Digital communication receivers P6 - P2

356 Performance Analysis of Synchronizers The feedback version of the NDA symbol synchronizer uses a timing error detector, whose output xc Ic; 2 is given by [eq.. 6.3 Tracking Performan

6.3 Tracking Performance of Carrier and Symbol Synchronizers 355

The M&M synchronizer needs only one synchronized sample per symbol, taken

at the estimated decision instant

Let us investigate the tracking performance of the M&M synchronizer As- suming that tik = al, for all !Z and 6 = 00, it follows from (6-68) that

=Re c ((&arc-pn - a;ak-I-)h(mT-eT) + (azBINk - aiNk-1) 1 (6- 143) n-6

where Nk is defined by (6-103) This yields the following timing error detector characteristic:

g,(e) = E[zE(k;2)] = h(T - eT) - h(-T - eT) (6-144)

As h(t) is an even function of time, it follows that g (0) = 0, so that e = 0 is a stable equilibrium point The resulting timing error detector slope at e = 0 is

The loop noise NC (k; ~0) at e = 0 is given by

Note that N, (Ic; ~0) contains no self-noise; this is due to the fact that the matched filter output pulse h(t) is a Nyquist-I pulse, i.e., h(mT) = 6, As both (ak) and (Nk} are sequences of statistically independent random variables, it follows that the loop noise N,(lc; ea) is white The loop noise power spectral density

SE (exp (jwT); 0) equals

$(exp (jwT); 0) = $

The corresponding tracking error performance is given by

Numerical performance results will be presented in Section 6.3.7

6.3.6 Non-Decision-Aided Symbol Synchronizer

The feedforward version of the non-decision-aided maximum-likelihood (NDA-ML) symbol synchronizer maximizes over E the function L(E), given

by (Section 5.6.2)

L(e) = c Iz(kT + ET)I~

356 Performance Analysis of Synchronizers

The feedback version of the NDA symbol synchronizer uses a timing error detector, whose output xc (Ic; 2) is given by [eq (6-106)]

x,(k; 2) = T Re[z*(lcT + E”T)z’(H’+ U)] (6-150) where z’(t) is the derivative (with respect to time) of z(t) The timing error detector output xC (L; i) is proportional to the derivative, with respect to E, of the lath term in (6-149) The feedback NDA symbol synchronizer needs two synchronized samples per symbol, taken at the estimated decision instants: one at the matched filter output and one at the derivative matched filter output

In the following, we will investigate the tracking performance of the feed- back synchronizer; from the obtained result, the performance of the feedforward synchronizer can be derived

Taking (6-68) into account, (6-150) yields

=T Re

I(

c a$+ h(mT-eT) + A$

)(

c ulc-na h’(mT-eT) + Ni (6-151)

where

Nk = N(kT + dT) exp (-joa) (6- 152)

Ni = N’( IcT + ZT) exp (-j&) (6-153) and h’(t) is the derivative of h(t) with respect to time For later use, we derive from (6-152) and (6-153) the following correlations:

NO E[N*, Nk+,] = -&,n

a

(6-154)

(6- 155)

a

(6- 156) where h”(t) is the second derivative of h(t) with respect to time, The timing error detector characteristic g,(e) is given by

g,(e) = E[x,(k; t)] = C h(mT-eT) h’(mT-eT) T

tn

(6-157)

which can be shown to become sinusoidal in e with period 1 when H(w) = 0 for Iw] > 27r/T, i.e., a < 1 excess bandwidth As h(t) and h’(t) are even and odd functions of time, respectively, it follows that g, (0) = 0, so that e = 0 is a stable

4.3 Tracking Performance of Carrier and Symbol Synchronizers 357 equilibrium point The resulting timing error detector slope at e = 0 is

Kc = (-h”(O) T2) - c (h’(mT) T)2

m

The loop noise NE (k; ~0) at e = 0 consists of three uncorrelated terms:

Ndk Ed = %N~N@; EO) + Nc,wv(k; eo) + Nc,sxs(k; go) (6 159) where

N r,~x~(k;~~) = T Re[N,” NL] (6-160)

N ~,sxN(~;Eo) = T R e ai Ni + c akwrn h’(mT)N,* 1 (6-161)

m

N ~,sxs(~ EO) = T Re a; C ak-m h’(mT) 1 (6-162)

m

and the subscripts N x N, S x N, and S x S refer to noise x noise term, signal

x noise term, and signal x signal term, respectively

The autocorrelation function R, ,N~N(IC; ~0) of the N x N contribution to the loop noise is given by

R ~,N~N(O) = f (-h”(O) T)2 [$I’ (6- 163)

s

R,,wv(nT) = -f (h’(nT) T)2 [2] 2 (6-164) The corresponding loop noise power spectral density SE,NxN(exp (jwT); 0) is

