Advantage of Using Permutation Trellis Codes andM-FSK Modulation for Power-Line

Communications ChannelTedy Mpoyi Lukusa, Khmaies Ouahada, Member, IEEE and Hendrik C. Ferreira, Member, IEEE

Department of Electric and Electronic Engineering Science, University of Johannesburg, South AfricaEmail: lukusa.tedy@gmail.com, {kouahada, hcferreira}@uj.ac.za

Abstract—The technique of Distance-Preserving Mappings(DPM) which maps the outputs of convolutional codes ontopermutation sequences has shown interesting application in pow-erline communication channel. The M-FSK modulation schemehas also shown its robustness when used for hostile channel asthe power line channel. The Combination of Distance-preservingmappings and M-FSK modulation scheme, achieves good resultswhen applied for narrowband interference. A simulation analysisof the importance of these two techniques is discussed andpresented in this paper. The obtained results have shows thebenefit and the advantage of the use of both techniques.

Index Terms—Distance-Preserving Mappings, convolutionalcodes, M-FSK modulation, powerline communication channel.

I. INTRODUCTION

The idea of representing permutation sequences with fre-quency sequences as introduced by Vinck [1] has stimulatedresearch into combining non-binary codes with the M-FSKmodulation scheme.

We can consider the designed permutation trellis codes(PTC) [2] based on the use of the distance-preserving map-pings which are coding techniques that map the outputs ofa convolutional code to other codewords from a permutationcodebook of less error-correction capabilities. We investigatethe importance of distance preserving mapping codes or per-mutation trellis codes comparing to the base codes, the binaryconvolutional codes. The number of states, th rate and thefree distance of the new obtained code. Also, we investigatein this paper the importance of M-FSK modulation schemecomparing to other modulations when used in powerline com-munication channel. The combination of these two techniquesand the performance of th e combined permutation trellis codeswith M-FSK is also investigated when used on the AWGNchannel with narrowband noise.

The paper is organized as follows. In Section II we presentbriefly the construction of permutation trellis codes (PTC) andthe design of distance-preserving mappings construction algo-rithms and some results of the advantages of these techniques.Section III introduces the M-FSK modulation scheme used inthis paper and present some of its advantages. The use ofpermutation trellis codes combined with M-FSK to combatpower line communications channel noise as the narrowband

interference is presented in Section IV. Finally in Section Va conclusion of the performed analysis will be drawn.

II. PERMUTATION TRELLIS CODES

As we have mentioned earlier, using distance-preservingmappings, the output of a convolutional encoder can bemapped to permutation codes, creating a permutation trelliscode [3], thus having the option to use the well known Viterbialgorithm for decoding. However, finding mappings can be adifficult and time consuming task. A mapping algorithm orconstruction to generate such mappings is then preferable. Welook in this section at how permutation trellis codes are createdand designed as well as presenting briefly distance-preservingmappings.

A. Distance Preserving Mappings Technique

The outputs of a binary convolutional code (BCC) can bemapped onto other codewords, which can be either binary ornon-binary [4], from a code with lesser error-correction capa-bilities. The purpose behind this mapping is to firstly obtainsuitably constrained output code sequences and secondly toexploit the error correction characteristics of the new codewith the use of the Viterbi algorithm [3]–[5].

Fig. 1. Encoding process for a distance-preserving spectral nulls code

Fig. 1 shows the mapping process where we can see theoutput binary M-tuples code symbols from an R = m/nconvolutional code are mapped into binary M-tuples. To ex-plain better the idea of mapping, we make use of a publishedexample of a four-state binary convolutional code with a rateof R = 1/2 [6].

