Digital communication receivers P7

Chapter 7 Bit Error Rate Degradation Caused by Random Tracking Errors 7.1 Introduction For coherent detection of digitally modulated signals, the receiver must be provided with accurat

Chapter 7 Bit Error Rate Degradation

Caused by Random Tracking Errors

7.1 Introduction

For coherent detection of digitally modulated signals, the receiver must be provided with accurate carrier phase and symbol timing estimates; these estimates are derived from the received signal itself by means of a synchronizer The bit error rate (BER) performance under the assumption of perfect synchronization

is well documented for various modulation formats [l-5] However, in practice the carrier phase and timing estimates exhibit small random fluctuations (jitter) about their optimum values; these fluctuations give rise to a BER degradation as compared to perfect synchronization It is important to know this BER degradation

in terms of the accuracy of the estimates provided by the synchronizer, so that the synchronizer can be designed to yield a target BER degradation (which should not exceed about 0.2 dB for most applications)

For various linear modulation formats (M-PSK, M-PAM, and M2-QAM) we evaluate the BER degradation caused by random carrier phase and timing errors

In Section 7.7 we show that the results also apply for the practically important case of coded transmission For nonlinear modulation and coded transmission we refer to the bibliographical notes in Section 7.9

7.2 ML Detection of Data Symbols

Figure 7-l conceptually shows how a maximum-likelihood (ML) decision about the symbol sequence {ak} is obtained The matched filter output is sampled

at the instant IcT + iT where C denotes the estimate of the normalized time delay

~0 The matched filter output samples are rotated counterclockwise over an angle 8 which is an estimate of the unknown carrier phase 0 The receiver’s decision about the transmitted sequence is the data sequence which maximizes the ML function [eq 4-841 when the trial parameters E, 0 are replaced by their estimates

The last equation shows that in general the ML symbol &a cannot be obtained by

a symbol-by-symbol decision but that the entire sequence must be considered due

419

Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing

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

420 Bit Error Rate Degradation Caused by Random Tracking Errors

Figure 7-l Conceptual Receiver Performing Maximum-Likelihood Decision

to intersymbol interference (nondiagonal H) The decision rule also applies when

in addition the data sequence is convolutionally encoded If the real-valued pulse g(t) is selected such that g(t) and g(t-mT) are orthogonal (Nyquist condition), then the matrix H becomes diagonal and a symbol-by-symbol decision is possible Using the normalization of Section 4.3.6 the matrix H is the identity matrix The receiver’s decision & about the transmitted symbol ak then is the data symbol which has the smallest Euclidean distance to the sample Zke-” at the input of

the decision device

Assuming perfect synchronization, the BER performance of the ML receiver is we11 documented in the literature for various modulation formats [l-5] However,

in the presence of synchronization errors, the BER performance deteriorates In the following we determine the BER degradation caused by random synchronization errors

for BER Degradation

In this section we derive an expression for the BER degradation caused

by synchronization errors, which is valid irrespective of the specific modulation format We first restrict our attention to either carrier phase errors in the presence of perfect timing recovery, or timing errors in the presence of perfect carrier recovery

At the end of this section, we consider the BER degradation when both carrier phase errors and timing errors are present

Let us introduce the notation $ for the synchronization error: $J = 4 = 00 - 4

in the case of a carrier phase error, whereas $ = e = EO - 2 in the case of a timing error The BER degradation D (detection loss), measured in decibels, is defined as the increase of E, /NO, required to maintain the same BER as the receiver without synchronization errors Hence, the BER degradation D at a bit error rate value BERo is given by

7.3 Derivation of an Approximate Expression for BER Degradation 421

where a2 and a$ are determined by

BERo = P(0; ~a) = &[P($; u)] (7-4)

In (74 w; 0 re resents the conditional BER, corresponding to a synchro- > P

nization error $, at E, /No = 1/ (2a2), and E+ [v] denotes averaging over the

synchronization error I/J Brute force numerical evaluation of the BER degradation

can be quite time consuming, because of the averaging over the synchronization

error $J, the computation of P(+; 0) for many values of $, and the iterations re-

quired to obtain u at a given value of 60 However, in most applications the BER

degradation caused by synchronization errors should not exceed about 0.2 dB; in

the following we will derive an approximate expression for the BER degradation,

which is very accurate for small degradations

For small synchronization errors, P(@; 0) can be

Taylor series expansion around $J = 0 This yields

approximated bY a truncated

where P(@)(.; ) and P(q$)(.; ) d eno e t single and double differentiation with

respect to $, and var[$] is the variance of the synchronization error $; the second