S E,NXA+XP &T); 0)

(-h”(O) T2) -c m (h’(mT) T)2 cos (muT) 1

(6-165)

Note that S,,NxN(exp (jwT); 0) is not flat However, as &,~~~(exp (jwT); 0)

is nonzero at w = 0, it can be approximated by its value at w = 0, provided that the loop bandwidth is so small that the variation of Sr,~x~(exp (jwT); 0) within the loop bandwidth can be neglected This yields

(-h”(O) T2) -‘x (h’(mT) T)2 [ 1 No 2

V+N,N] = (2&T)

2E, (6- 166)

m

where eNxN denotes the timing error caused by the N x N contribution to the

358 Performance Analysis of Synchronizers

loop noise The performance of the feedforward version is obtained by replacing 2BLT by l/K

The autocorrelation function RE,,sx~( Ic; eo) of the S x N contribution to the loop noise is given by

R E,SXN(O) = f (-h”(O) T2) + c (h’(mT) T)2 $ 1 (6-167)

s The corresponding loop noise power spectral density SC ,SxN (exp (jwT); 0) is Ss,sxN(exp (GT); 0)

=f $! (-h”(O) T2) + f $? ~(h’(rr~T)T)~ (l-2cos(mwT)) (6-169)

Approximating S&N(exp (jwT); 0) by its value at w = 0, we obtain

(-h”(O) T2) - c (h’(mT) T)2 2E,

where eSxN denotes the timing error caused by the S x N contribution to the loop noise The performance of the feedforward version is obtained by replacing 2BLT by l/K

The self-noise term NC ,sx s (Ic; ~0) from (6- 162) is the same as the self-noise term NE,s (Ic; ~0) given by (6-134) Hence, denoting by esx s the timing error caused by the S x S contribution to the loop noise, var[esxs] is given by the right-hand side of (6-140), with K, given by (6-158) Notice that (6-140) has been derived for complex-valued data symbols with E [ui] = 0; for real-valued data symbols the result from (6-140) should be multiplied by 2

The total timing error variance equals the sum of the N x N, S x N and S x S contributions, given by (6-166), (6-170) and (6-140) with Kc given by (6-158) Numerical performance results will be presented in Section 6.3.7

Other NDA symbol synchronizers, such as the Gardner synchronizer and the digital filter and square synchronizer, are briefly discussed in Section 6.3.10

6.3.7 Tracking Performance Comparison

Carrier Synchronizers

For moderate and large E, /NO, the linearized tracking error variances (6-108) and (6- 122) resulting from the DD and the NDA carrier synchronizers, respectively,

6.3 Tracking Performance of Carrier and Symbol Synchronizers 359 are well approximated by

var[+] M (ZBLT) A

The approximation involves neglecting the effect of decision errors (for the DD synchronizer) and of higher-order noise terms (for the NDA synchronizer) For the

DD synchronizer, A = 1 and B = 0, in which case (6-171) equals the Cramer-Rao bound For the NDA synchronizer, A and B are given by (6-123) and (6124), respectively

Figure 6-l shows the linearized tracking performance of the DD carrier synchronizer for QPSK, taking decision errors into account, and compares this result with the Cramer-Rao bound (CRB)

Figure 6-l Linearized Tracking Performance of DD-ML

Carrier Synchronizer for QPSK

360 Performance Analysis of Synchronizers

0.8

-0.8 [ - - - - Es/NO=OdB - - - Es/NO=lOdB - EslNO=POdB - Es/NO=inflnity

Figure 6-2 DD Phase Error Detector Characteristic

Taking Decision Errors into Account

The difference between both curves is caused only by the additive noise affecting the receiver’s decisions Figure 6-2 shows the phase error detector characteristic go($) for QPSK, with the effect of decision errors included

For small E, /No, go (4) becomes essentially sinusoidal; this agrees with the observations made in Section 3.2.2 of Volume 1 Also, we notice that ge(4) for QPSK is periodic in 4 with period 7r/2 rather than 27r Indeed, when the signal constellation has a symmetry angle of 27r/M (for QPSK, we have M = 4), the statistics of the received signal do not change when the data symbols al, are replaced by al, exp (j2n/M) and the carrier phase 60 is replaced by 80 - 2n/M Hence, the statistics of any phase error detector output are periodic in 4 with period 27r/M, when for each valid data sequence {ok} the sequence {uk exp (j27r/M)}

6.3 Tracking Performance of Carrier and Symbol Synchronizers 361

also represents a valid data sequence We observe from Figure 6-2 that the decision errors reduce the phase error detector slope at 4 = 0, which in turn increases the tracking error variance Denoting this slope corresponding to a given E,/NIJ by KQ(E~/N~), Figure 6-3 shows the ratio Ke(E,/lV~)/Ke(oo)

