unger boeck

8/12/2019 Unger Boeck

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IEEE TRANSACTIONS ON INFORhtATION THEORY, VOL. IT-28, NO. 1, JANUARY1982

Y)sin(x+y)+(x-y)(siny-sinx)isalwayszOforxartdyin this region. Thusf(x, y) 5 4 and dg I 4 for all h,, h, 5 1.5.

REFERENCES

[II

PI

131

141

[51

[61

[71

PI

[91

H. Miyakawa, H. Harashima, and Y. Tanaka, A new digitalmodulation scheme, multi mode binary CPFSK, in Proc. Third Int.Conf. on Digital Satellite Commun., Nov. 1975, 11-13, Kyoto,Japan, pp. 105-I 12.J. B. Anderson and D. P. Taylor, A bandwidth-effi cient class ofsignal space codes, IEEE Truns. Inform. Theory, vol. IT-24, no. 6,pp. 703-712, Nov. 1978.J. B. Anderson and R. de Buda, Better phase-modulati on errorperformance using trellis phase codes, El ectron. Lett., vol. 12, no.22, pp. 587-588, Oct. 28, 1976.J. B. Anderson, Simulat ed error performance of multi-h phasecodes, in Conf. Rec. Int. Conf. on Commun., Seattle, WA, June1980, pp. 26.4.1-26.4.5.J. B. Anderson, D. P. Taylor, and A. T. Lereim, A class of trellisphase modulation codes for coding without bandwidth expansion,in Conf. Rec. Int. Conf. on Commun., Toronto, ON, Canada, June1978, pp. 50.3.1-50.3.5.

A. T. Lereim. Snectral nronerties of multi-h chase codes. M. Ena.Thesis, M&a&r Uni;., Techn. Rep. no.& CRL-57, July 1978,Communicati ons Research Laboratory, Hamilton, ON, Canada.J. B. Anderson, C-E. Sundberg, T. Aulin, and N. Rydbeck, Power-bandwidth performance of smoothed phase modulation codes,IEEE Trans. Commun., vol. COM-29, no. 3, pp. 187-195, Mar.1981.T. Aulin and C. E. Sundberg, Continuous phase modulation (CPM),part I, full response signalling, IEEE Trans. Commun., vol. COM-29, no. 3, pp. 196-209, Mar. 1981.T. Aulin, N. Rydbeck, and C. E. Sundberg, Continuous phasemodulation (CPM)-Part II: Partial response signalling, IEEETruns. Commun., vol. COM-29, no. 3, pp. 210-225, Mar. 1981.

1101

[Ill

[I21

[I31

[I41

[I51

[I61

[I71

[If31

[I91

PO1

PII

55

J. M. Wozencraft and I. M. Jacobs, Principles of Communicati onEngineering. New York: Wil ey, 1965.W. P. Osborne and M. B. Luntz Coherent and Noncoherentdetection of CPFSK, IEEE Trans. Commun., vol. COM-22, pp.1023-1036, Aug. 1974.R. de Buda, About optimal properties of fast frequency-shiftkeying, IEEE Trans. Commun., vol. COM-22, pp. 1974-1975, Oct.1974.T. A. Schonhoff, Symbol error probabili ties for M-ary CPFSK:coherent and noncoherent detecti on, IEEE Trans. Commun., vol.COM-24, no. 6, pp. 644-652, June 1976.T. Aulin, N. Rydbeck, and C. E. Sundberg, Performance ofconstant envelope M-ary digit al FM syst ems and their implementa-tion, in Conf. Rec. Nat. Telecommun. Conf., Washington, DC, Nov.1979,~~. 55.1.1-55.1.6.T. Aulin, N. Rydbeck, and C. E. Sundberg, Transmitter andreceiver structures for M-q partial response FM, in Proc. 1980Int. Zii rich Seminar on Digital Commun., Mar. 1980, pp. A.2.1-A.2.6.T. Aulin, Viterbi detection of continuous phase modulated signals,in Conf. Rec. Nat. Tel ecommun. Conf., Houston, TX, Nov. 1980, pp.14.2.1-14.2.7.T. Aulin and C. E. Sundberg, Mini mum Euclidean distance andpower spect rum for a class of smoothed phase modulation codeswith constant envelope, University of Lund, Nov. 1980 (submittedto IEEE Trans. Commun.).

S. G. Wilson, J. H. Highfill III, and C. D. Hsu, Error bounds formulti-h phase codes, Universit y of Virginia, Charlottesvill e (sub-mitted to IEEE Trans. Inform. Theory).J. K. Omura and D. Jackson, Cutoff rates for channels usingbandwidth efficient modulati ons, in Conf. Rec. Nat. Telecommun.Conf., Houston,TX, Nov. 1980, pp. 14.1.1-14.1.11.T. Aulin, N. Rydbeck, and C. E. Sundberg, Mini mum distance andspectrum for M-ary multi-h constant envelope digital FM signals,Tech. Rep. TR-129, May 1979, Telecommun. Theory, Univ. Lund,Sweden.T. Aulin and C. E. Sundberg, Further results on M-ary multi-hconstant envelope digital FM signals, Tech. Rep. TR-139, Apr.1980, Telecommun. Theory, Univ. Lund, Sweden.

