Design of optimum M-PSK codes for Rayleigh fading channel

A.G.Burr

Abstract: A i nc t l i odoh~ for t h e design of M-PSK codcs for thc Raplcigh Fading channel optimiscd for a specific target BER is piqioscd, using the framework of tnultilevel coded moclulation. The principle is to equalise the BER at cach s t a g or a multistage decoder. An 8-PSK code is desi ned which cxcccds the pcrrormaiice of the code of Sestiadri and Sundbcrg by 2.4dB a1 BER = IO9, as wcll as I6-PSIC codes optimised for BERs or IO and In b . No additional interlenver delay is i n " d hcyond that required by the ciiunncl.

1 Introduction

It has been kiiowii for some time [ I , 21 that schemes such as Ungerboeck's trcllis-coded modulation (TCM) [3] which arc optiniiscd Tor rhc AWGN chatlnel do not perform wcll on fitding channels, arid cspccially 011 the Rayleigh f d n g chilnnel. Ungerboeck's codes maximisc rnininiuni Eucli- dean distance bctwccii codcd sequences; D i v s h r ;ud Simon [ 13 identified it number of parainclcrs that should be maximiscd in preference, notably the minimum [Hamming dislancc and the product distance. Schiegel and Coskllo [2] found new TCM scliccmcs which pctfortned signifimtitly better 011 thc racling channel.

More rewntly, il has bccn suggested [4] that the frainc- work of multilevel codcd modulation (MLCM) forms a suitable basis for the design of such codcs sincc it enables the Hamming distance rearlily to be maximiscd. MLCM, first described by Iinai and Hiriikiwa 151, Ibllows Ungerb- occk's schcmc [3] in cstablishing aii tn-level partition chain for :in hf-point signalling constellittion, w b r c )PI = log@). The principle then is to cncodc cach partitioii with A differ- cnt binary codc. Coriveiitioiially for the AWGN channel tlic codc ralcs and rnininiutn distances are chosen l o cqual- ise minimum Euclidean distancc in tlic coded partitions on each of the partition Icvcls, and hcncc to maximise overall miniinum Euclidcan distance. Seshadri and Sundbcrg [4], howcver, developed multilevel codes in which Hamming distance was maximised by equalising miniinuni Hamming distance in each of the component codcs. This paper con- sidcrs M LCM schenies usiiig convolutioiial codcs only, allhough il can also iise block codes (and indccd was origi- nally eiivisagd in thcsc Lcnns).

Multistage decoding is norrnally iiscd for MLCM to con- stril in decoding complexily. Thc principle is to decode each partition of the signalling constellation separately, trcating the remaining piirtitions as uncodd This is of course sub- optimum, but not excessivcly so. The MLCM methodoiogy

0 IEE, 2000 TKE fwwcrbg.~ oiilinc no. 2oooO229 /XX 10.1049ii~ni:2Mxx122~ hpcr first irccival 16111 June 1998 and in revid f o m 1st kptcnilxr I9W llm author is with the DepHrimneul or Electmiiics, Univci?ity of York, IIding- ton, York YO1 SOD, UK

applied to convolutional codes has the fiirther advantagc [6] that decoders may be implemented using redily availa- ble ASICs for binary convoliitional codcs, since thc conipo- iient codes iIre binary. The iise of piinctured cotlcs also letds to veiy flcxiblc, vtiriiiblc-rfik schcmcs.

A diffcrcnt approach has been employed by Zeliavi [7]. Here, a pragmatic approach is useci in which separatc bit interleaveis are employed in the eiicodi~ig of. each or Ihc M levels of the partition. This rcsults in il potential iiicrcase in the ovcrall Hamining distaiicc by a factor log2(M). Reccntly, this approach has bccn cxlendcd by Hansson and Aulin [XI by incrcasiig thc conslcllation expansion and hcncc M. Thc disadvantage of the Zehavi approach, how- cvcr, is h a t siticc cacli of the interleavets must result iti an uiicorrelated channel, the delay introduced inust also bc itzcretised by the factor log2(M). Sincc this work is aimcd primarily at speech applications, it was considered impor- tant to keep interleaving delay to a minimum, and hcnce this approach has not been used.

In most of this work the aim has been to opliinise asymptotic perrormancc at high sig~ial-to-n& ratio (SN R), and the choice of code parameters has been madc accordingly. However, it is well-known (e.g. [9]) that this inay not optimise perfoi-mttnce iIt pnictiml SNR. This paper presents a design technique to initiitnise rcquircd SNR perforinillice Tor ii spccific targct bit crror ratio (DER). Thc design technique makes iise of B simplified esti- mation technique for the BER of MLCM with mu1htngc decoding on a Kiayleigli fiiding channd, avoiding the iisc of Chernoff bounds.