Let us consider the NDA carrier synchronizer for two signal constellations

of practical interest, i.e., the M-PSK constellation and the square N2-QAM constellation These constellations have symmetry angles of 2n/M and n/2, respectively, so the corresponding synchronizers use the Mth power (for M-PSK) and 4-th power (for QAM) nonlinearities, respectively For M-PSK, (6-123) and (6-124) yield A = 1 and B = 0, so that (6- 17 1) reduces to the CRB This indicates that, for large E, /No, the NDA synchronizer yields optimum tracking performance

0.9

0.8

0.7

0.8

0.5

0.4

0.3

0.2

0.1

Figure 6-3 Phase Error Detector Slope

362 Performance Analysis of Synchronizers

lo4

102

100

1 o-2

1o-4

- ‘\ ‘

‘\

‘.,

‘\

E Z-PSK

B 4-PSK

B 8-PSK

-

1""

2b

3b,

Es/No [dB]

Figure 6-4 Linearized Tracking Performance of NDA

Carrier Synchronizer for M-PSK

in the case of M-PSK constellations Figure 6-4 shows the tracking performance for M-PSK, taking into account all terms from (6120), and compares it with the CRB

The degradation with respect to the CRB is caused by the terms with m 2 2

in (6-120) It increases with increasing size of the constellation and decreasing Es/No The tracking performance for N2-QAM, taking into account all terms from (6-120), is shown in Figure 6-5

The performance for 4-QAM is the same as for 4-P%, because the constel- lations are identical For N2 > 4, the tracking performance is considerably worse than the CRB [basically because A > 1 and B > 0 in (6-171)], and worsens with increasing constellation size Note that a limiting performance exists for N2 -+ 00 Symbol Synchronizers

The tracking performance of symbol synchronizers depends on the shape of the received baseband pulse g(t) In obtaining numerical results, it will be assumed that g(t) is such that the pulse h(t) at the output of the matched filter is a raised

6.3 Tracking Performance of Carrier and Symbol Synchronizers 363

vd W2B $)I

lo2

100

1 o-2

10-4

1

B 16-QAM

B N 2->infinity

.,.- w _ _ _ _ _ -._.- - -

-

-?:

q

1""

lb

2b

Es/No [dB]

Figure 6-5 Linearized Tracking Performance of NDA-ML

Carrier Synchronizer for N2-QAM

cosine pulse, i.e.,

h(t) = sin (7rt/T) cos (curt/T)

where a E (0,l) represents the rolloff factor

For moderate and large E, /No, the timing error variance is well approximated

bY

var[e] ~tl (2ll~T)A(a)~ + KF(~BLT)~B(~) (6-173)

s The first term in (6-173) is proportional to the synchronizer bandwidth and repre- sents an approximation of the contribution from additive noise; the approximation involves neglecting the effect of decision errors (DD-ML and M&M synchroniz- ers) or of the noise x noise contribution (NDA synchronizer) The second term in (6-173) is proportional to the square of the synchronizer bandwidth and represents the self-noise contribution; the quantity KF depends on the closed-loop transfer

364 Performance Analysis of Synchronizers

1

rolloff

Figure 6-6 Quantity A(a) for the DD-ML, M&M, and NDA Synchronizers

function The factors A(a) and B( ) CY incorporate the effect of the shape of the matched filter output pulse g(t)

Figure 6-6 shows the quantity A(Q) for the DD-ML, the M&M and the NDA symbol synchronizers

10

1

0.1

0.01

0.001

l

6.3 Tracking Performance of Carrier and Symbol Synchronizers 365

rolloff

Figure 6-7 Quantity B(a) for the DD-ML and NDA-ML Synchronizers

The DD-ML synchronizer yields the smallest Ala), which corresponds to the CBR The M&M and NDA synchronizers behave poorly for large and small values of

cy, respectively The quantity B(a) is shown in Figure 6-7 for the DD-ML and the NDA symbol synchronizers; for the M&M synchronizer, self-noise is absent, i.e., B(a) = 0

366 Performance Analysis of Synchronizers

lE-1

T

variance

- - - DD M&M

- NDA

- CRB

Es/NO [de]

Figure 6-8 Tracking Error Variance for DD-ML, M&M, and NDA Synchronizers

The NDA synchronizer yields more self-noise than the DD-ML synchronizer, especially for small values of a; for both synchronizers the self-noise decreases with increasing rolloff, due to the faster decay of the baseband pulse h(t) at the matched filter output

The approximation (6-173) of the timing error variance ignores the effect of decision errors and of the noise x noise contribution, so that (6-173) underestimates the tracking error variance at low E,/Nc Figure 6-8 shows the tracking error variance when these effects are taken into account, along with the CBR; it is assumed that a = 0.2, 2B~2’ = 10w2, and KF = 2

We observe that the timing error variance of the DD-ML synchronizer exceeds