Channel Coding with Multilevel/PhaseSignals

GOTTFRIED UNGERBOECK,MEMBER, IEEE

A bstruct- A coding technique i s described which improves error perfor-mance of synchronous data links without sacrifici ng data rate or requiringmore bandwidth. This i s achieved by channel coding wit h expanded sets ofmultil evel/phase signals in a manner which increases free Euclidean dis-tance. Soft maximum-li keli hood (ML) decoding using the Viterbi algo-rithm is assumed. Following a discussion of channel capacity, simplehand-designed trellis codes are presented for 8 phase-shift keying (PSK)and 16 quadrature amplitude-shift keying (QASK) modulation. Thesesimple codes achieve coding gains in the order of 3-4 dB. It is then shownthat the codes can be interpreted as binary convolutional codes with amapping of coded bits into channel si gnals, which we call mapping by setpartitioning. Based on a new distance measure between binary codesequences which efficiently lower-bounds t he Euclidean distance between

Manuscript received May 24, 1977; revised January 7, 1981.The author is with t he IBM Zurich Research Laboratory, 8803

Rtischlikon, Switzerland.

the corresponding channel signal sequences, a search procedure for morepowerful codes is developed. Codes with coding gains up to 6 dB areobtained for a variety of multil evel/phase modulation schemes. Simulati onresults are presented and an example of carri er-phase t racking is discussed.

I. INTRODUCTION

I N CHANNEL CODING of the algebraic codingtype, one is traditionally concerned with a discretechannel provided by some given modulation and hard-quantizing demodulation technique. Usually, inputs andoutputs of the channel are binary. The ability to detectand/or correct errors can only be provided by the addi-tional transmission of redundant bits, and thus by l oweringthe effective information rate per t ransmission bandwidth.

001%9448/82/0100-005X$00.75 01981 I EEE


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56 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL.IT-28, NO. 1, JANUARY 1982

In additi on, hard ampli tude or phase decisions made in thedemodulator prior to fi nal decoding cause an irreversibleloss of information. In the binary case, this amounts to lossequivalent to approximately 2 dB in signal-to-noise ratio(SNR).

In this paper, we take the viewpoint of probabilisticcoding and decoding and regard channel coding and mod-ulation as an entity [l]. Comparisons are strictly made on

the basis of equal data rate and bandwidth. A possibilityfor redundant coding can therefore only be created byusing larger sets of channel signals than required for nonre-dundant (uncoded) transmission. Maximum-likelihood(ML) soft decoding of the unquantized demodulator out-puts is exclusively assumed, thus avoiding loss of i nforma-tion prior to final decoding. This implies that codes formultilevel/phase signals should be designed to achievemaximum free Euclidean distance rather than Hammingdistance. For coded 2-amplitude modulation (AM) and4-phase-shift keying (PSK) modulation, this was never aproblem because in this case, binary Hamming distance(HD) and Euclidean distance (ED) are equivalent. The

situation is different, however, if signal sets are expandedbeyond two signals in one modulati on dimension.The investigations leading to this paper started some

time ago with the heuristic design of simple trellis codes for8-PSK modulation conveying two bits of information permodulation interval. When soft ML-decoded by the well-known Viterbi algorithm (VA) [2], coding gains of 3-4 dBwere found over conventional uncoded 4-PSK modulation.The investigations were then extended to other modulationforms. First results were presented in [3]. A much betterunderstanding of the subject was later obtained by inter-preting the hand-designed codes as binary convolutionalcodes with a mapping of coded bits into multilevel/phase

channel signals called mapping by set partitioning.Related work was reported in [4]-[8]. The approachestaken in [6]-[8] aim at constant-envelope modulation whichis in contrast to the present paper. In [9], a comparison ofvarious bandwidth-efficient modulation techniques bycomputer simulation is presented which includes codes ofthis paper.

In Section II, we investigate the potential gains in termsof channel capacity obtained by introducing more signallevels and/or phases. The results are similar to t hose ofWozencraft and Jacobs [IO] on the exponential boundparameter R,, and suggest that for coded modulation, itwill be sufficient to use twice the number of channel

signals than for uncoded modulation. In Section III, heur-istically designed trellis codes for coded 8-PSK and 16-QASK modulati on are presented and the concept of map-ping by set partitioning is introduced. The codes are inter-preted in Section IV as binary convolutional codes of rateR = m/(m + 1) with the above mapping of coded bitsinto channel signals. Preference is given to realizations inthe form of systematic encoders with feedback. The map-ping rule allows the definition of a new distance measurethat can easily be applied to binary code sequences andefficiently lower-bounds the ED between the correspond-

ing channel-signal sequences.Based on this distance mea-sure, a search procedure for more powerful codes is devel-oped in Section V, and codes with coding gains up to 6 dBare obtained for a larger variety of coded one- and two-dimensional modulation schemes. n Section VI simulationresults are presented. Finally in Section VII, the problemof carri er-phase tracking, which can play an important rolein the practical application of coded two-dimensional mod-

ulati on schemes, s discussed for one specific case of coded8-PSK modulation.

