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Detector characteristic for decision-directedcarrier phase recovery of 16=32-APSKsignals

W. Gappmair and J. Holzleitner

Usually implemented as second-order loops, decision-directed (DD)

feedback devices are frequently applied for carrier phase control.

Using 16-ary and 32-ary amplitude phase-shift keying (APSK), recom-

mended as modulation schemes in the new digital video broadcasting

standardfor satellite communications (DVB-S2), it is shown that the DD

detector characteristic (S-curve) can be derived in closed form. In the

sequel, theslope ofthe S-curve atthe stableequilibriumpoint,as it would

be required to design the linearised recovery loop, is computed.

Introduction: In the new digital video broadcasting (DVB) standard for

satellite communications[1], frequently denoted by DVB-S2, 16-ary and

32-ary amplitude phase-shift keying (APSK) are specified as modulation

techniques. As for any other linear scheme, decision-directed (DD)

tracking devices might be implemented in order to follow the carrier

phase [2], usually established as second-order loops. During normal

operation, the deviations from the stable equilibrium point are sufficiently

small such that the loop can be described appropriately by a linearised

model. In this context, the knowledge of the detector characteristic

(S-curve) is of particular interest, which will be subsequently developed

in closed form for 16=32-APSK signals.

Equivalent baseband model: Throughout this Letter, it is assumed

that perfect symbol timing has been achieved. Furthermore, let the

independent and identically distributed 16=32-APSK symbols ck be

normalised to average unit energy E[jckj2] 1, i.e. with cks arrangedon two=three rings, as shown in Fig. 1; the radii R1, R2 and R3 are

established as soon as the ratios b1: R2=R1 and b2: R3=R1 aregiven. In the sequel, the receiver samples at the output of the matched

filter are provided by

rkejykck nk 1

where yk denotes the carrier phase. Real and imaginary parts of the

zero-mean AWGN samples nk are assumed to be independent, each

with the same variance of 1=2gs, where gs: Es=N0 is the meansignal-to-noise ratio (SNR) per symbol.

Fig. 1 Symbol constellation for 16=32-APSK schemes

The recovery scheme to be investigated in detail is shown in Fig. 2.As already mentioned in the Introduction, it is usually designed as a

second-order loop, specified through noise bandwidth and damping

factor. Whenckdenotes the decision on ckandykthe estimate ofyk, the

DD detector delivers an error signal

uk Imc*krkejyk 2

Fig. 2 Feedback recovery of carrier phase with DD detectors

Detector characteris tic: The detector characteristic or S-curve is

defined as the expected value of uk for the open loop, i.e. S(gs,

y): E[ukjyk y, yk 0]. With ck: akjbk and the quadraturesymmetry of 16=32-APSK constellations taken into account, it can

be shown [3] that lc(gs, y): E[akak] E[bkbk] and ls(gs, y): E[bkak] E[akbk]. Therefore, after some lengthy but straightfor-ward manipulations, the S-curve develops as

Sgs; y 2gcgs; y sin y lsgs; y cos y 3

What remains is the evaluation of the symbol statistics lc() andls().To this end, it is assumed in the following without loss of generality that

the APSK constellation is rotated by p=4, which simplifies the analy-

tical work considerably.

First, 16-APSK is to be investigated in detail. By inspection ofFig. 1,it is clear that this consists of a 4-PSK and a 12-PSK ensemble,

separated by R12. Motivated by the fact that, forM-PSK schemes, the

S-curve is available in closed form[4], the developed relationships can

be adjusted appropriately. As a result, the partial symbol statistics of

lc(), i.e. lc() conditioned on bothjckj Rm andjckj Rn, where m,n 2 {1, 2}, appears as

lc;mngs; y 2RmRn

M

PMm=21i0

PMn1l0

cos 2pi

Mm

cos

2pl

Mn

Pil;mngs; y 4

where M 16, M1 4, M2 12 and

Pil;mn

gs

; y

jil;mn;2y

jil;mn;1yp

FR

m

;R12

; gs

;j

dj

5

Upper and lower limits are given by

jil;mn;1y l2p

Mn p

Mn i2p

Mn y

pp

jil;mn;2y l2p

Mn p

Mn i2p

Mn y

pp

6

where [j]pp means thatj has to be wrapped aroundp. Using the

joint probability function for complex Gaussian noise, suitably estab-

lished in polar co-ordinates [5, eqns. 4.2.100102] and denoted by

pR,F(S,gs,r,j), the angular density in (5) is, after some algebra applied

to [6, eqns. 3.322=12], immediately provided as

pFs;R; gs;j R

0

pR;Fs; gs; r;j dres

2gs

2p 1 egsR22sR cosjn

ffiffiffiffiffiffiffiffiffiffiffips2gs

p cos jes

2gscos2 j erfc ffiffiffiffigsp s cos j R

erfcffiffiffiffiffiffiffiffis2gs

p cos j

o 7

where erfc(x) : (2=pp)x1et2dtsymbolises the complementary errorfunction. Note that (7) is only valid forn 1, whereas forn 2 it has to

be replaced by

pGs;R; gs;j 1

R

pR;Fs; gs;r;jdr es

2gs

2p

egsR

22sR cosj

ffiffiffiffiffiffiffiffiffiffiffips2gs

p cosjes

2gscos2 jerfc ffiffiffiffigsp R s cosj

8

Following the procedure in [4], the components of ls() are simplyexpressed through