Emphasizing the mapping technique in a better way, wepresent an example, where we use the convolutional code withhalf rate and constraint length K = 3 [6] as a base code. Theoutput of the encoder, which is a set of binary 2-tuple codesymbols, can be mapped to a set of permutation M -tuples.Note that in general the information transmission rate of the

TABLE ISOME CONSTRUCTED PERMUTATION TRELLIS CODES

Q(M,n, δ) |Emax| |E| Mapping

Q(4, 3, 1) 192 192{

1243, 1342, 4213, 4312, 1234, 1432, 3214, 3412}

Q(4, 4, 0) 768 768{

1243, 1342, 4213, 4312, 1234, 1432, 3214, 3412, 2143, 2341, 4123, 4321, 2134, 2431, 3124, 3421}

Q(5, 5, 0) 4090 3712

12345, 52341, 14325, 54321, 32145, 52143, 34125, 54123, 12435, 52431, 13425, 53421, 42135, 52134,

43125, 53124, 21345, 51342, 24315, 54312, 31245, 51243, 34215, 54213, 21435, 51432, 23415, 53412,

41235, 51234, 43215, 53214

Q(6, 7,−1) 81912 77824

123456, 163452, 523416, 563412, 123465, 153462, 623415, 653412, 143256, 163254, 543216, 563214,

143265, 153264, 643215, 653214, 321456, 361452, 521436, 561432, 321465, 351462, 621435, 651432,

341256, 361254, 541236, 561234, 341265, 351264, 641235, 651234, 124356, 164352, 524316, 564312,

124365, 154362, 624315, 654312, 134256, 164253, 534216, 564213, 134265, 154263, 634215, 654213,

421356, 461352, 521346, 561342, 421365, 451362, 621345, 651342, 431256, 461253, 531246, 561243,

431265, 451263, 631245, 651243, 213456, 263451, 513426, 563421, 213465, 253461, 613425, 653421,

243156, 263154, 543126, 563124, 243165, 253164, 643125, 653124, 312456, 362451, 512436, 562431,

312465, 352461, 612435, 652431, 342156, 362154, 542136, 562134, 342165, 352164, 642135, 652134,

214356, 264351, 514326, 564321, 214365, 254361, 614325, 654321, 234156, 264153, 534126, 564123,

234165, 254163, 634125, 654123, 412356, 462351, 512346, 562341, 412365, 452361, 612345, 652341,

432156, 462153, 532146, 562143, 432165, 452163, 632145, 652143

resulting permutation trellis coded scheme will be bits perchannel use.

Definition 1 The Hamming distance dH(xi, xj) is defined asthe number of positions in which the two sequences xi and xjdiffer. 2

Applying the definition of the Hamming distance to binaryand non-binary sequences, we can denote by D = [dij ] thedistance matrix whose entries are the Hamming distancesbetween two binary sequences xi and xj defined as follows:

D = [dij ] with dij = dH(xi, xj). (1)

Similarly for permutation sequences we denote by E = [eij ]the distance matrix whose entries are the Hamming distancesbetween two permutation sequences yi and yj defined asfollows:

E = [eij ] with eij = dH(yi, yj). (2)

The sum of all the distances in E, which is denoted by |E|plays a role in the error correcting capabilities, as was shownin [7].

In general, three types of DPMs can be obtained, dependingon how the Hamming distance is preserved as depicted in thefollowing definitions.

Definition 2 Distance-increasing mappings (DIMs), whereeij ≥ dij + δ, δ ∈ {1, 2, . . .}, ∀i 6= j. 2

Definition 3 Distance-conserving mappings (DCMs), whereeij ≥ dij , ∀i 6= j and equality achieved at least once. 2

Definition 4 Distance-reducing mappings (DRMs), whereeij ≥ dij + δ, δ ∈ {−1,−2, . . .}, ∀i 6= j. 2

Table I shows few example of constructed permutationtrellis codes using the cube construction [8]. The sum on theHamming distances for each codebook and the maximum sumfor each example are presented.

Example 1 Applying (1) and (2) for the mapping of{00, 01, 10, 11} → {1243, 1342, 4213, 4312}, the distancemetrics matrices of each set and their corresponding summa-tions on the Hamming distances could be presented as follows,

00 01 10 11

D =

00011011

0 1 1 21 0 2 11 2 0 12 1 1 0

⇒ |D| = 16,

1243 1342 4213 4312

E =

1243134242134312

0 2 2 42 0 4 22 4 0 24 2 2 0

⇒ |E| = 28,

Taking into consideration the fact that the entries on themain diagonals are all zeros, we have eij = dij + δ, withδ ≥ 1 for all i 6= j. The mapping of the outputs of thebase code {00, 01, 10, 11} to the permutation set {1243, 1342,4213, 4312}, guarantees an increase of at least one unit ofdistance per step between any two unremerged paths in thetrellis diagram of the resulting permutation trellis code, whencomparing it to the base code.

