Koudelka'. ' Institute of Communication Networks and Satellite Communications. Graz University ofTechnology, 8010 Graz, Austria, E-mail: {gappmair, koudelka} ...
Symbol-Timing Recovery Modified Gardner
with
Detectors
W. Gappmairl, S. Cioni2, G. E. Corazza2, 0. Koudelka' Institute of Communication Networks and Satellite Communications Graz University of Technology, 8010 Graz, Austria, E-mail: {gappmair, koudelka} inw.tugraz.at 2 Department of Electronics, Computer Science and Systems University of Bologna, 40136 Bologna, Italy, E-mail: {scioni, gecorazza}@deis.unibo.it '
Abstract-For symbol timing recovery in digital receivers, a modified form of the Gardner detector is introduced. If applied to tracking loops for M-PSK modulated signals shaped as raised cosines, it is shown that the suggested synchronizer achieves a definitely better jitter performance (Gaussian and self-noise domain) than the original proposed by Gardner. In particular, this turns out to be true for baseband shapes with small roll-off factors as it is required for bandwidth-efficient systems.
II.
EQUIVALENT BASEBAND MODEL Let the independent and identically distributed M-ary symbols Ck = ak +jbk be normalized to the expectation E[ICkl2] = 1. Further, let the complex zero-mean white Gaussian noise be denoted by w(t) with independent real and imaginary parts, each of power spectral density 1/(2X), where X = E,/N0 is defined as the mean signal-to-noise ratio (SNR) per symbol. The unit-energy baseband pulse h(t) is assumed to be a root-raised cosine with roll-off c 0 < a< 1. Therefore, if affected by time shift ;I1r < T/2, and carrier phase offset Se [-x r), the received signal can be written as
Keywords-Digital receiver, non-data-aided symbol-timing recovery, Gardner detector.
I.
INTRODUCTION
Powerful feedforward estimators for non-data-aided symbol-timing recovery are well-known from the open literature [1], [2]. They are required for time-division multiple access (TDMA) systems, where efficient acquisition procedures are mandatory. Feedback solutions, on the other hand, are often much less complex from the computational point of view. The latter represent an attractive alternative for continuous transmission of data, where rapid acquisition is not that important as with TDMA, e. g., DVB receivers running at rather low signalto-noise ratios as it is typical for powerful coding techniques
r(t) = ej'
I cih(t - iT - r) + w(t)
(1)
Usually, t is normalized to the symbol period T and as such denoted by £. In the following, r(t) passes the matched receiver filter h*(-t) as shown in Fig. 1. The corresponding output signal x(t) is sampled appropriately for subsequent processing, i. e.,
[3], [4].
Xk = r(t)
Symbol-timing recovery via feedback loops, based on the maximum-likelihood principle and as such related to the first derivative of the matched receiver filter, is in many cases also too complex to be implemented in practice. The problem is usually circumvented by simple ad-hoc algorithms like decision-directed zero-crossing (ZC) or non-data-aided Gardner (GA) synchronizers as most prominent examples. Subject of this paper is a modified form of the GA detector. If applied to tracking loops for M-PSK modulated signals shaped as raised cosines, part of the gap between the GA jitter variance and the Cramer-Rao lower bound as the theoretical limit [5], [6] can be closed. To this end, the equivalent baseband model is introduced in Section 2. In Section 3, the detector characteristic (S-curve) is presented and verified by simulation results. In the sequel, the slope of the S-curve in the stable equilibrium point is used to evaluate in Section 4 the jitter variance (Gaussian and self-noise domain) of the linearized recovery loop. Conclusions are found in Section 5.
*((-t)|
h
=
ejsk
+
(2)
where ® symbolizes the convolutional operation and nk is the Gaussian noise sample at the matched filter output. With g(t) = h(t) ® h*(-t), the signal samples Sk are given by sk
=Zcjg[(k-i-e)T] i
(3)
Gardner [7] suggested a detector generating an error signal Uk = Re[(xkl -Xk) x k-1/2]. In the original article, the algorithm is analyzed for binary as well as quatemary PSK signals, but the relationship applies to all linear modulation schemes in the same manner [1], [8]. Nevertheless, the Gardner detector may be modified suitably if xk is expressed in generalized polar form, i. e., Xk -+ pei g(xL), Pk = [kI, such that Uk develops as j I) _pa ejarg(xk))x;] uk = Re[(p Ijarg(x
0-7803-9206-X/05/$20.00 ©2005 IEEE
831
(4)
which describes the modified Gardner (mGA) detector, henceforth. Note that x k-1/2 is not subject of the modification introduced previously. Note also that, with u = 1, the error signal converges to its initial shape.