2 channel

The mhiiiquc is based on the fact that each stage of the multistkige decoding pi+ocess is effcctivcly bitiary. Onc may then iise ailalylical cxprcssions for Ihc BER of binary sig- nalling on a IZayleigh fading channel [IO]. Assume that interleaving is applied of sufficient depth to enstire thnl a11 symbols RIC subject to independent fading. (On very slowly fading chatineh slow fi-equmcy hopping inuy Lx rcquired lo achicvc this within R realistic delay).

An error evetit occxirs in the Viterbi decoder for ii givcn stage wlieii two code sequelices diffei-iiig in d symbols arc incorrectly discrimitrated. On the :issumption of iiidepcnd-

Estimation of BER of MLCM on a Rayleigh

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cnl t’ading of cach symbol, this is cquivalent to binary sig- nalling with J branch divcrsily. This is because the decision metric used by the dccodcr niakcs iisc of thc signals fi-om all d symbols in order to d ihcnt ia lc bchvccn thc two codc sequences. From [IO] p.474, thc m o r probability is

whcrc p = d((ELINo)/(l + E,JNo)), and EJN, is the ratio of cncrgy per codc symbol to noise power spectral density. This uses the formula for maximal-r:itio combing (MRC), which is equivalent to ML decoding, and awnies that each codc symbol i s weighted by its fading amplitude.

For binary coiivolutional codes Rn lipper bound on the BER may then be obtained using a union bounding technique

pe& 5 e(d)P,, (d, $1 (2)

where e(d) is the cimr-wcightcd distance spectrum [ I 1 J of the code: the number of it-merging trellis paths at l-lain- ming distancc d multiplied by the average number of dmdtxt bit errors corresponding lo such paths. 11 has bccn shown by comparison with simulations [I l l that a better approximation to the Inie DER is clblaincd by taking (Iic partial sum in eqn. 2

W

d=rlfg,c:,,

For MLCM this approach is adapted by defining the stage BEE for each stage of the decodirig process. The ith stage BEli PCi is the ratio of the number of data bit errors, miirring in the ith and all subsequent shge decoders, to thc total numbcr of data bits k “ m t c d on all lcvels. Eqn. 3 may then be adwpted to give the stage BER by ttik- ing three factors into account.

First, the effective signal ;unplitiide tit thc ith stag decoder depends on the constellation point spacing within the ith level subset, defined as Ai, where thc mcan conslclla- lion radius is taken as unity. Sccondly, in all but thc small- est subsets there are a, > 1 neighbouring conskllatim points. It has been shown [12] that this results in an inclwse in the effectivc number of trcllis paths at distancc il by thc [actor U!‘. For 8-PSK

m. = 3

{ a a , i = l ... m)={2,2;1} ( d i ) For IG-PSK

m = 4

{ U * , Z = I . . .In} = { 2 , 2 , 2 , I} (5)

Thirdly, crror propagation will occur from one stage to the next. This is modcllcrl here by means of an error inultipli- cation factor c,, which givcs the expected number o f errors causcd in subscquc~il stages by a n error occurring in an carlicr stagc. Ti is thc ratio or Lhc total number orbit crrors, in the it11 and in subsequcnt s l a p , 10 llic numbcr a1 thc it11 stage. An cstimalc or is obtained by considering an error event of length e code symbols occurring at the ith stage,

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This will spin Rie data bits, within which one can expect on average an error rate of 0.5, giving rise to on average 4 f / 2 data bit errors at that stage. Since the ( i + 1 )th code will in general have a smaller minimum distance, it is assumed that it will be unable to decode correctly during h i s burst and tlierc will similarly be Xi+1.f/2 data errors at this stage, and so on. Then

I n

whew Ri is the rate of the code on thc ith level. Then one may upper hound the ith stage BER 21s

anti the overall BER and rate is 7n m

(8) P,, = p e i R = ~i

Rmll that P , i incorporates the effects of error propagu- lion, since its defnitioii includes errors caused in subse- quent stages by errors in the ith. Note that EJN, = RE/,”, and thit K here is metmred in bits per symbol anti may thcrcrorc cxmd unity, unlikc thc lntc of’ binary codcs.

