Digital communication receivers P6 - P1

The signal samples sf kT, are given by sjkTd = C a,gkT, -nT-ET ejeejnkT* Since the parameters are unknown but deterministic, the lower bound on the variance is determined by the element

Chapter 6 Performance Analysis

of Synchronizers

During data transmission the synchronizer provides an estimate which most

of the time exhibits small fluctuations about the true value The synchronizer is operating in the tracking mode The performance measure of this mode is the variance of the estimate

In Section 6.1 and the appendix (Section 6.2) we derive a lower bound on the variance of these estimates This bound will allow us to compare the variance

of practical estimators to that of the theoretical optimum and thus assess the implementation loss

In Section 6.3 we compute the variance of carrier and symbol synchronizers of practical interest The tracking performance is first computed under the assumption that the parameters are constant over the memory of the synchronizer Later on

we relax this prescription to investigate the effect of small random fluctuations (oscillator phase noise) and of a small frequency offset

Occasionally, noise or other disturbances push the estimate away from the stable tracking point into the domain of attraction of a neighboring stable tracking point This event is called cycle slip (Section 6.4) Cycle slips have a disastrous effect since they affect many symbols Their probability of occurrence must be at least a few orders of magnitude less frequent than the bit error rate Cycle slipping

is a highly nonlinear phenomenon which defies exact mathematical formalism in many cases One must resort to computer simulation

At the start of signal reception the synchronizer has no knowledge about the value of the parameters During a start-up phase the synchronizer reduces the initial uncertainty to a small steady-state error This process is called acquisition

To efficiently use the channel, the acquisition time should be short In Section 6.5

we discuss various methods to optimize the acquisition process

In this section we compute bounds on the variance of the estimation errors of synchronization parameters These bounds will alldw us to compare the variance

of practical synchronizers to that of the theoretical optimum and thus assess the implementation loss

325

Heinrich Meyr, Marc Moeneclaey, Stefan A Fechtel Copyright  1998 John Wiley & Sons, Inc Print ISBN 0-471-50275-8 Online ISBN 0-471-20057-3

We consider the task of joint estimation of a frequency offset S& a phase 8 and a timing delay E All parameters are unknown but nonrandom The signal samples sf (kT,) are given by

sj(kTd) = C a,g(kT, -nT-ET) ejeejnkT*

Since the parameters are unknown but deterministic, the lower bound on the variance is determined by the elements of the inverse Fisher information matrix J-l which we introduced in Section 1.4 The elements of J equal

Jil = -E a2 ln P@j P)

We recall that E[ ] indicates averaging over the noise, and 6 is a set of parameters

e = (e,, ,eK)

For the log-likelihood function [eq (4-69)] we obtain, up to a constant,

In P(rj 10) = 2 Re{ rTQsf } - sy Qsj 64) Taking the derivative of (6-4) with respect to the trial parameter t9i we obtain

But since the right-hand side of (6-7) contains nonrandom quantities, the expec- tation operation is the quantity itself:

1

(6-9)

= -Jil

The variance of the estimation error 8i - tii(rf )] obeys the inequality

var [8” - &(rf )] 1 Jii

where Jii is an element of the inverse matrix J-l

It is, however, intuitively plausible that for high SNR or a large number of data symbols the performance of the synchronization parameter estimator should come close to the theoretical optimum We next want to prove this assertion and quantitatively investigate the asymptotic behavior of the joint synchronization parameter estimator

Consider again the general parameter vector 8 = (01, , t9~) For a high SNR or a large number of symbols the ML estimate 8 will be clustered tightly around the point 80 = (01, , 0~)~ of the actual parameter values We therefore develop the log-likelihood function into a Taylor series retaining terms up to second order:

because

Using

mom ln P(rj 10) = ln P(q I@) le (6-13)

In p(rf 10) = 2 Re{ ry Qsj} - s?Qsj (6-14)

(6-18)

We immediately recognize that the coefficients in the last term are the elements

of the Fisher matrix J (eq 6-9) so that

Solving the last equation for the ML estimate we obtain

k Hence, fik is a Gaussian random variable with mean &,a (and thus unbiased)

We now want to apply these general results to the problem of jointly estimating the synchronization parameter set 8 = 10, E, a} The computation of the elements

of the Fisher information matrix J (6-9) is conceptually simple but technically somewhat tricky and for this reason delegated to the appendix (Section 6.2)

In the following two examples we determine the Fisher information matrix for two different estimation problems While in the first example we consider the joint estimation of the synchronization parameter set 8 = (a, 0, E), we assume in the second example that the frequency offset R is known when jointly estimating the parameter set 8 = (0, E} The dimension of the Fisher information matrix

is determined by the number of unknown parameters Therefore, the Fisher information matrix is of dimension 3 x 3 in the first example and is reduced to a