line of (7-5) assumes that $J is a zero-mean random variable, i.e Eq [+I = 0 For

small degradations, the second line of (7-5) can be approximated by a truncated

Taylor series expansion about u = 60 Keeping only linear terms in (u - UO) and

Hence, for small var[$], the BER degradation is well approximated by

which indicates that the (small) BER degradation, measured in decibels, is essen-

tially proportional to the tracking error variance var[$] , and independent of the

specific shape of the probability density of the synchronization error $

422 Bit Error Rate Degradation Caused by Random Tracking Errors

When both carrier phase errors and timing errors are present, a similar reasoning can be followed to obtain an approximate expression for the BER degradation Assuming that the carrier phase error and the timing error are uncorrelated, it turns out that the BER degradation, measured in decibels, equals the sum of the BER degradations caused by carrier phase errors and timing errors individually

7.4 M-PSK Signal Constellation

In the case of M-PSK, the data symbols ak take values from the alphabet

A = (exp(j2rm/M)lm = 0, , M - 1) The ML receiver from Figure 7-l decides tin: = exp (j27rm/M) when the argument of the sample zk c-j’ at the input of the decision device is in the interval ((2m- 1) r/M, (2m+ l)?r/M) The BER for M-PSK depends on how the transmitter maps blocks of log@ bits onto constellation points The minimum BER is achieved when blocks of log&f bits that are mapped onto constellation points being nearest neighbors in the Euclidean sense, differ by one bit only; this is called Gray mapping Figure 7-2 shows an example of Gray mapping for 8-PSK

In the case of Gray mapping, the conditional bit error rate P($; U) is well approximated by

the approximation being that each symbol error gives rise to one bit error only; this approximation is accurate at moderate and large E,/Nc, where the major part

: Makl

I 011

010 0 0 001

Figure 7-2 Gray Mapping for 8-P%

7.4 M-PSK Signal Constellation 423

of the erroneous decisions of the receiver corresponds to a detected symbol being

a nearest neighbor of the transmitted symbol For M-PSK, the conditional symbol error rate is given by

to h(O) = 1 As a result, the conditional symbol error rate is approximately given

bY

where E, [*I denotes averaging

contribute to s($), and where

over all data symbols a, (with n # k) that

The result (7-12) is exact for M = 2, and a close upper bound for M > 2

at moderate and large Es/No Assuming that the probability density of the synchronization error $ is an even function, it can be verified that dl($) and

d2($) have identical statistical properties In this case the average bit error rate

E+ Cp($; a)] is not affected when dz($) is replaced by dl($) in (7-12) Hence,

424 Bit Error Rate Degradation Caused by Random Tracking Errors

t

Im[zkl

Figure 7-3 Illustration of dr ($) and dz($)

as far as the evaluation of E+ [P($; a )] is concerned, the following expression can

be used for P($; a):

7.4 M-PSK Signal Constellation 425 one obtains

(7-21)

where

and the superscripts r,l~ and $I+!J denote single and double differentiation with respect

to the synchronization error $

Although the BER degradation formula (7-21) for M-PSK has been obtained

in a rather formal way, an interesting interpretation is given below

Considering complex numbers as phasors, the average BER for M-PSK equals f( M)/log,M times the probability that the projection of the sample Xk e-j’ on

zero-mean and Gaussian with a variance a2; the projection of the signal component s(+) of zge-jJ equals d(q), which for small $ can be expanded in a truncated Taylor series expansion:

d($)=d(O) + $J d(@)(O) + 1 +2 cw)(O)

The first term in (7-23) is the projection of the signal component in the absence

of synchronization errors The second term is a zero-mean disturbance, with a variance equal to B var[$] [see (7-22)-j; this second term adds to the projection

of the noise component wk of Zke-j’, yielding a total disturbance with variance equal to u2 + B var[$] The third term has a mean equal to -Ad(O) var[$]/2 (see [7-22)] which reduces the effect of the first term of (7-23); for small I/J, the fluctuation of this third term can be neglected as compared to the second term of (7-23) When a2 >> B var[$], the total disturbance is approximately Gaussian, in which case the average BER for M-PSK is given by

f(W Q 40)(1 - (A/2) va441) (7-24)

where the square of the argument of the function Q( ) in (7-24) equals the signal- to-noise ratio of the projection of be-j’ on j exp (-jr/M) Taking into account that

it follows that the BER degradation D, defined by (7-3) and (7-4), is obtained

by simply equating the arguments of the function Q(e) in (7-24) and (7-25) For small values of var[$] this yields

q2 N 1

426 Bit Error Rate Degradation Caused by Random Tracking Errors

so that the BER degradation in decibels is approximated by

D=

(7-27)