Simulation has shown that for q > 1 , eqn. 7 is closes1 to llic iruc rcsull i T &r.&] = dfi& (i.c. onc lcini only of the partial bound is used), and reniains a substantial over- bound cvcn then. llcncc this value is used for i < m. The result is clearly only an estimate of the HER, but its accu- racy is adcquak Tor usc in tlic dcsigii lcchnique described

3 Design of optimum MLCM schemes for given BER

These bounds are now used to design oplimum MLCM schemes. Pmiously, two appronches have becn uscd. Con- vcntionally oiic scclcs 10 cqualisc tlic minimum Eiiclidean diskmcc of the codcd parlitions on cach levcl A,d’Jz? and thus to inaxiinisc tlic ovcrall minimum Euclidean distance. However this is optimum only for the AWGN channel. Seshadri and Sundbcrg’s approach was to cqualisc the Hamming distances on cach Icvcl, and hcncc thc overall miniriium T-Iamniing discancc or thc schcmc, which maxim- iscs tlic asymptotic coding gain on a ttayleigh fading chan- ncl. OIIC such schcmc uscs 8-PSK arid ratc 213 punctured convolutional codes on all levels, resulting in an overall rate of 2 bits pcr symbol. A plot of tlic stage BERs for this scheme is stiowti in Fig. 1, which shows that at practical

i= l i.. 1

as r~iiows.

I O ‘ h 1yI: 1 o 3

SNR the first stage BER is grcater than d l the others by scvcral ordcrs of magnitude, and hence will doininak the overall BBR.

Thus it should be possiblc lo rcducc the overall BER by reducing the code rate Ibr that stage (and incre~ing its minirnuni I-lammiiig distance), and vice W'L'I'SCI for onc or morc ol' thc others. One expects p e r i ' o ~ m i ~ n ~ to bc opti- mised whcn all stage BElh are approximately qua l , because if a significant differcncc cxistcd performance could be improved by redistributing codc ratcs between the levels so as to reduce the highcst BER al thc cxpense of the low- est. However, it is not possible to cqiialisc thcse for all SNRs, and therefore one must optimise for a spccific SNR, or cquivalcntly for a specific BER. Note that schcincs opti- rnised in this respect will h a w R inuch poorer asymptotic performance than Seshiidri and Suidberg's.

The goal, then, is to eqiialise s ~ g c nERs to yield a given overall BER a: i.e. to :diieve the stage RER ahn at each stage for minimum Ed&, The design procedure is an ilcra- tive process, in which codes are chosen for each svdge in succession. In priticiple, these codes inay bc choscn in any order, but it is convenient to start cilhcr with the first or the last. The procedure ruiis as follows: (i) Select sigiinlling cons~cllation, tar@ BER a, and target overali rate R. (ii) Select lint level code. Use eqn. 7 to find E,,N,, to give Pt!, = or/rrt. Note that EILi+l Rk = X - ELT, R,, which may bc uscd in q n . 6. (jii) For the second and subscqucnt lcvcls, use eqn. 7 to sclccl by trial and error a code which gives Ppi as closc as possihlc to dr17 at the value of Ei,ljVfl deterniinctl a1 stcp (ii). (iv) Find oveiall BER and rate using eqn. 8. If thc target is not met, iterate the procedurc from slcp (ii) using a first level code of diffcrent ratc. As noted, the vttlue for the sti~gc RERs ohtained from eqn. 7 is ;in overeshxitc, which might suggest that it will not give ii sutkient basis for cqualising the shge BERs. Howevei-, in practical caw on the Rayleigh f>iditig chatincl the error in the estimate is approxiinatcly lhc same for each stage (SBctioii 5). Thus despite the estitnaticm crror the pro- cedure still finds optiinum lcvcl codes to equulise stage BDR.

4

This procedure has bccn used Lo dcsign 8-PSK codes with i-tite 2 bits per symbol for largct UERs IOJ and IU'. (These wlues were obtained from the draft standard Ibr the IMT2000 third-generation mobile r d i o stunciard [ 131 as the target BERs for speech and for data serviccs, rcspec- tively). The resulting schemes use the punctured convolu- tional codes listcd in Tablc I. All codcs have constraint Icnglh 7 (64 stales). Codes for 16-PSK with target rate 3 bits/syibol arc also included. The Tahle gives the estimate on rcquircd Ed& 10 achicvc thc target BER oblained from cqns. 7 and 8.