2 x 2 matrix in the second example For the special case that only one parameter requires estimation, the Fisher information matrix degenerates to a scalar, and the bound is the Cramer-Rao bound known from the one-dimensional problem Example I

We consider the Fisher information matrix for the joint estimation 8 = {Q, 8, E}

We assume

(i) Independent noise samples n(lcT,) The inverse covariance matrix Q is thus

a diagonal matrix with elements l/o2 = (IVO/A~T~)-~

(ii) The signal pulse g(t) is real

(iii) Random data

Condition (i) is merely for convenience; all results also apply for colored noise Due to conditions (ii) and (iii) the cross terms Jaa and J,e vanish (see Section 6.2) Notice that g(t) stands for the overall pulse shape of the transmission path, The requirement for g(t) to be real therefore imposes demands on the transmitter as well as receiver filter and on the other units such as mixers and oscillator accuracy Under these conditions, the Fisher matrix equals

It can be verified that the

Since J&/Jan 2 0 and J&/ Jee 2 0, the expressions for J”” (6-22) and Jee

(6-23) exemplify that the estimation error variances for 0 and L! are always larger than the variance obtained when only a single parameter is estimated, because in this case the variance of the estimation error is the inverse of the corresponding entry in the main diagonal of J Given that the Fisher information matrix is diagonal, the estimation error variances of a joint estimation are equal to those obtained when only a single parameter requires estimation It is evident from the above consideration that we should try to force as many cross-terms of J to zero

as possible This can be accomplished, for example, by a proper selection of the pulse shape

The lower bound on the timing error variance [eq (6-24)] is not influenced by the fact that all synchronization parameters have to be jointly estimated, since the

cross coupling terms JaE and Jet vanish under the above conditions Therefore, the timing error variance is the same as the variance we would expect to find if

we had to estimate the timing when phase and frequency were known Jet is strongly dependent on the pulse shape, as we would expect for timing recovery The variance of the timing error increases with decreasing excess bandwidth

A last comment concerns the dependency of var 80 { - 8} on E This de- pendency expresses the fact that there exists an optimal timing offset e for the estimation of this parameter However, the influence is weak since it decreases with O(Nm2)

Example 2

Now we assume that the frequency offset Re is known Then the Fisher information matrix is reduced to a 2 x 2 matrix Applying the same conditions as in the previous example and using the results of the appendix (Section 6.2), we get the following diagonal matrix:

J = J;e Jo

[ 1 cc

and for the lower bounds it follows

var{ee - j} 2 Jee = $ = [$I-‘+

and

2 E, -’ 1 var{e, - e) >, JEc = f = [ 1 - oo ‘s” lG(w)j2 dw

at least in the steady state, if the synchronizer structure in the joint estimation case meets the lower bounds determined by (6-28) and (6-29)

The Fisher information matrix is given by

(6-30)

with entries Jeiol [eq (6-g)]:

Joie, = -E a2

where t9d, 01 are elements of the synchronization parameter set 8 = { fl, 0, E} From (6-31) it follows that Je,e, = Jelei Throughout this appendix we assume that the second condition given by (4-127) [symmetric prefilter IF( is satisfied so that

Q is a diagonal matrix with entry A2T,, /No:

n E [-(N-1)/2, (N-1)/2] for the transmission of the (N+l) symbols We discuss the case of statistically independent known symbols ’ -

Performing the operations in (6-34) for the various combinations of parameters yields the elements Joie1 = Jelei We begin with Joa

Using the convolution and differentiation theorem of the Fourier transform, the

right-hand side of the above equation can be expressed in the frequency domain as

ccl

s t2 g(t - nT - ET) g*(t - mT - ET) dt

-ca

= & / [ _ & (+,) e-ju(nT+rT))] [G*@) ejw(mT+rT)] du (6-38)

Partial integration yields

Since G(w) and d/dw G( u ) vanish at the boundaries,’ inserting (6-39) into (6-35) yields for statistically independent symbols

No/A2Ts T, 27r

+ 5 n2 T2 / jG(w)j2 dw

n=l + N &2 T2

Jan can be written as

With E, denoting the symbol energy and with the abbreviation E, for the energy

The dependency of Jan on E expresses the fact that there exists an optimal time offset for the estimation of 0 Jan depends on both the magnitude and on the sign of e If we demand that g(t) is real and even, then the sign dependence vanishes as expected in a symmetrical problem where a preferential direction cannot exist The E dependence is of order N and therefore becomes rapidly negligible compared to the term of order N3

The dominant term of 0 (N3) is independent of the pulse shape Only for small N the pulse shape comes into play with order O(N) The term belonging to 8G(w)/B w is proportional to the rolloff factor l/o (for Nyquist pulses) as can readily be verified

