freedom in deriving an algorithm

is much larger Most importantly, the sampling rate l/T8 to compute z~(E) can

be chosen independently of the symbol rate At the one extreme the samples G&) = z(nT + ET) can be obtained by synchronously sampling the output of

an analog matched filter z(t) at t = nT + ET A digital error feedback algorithm running at rate l/T is used to generate an error signal for the control of an analog voltage-controlled oscillator (VCO) in this hybrid timing recovery system Using

a higher sampling rate l/T, > l/(T(l + CX)) (a: excess bandwidth), the matched filter can be implemented digitally The samples ~~(6) are then obtained at the output of a decimator zla (E) = z( m, Tj + pnTJ) Timing recovery is performed

by a digital error feedback system (FB) or direct estimation (FF) of the timing parameter e and subsequent digital interpolation All of them - DD, DA, and NDA - methods are of practical interest

5.2 Estimator Structures for Slowly Varying

Synchronization Parameters

5.2.1 Time Scale Separation

Discussing the optimal receiver in Section 4.3.1 we modeled the synchroniza- tion parameters as random processes The time scale on which changes of these parameters are observable is much coarser than the symbol rate l/T It is therefore reasonable to separate these time scales We consider a first time scale which op- erates with units of the symbol duration T to deal with the detection process The second (slow) time scale deals with the time variation of the parameters E(nT)

and O(nT) with time constants being much larger than T We can thus consider the synchronization parameters as approximately piecewise constant and estimate these parameters over segments of M >> 1 symbols The number M of symbols over which the synchronization parameters can be assumed to be approximately constant depends on the stability of the oscillators and the frequency offset

If the variance of the short-term estimate is much larger than the variance of the synchronization parameter, processing the short-term estimate in a postproces- sor which takes advantage of the statistical dependencies between the estimates yields a performance improvement We have thus arrived at a first practical two- stage structure as shown in Figure 5-2

276 Synthesis of Synchronization Algorithms

> Estimator _

* Post- D

* Processor D MT-interval

b) - 8 (kT)

8;kMT) - &nT)

D

kMT (k+l)MT (k+2)MT nT

Figure 5-2 (a) Two-Stage Estimator Structure, (b) Time Variation of Parameter

The two-stage estimator has several advantages from a digital signal process- ing point-of-view Computation of ML estimates basically is an inner product (feedforward) computation which can be processed in parallel and thus is suitable for VLSI implementation The more complex processing in the postprocessing stage then runs at a lower rate of l/(MT) compared to l/T8 > l/T in the first stage

An alternative to estimating a slowly varying parameter is to generate an estimate of the error and use this estimate as an error signal in a feedback system (Figure 5-3)

kTs

error signal Interpolator e(kT, )

> hl (kT, ,Q 1

y Error ) Loop _

’ Detector Filter

A A a&) &kT, 1

Figure 5-3 Timing Error Feedback System

5.2.2 Truncation and L Symbols Approximation

of the Likelihood Estimates

In this section we consider approximations to the optimal estimator with respect to (i) Estimation of the synchronization parameters using only a subset of L < N symbols out of N transmitted ones

(ii) Estimation of the synchronization parameters using a finite observation inter- val {rf(kT,)}, a 5 k L b

In many practical applications we estimate the parameters (6, E) over L symbols

in a continuous stream of N symbols, L << N [case (i)] The observation interval

L is limited by the fact that the parameters are not strictly constant but slowly time variable L is then chosen such that

LT < r,j,rc (S-12) where ~6, rC are the correlation time of the respective processes Following (4-83) the log-likelihood function for L symbols reads

L(rf 1 aL, 0, E) =2 Re C ai ~~(a) e+

J+(L-1) J+(L-1)

- C C 4 al hl,n n=J l=J

(5 13)

Note that the entire received signal rf is processed Only the data vectors

a and z are replaced by aL and ZL, respectively This remark is important since truncation of the received signal rf leads to entirely different degradations [case (ii)] as will be discussed shortly