For the base code, the shortest re-merging paths in the trellisdiagram, which are known to determine the free distance [6],

TABLE IICOMPARISON OF DISTANCES FOR DISTANCE-CONSERVING MAPPINGS

M |Emax| Prefix [2] Chang [10] k-Cube [8]

4 768 732 768 768

5 4090 3613 4020 3712

6 20472 17072 18432 19456

7 98294 78528 88064 94208

8 458752 355840 413312 458752

9 2097144 – 1802240 1982464

10 9437160 – 8110080 9043968

12 184549344 – 154927104 180355072

have different distances between each pair of branches. Thesebranch distances have been changed with the code obtainedafter the mapping. It can be seen that the distances haveincreased, which makes our example represent a distance-increasing mapping. 2

B. Distance Preserving Mappings Construction

The technique of distance-preserving mappings, which hasthe purpose of using the existing error correcting capabilityof the base convolutional code to the permutation codes,needs an algorithm that help choosing the best combinationof permutation sequences for the mapping. This algorithm iscalled construction. Many contributions in this field basicallyon the search for the optimum mapping or construction thatgives better error correction results.

We now provide a brief synopsis of some of the previouswork. Ferreira and Vinck [2] proposed the prefix construction,which they applied for 4 ≤ M ≤ 8, and it was latergeneralized by Chang et al. [9] to all values of M . Since thenseveral research papers proposing different constructions werepresented in [9]–[10] and [11]–[12]. All the work used theHamming distance as the distance metric for the permutationscodes.

Initially, Chang [10], [13] studied distance-increasing map-pings (DIMs) and later on, Lee [12] presented new construc-tions for DPMs of odd lengths and investigated the distanceincrease achieved by the mappings. He proved that the distanceincrease in his constructions is more than what Chang achievedwith his mappings.

Lee [11], [12] then generalized Chang’s work to all valuesof M and was the first to introduce the graph concept inthe mapping to illustrate and explain the functioning of hisalgorithms.

Ouahada and Ferreira [8] have used graph theory to design aconstruction based on the properties of k-cube graph. The newdesigned cube graph multilevel construction is considered tobe robustly optimum on the sum of distances since it reachedthe upper bound on the sum of the Hamming distances forany value of M = 2k. It also achieved better results for othervalues of M than previous constructions for most values ofM . Table II compares the sum on the Hamming distances for

Fig. 2. BER performance of PTC codes from Cube and Prefix constructions

distance conserving mapping codes from different construc-tions.

C. Simulation Results

To see the advantage of the technique of distance preservingmappings, we have taken a look at the DPM code rate inits binary form and compare it to different other similar-rateconvolutional codes. With trial and error search, we couldhave found different convolutional codes (BCC1 → BCC4in Table III), with similar rate and similar constraint length,which means that we have similar number of states.

Since we are using M-FSK modulation scheme, then weneed to be sure that our transmitted signals are transmittedwith all M frequencies. Unfortunately this is not guaranteedwhen we use similar rate and constrain length codes since thefree distance is becoming the criteria. If we need a good freedistance then we have to increase the generator function entriesand this will minimize the chance to cover all the frequenciessince the outputs of the designed convolutional code will havesimilar codewords and this also can lead to a catastrophic code.

Daut and Lee [14], [15] in their papers and with a better andwell structured search for codes with better error correctioncapabilities, have found very good error correcting convolu-tional codes (BCC5 → BCC8 in Table III), for different rateand different constraint lengths that can give very good freedistance. Here the trade off is with the increase of the constrainlength and therefore the number of states which will makethe design and the implementation of the Viterbi decodingalgorithm very complicated.

Based on previous researches, we can see the advantagesand the benefits of using distance preserving mappings tech-niques that keeps the number of state as it was with the basecode and increase the free distance and cover all frequencieswith a organized order to match wit the frequency modulation.All the results are depicted in Table III.