2 Calculation of JEE: JEE is given by

(6-47)

Notice:

a In obtaining (6-47) we assumed statistically independent data If this as- sumption is violated, the above bound does not hold Assuming the extreme case of identical symbols and N + co, the signal sf (LT,) (6-33) becomes

a periodic function with period l/T But since the signal is band-limited to

B < l/T, sf (ICT,) becomes a constant from which no timing information can

be obtained As a consequence J EE equals zero, and the variance becomes infinite.2

b JEE is independent of s2 and 8

C In the case of Nyquist pulses, JEE is minimum for zero rolloff (CY = 0) This

is as expected as the pulses decrease more rapidly with increasing excess bandwidth

2 In most cases cross terms can be neglected

3 Calculation of Joe : Joe is given by

a While for Jan and JcI random data were assumed, Joe exists for any sequence

{an} This is a consequence of the fact that the quadratic form (6-50) is positive definite

b This is most clearly seen if hm,n has Nyquist shape Then

(6-52)

where Es and E, are the pulse and symbol energies, respectively

C For random data, by the law of large numbers we obtain

which is the true irrespective whether hm,, is of Nyquist type or not However, note the different reasoning which lead to the same result:

E, Joe = - 2N

5 Calculation of the cross terms JEe, JEn and Joa: The details of the calculations are omitted, since the underlying computational principles are identical to those

of Jslsl, J,, , and Jee

a Calculation of the cross term Jee :

J Ee = 2 Re as? &3f 1

WY& a,}, m # n can be neglected With

JEe = 2NE [ IanI’] g 7 (jw) IG(w) I2 dw

-CXl

= 2NE[la,12] T Re{j i(O)}

= -2NE [la, 12] T Irn{ h(O)}

b Calculation of the cross term Jcn:

we can write J,ga as

Jm = 2N No,A2 -(ET+ & Im[[ (&G*(u)) G(u) du]) (6-65)

or in the time domain as follows:

co

Jest =2N$ eT + +

9 -CO

If Is(t)I = Id-t) I or equivalently g* (-t) = g(t), then Joa vanishes for E = 0

Bibliography

[l] D C Rife and R R Boorstyn, “Multiple-Tone Parameter Estimation from Discrete-Time Observations,” Bell Syst Tech J., vol 55, pp 1121-1129, Nov 1976

[2] M H Meyers and L E Franks, “Joint Carrier Phase and Symbol Timing Recovery for PAM Systems,” IEEE Trans Commun., vol COM-28, pp 1121-

1129, Aug 1980

[3] D C Rife and R R Boorstyn, “Single-Tone Parameter Estimation from Discrete-Time Observations,” IEEE Trans In8 Theory, vol IT-20, pp 591-

599, Sept 1974

[4] M Moeneclaey, “A Simple Lower Bound on the Linearized Performance

of Practical Symbol Synchronizers,” IEEE Trans Comrnun., vol COM-31,

Assuming unbiased estimates, a convenient measure of the tracking performance

is the synchronization error variance, which should be small in order to keep the associated degradation of other receiver functions such as data detection within reasonable limits As synchronization errors are small during tracking, the system equations describing the synchronizers can be linearized with respect to the stable operating point This simplifies the evaluation of the synchronization error variance, which is then referred to as the linearized tracking performance We will evaluate the linearized tracking performance of various carrier and symbol synchronizers operating on the complex envelope TJ (t) of a carrier-modulated PAM signal given by

am = Cam g(t-mT-eoT) ejeo + F

m

(6-67)

The operation to the synchronizer is discussed in terms of the matched filter samples

z(nT+ iT) = c amh [(n-m)T + (EI EO)T] ejeo + N(nT) (6-68)

m

We use the normalized form of Section 4.3.6 (appendix), eq (4-159, which holds for a symmetric prefilter and Nyquist pulses In this case the symbol unit variance

El [ a, I] 2 = 1, h&-e0 = 0) = 1, and the complex noise process iV( nT)

is white with independent real and imaginary parts, each having a variance of

NO /2 E;, Since

hm,n(0) = { A zs’ n (6-69)

the matched filter output (ideal timing) exhibits no ISI:

The likelihood function [see eq (4-157)] for the normalized quantities reads

y (Ian,” - 2 Re[aiz,(E) eaje])]} (6-71) n=O

The linearized tracking performance will be computed under the assumption that the synchronization parameters are constant over the memory of the synchronizers This restriction is relaxed in Section 6.3.8, where the effects of a small zero-mean random fluctuation (oscillator phase noise) and of a small frequency offset are investigated

6.3.2 Tracking Performance Analysis Methods

Here we outline the methods for analyzing the linear tracking performance of feedback and feedforward synchronizers