The approximation error of choosing a finite interval L among a number of

N transmitted symbols is determined next The received signal equals the sum of useful signal s( t ,&Jo) and noise n( t ) Notice that the parameter values 80 = (00, eo ) are the true values to be clearly distinguished from the trial parameters (0, e) Inserting Sf (t) + n(t) into (4-71) yields

z(nT + ET) = a, g( kT, - mT - QT) ejeo + n(kT,) 1

5 Can be omitted at first reading

278 Synthesis of Synchronization Algorithms

Interchanging the summation over k and m yields [again with the short-hand notation Z, (E) = z(nT + ET)]

%&)==~LI, 2 g kT, -mT-e*T) ( gMF(-kTs + nT + ET) ejeo +n,(nT)

m=O k=-co

(5-15) with the additive noise term

n=J m=O J+(L-1) N + C C aiamhm,n(&-&O) e-j(e-eO) n=J m=J+L

(5- 19)

We recognize in the first double sum exactly the term that would appear in the log-likelihood function if L symbols were transmitted in total The second two sums are interfering terms caused by symbols outside the considered L interval They represent self-noise which is present even in the absence of additive noise

*L 81

Figure 5-4 Symbol Vector a

Using vector matrix notation the contribution of the self-noise can be seen

of two disjoint vectors,

noise contribution depends on the amount of intersymbol interference (ISI) as evidenced by the nonzero off-diagonal terms in the matrix Let us denote by D

J J+(L-1)

J+(L-1) -

Figure 5-5 Projection of Hal on a;

280 Synthesis of Synchronization Algorithms

the number of symbols which essentially contribute to the self-noise This number

is independent of L For sufficiently large L the self-noise becomes negligible and the performance approaches asymptotically that of the optimal ML estimator Further, one recognizes

bution for E = eo

that for Nyquist pulses there is no self-noise contri-

In case (ii), only a subset of samples {rf (ITS)}, J < k 5 (J+ ML - 1) is processed The factor A4 is the integer of the ratio [T/TslINT = M The interval then approximately equals that of received L symbols

The truncated log-likelihood function is obtained from (4-95) by restricting the summation from J to (J+LM-1):

c rj (kG) d(kT,) - c

E=J 1 J+(L-1) J+(L-1)

c a: ai h,, n=J l=J

It is important to realize that the L-segment correlation is not equivalent to the matched filter method [compare with (513)] While in both cases a subset of L symbols is considered, in the matched filter approach the entire received signal is processed Performing the correlation operation defined above, an additional term

is introduced by truncating the received signal to LM samples Thus, correlation

in general is inferior to matched filtering

5.2.3 Maximum Search Algorithm

There exist a variety of algorithms for maximum search of the objective function The choice depends mostly on the bit rate and technology available

Parallel Search Processing

Today’s technology allows the integration of highly complex digital signal processors The computational load can be managed by parallel processing rather then using an extremely fast technology

Iterative Search Processing

The maximum search operation can be performed serially A necessary, but not sufficient condition for the maximum of the objective function is

Notice that the received data over a segment of L symbols is processed repeatedly which requires that the data are stored in memory This places no problem with today’s digital technology The iterative search is of particular interest to achieve acquisition with known symbols during a training period

5.2.4 Error Feedback Systems

Error feedback systems adjust the synchronization parameters using an error signal The error signal is obtained by differentiating an objective function and computing the value of the derivative for the most recent estimates 8, , in,

dL 3-r ( 6,e=8,,&=& > (5-26)

dL

ae ( i&e=&,&=& >

For causality reasons, the error signal may depend only on previously re-

282 Synthesis of Synchronization Algorithms

viewed symbols a,, (which are assumed to be known) The error signal is used

to predict the new estimate:

(5-27)