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ls;mngs;y 2RmRn

M

PMm=41iMm=4

PMm1l0

sin 2pi

Mm

cos

2pl

Mn

Pil;mngs;y 9

Finally, the symbol statistics develop as

lcgs; y Pm;n

lc;mngs; ylsgs; y Pm;n

ls;mngs; y 10

Although not shown due to limited space, (3) has been verified by

extensive simulation work, confirming also that the detector is unbiased,

i.e.S(gs, 0) 0 irrespective of the selected value ofgs. For the linearisedtracker model, however, knowledge of the full S-curve is of lessinterestin contrast to the slope in the stable equilibrium point, defined

as the first derivative of (3) with respect to y evaluated aty 0, i.e.

Kd : @

@ySyjy02lcgs; 0 l0sgs; 0 11

where ls0(gs, 0) : (@=@y)ls(gs,y)jy0. It is easily checked that the

probabilities Pil,mn(gs, y) are the only terms in ls( ) that are a functionofy. Hence, the first derivative of (5) with respect to y appears as

P0il;mngs; y pFRm;R12; gs;jil;mn;1y pFRm;R12; gs;jmn;2y;

n1pGRm;R12; gs;jil;mn;1y pGRm;R12; gs;jil;mn;2y;

n2

8>:

12Forb1 2.5 andR12 (R1R2)=2,Fig. 3 exemplifies the evolution of(11) as a function of gsEs=N0. As expected, Kd! 0 for gs ! 0,whereasKd! 1 forgs!1. However, the computational complexity isconsiderable. Fortunately, at larger SNR values, it turns out that the

contribution oflc,mn() andls,mn(),m 6 n, is negligible compared to thecase with m n. In addition, (7) and (8) can be approximated byp

(p1s2gs)es2 gsj

2

such that (5) reduces to

Pil;mnls; y 1

2 erfc

ffiffiffiffiffiffiffiffiffiffiR2mgs

p jil;mn;1

erfc

ffiffiffiffiffiffiffiffiffiffiR2mgs

p jil;mn;2

h i 13

Computing (11) with lc,mn() and ls,mn(), m 6 n, omitted as well as(5) replaced by (13), the result is shown inFig. 3. As can be seen, the

simplified analysis approaches the exact solution as soon as gs >

12.5 dB.

Fig. 3Evolution of detector slope Kdfor 16=32-APSK (b1 2.5, b2 4.3)

For 32-APSK schemes, S-curves and slopes are achievable in the

same manner as just demonstrated with 16-APSK. Of course, M 32and m, n 2 {1, 2, 3}; cardinality index M3 16 and radius R3 char-acterise the third ring. Again, pF(Rm, R12, gs, j) is used in (5) ifn 1,while replaced by pF(Rm, R23, gs, j) pF(Rm, R12, gs, j) ifn 2 and

pG(Rm, R23, gs, j) if n 3. Forb1 2.5, b2 4.3, R12 (R1R2)=2and R23 (R2R3)=2, Fig. 3 illustrates the evolution of the detectorslope. Applying the simplifications introduced for 16-APSK, the exact

results are suitably approximated for values ofgs > 15 dB.

Conclusions: Using 16=32-APSK as modulation schemes, the S-curveforDD recoveryof the carrier phasehas been developedin closedform.

Since it is rather complex from the computational point of view,

appropriate simplifications are introduced such that the exact solution

is conveniently approximated for larger SNR values. The availability of

the S-curve allows the derivation of the slope in the stable equilibrium

point, which is required to design the linear ised recovery loop.

# The Institution of Engineering and Technology 2006

9 August 2006

Electronics Letters online no: 20062489

doi: 10.1049/el:20062489

W. Gappmair and J. Holzleitner (Institute of Communication Networks

and Satellite Communications, Graz University of Technology,

Austria)E-mail: gappmair@tugraz.at

References

1 ETSI EN 302 307 (V1.1.1): Digital Video Broadcasting (DVB); Secondgeneration framing structure, channel coding and modulation systems forBroadcasting, Interactive Services, News Gathering and other broadbandsatellite applications http://www.etsi.org, 2004

2 Mengali, U., and DAndrea, A.N.: Synchronization techniques in digitalreceivers (Plenum Press, New York, USA, 1997)

3 De Gaudenzi, R., Garde, T., and Vanghi, V.: Performance analysis ofdecision-directed maximum-likelihood phase estimators for M-PSKmodulated signals,IEEE Trans. Commun., 1995,43, pp. 30903100

4 Gappmair, W.: Open-loop characteristic of decision-directed maximum-likelihood phase estimators for MPSK modulated signals, Electron.Lett., 2003, 39, pp. 337339

5 Proakis, J.G.: Digital communications (McGraw-Hill, New York, USA,1989)

6 Gradshteyn, I.S., and Ryzhik, I.M.: Table of integrals, series, andproducts (Academic Press, London, UK, 1994)

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