II. CHANNEL~APACITYOF MULTILEVEL/PHASEMODULATIONCHANNELS

Before addressing the code-design problem with ex-panded channel-signal sets, t is appropriate to examine interms of channel capacity the limits to performance gainswhich may thereby be achieved. Because of our intendeduse of soft ML-decoding in the receiver, we must studymodulation channels with.discrete-valued multilevel/phaseinput and continuous-valued output. One- and two-dimensional modulation is considered and intersymbol in-terference-free signaling over bandlimited channels withadditive white Gaussian noise (AWGN) is assumed. Withperfect timing and carrier-phase synchronization, we sam-ple at time nT + rs where T is the modulation interval and7s the appropriate sampling phase. The output of themodulation channel becomes

zn = a, + w,, 0)

where a,, denotes a real- or complex-valued discrete chan-nel signal transmitted at modulation timenT, and w, is anindependent normally distributed noise sample with zeromean and variance a* along each dimension. The averageSNR is defined as

. (a) one-dimensional modulation

(b) two-dimensional modulationI(2)

Fig. 1 illustrates the channel-signal sets considered in thispaper. Normalized average signal powerE{ I a, I} =1 isassumed.

Extension of the well-known formula for the capacity ofa discrete memoryless channel [ 1 ] to the case of continu-ous-valued output yields

N-l

log2~~~~(Y~~~f,z 3)in bit/T. N is the number of discrete channel input signals


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UNGERBOECK: CHANNEL CODING 57

2.AM

(4

32-AMPM 64.QASK

0)

Fig. 1, Channel-signal sets considered in this paper. (a) One-dimensionalmodulation. (b) Two-dimensional modulation.

c~O...~N- } and Q(k) denotes thea priori probabilityassociatedwith ak. Becauseof AWGN, we can substitutein (3)

P(z/ak) = exp[-(z - ak(*/2a2]

. (27ra*)-*

1

..a (a)

(2na2)- 1s- (b) (4)

With the further assumption that only codes withequiprobable occurrence of channel input signals are ofinterest, the maximization over the Q(k) in (3) can beomitted. Equation (3) can now be written in the form

C&,=,/N = log, (N) - +

(5)

In (5) we have integration replacedby expectation over thenormally distributed noise variable w which is real withvariance a2 for (a), and complex with variance 2a2 for (b).Using a Gaussian random number generator, C* has beenevaluated by Monte Carlo averaging of (5). In Figs. 2(a)and 2(b), C* is plotted as a function of SNR for the signalsets depicted in Fig. 1. The value of SNR at which inuncoded t ransmission symbol-error probability Pr( e) =

1O- is achieved s also ndicated.

(b)

In order to interpret Figs. 2(a) and 2(b), we consider asan example transmission of 2 bit/T by unc oded 4-PSKmodulation where Pr(e) = 10e5occurs at SNR = 12.9dB.If the number of channel signals is doubled, e.g., bychoosing 8-PSK modulation, error-free transmission of 2bit/T is theoretically possible already at SNR = 5.9 dB(assuming unlimited coding and decodi ng effort). Beyondthis- with no constraint on the number of signallevels/phases except average signal power-only 1.2 dBcan further. be gained. Similar proportions hold for theother modulation schemes. t can be concluded that bydoubling the number of channel signals, almost all is

Fig. 2. Channel capacity C* of bandlimit ed AWGN channels withdiscrete-val ued input and continuous-valued output. a) One-dimensionalmodulation. b) Two-dimensional modulation.

signal-set expansion if at given SNR satisfactory errorperformance can no longer be achievedby uncoded modu-lation.

III. SIMPLETRELLIS CODES

Coding gains can be realized either by block coding orby state-oriented trellis coding. There exists also the possi-bility of concatenation, e.g., by assigning short block-codewords to state transitions in a trellis structure. Note that

gained in terms of channel capacity that is achievable by the choice of a signal set for two-dimensional modulation

(4


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2)

3)

4)

OOOO 1000 0100 1100 0010 1010 0110 1110 DO01 1001 0101 1101 0011 1011 0111 1111 51

Fig. 5. Partitioning of 16-QASK channel signals into subsets with increasing mi nimum subset distances (A O < A, < A z < As;E{l4I, = 1).

, TRELLIS STATE

Fig. 6. Uncoded 4-PSK modulation, 2 bit/T.

and with a fair amount of regularity and symmetry,transitions originating from the same state receivesignals either f rom subset BO or Bl,transitions joining in the same state receive signalseither from subset BO or Bl,parallel transitions receive signals either f rom subsetCO or Cl or C2 or C3.

Rule 1) reflects the intuition that good codes shouldexhibit regular structures, and rules 2), 3), and 4) guaranteethat the ED associatedwith all single and multiple signal-error events exceeds he free ED of uncoded 4-PSK modu-lation by at least 3 dB. We have seen that with only twostates, rules 2) and 3) cannot be simultaneously satisfied.Since this is not unique to coded %PSK, we shall not befurther interested in two-state codes.

Note that parallel transiti ons imply that single signal-error events can occur. This limits achievable free ED tothe minimum di stance n t he subsetsof signals assigned oparallel transitions. On the other hand, parallel transitionsreduce the connectivi ty in t he trellis and thus allowextension of the minimum length of multiple signal-errorevents. With four states, he trade-off still worked in f avorof parallel transitions; the best 4-state 8-PSK code gains 3dB over 4-PSK, with single signal errors being most likely,whereas codes with distinct transitions to all successorstates remained inferior because ules 2) and 3) could notbe satisfied simultaneously. With eight and more states,only trellis structures with distinct transitions can be ofinterest because otherwise free ED gains would remainlimited to 3 dB.