0.4
Loop filter Timing errordetector
NCO
I) F(:)
-4
GA/rnGA -0.4
Matched filter
r(t)
1. S
n r
r
hf
\V_
*
c lx -0.4
-0.2
0.2
0
E=
Fig. 1. Symbol-timing recovery with feedback loops
mGA(A-=-I
GA.mGA(,u=I) 0.4
rIT
Fig. 2. S-curves for GA and mGA detectors: 4-PSK, a= 0.1, = 7 dB ,
DETECTOR CHARACTERISTIC In order to study the recovery process, the error detector characteristic (S-curve) must be derived. The latter is defined as the expected open-loop behavior of the error signal, i. e., III.
q(E) = E[Uk
Vk: E = const, e I
-3. A detailed inspection (6) reveals also that the modified GA detector exhibits no bias, i. e., iX() = 0, which can be observed in Fig. 2 as well, where the S-curve is checked by simulation results for 4-PSK, a= 0.1, X = 7 dB and ,u E {-l, 0, I}. For comparison purposes, the characteristic of the Gardner detector [7] is included. Also verified by simulation results, Fig. 3 shows the evolution of the S-curves for 16-QAM. Compared to Fig. 2, it can be seen that the GA performance is the same in both cases, but the amplitude of the S-curves decreases for ,u E {-1, 0} such that the related slopes are smaller at e= 0.
832
the mGA detector does not work satisfactorily if applied to modulation schemes with non-constant envelope as exemplified in Figs. 4 and 5 for 16-QAM and 16-APSK, where the performance degrades due to self noise, in particular.
ft A ft
0.001 0)
0.
0001
O>
CAl
-. A
106
S.__
0
z
10
15
20
A
25
a
30
0.01
A
_a i
*A
A
A
' °
z.
35
a=
* A O A
GA: 8-PSK GA: 16-APSK mGA: 8-PSK mGA: 16-APSK
A
A
A
A
A
A
A
*
t *
*
*
*
U
v
5
10
15
20
25
30
Fig. 5. Jitter perform. for GA and mGA (,u= 0) trackers:
A
A
A
A
A
f
6
A
A
A
-1
A
a
A
-0.5
0
0.5
1
1.5
35
a=
40
0.1, BLT= 10-3
[2 1: g[(m - Y2)T]}2, i=0O g[(m+ )T]g[(m Y2)T], i ±1 M;--gti+ ,)]}2, i2±1 -
=
R() + 2E (1- 4iBLT)R(i)I
2
i21
(10)
a. For M-PSK schemes operated at a= 0.1, it is obvious that the mGA (A = 0) performance is better by about two orders of magnitude if compared to the GA tracker. Fig. 7 confirms also that mGA loops are less attractive for M-QAM signals since they exhibit much more self noise as the corresponding GA algorithm.
,
- [g(T)]2
A
°
as it is the case with (9), the relationship may be simplified accordingly. For different modulation schemes and BLT = 10-3, Fig. 7 shows the evolution of the self-noise variance, checked by simulation results, as a function of the roll-off
Fig. 6 illustrates the evolution of the mGA variance versus , for 4-PSK, a= 0. 1, BLT= 10-3, and different values of r. For X as low as 5 dB, a more or less pronounced minimum around ,u = 0 is identified. With X increasing, the variance begins to flatten towards smaller values of u such that p = 0 seems to be a good choice not at least from the computational point of view. Only if X >> 1, where self-noise effects emerge, negative values ofp might be useful. Because of intersymbol interference (ISI), the jitter variance exhibits a floor with increasing values of X, (see Figs. 4 and 5). For analysis, the ISI-dominated branch requires the auto-covariance of uk to be evaluated, i. e., R(i) = E[uk Uk+i]. Due to clarity and limited space, the derivation of R(i) is not provided for the general case. With M-PSK, however, the relationship simplifies to [1 12]
A -
A
*0
°
If Xi R(i) = 0,
Es/No [dB]
R(i) =A
o
A
s
-
MCRLB
o
0.1, BLT= 10-3
z o
ol
o
U*****
With K. as the slope in the asymptotic case, the ISI or selfnoise variance appears for first-order loops as [2]
40
0.00001
10
o
=2OdB 40 dB
Design parameter ,u
0001
._,n
o
A
0.001
1.