We readily recognize in eq (5-27) the estimate of a first-order discrete-time error feedback system where (cy,, cre) determines the loop bandwidth Higher-order tracking systems can be implemented employing appropriate loop filters

The error signal can always be decomposed into a useful signal plus noise For E and similarly for 19 we obtain

zero-mean

v

noise procese

is sufficiently small The useful signal in (5-28) must be zero for zero error in order to provide an unbiased estimate The process of bringing the system from its

phenomenon (see Volume 1, Chapter 4)

We observe some similarities between the maximum search and error feedback systems In both cases we use the derivative of the objective function to derive an error signal However, we should be aware of the fundamental differences The

the final estimate The feedback control systems, on the other hand, operate in real time using only that segment of the signal which was received in past times

5.2.5 Main Points

0 ClassiJication of algorithms

known and the obverse: The first class is called decision-directed (DD) or data-aided (DA), the obverse NDA We further distinguish between feedfor- ward (FF) and feedback (FB) structures

one of the two structures:

Two-stage tracker

The synchronization parameters are approximately piecewise constant

Postprocessing of the short-term estimates is done in a second stage This stage exploits the statistical dependencies between the short-term estimates of the first stage

Error feedback system

An automatic control system is used to track a slowly varying parameter

We process the entire received signal rj (kT,) but use only a subset of L << N symbols out of N transmitted ones for estimation This produces self-noise

If we process only a segment of samples T-I(~&) we generate additional noise terms

Bibliography

[l] M Oerder, Algorithmen zur Digitalen Taktsynchronisation bei Dateniibertragung Dusseldorf: VDI Verlag, 1989

5.3 NDA Timing Parameter Estimation

The objective function for the synchronization parameters (0, E) is given by

In a first step we derive data and phase-independent timing estimators The estimate

of E is obtained by removing the unwanted parameters a and 0 in eq (S-29)

To remove the data dependency we have to multiply (5-29) by P (ia), where (iu) is the ith of M symbols, and sum over the M possibilities Assuming independent and equiprobable symbols the likelihood function reads

L(O,e) = Nfi1 Fexp { -$ Re[‘az Zn(E) e-je] P(‘a)}

n=O i=l

(5-30)

There are various avenues to obtain approximations to (5-30) Assuming M-PSK modulation with M > 2, the probabilities

P(‘a) = $ for ia=ej2*ilM i= l, ,M (5-3 1)

can be approximated by a continuous-valued probability density function (pdf) of

284 Synthesis of Synchronization Algorithms

eia where CY has a uniform distribution over (-X, r):

Zn(E) GcBaBe)

11 dell

= n j exP \ -$ lM)I COS (-a - 0 + arg Z,(E)) da,

(5-34)

Notice that the result is the same for all phase modulations (M-PSK) (since

l%l = const.), but not for M-QAM In order to obtain an NDA synchronization algorithm for M-QAM, we would have to average over the symbols which is not possible in closed form

But the objective functions of (5-33) and (S-34) can be further simplified by a series expansion of the modified Bessel function, Taking the logarithm, expanding into a series

XL IO(X) Iii 1 + -

and discarding any constant irrelevant for the estimation yields

NDA: t = arg max Lr(E)

cz arg max Nc Iz,(E)12

c n=O

(5-36)

DA: t= arg max &(a,&)

2 arg max Nc I%n(&)121Un12

c n=O

For M-PSK (ian I2 = const.) both algorithms are identical

Let us now explore a totally different route We want to eliminate the data dependency in (5-29) This requires averaging over the symbols, if possible at all Furthermore, it requires knowledge of 0: (operating point) which is not available

in general The algorithm would possibly be sensitive to this operating point Both problems can be circumvented by considering the limit of the likelihood function

(5-29) for low signal-to-noise ratio (SNR), ui >> 1 For this purpose we expand the exponential (5-29) function into a Taylor series:

1 2 (5-37)