The i deas can also be applied to t he other modulationforms. As a further example, we consider transmission of 3

2 TRELLIS STATES

dfree = p$ 1.606

(1.1 dS GAlN OVER 4 -PSK).

Pr(e) E2.0(d,,ee/2e)

4 TRELLIS STATESco c2

g=$*Cl c3

733-i2604

3715

-7, dfree = A2 = 2.000

a(3.0 dS GAIN OVER 4-PSK).

0Prk) 2

0 0l.Dtd,,&Zal.

6 TRELLIS STATES

0426

1537 0

406 2 7

dlree =jm = 2.141

0 (3.6 dS GAIN OVER 4-PSK)5173 6 02604 0

Prk) E 2.D(df,ee/2a).

3715 06240 07351 0

16 TRELLIS STATES

0426153740625173260437156240735 140625173042615376 240735126043 715

d ,ree =jm E 2 274

14.1 dB GAIN OVER 4-PSK).

Pr(e)r13?)01d,ree/20)

Fig. 7. Coded 8-PSK modulation, 2 bit/T.

bit/T by coded 16-QASK modulation. Uncoded 8-PSK or8-AMPM modulation is r egarded as a reference system.From the preceding discussion and the partitioning of the16-QASK signal set into subsets shown in Fig. 5, codesfollow easily. In the 8-PSK codes of Fig. 7, the 8-PSKsignals must only be replaced by the corresponding 16-QASK signal subsetsDO-D7 of F ig. 5. An g-state 16-QASKcode obtained in this manner is presented n Fig. 8. The

reader must be cautioned, however, about this approach. Asimilarly obtained 16-state 16-QASK code turned out to


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60 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL.IT-28, NO. 1, JANUARY 1982

8 TRELLIS STATES

0, De 4 DC

0, J,D,DI

4 Do % 02

D,D, D,D,

D,D,DoD,

DO-DOd ,ree=~A~+A:,+A~=1414

4.0 dB GAIN OVER 8 -AMPM

( 5 3 dB GAI N OVER ZI-PSKd )

Pr,e) 5 ,?I 0 (dlree/2n)

Fig. 8. Coded I6-QASK modulation, 3 bit/T.

have only the same free ED as the 8-state 16-QASK codes.Partit ioning of the one-dimensional N-AM signal sets

results in minimum subset distances A,,, = 2 . A,,i =O,l, ..* (6 dB steps), whereas for two-dimensional N-AMPM and N-QASK signal sets we have Ai+, = fi * Ai .j XI (),I . . .(3 dB steps). Numerical values are given in thecode tables of Section V for all signal sets of Fig. 1 andnormalized signal power. Partit ioning of larger signal setswill soon lead to values of Ai that exceed the free ED that

one can ever expect to achieve at given code complexity.Therefore it will usually be sufficient-even for very com-plex codes-to partiti on a signal set two or three times,and associate the signals of the subsets not further parti-tioned with parallel transitions in the code trellis.

One further remark is necessary. The specific mappingof coded bits into channel signals indicated in Figs. 4 and 5-which in t he case of Fig. 4 leads to a straight binarynumbering of the 8-PSK signals-is not important. Bypermuting subsets, other mappings can be obtained withthe same pattern of increasing minimum subset distances.Only this latter property is significant.

IV. MULTILEVEL/PHASE CODES VIEWED ASBINARY CONVOLUTIONAL CODES WITH

MAPPING BY SET PARTITIONING

The codes presented in Figs. 7 and 8 have been selectedamong equivalent codes because with some effort they canbe identif ied as binary convolutional codes of rateR =m/( m + 1) generated by feedback-free encoders n cascadewith the mapping by set partit ioning il lustrated by Figs. 4and 5. Fig. 9 shows the encoders in this form of implemen-tation for constraint lengths v = 2,3,4, corresponding to4,8, 16 states.

Using polynomial notation, the binary output sequencey(D) must satisfy the parity check equation:

[y(D) . . . Y(D)> Y(D)]-[H(D) ... H(D), H(D)]= 0. (8)

For the encoders of Fig. 9, the parity check matrices arev = 2: H(D) = [(0), 0, D, D2+ 1 12v = 3: H(D) = [(0), D, D*, D3 + 1 1,v=4: H(D)=[ D, D3 + D2, D4 + D3 +11,

(9)Among the codes presented in Section V, there will be a better 16-state

16-QASK code.

L2

Fig. 9. Realization of I-PSK and 16.QASK codes by means of minimalconvolutional encoders.

where the trivial entry (0) accounts for the addit ionalunchecked bit in coded 16-QASK modulation. Note thatthe encoders are minimal [ 131 since Y = maximum degreeof H(D), 0 I j 5 m. Considering (9), we observe that theparity-check polynomials have the form (v 1 2):

H(D) = 0, ticjIm, (104

Hi(D) = 0 + h;-,D- + 0.. h{D + 0, 1


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Fig. 10. Systemati c convolutional encoder structure with f eedbacksatisfying condition (I 1).

x(D) to the feedback-free encoders. Becauseof [l 11, herational functions in (12) are realizable. A general realiza-tion of (12) with v delay elements s depicted in Fig. 10,and the specific systematic encoders corresponding to theminimal encodersof Fig. 9 are shown n Fig. 11.