Codes for 8-PSK and 16-PSK

10.1

Table 1: Multilevel codes for 8-PSK and 16-PSK, with estimated €dN, for given target BER

Puncture pattern Code Generator E b IIP:Rgst 'eve1 rate polynomials

8-PSK 2 0.001 1 318 133/145/175 1 1 1 110 1 1 1 11.2 2 314 1331171 1 1 1001 3 7/8 133/171 1 1 10 10 10 10 10 10

2 2/3 133/171 01 1 1 3 5/6 133/171 01101101 10

2 3/4 1331171 1 1 1001

8-PSK 2 10-6 1 1/2 1331171 15.4

16-PSK 46/15 0.001 1 112 1331171 16.1

3 9/10 1331171 11 01 01 IO io 01 IO 01 i o 4 11/12 133/171 11 IO 1000 1 1 ioioioio 1001

16-PSK 3 10-6 1 1/2 1331171 21.2 2 3/4 1331171 11 1001 3 516 133/171 01 10 1 1 01 10 4 11/12 1331171 11 I O I O O O 1 1 ioioioio 1001

Figs. 2 and 3 show thc slagc arid total BERs for the 8-PSK and 16-PSK sclicmcs designed for target BER of i t 3 . These show that il is difkult to achieve accurate matching of stage BERs at the target BER bccausc of the limited choice of code rates availablc. Also one of the 16- PSI< codes exceeds its target rate (by 1/15 bitisymbol), again bocausc or lhc difficulty of choosing codes for each level. Nevcrtliclcss the total BER in the range of interest is not strongly dominated by one stage. The curves for total BER show it marked concavity around the target BER, indicating that pzrfoimance is optiinisod at this point.

Fig. 4 shows the estimated total BERs Tor thc two 8-PSK codes, compared with the Seshadii and Suiidberg 8-PSK code. It shows lhal thcy achieve optimum perform- ance over different RER ranges. At their target BERs they improve on the Seshadii and Sundberg code by 2.9 and I AdB, respctively. However, llic asymptotic performance of both codes is clrady much poorer.

10 12 14 16 18 blt energy to noise denslly ratlo. d8

5 Comparison with simulation

Estimated BERs have been coinpared with simulation results for the (3/8, 3/4, 7/8) coded 8-PSK schcmc. Tlic simulalion implements B Rayleigh fading channel, and assumes ideal coherent detection with peifect sidc inrorma- tion. Thc structure of the decoder is shown in Fig. 5. Error propagation omurs in this decoder through incorrect deci- sions being fed foward to the next stage demodulator. To study this effect, a duplicate decoder was simublcd in which correct decisions taken from the encoder were fed dircctly to thc demodulators. This also enables the stagc: BERs as defined in Section 2 lo bc dctcrmined.

Fig. 6 shows tlic stagc and total BERs as obtained from the simulation, compared to the eslimatcd total BElt. Comparison with Fig. 2 shows that the estimate is in fact a substantial overbound, but that all stage BERs arc ovcr- bounded by approximately the same factor, about I .GdB.

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Thus thcir i.elationsliip in the vicinity of lhe target HER is as predicted, and hence the design procedure remains valid. However, the estimate of required BER obtaincd in Ilie course of the design procedure is pessimistic. Thc simula- tion shows for this code lhal thc largcl DER i s attained for a bit energy to noise densily ratio of 8.7dB. This compares very well with the best k i ~ o w i ~ 8-PSK coda for the fading channel, for example those proposed in [14], for which the rcquircd E'div, is I 1.1 dB (interpolating between values givcn to estimate the figure for a 64 state code). It is neaily identical to the result of [ I , after extrapolation for the usc of constraint lciigth scven codes, but interleaving delay is much smallcr.

' t

There is clearly scope for iniproving the lightness of the bound of cqn. 7. Thc iiiaccuracy of this bound appears to be H conscquencc or the slackness of tlic union bound itself whcn the cl'fcclivc numbcr of neighbours (which in this case is incrcascd by the factor a/q is very high. Other approaches, such BS those described in [15], might also bc investigated.

plot- ted against EIJN,,. The corresponding bounds, obtaincd from eqn. 6, art: 5.33 and 2.17. Thc graph shows that the error propagation factors are approximately coiistanl in the range or &INo of inlcrcst, and hencc thc concept i s useful. Thc values also fall within the bounds, but the bound is much tighter for E~ than for E , . This appears to bc mainIy bccausc propagation froin stage 1 to stage 2 is smaller than the bound suggests.

Fig, 7 shows the error propagalion [actors el and

IIX Proc.-Comrwn.. Vd. 147. h'o. 1. ~FiJIrmry . W O

6 Conclusions

A ncw code design procedure bas heen presented for multi- lcvcl M-PSI< codcs on the Rayleigh fading channel. This procedure generates codcs optiiniwd for a specific target BER, rather than for maximum tisymptotic coding gain. New codes are presented for 8-PSK at rate 2 bits per sym- bol, and for 16-PSK ltt rate 3 bits per symbol, for targcl BEHs of and lw. Simulation i-esulls are also given for the 8-PSK code with target BER W3, and it is shown that the required bit energy to noisc dmsily ratio for this DER is 8.7dB, which is better by approximately 2.4dB than ttic bcst previously-known code with minimal interleaving delay.

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