10
y=
Fig. 6. Jitter performance for mGA trackers: 4-PSK, a= 0.1, BLT= 10-3
Fig. 4. Jitter perform. for GA and mGA (,u= 0) trackers:
9
1.'
A
Oo
-1.5
E/lNo [dB]
0.
0.00001
A
-
MCRLB 5
08
0.0001
. >o AX AZ X f f l f~~~~~~~~~~0 !0o!
z
08
0.001
=IOdB
O A * A
*
A
0.00001
1.
et
GA: 2-PSK GA: 4-PSK GA: 16-QAM mGA: 2-PSK mGA: 4-PSK mGA: 16-QAM
A * 5 A
0.01
Ce
X=5dB
*
0.01
(9)
833
0)
08
EU
1.
EL GA16QM ~~~GA: 2-PSK
lo-,
GA: 16-QAM o Z
A O
mGA: 2-PSK mGA: 4-PSK mGA: 16-QAM 0.2
0.4
0.6
0.8
1
Roll-off a Fig. 7. Self-noise performance for GA and mGA (,u= 0) trackers: BLT= 10-3
CONCLUSIONS For symbol-timing recovery, a modified form of the Gardner detector has been studied. With regard to raised-cosines for baseband shaping and M-PSK as modulation scheme, it could be shown that the suggested synchronizer performs definitely better than the original proposed by Gardner. In particular, this tums out to be true if applied to small roll-off factors. The recovery of the symbol timing has been verified against frequency errors AlT, too. No significant degradation of the jitter variance for both GA and mGA algorithms is obV.
served as long as IA!I < 10-2. But if AJ7]> 10-2, mGA synchronizers seem to be less sensitive, especially for smaller values of a.. In order to minimize the jitter variance, it is suggested that the design parametery = 0 with an additional benefit from the computational point of view; only for very large SNR values, where self-noise effects dominate, smaller values of ,u might be helpful. Unfortunately, because of increased self-noise jitter, mGA detectors are less attractive for signals with non-constant envelope.
Signal-to-Noise Ratios", IEEE J. Select. Areas Commun., vol. 19, pp. 2320 2330, Dec. 2001. [4] E. Casini, R. De Gaudenzi, and A. Ginesi, A. "DVB-S2 Modem Algorithms Design and Performance over Typical Satellite Channels", Int. J. Satell. Commun. Network., vol. 22, pp. 281 318, May-June 2004. [5] M. Moeneclaey, "On the True and the Modified Cramer-Rao Bounds for the Estimation of a Scalar Parameter in the Presence of Nuisance Parameters", IEEE Trans. Commun., vol. 46, pp. 1536 1544, Nov. 1998. [6] H. Steendam and M. Moeneclaey, "Low-SNR Limit of the Cramer-Rao Bound for Estimating the Time Delay of a PSK, QAM, or PAM Waveform", IEEE Commun. Lett., vol. 5, pp. 31 33, Jan. 2001. [7] F. M. Gardner, "A BPSK/QPSK Timing-Error Detector for Sampled Receivers", IEEE Trans. Commun., vol. 34, pp. 423 429, Oct. 1986. [8] R. De Gaudenzi and M. Luise, "Analysis and Design of an All-Digital Demodulator for Trellis Coded 1 6-QAM Transmission over a Nonlinear Satellite Channel", IEEE Trans. Commun., vol. 43, pp. 659 667, Feb. 1995. [9] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover Publications, 1970. [10] S. Wolfram, Mathematica: A System for Doing Mathematics by Computer. Reading, MA: Addison-Wesley, 1997. [11] W. G. Cowley and L. P. Sabel, "The Performance of Two Symbol Timing Recovery Algorithms for PSK Demodulators", IEEE Trans. Commun., vol. 42, pp. 2345 2355, June 1994. [12] W. Gappmair, "Self-Noise Performance of Zero-Crossing and Gardner Synchronisers Applied to One/Two-Dimensional Modulation Schemes", Electron. Lett., vol. 40, pp. 1010 1011, Aug. 2004.
ACKNOWLEDGMENT Part of the work has been done in SatNEx (Satellite Communications Network of Excellence, IST NoE No. 507052) launched by the European Commission for advanced research in satellite communications within the Sixth Framework Programme. REFERENCES [1] [2]
[3]
U. Mengali and A. N. D'Andrea, Synchronization Techniques for Digital Receivers. New York: Plenum Press, 1997. H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing. New York: Wiley, 1998. A. A. D'Amico, A. N. D'Andrea, and R. Reggiannini, "Efficient NonData-Aided Carrier and Clock Recovery for Satellite DVB at Very Low
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