10

which is the same as (7-21) Hence, we conclude that the BER degradation caused

by synchronization errors is due to the combined effect of a reduction of the useful signal (this is accounted for by the quantity A) and an increase of the variance (this

is accounted for by the quantity B) at the input of the decision device The latter effect becomes more important with increasing E,/No, i.e., at smaller values of the ideal BER

In the following, we evaluate the quantities A and B from (7-22), in the cases

of carrier phase errors (1c) = 4) and timing errors ($ = e), respectively

7.4.1 Carrier Phase Errors

When the synchronization error 1c) equals the carrier phase error 4, and timing

is perfect, we obtain

44) = exP (j$>

44) = sin (4 + (+w)

= cos q5 sin (7r/M) + sin 4 cos (r/M)

Using the above in (7-22) yields

(7-29)

Note that B = 0 for M = 2

It follows from (7-28) that a carrier phase error affects the signal component

~(4) at the input of the decision device by a reduction of the signal component (cos 4 5 1) and the introduction of a zero-mean disturbance j sin 4 The useful component and the disturbance are along the real axis and imaginary axis, respec- tively The BER degradation is determined by d(4) from (7-29), which is the projection of s(4) on j exp (-jr/M)

7.5 M-PAM and WQAM Signal Constellations 427 Using the above in (7-22) yields

c (h’(mT)q2 M= 2

m where h’(x) and h” (x) denote the first and second derivative of h(z), respectively For M = 2, the value of B is twice as large as for M > 2, because E [a;] = 1 for M = 2, but E [ui] = 0 for M > 2

Equation (7-31) shows that a timing error yields a reduction of the useful com- ponent (h( eT) 5 1) at the input of the decision device, and introduces intersymbol interference (ISI), which acts as an additional disturbance

In the case of M-PAM, the data symbols ak are real-valued and are denoted as ck =cR,k+jO The symbols c&k take values from the alphabet

EL 1

f (M - l)A}, where the value of A is selected such that

‘i,k = 1 Taking into account that

428 Bit Error Rate Degradation Caused by Random Tracking Errors

In the case of M2-QAM, the data symbols are complex-valued and are denoted as ak=aR,k+jaI,k ; a&k and aI,& are the in-phase symbols and quadrature symbols, respectively The symbols a&k and al,l, are statistically independent, and both take values from the M-PAM alphabet; this yields a square M2-QAM

COnStellatiOn for the Complex symbols ak

is the same as for the in-phase symbols a,, but with Re zk e-j’ [ 1 replaced by

Im %ke-je’ In the following we will assume that blocks of 2 log2M bits that are

mapped onto constellation points being nearest neighbors, differ by one bit only (Gray mapping); Figure 7-5 illustrates the Gray mapping for 16-QAM Under this assumption, the in-phase decisions tiR,k and quadrature decisions til,k yield the

same bit error rate, which

system Hence, we have

decision &~,k

7.5 M-PAM and Mz-QAM Signal Constellations 429 equals the bit error rate of the M2-QAM transmission

to consider only the bit error rate resulting from the

Taking the Gray mapping into account, the conditional bit error rate for both M-PAM and M2-QAM is approximately given by

denotes the symbol error rate, conditioned on the transmitted in-phase sym- bol and on the synchronization error Equation (7-37) takes into account that

itive in-phase symbols

Let us denote by s( $J; 2m- 1) the signal component of the sample zke-je^ at the input of the decision device, corresponding to a synchronization error TJ and a transmitted in-phase symbol a~,$ = (2m- l)A; the additive noise component u&

of ~ke-j’ is Gaussian, with zero-mean independent real and imaging parts, each having a variance tr2 = No/(2E,) Hence,

430 Bit Error Rate Degradation Caused by Random Tracking Errors

d(\y; 2m-1) 2A - d(\Cr; 2m-1)

Figure 7-6 Illustration of cZ($; 2m- 1)

In (7-39) and (7-40), &Je] denotes averaging over all in-phase and quadrature symbols that contribute to s( $; 2m- 1), with the exception of the in-phase symbol a~+, = (2m-1)A Note that Q(0; 2m-l)= A, because

~(0; 2m-1) = (2 m- l)A The bit error rate in the case of perfect synchroniza- tion (i.e., 1c) = 0) is obtained from (7-37), (7-39), and (7-40) as

P(O;a) = & 2(Mi1) Q( > $

Evaluating the BER degradation in decibels, given by (7-Q, one obtains

M-l (d(“)(O; M-l))‘]+FEIEa[ m=l (d(‘)(O;2m-1))2]

and the superscripts V/J and J,!J$ denote single and double differentiation with respect

to the synchronization error 4 Using a similar reasoning as for M-PSK, it can be verified that the quantities A and B reflect the reduction of the useful component

and the increase of the variance of Re zk e-j’ [ 1 , respectively