We next average every term in the series with respect to the data sequence For

an i.i.d data sequence we obtain for the first term

286 Synthesis of Synchronization Algorithms

since ~!?[a~] = 0 For the second term we obtain

[

5 Re [u;", z,(E) e-je]

Y&=0 1 [ ' = t E uI", zn(E) e-je + a, z;T(e) ej'] 2

n=O

+ aNc [(U~)2(Zn(E))2fZw'2e + (Un)2(.Z~(E))2e+52e + 21a,121&b(E)12]

n=O

(5-39) Now, taking expected values of (5-39) with respect to the data sequence assuming i.i.d symbols yields (E[cn] = 0):

which is the same as equation (5-36) (NDA)

Equation (5-41) serves as a basis for the joint non-data-aided estimation of phase and timing The phase estimate

BA = $ arg Ne E[ui] (~i(e))~

(5-43)

maximizes the second sum of (5-41) for any c, since the sum

becomes a real number

of the absolute value

5 E[uX] (%;(e))2 e-jae^

L(E) = Ne E[1”“(2] 1zn(E)12 +

Comparing (5-46) and (5-42) we observe that the noncoherent (NC) timing estimator does not depend on the signal constellation while the implicitly coherent (IC) estimator does so In the following we restrict our attention to two classes of signal constellations of practical importance The first contains the one-dimensional (1D) constellation comprising real-valued data for which E [ ui] = 1 (normalized) The second class consist of two-dimensional (2D) constellations which exhibit

a 7r/2 rotational symmetry, this yields E [uz] = 0 Hence, for 7r/2 rotational symmetric 2D constellations, the IC and NC synchronizers are identical, but for 1D constellations they are different

timing estimators The practically most important result is a phase-independent algorithm

e = arg m.m C Izn (&)I”

n

The algorithm works for M-QAM and M-PSK signaling methods

5.4 NDA Timing Parameter Estimation by Spectral Estimation

In the previous section the unknown timing parameter was obtained by a maximum search (see Figure 5-6) It is interesting and surprising that one can circumvent this maximum search

We consider the objective function (5-42),

L(e) = 5 Iz(lT+ &T)12 (S-48) 1=-L

We assume a symmetric observation interval [-L, L] This choice is solely to simplify some of the following computations For a sufficiently large number N

of transmitted symbols (N >> L) the process ]z( IT + ET) I2 is (almost) cyclosta- tionary in the observation interval A cyclostationary process has a Fourier series representation

n=-a3

=

c cg) ej2rnc

n=-co

where the coefficients cn (I) are random variables defined by

cg) = + ] 1%(/T + eT)12 e-j(2+lT)nTc Q(&T)

0

1

=

s Iz(lT + eT)12 e-j2rnr dc

0

(5-50)

The objective function (5-48) equals the time average of the cyclostationary process ]z(lT + ET) la over the interval [-L, L] This time average can be obtained by

290 Synthesis of Synchronization Algorithms

(1) * computing the Fourier coefficients cn m every T interval and averaging over

By definition, the ML estimate g is that value of E for which (5-48) assumes its maximum:

Since CO and the absolute value Icr I are independent of E (to be shown later) the

maximum of (5-55) is assumed for

1 if= arg cl 2n

It is quite remarkable that no maximum search is needed to find t since it is

explicitly given by the argument of the Fourier coefficient cl The coefficient

cl is defined by a summation of (2L + 1) integrals The question is whether the integration can be equivalently replaced by a summation, since only digital

algorithms are of interest here This is indeed so, as can be seen from the following

plausibility arguments (an exact proof will be given later)

The output of the matched filter is band-limited to B, = (1/2T)( 1 + CX)

Since squaring the signal doubles the bandwidth, the signal (z(t) 1” is limited to

twice this value Provided the sampling rate l/T, is such that the sampling theorem

is fulfilled for ]z(t)12 [and not only for z(t)],

BI4’ =~(l+ck).&

the coefficients cl, CO can be computed by a discrete Fourier transform (DFT)