Let y(D) and y(D) = y(D) @ e(D) be two similar bi-nary code sequenceswhere @ denotes modulo 2 addition.Since we deal with linear codes, the binary error sequence

e(D) = e,Dk + ek+,Dk+ + -. . ek+LDkfL,

ek,ek+L#O, L 20 (13)

belongs to the set of code sequences. n order to lower-bound the ED between the channel-signal sequences (D)and a(D) obtained from y(D) andy(D), we define theEuclidean weights w(e,) = min d[a(z), a(z @ e;)], whereminimization goesover all z = [z . . . zl, z], and d[ . . . ]is the ED between the channel signals specified. For thesquared ED between a(D) and u(D),we then obtain

kfL kfL

izk d2[u( Yi>, a( S: @ et)] 2 jzk W(ei> w2[e(D)].

(14)

Theorem: For each e(D) there exists a pair u(D) andu(D) for which (14) is satisfied with equali ty.

Pro08 Becauseof symmetry in the channel signal setsw(ei) = mind[a(z), u(z @ ej)] is already achieved by let-ting the last el ement z in z be arbitrarily 0 or 1, andcarrying out the minimization only over [z* . . * z]. Sinceencoding at rate R = m/(m + 1) allows any successionofelements [ yj* . .. v,] to occur (best seen from Fig. lo),there exists for each e(D) a code sequence (D) that leadsin (14) to equality for each ndividual term of i.

The risk taken in settingd, = Afree s very small be-cause the minimum in (15b) is usually achieved by morethan one e(0). This is especially true if fi =Cm, i.e., higherorder bits of e, are not i nvolved in the parity-check opera-tion. We could never find a code wheredfreewas not equal

to A r&Therefore we adopt this latter definition of f ree

ED in terms of mini mum subset distances, which makesthe calculation independent of the exact mapping of codedbits into channel signals as long as the same pattern ofthese mini mum distances results.

Free ED between channel-signalsequences an thereforebe determined n an analogousmanner to finding free HDbetween the binary code sequencesy(D). In the ap-propriate search algorithm that examines all nonzero codesequences (D) [or y(D)], the Hamming weights of e, mustonly be replaced by the squared Euclidean weights w2(ei).Hence

To conclude this section, we note another importantproperty of conditi on (11). From Fig. 10, one can see hatmultiple signal errors (L > 0) must start with ek =[er**. eL,O] and with ektL = [er+L.. . eL+,,O]. Thesquared ED associated with these errors is therefore atleast 2 . A:. In other words, the condition guarantees hattransitions originating from one state or joining in onestate can have signals only from subset BO or. Bl (cf.Section-III, conditi ons 2) and 3)). We note further thatfi -Cm implies dfree 5 A,, since then single-signal errors(L = 0) with ED equal to An are possible.

d&, = minw2[e(D)] (154over all code sequences (D) # 0.

Fig. 1 . Equivalent realization of 8-PSK and I6-QASK codes by meansof systemati c convolutional encoders with feedback.

Let q(ei) be the number of trailing zeros n ei, e.g., fore, = [e, . - .e?, ,O, 0] we have q(e,) = 2. From the map-ping by set partitioning, one can see that w(e,) L AqCe,),and that equality holds for almost all ej. In the 8-PSKmapping illustrated by Fig. 4, we have only found thatw([lOl]) > A,, and that in the 16-QASK mapping in Fig5, only w([ OOl]), w([ 1 oll), and w([ 11111) xceedA,. In allother cases, w(ej) = AqCe,) note w(0) = A4(s)= 0). Thismotivates us to write

k+L

over all code sequences (D) # 0.


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62 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL.IT-28, NO. 1, JANU ARY 1982

V. SY~TJ~MATIC SEARCH FOR MORE POWERFULCODES

The problem to be solved is to find for given constraintlength v and given mi nimum subset distances A0 c A,< . * . Am, the systematic encoder with feedback that maxi-mizes d, = Arm. Considering (lo), there are (& + 1) - (Y- 1) coefficients in the parity-check polynomials to bedetermined. For a feedback-free encoder with a restrictionequivalent to (1 ), one can show that there would be(tfl+ 1) - (v - 1) + ti2 + 1 generator-polynomial coeffi-cients to decide on. The difference of fi2 + 1 accounts forthe fact that with the systematic encoder structure,catastrophic encoders are automatically excluded. Anotheradvantage of the systematic encoder structure is that noprior assumptions are necessary on the individual degrees9 of the generator polynomials, which for a minimal en-coder must satisfy ZT ,vj = v.

The fi rst step is to determine the value of 6~. Fromknowledge of d, already achieved by simpler codes, andfrom the limitation d, I A,, the appropriate value fol-

lows easily. In fact, we shall see i = 1 or 2 only.The search procedure developed below is basically anexhaustive search with a number of rejection rules. Weassume that for good codes (11) is always satisfied. TheY - 1 remaining coefficients in each parity-check poly-nomial (lob) and (10~) are represented in the form ofbinary integers

0 s ih = (hi-,2-* + h -,Y3 + * *. h{ZO)

12- - 1, OIjSt?i. (16)

Definition: An incomplete code at level I, 1 5 I < fi isspecified by I% =[ih, ihI-, * * . iho].We let this be equiva-

lent to setting encoder inputsx+(D), . . * ~~(0)

to zerofor a complete code specified by a.In the search program, we increment in the outermost

loop iho from 0 to 2- - 1, n the next inner loopih from0 to 2- - 1, etc. Whenever at level I a code is rejected byone of the following rules, we skip inner loops at levels1+ 1 to&.