Example 2 In this example we take two different codebooksgenerated from two different constructions, the cube construc-tion represented by the codebook Q(4, 4, 0) and the prefix

TABLE IIICOMPARISON OF DISTANCES FOR DISTANCE-CONSERVING MAPPINGS

Convolutional Code Binary Rate dfree Constraint Length, K Number of States Code Generator in Octal

Base Code 1/2 5 3 4 7, 5

PTC Code 1/8 20 3 4 7, 5

BCC1 1/8 13 3 4 1, 4, 7, 2, 4, 5, 3, 6

BCC2 1/8 14 3 4 6, 4, 3, 1, 7, 5, 2, 5

BCC3 1/8 16 3 4 7, 5, 1, 3, 6, 5, 4, 7

BCC4 1/8 18 3 4 7, 5, 7, 3, 6, 5, 4, 7

BCC5 1/8 44 8 128 371, 353, 331, 323, 275, 267, 237, 225

BCC6 1/8 48 9 256 767, 735, 665, 637, 571, 551, 461, 453

BCC7 1/8 52 10 512 1731, 1621, 1575, 1433, 1327, 1277, 1165, 1123

BCC8 1/8 57 11 1024 3651, 3453, 3375, 3167, 2763, 2361, 2265, 2155

(a) BER for M-FSK (b) SER for M-FSK

Fig. 3. Theoretical Performances of M-FSK modulation

construction represented by the codebook P(4, 4, 0). As wecan see the δ = 0 which means that we are in the caseof distance-conserving mappings. The codebooks and theircorresponding sums on the Hamming distances are presentedbelow.

Q(4, 4, 0) =

1243, 1342, 4213, 4312,1234, 1432, 3214, 3412,2143, 2341, 4123, 4321,2134, 2431, 3124, 3421

, |EQ(4,4,0)| = 768

P(4, 4, 0) =

1234, 1243, 1324, 1342,1423, 1432, 2134, 2143,3214, 3241, 2314, 2341,3421, 3412, 3124, 3142

, |EP(4,4,0)| = 732

2

Fig. 2 shows that the cube constructed codebook outperformthe prefix constructed codebook. This can be clearly seen evenfrom their corresponding sums on the Hamming distances

where |EQ(4,4,0)| > |EP(4,4,0)|. The higher sum on theHamming distances, the better error correction capability.

III. M-FSK MODULATION SCHEME

A. Description

The discussion in this section revolves around a few ofthe major properties of the M -ary frequency shift keying(FSK) modulation scheme [17]–[18] that has been used inthe telecommunication systems, especially in power-line com-munications. Combined with other codes like permutationcodes, M -ary FSK has shown robustness against permanentfrequency disturbances and impulse noise [1]. The modulationscheme with its constant envelope signal is in agreement withthe European Committee for Electrotechnical StandardizationNorms (CENELEC) [19].

The M -ary FSK is considered to be an orthogonal frequencymodulation scheme, the same as OFDM modulation. In thiscommunication system we consider using M orthogonal wave-

forms to transmit information, presented as follows:

s1(t), s2(t), . . . , sM (t).

The signal space has a dimension M and the received vectorr is given by

r = [r1, . . . , rM ], with

ri =

{√Ese

jφ + ni,c + jni,s, for i = k,

ni,c + jni,s, for i 6= k,

p(r|sk) =(

12πσ2

v

)Ne(|r|

2+E2s )/2σ2

vI0

(√Es|rk|2σ2v

),

where, Es represents the energy per symbol, φ, the phase shiftof the signal and I0 is the modified Bessel function of orderzero.

At the demodulator the optimum detector computes themagnitude of the different coordinates of the received vectorand chooses the maximum as depicted in the following:

max{|r1|2, |r2|2, . . . , |rM |2

}.

Fig. 3(a) shows that the increase of the value of M improvethe performance of the code when combined with M -FSKsince we are using the measure of the bit error rate (BER). Onthe other hand, Fig. 3(b) shows that the increase of the valueof M decreases the performance of the code when combinedwith M -FSK, since we are then using symbol error rate (SER).This is confirmed with the published theory in [16].

This property of orthogonality and the use of the non-coherent demodulation will have great importance in thecorrection of the narrow band interferences in a power-linechannel using different detectors as it will be presented in thefollowing section.