Let us denote by the integer Md the (nominal) ratio between sampling and symbol

rate, AI8 = Z’/T8 For the samples taken at kT, we obtain

I%( [MS + k]T,) 12e-j(2*lM,)k

I

A particularly simple implementation is found for Mb = 4 (Figure 5-7) For this

value we obtain a multiplication-free realization of the estimator:

Cl = I~([41 + k]T,)12(-j)k 1 d?T-k = (-jJk (s-59)

z-l : Delay of Ts

h

c iif -18

I

8 +

Figure 5-7 Timing Estimation by Spectral Analysis; M, = T/T’ = 4

292 Synthesis of Synchronization Algorithms

Splitting up in real and imaginary part yields

We next prove that E^ = -1/27r arg cl is an unbiased

rather technical and may be omitted at first reading

(ii) The ratio T/T3 is an integer MS = T/Td

(iii) i.i.d data

(iv) g(4) = g(t) (real and symmetric pulse)

The matched filter output z(lT,) equals

k=-00 Replacing in the previous equation the samples rj (kT,) by

(5-W

The last line of the previous equation is a consequence of the equivalence theorem

of analog and digital signal processing [assumption (i)]

data and noise yields

with Pn 2 0 the additive noise contribution being independent of eo

If the number of symbols is sufficiently large, we commit a negligible error when the summation in the previous equation runs from -co to +oo (instead of [-IV, IV]) The infinite sum

hg(t) = 5 Ih(t - nT - E~T)I~ (5-66)

n=-00

represents a periodic function hp(t) which can be expanded into a Fourier series

T J lWl2 e -j2dt/T &

Since the pulse g(t) is band-limited to

Ih(t is limited to twice this bandwidth:

294 Synthesis of Synchronization Algorithms

Thus, only three coefficients d- 1, de, dl are nonzero (a < 1) in the Fourier series (Figure 5-8):

E[Iz(t)12] = E[lan12] 2 Ih(t - nT- EoT)12

n=-03

= 2 E[lan12]di emww i=- 1

(5-7 1)

with do, dl , d- 1 given by (5-68) Due to the band limitation of E [ ~z(t) 1’1 the Fourier coefficients cli and E [ IZ(ATT,) 12] are related via the discrete Fourier transform The coefficients do and dl, respectively, are given by

do = - ; Mgl E [l+T.)12]

’ k=O

(5-72)

Comparing with the coefficient c,, of (5-58) we see

4 = const.E[q] do = const.E[co] d, = E[c,J = 0

(5-75)

If the pulse g(t) is real and symmetric, then Ih( = Ih( and

ThUS

is an unbiased estimate

Main Points

One can circumvent the maximum search for 2 by viewing I zn (E) I2 as a

periodic (l/T) timing wave The argument of the first coefficient cl of the Fourier series representation of ]zn (E) I2 yields an unbiased estimate

In the inner sum the Fourier coefficient c(II) for the Ith time interval of length T

is computed The final estimate for cl is obtained by averaging the result over (2L + 1) intervals

A particularly simple realization is obtained for four samples per s m- bol, MS = 4 Because in this case we have exp (-j27rk/M,) = (-j) r , a multiplication-free realization is possible

2% Synthesis of Synchronization Algorithms

5.5 DA (DD) Timing Parameter Estimators

Replacing in eq (5-29) the trial parameter a,, and the phase 8 by their estimates yields a DD phase-independent algorithm:

= This algorithm finds application when phase synchronization is done prior to timing recovery The computation of the objective function can again be computed

in parallel Instead of the nonlinearity to remove the data dependency we now have the multiplication by the symbols (see Figure 5-9)

We next consider the joint estimation of (0, E),

L(ii, E, 0) = exp a; z,(e) e+ (5-81)

The two-dimensional search over (0, e) can be reduced to a one-dimensional one