Rule 1: Reversing time. does not change the distanceproperties of a code. Let ih&, be the bit-reversed binaryinteger ihj, e.g., (01 l), e, (11 o),. It follows f rom theorder in which theihj are varied in the search program thata code can be rejected at level 1 f ih,,, < ih*and ihie, = ihjforOsj 1. i h$ ih- < ih-152; ih ~ih/-2 < ihI-

ih~ ihI- ~ih-2 < ihI-2(20)

123: ih ~ih-3 < ihI-

ih ~ihI2 $ihI-3 < ihI-

ih G3 h- $ihI-2 < ihI-

ih CB h- $ih-* @ihIm < ihle3,etc.H(D) Here, ih @ h denotes the integer obtained by modulo2=[Nm(D);..H(D);..H(D)~HS(D);.. ~a( D)] , addition ofih and ih bit by bit.

Ols


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UNGERBOECK:CHANNELCODING 63

TABLE 1CODESFORAMPLITUDEMODULATION

m= 1 In=2 m=3" Fi Ho(D) t+(D) H2(D) d;ree/b; G4m,2AM%An/m G16AM/aAM

2 1 58 28 - 2.25 -2.5 dB -3.3 dB -3.5 dB

3 1 13 04 - 2.50 3.0 3.8 3.9

4 1 23 04 - 2.75 3.4 4.2 4.3

5 1 45 10 - 3.25 4.2 s.0 5.1

6 1 103 024 - 3.50 4.5 5.2 5.4

7 1 235 126 - 4.00 5.1 5.8 6.0

8 1 515 262 - 4.25 5.3

9 1 1017 0342 - 4.50 5.6

* 10 1 2051 1536 -

* 11 1 4017 1602 - 5.00 6.0

A;@AM)/A;(4AM) = 20/21 (-0.21 dB)

*Search not completed.~NO improvement obtained.

Further rules are conceivable; for example, a code isrejected f

GCD{Hfi(D);--H(D)} # 1. (21)

This latter rule was not i mplemented.A block diagram of the code-search program is pre-

sented n Fig. 12. For the calculation of d$Le, he bidirec-tional search algorithm i n the form discussed by Larsen[ 141was adopted. For computing in this algorithm dist anceincrements of state extensions; the systematic encoderstructure offers the same computational advantageswhichPaaske [12] obtained by regarding the syndrome formerstates instead of the ordinary states of feedback-free en-coders.

The codes obtained by the program are listed in TablesI-III. For each constraint length v, a limit of 30 min CPUtime was set, and no particular effort for program optimi-zation was made (IBM/370-158, PL/I program withoutspecial assemblerwritten routines). Parity check polynomi-als are presented n the Tables in octal form, e.g., Ho(O)= D6 + D2 + 1 4 (001000 Ol), g (105)s. Free ED isfirst gi ven in the normalized form d&/A:(coded). Notethat for normalized signal power E{ ( u, I} = 1, we haveA,(coded) = A,(uncoded). The asymptotic codi ng gain foreach specific coded/uncoded compari son s then computedby replacing A,(coded) by the appropriate A,(uncoded)

TABLE IICODESFOR&PSKMODULATION

v ;;; Ho(D) H1 K') H'(D)m=2

%SK/4PSK

2 1 58 20 - 2.000 - 3.0 dB

3 2 11 02 04 2.293 3.6

4 2 23 04 16 2.586 4.1

5 2 45 16 34 2.879 4.6

6 2 105 036 074 3.000 4.8

7 2 203 014 016 3.172 5.0

a 2 405 250 176 3.465 5.4

9 2 1007 0164 0260 3.758 5.7

210 2 2003 0164 0770 3.758 5.7

E(/afjl - 1:

A0(4PSK) = 5, A,(SPSK) = 2sin(rr/S),

Al (BPSK) = fi, A2(8PSK) = 2

A; (SPSK) /A; (4PSK)

= 1 (0 dB)

*Search not completed.~NO improvement obtained.

and expressing he above ratio in decibels. For example,

= 10 * log,, (&,( N - AM)/Ai[( N/2) - AM]}.

(24

The tables show that i ncreasing v does not always ead toa code with larger dmree,specially if coding gains arealready in t he order of 6 dB. The code search could havebeen extended to larger values of v by optimizing criticalprogram sections, allowing l onger program runs, and per-haps by including additional rejection rules. In view of theexponential i ncrease in code complexity and t he smalladditi onal gains in free ED to be expected, we were con-tent with the codes already found.