B. advantage

The combination of M-FSK modulation and coding fora constant envelope modulation signal has the advantage ofhaving a frequency spreading that can help us avoiding somebad parts of the frequency spectrum to facilitate correctionof frequency disturbances and impulse noise simultaneously.M-FSK has the advantage of a constant envelope signalmodulation and a demodulation in a coherent as also in anon-coherent way. as it is known, in a M-FSK modulationscheme, symbols are modulated as one of the sinusoidal wavesdescribed by the following description [1],

si(t) =√

2EsTs

cos(2πfit) ; 0 ≤ t ≤ Ts (3)

taking 1 ≤ i ≤M and Es as the signal energy per modulationsymbol and fi = f0 + i−1

Ts, 1 ≤ i ≤M .

The correspondence between symbols generated by thedistance preserving mappings and the frequencies from theM-FSK modulation make this combination very practical in aenvironment like the powerline communication channel.

Fig. 4. Performance of combined PTC and 8-FSK with ED detector in thepresence of NBI

Fig. 5. Performance of combined PTC and 8-FSK with TD detector in thepresence of NBI

Finally we can say that the modulation scheme may playan important role in the error correction process. As we cansay that the M-FSK is the best candidate for power linecommunications channel.

IV. APPLICATION: COMBINED PERMUTATION TRELLISCODES AND M-ARY FSK FOR THE NARROWBAND

CHANNEL

In this section, we make use of different detectors thatM-FSK modulation can use and see the performance ofour permutation trellis codes when narrowband noise [20] ispresent.

We can describe the narrowband noise as a continuousdisturbing source that effects one of the transmitted frequencysignals. This source can have a probability of presence denotedby pn for a duration of Tn symbols, where Tn ∈ [0,∞). In thisexample of application, we assume and model the narrowbandnoise signal as a signal with an energy Ens = 4Es, which ischosen to reflect the fact that it is much higher than the energy

Fig. 6. Performance of combined PTC and 8-FSK with VRTT detector inthe presence of NBI

of the transmitted symbols. This will cause a total saturation ofthe signal at the corresponding frequency. The correspondingfrequency of the effected signal by the narrowband noise isdenoted by fns = fi, where fi corresponds to one of thetransmission frequencies (in our simulation we just add thevalue two to the symbols’ energy).

Example 3 As discussed previously, when combining PTCcodes with an M-FSK modulator, every symbol, 1, 2, . . . ,Mcorresponds uniquely to one of the M frequencies. The M-arysymbols are transmitted in time as the corresponding frequen-cies, thus the transmitted signal has a constant envelope.

In this example, we make use of the case of M = 8 and alsomake use of the cube construction to design our permutationtrellis codes. In this case, if the codeword (1, 3, 4, 2, 6, 5, 8, 7)from the codebook Q(8, 2, 6) is sent, then the frequencies(f1, f3, f4, f2, f6, f5, f8, f7) are sent consecutively over thechannel. We make use here of distance-increasing mappingsto design our permutation trellis codes.

We run our simulation for uncoded data (UC) and our PTCcodes for different detectors as the envelope detector (ED), thethreshold detector (TD) for different values of its threshold τ .And also for the Viterbi’s ratio threshold detector (VRTT) fordifferent values of λ. Although TD and VRTT detector aredesigned for erasure channels, we make use of them of theirsoft decision properties and consider our channel as not anerasure channel. The obtained results for the ED detector, theTD detector and the VRTT detector are respectively presentedin Figures 4, 5 and 6. 2

V. CONCLUSION

We have shown from the investigation conducted in thispaper and the obtained results, that the technique of distance-preserving mappings to design permutation trellis codes is agood choice for power line communication when combinedwith M-FSK modulation. This technique helped allocate sym-bols to corresponding frequencies when combined with M-FSK modulation. And also helped increasing the free distance

and the sum on the Hamming distance when using the optimalconstruction. The distance-preserving mappings technique alsokept the number of state the same as the base codes whichmakes Viterbi decoding algorithm simple and more applicable.

The M-FSK modulation scheme has shown also its robust-ness in a hostile environment as the power line channel in thepresence of permanent frequency disturbances which are alsocalled narrow band interferences.

All these advantages make of the combined permutationtrellis codes to M-FSK modulation scheme to be the bestchoice for power-line communications channel.

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