VI. SIMULATION RESULTS

Coding gains have so far been expressed n terms of

larger free ED, which is adequat e for high SNR. In thissection, we show simulation results obtained at moderateSNR. We concentrate mainly on error-event probabilitywhich is relevant for most blockwise data communication.This makes the result independent of the specific mappingof information bit sequencesnto channel signal sequences.In a practical system, several factors can inf luence thismapping: the particular bit labeling of branches n signal-set partitioning, the realization of t he encoder n the formof a feedback-freeminimal encoderor a systematic encoderwith feedback, phase-differential coding in order to resolvephase ambiguity, and scrambling Results on bit-error


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4 IEEE TRANSACTIONS ON INFORMATION THEORY,VOL. ~~-28, NO. 1,JANUARY 1982

TABLE IIICODESFORAMPLITUDE/PHASEMODIJLATION

Y ii H(D) H1(D)H'(D) d;&A; $I = 2 3'Y.~~ASK/m=4

G32AMpM/m=5

MMPMI G64QASK/4PSK BAM~M(SPSK) 16QASK 32AtE'M

2 1 %3 2 11

4 2 23

A5 2 41

6 2 ,101

7 2 203

A8 2 401

9 2 1001

*10 2 2003

2a02

04

06

016

014

056

0346

0164

04

16

10

064

042

354

0510

0770

2.0

2.5

3.0

3.0

3.5

4.0

4.0

4.5

5.0

-2.0 dB

3.0

3.8

3.8

4.5

5.1

5.1

5.6

6.0

-3.OC4.4) dB -2.8 dB -3.0 dB

4.OC5.3) 3.8 4.0

4.8C6.1) 4.6 4.8

4.8C6.1) 4.6 4.8

5.4C6.8) 5.2 5.4

6.OC7.4) 5.8 6.0

6.OC7.4) 5.8 6.0

E(lap= 1: A;(BAMPM)/A;(4pSK) = 415 (- 0.97 dB)

A0C3AMPM) = 2/h, A0(16QASK) = h/5, A;(lCQASK)/A; (L)AMPM) = 1 ( 0.0 dB)

A0(32AMPM) = 2/&,A0(64QASK) = J2/21, Af(16QASK)/Af(8PSK) - l/(5sin2(n18)) (+ 1.35 dB)

Ai = fi.Ai-l(x), i = 1, 2, . . A;(32AMi'M)/A;(16QASK) = 20/21 (- 0.21 dB)

A;(64QASK)/A;( 32AMPM) = 1 ( 0.0 dB)

*Search not completed.ANo improvement obtained.

SNR [dB] SNR [da]

5 6 7 6 9 IO II 12 (3

g. 13. Error-event performance of coded &AM versus uncoded 4-AM,2 bit/T.

obability are therefore given only for one particularample.In the simulation programs, the length of the pathstories in the Viterbi decoding algorithms wasM = 6vecision delay). An error event was counted when a falseannel signal was decoded following a state that stilllonged to the correct path through the code trellis. Figs.

3-15 show error-event frequency versus SNR for codedAM, 8-PSK, and 16-QASK in comparison with com-ted error-event probability of 4-AM, 4-PSK, and 8-PSK

COOED I-P%

SlMULATlON -lMULATlON -\\ \

Fig. 14. Error-event performance of coded 8-PSK versus uncoded 4-PSK2 bit/T.

or 8-AMPM, respectively. The simulation results are givenwith 90 percent confidence intervals. Systematic encoderswith the parity-check polynomials given in Tables I-III forv = 2, 3, and 4 were used. For the simpler codes, whereiV(d,,) could still be determined by inspection of thetrellis diagrams, lower bounds on error-event ,probabilityare included. Also indicated in Figs. 13-15 is the SNR atwhich ,channel capacity of the expanded signal sets (cf;Figs. 2(a) and (b)) equals the transmission rate in bit/T.This illustrates that by simple 4-state codes a significant


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- SNR=9.3dE: v=Z: Nldfree)2

CCR -PAW = 3 ElllT

d

Fig. 15. Error-event performance of coded I4-QASK versus uncoded8-PSK and 8-AMPM, 3 bit/T.

UNCOOEO -P%

(EVENT 6 BIT)

ECOOED -P% \ \t\

Fig. 16. Error-event and bit-error performance of coded 8-PSK (v = 2,minimal encoder) and uncoded 4-PSK, 2 bit/T.

portion of the totally possible mprovement of perfor manceis al ready achieved, and that further improvements byusing more complex codes (v = 3,4, * * . ) t end to be rela-tively expensive.

Fig. 16 shows -error-event and bit-error frequency ofcoded 8-PSK using the 4-state minimal encoderdepicted inFig. 9. The simplicity of this code permitted t he calculationof lower and appropriate upper bounds on error-event and

Fig. 17. Coded 8-PSK transmission syst em with carrier-phase trackingloop and differential coding/decoding to resolve 180 phase ambiguity.

bit-error probabilities (upper-boundswill not be discussed).

It can be observed that the two error probabilit ies becomeidentical for hi gh SNR because n this particular case, hefree ED occurs for single-signalerror events with only onewrong bit per event, and because no differential codingand scrambling s used.

VII. CARRIER-PHASE TRACKING

In the preceding sections, deal timing and carrier-phasetracking wer e assumed to be available in the receiver.Whereas timing synchronization can be accomplished inthe usual manner by signal squaring and filtering with anarrow bandpass filter tuned to l/T Hz, carri er-phasesynchronization can pose a somewhat harder problem. Inthis section, we restrict attention to coded 8-PSK modula-tion using the 4-state encoder shown in Fig. 9, and measurethe phase tracking performance of a simulated receiver.The problems encountered may be regar ded as typical ofother casesof coded carrier modulation. Instead of (l), weassume hat the receiver deals with

z, = a, . ,A + wn (23)where p, is a slowly varying carrier phasea priori unknownin the receiver. Fig. 17 illustrates the tracking of p, by adecision-directed carrier-phase oop that uses tentative de-

cisions of the Viterbi decoding algorithm: $J~epresentsanestimate of p,, and Li,-, denotes a tentative signal deci-sion that is obtained bi backtracking in the decodingalgorithm the survivor path from the most likely survivorstate at time n, Mp I M times. The input to the decodingalgorithm becomes

z; = a, *e (lp,-tL) + w;. (24)A measuredphase error

Aq+,= L (z& . a,-,) r= h( z& . cinlMp),

(I a, I= 1) (25)

is used to predict QJ,+,by a first-order filter&+, = %, + Y . Ap,, O


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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL.~~-28,NO. 1, JANUARY 1982 61

[IO] .I. M. Wozencraft and I. M. Jacobs, Principles of Communication IT-20, pp. 683-689, Sept. 1974.Engineering. New York, Wiley, 1965, pp. 318-319. [ 131 G. D. Fomey, Jr., Convolutional codes I: algebraic structure,

[ll] R. G. Gallager, Information Theory and Reliable Communication. IEEE Trans. Inform. Theory, vol. IT-16, pp. 120-138, Nov. 1970.New York, Wiley, 1968, p. 74. [ 141 K. J. Larsen, Comments on An efficient algorithm for computing

[12] E. Paaske, Short binary convolutional codes with maxi mal -free free distance, IEEE Trans. Inform. Theory, vol. IT-18, pp. 431-439,distance for rates 2/3 and 3/4, IEEE Trans. Inform. Theory, vol. May 1972.

Finite Sampling Approximations forNon-Band-Limited Signals

STAMATIS CAMBANISMEMBER, IEEE, ANDMUHAMMAD K. HABIB

Abstract-Finite sampling approximations, along with bounds on theapproximation error, are derived for certain deterministic and randomsignals which are not band-limited.

I. STATEME NT AND DISCUSSION OF RESULTS

T HIS PAPER derives finite sampling approximationsand their rates of convergence or deterministic andrandom signals which are not band-limited. The merit ofthese results lies in the fact that in many practical situa-tions the signals under consideration are not band-limited,and only a finite number of samplesare available. We thus

study conditi ons under which a finite sum of the formNF) f(sj si;$;;L--;)

(1)n= -N(W)

converges to the non-band-limited function f(t) as thesampling rate W tends to infinity, and as the number2N(W) + 1 of samples used, which depends on the sam-pling rate W, also tends to infinity with W. We alsodetermine the speedof the convergence.

The problem of approximating a non-band-limited sig-nal by an infinite series of the form (I), i.e.,N(W) = 00,has been considered for deterministic signals in [2], for

random stationary signals n [5], for time-limited determin-istic signals in [6], and for certain analytic si gnals in [21]and [14].

Manuscript received April 22, 1980; revised January 13, 1981. Thiswork was supported in part by the Air Force Office of Scientific Researchunder Grant AFOSR-80-0080 and in part by a scholarship of the ArabRepublic of Egypt. This paper was presented at the 14th Annual Con-ference on Information, Sciences and Systems, Princeton, NJ, March26-28, 1980.

S. Cambanis i s with the Department of Statistics, University of NorthCarolina, Chapel Hill, NC 27514.

M. K. Habib is with the Department of Biostatistics, School of PublicHealth, University of North Carolina, Chapel Hill, NC 275 14.

There is, of course,extensive iterature on bounds for thetruncation error in the approximation of a band-limitedsignal by a finite sum of the form (l), in chronologicalorder [22], [lo], 1181, 3], [4], [17], [l]. In this case,W isfixed, and N is independent ofWand tends to infinity. Fora review of the sampling theorem, see 111.

Here we consider the approximation of non-band-limitedsignals by finite sums of t he form (1). This problem hasbeen considered or certain analytic deterministic signals n[21], [14]; for certain deterministic as well as randomsignals which have at least two derivatives in [13]; for

certain deterministic signals whose derivatives satisfy aLipschitz condition in [7]; for certain deterministic signalswith multi dimensional parameter which have mixed partialderivatives in [19]; and for deterministic signals which areintegrable, have integrable Fourier transform, and fall offfaster than const. ] t ]-a, (Y> 2, in [9]. The deterministicand random signals considered n this paper satisfy muchless stringent conditions. Al though we consider only signalswith a one-dimensionalparameter, generalization to multi-dimensional parameter signals should be feasible.

The results for deterministic signals are stated in Theo-rem 1 and Theorem 2 and its corollary. The results forrandom signals are stated in Theorems 3-5. The deriva-tions of the results are given in Section II.

We begin by considering a (Cezaro) version of (I), givenby (4) (see also [16]). The (CezBro) coefficients ensure hat(4) converges to f(t), provi ded a sufficient number ofsamples is used (Theorem 1). In contrast to this generaland simple result, the convergenceof (1) to f(t) is a morecomplicated matter requiring additional assumptions onthe signal and the number of samples required. The dif-ference between (1) and (4) is that while (4) leads to thenonnegative Fejer kernel, (1) leads to the more intractableDirichlet kernel (cf. (19) (20) and (28), (29)).

0018-9448/82/0100-0067$00.75 01981 IEEE

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