Advantage of Using Permutation Trellis Codes and M

Advantage of Using Permutation Trellis Codes and M-FSK Modulation for Power-Line Communications Channel Tedy Mpoyi Lukusa, Khmaies Ouahada, Member, IEEE and Hendrik C. Ferreira, Member, IEEE Department of Electric and Electronic Engineering Science, University of Johannesburg, South Africa Email: [email protected], {kouahada, hcferreira}@uj.ac.za

Abstract—The technique of Distance-Preserving Mappings (DPM) which maps the outputs of convolutional codes onto permutation sequences has shown interesting application in powerline communication channel. The M-FSK modulation scheme has also shown its robustness when used for hostile channel as the power line channel. The Combination of Distance-preserving mappings and M-FSK modulation scheme, achieves good results when applied for narrowband interference. A simulation analysis of the importance of these two techniques is discussed and presented in this paper. The obtained results have shows the benefit and the advantage of the use of both techniques. Index Terms—Distance-Preserving Mappings, convolutional codes, M-FSK modulation, powerline communication channel.

I. I NTRODUCTION The idea of representing permutation sequences with frequency sequences as introduced by Vinck [1] has stimulated research into combining non-binary codes with the M-FSK modulation scheme. We can consider the designed permutation trellis codes (PTC) [2] based on the use of the distance-preserving mappings which are coding techniques that map the outputs of a convolutional code to other codewords from a permutation codebook of less error-correction capabilities. We investigate the importance of distance preserving mapping codes or permutation trellis codes comparing to the base codes, the binary convolutional codes. The number of states, th rate and the free distance of the new obtained code. Also, we investigate in this paper the importance of M-FSK modulation scheme comparing to other modulations when used in powerline communication channel. The combination of these two techniques and the performance of th e combined permutation trellis codes with M-FSK is also investigated when used on the AWGN channel with narrowband noise. The paper is organized as follows. In Section II we present briefly the construction of permutation trellis codes (PTC) and the design of distance-preserving mappings construction algorithms and some results of the advantages of these techniques. Section III introduces the M-FSK modulation scheme used in this paper and present some of its advantages. The use of permutation trellis codes combined with M-FSK to combat power line communications channel noise as the narrowband

interference is presented in Section IV. Finally in Section V a conclusion of the performed analysis will be drawn. II. P ERMUTATION T RELLIS C ODES As we have mentioned earlier, using distance-preserving mappings, the output of a convolutional encoder can be mapped to permutation codes, creating a permutation trellis code [3], thus having the option to use the well known Viterbi algorithm for decoding. However, finding mappings can be a difficult and time consuming task. A mapping algorithm or construction to generate such mappings is then preferable. We look in this section at how permutation trellis codes are created and designed as well as presenting briefly distance-preserving mappings. A. Distance Preserving Mappings Technique The outputs of a binary convolutional code (BCC) can be mapped onto other codewords, which can be either binary or non-binary [4], from a code with lesser error-correction capabilities. The purpose behind this mapping is to firstly obtain suitably constrained output code sequences and secondly to exploit the error correction characteristics of the new code with 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 the output binary M-tuples code symbols from an R = m/n convolutional code are mapped into binary M-tuples. To explain better the idea of mapping, we make use of a published example of a four-state binary convolutional code with a rate of R = 1/2 [6]. Emphasizing the mapping technique in a better way, we present an example, where we use the convolutional code with half rate and constraint length K = 3 [6] as a base code. The output of the encoder, which is a set of binary 2-tuple code symbols, can be mapped to a set of permutation M -tuples. Note that in general the information transmission rate of the

TABLE I S OME C ONSTRUCTED P ERMUTATION T RELLIS C ODES Q(M, n, δ)

|Emax |

|E|

Q(4, 3, 1)

192

192

Q(4, 4, 0)

768

768

Q(5, 5, 0)

4090

3712

Q(6, 7, −1)

81912

77824

n

Mapping o 1243, 1342, 4213, 4312, 1234, 1432, 3214, 3412

n o 1243, 1342, 4213, 4312, 1234, 1432, 3214, 3412, 2143, 2341, 4123, 4321, 2134, 2431, 3124, 3421       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   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 per channel use. Definition 1 The Hamming distance dH (xi , xj ) is defined as the number of positions in which the two sequences xi and xj differ. 2 Applying the definition of the Hamming distance to binary and non-binary sequences, we can denote by D = [dij ] the distance matrix whose entries are the Hamming distances between 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 distances between two permutation sequences yi and yj defined as follows: 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 shown in [7]. In general, three types of DPMs can be obtained, depending on how the Hamming distance is preserved as depicted in the following definitions. Definition 2 Distance-increasing mappings (DIMs), where eij ≥ dij + δ, δ ∈ {1, 2, . . .}, ∀i 6= j. 2 Definition 3 Distance-conserving mappings (DCMs), where eij ≥ dij , ∀i 6= j and equality achieved at least once. 2 Definition 4 Distance-reducing mappings (DRMs), where eij ≥ dij + δ, δ ∈ {−1, −2, . . .}, ∀i 6= j. 2

Table I shows few example of constructed permutation trellis codes using the cube construction [8]. The sum on the Hamming distances for each codebook and the maximum sum for each example are presented. Example 1 Applying (1) and (2) for the mapping of {00, 01, 10, 11} → {1243, 1342, 4213, 4312}, the distance metrics matrices of each set and their corresponding summations on the Hamming distances could be presented as follows,  00 01 0 1 00 01   1 0 D= 10  1 2 2 1 11 1243 1243 0 1342   2 E= 4213  2 4312 4

1342 2 0 4 2

10 11  1 2 2 1   ⇒ |D| = 16, 0 1  1 0 4213 2 4 0 2

4312 4 2   ⇒ |E| = 28, 2  0

Taking into consideration the fact that the entries on the main diagonals are all zeros, we have eij = dij + δ, with δ ≥ 1 for all i 6= j. The mapping of the outputs of the base code {00, 01, 10, 11} to the permutation set {1243, 1342, 4213, 4312}, guarantees an increase of at least one unit of distance per step between any two unremerged paths in the trellis diagram of the resulting permutation trellis code, when comparing it to the base code. For the base code, the shortest re-merging paths in the trellis diagram, which are known to determine the free distance [6],

TABLE II C OMPARISON OF DISTANCES FOR DISTANCE - CONSERVING MAPPINGS M

|Emax |

Prefix [2]

Chang [10]

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

k-Cube [8]

9

2097144

–

1802240

1982464

10

9437160

–

8110080

9043968

12

184549344

–

154927104

180355072

have different distances between each pair of branches. These branch distances have been changed with the code obtained after the mapping. It can be seen that the distances have increased, which makes our example represent a distanceincreasing mapping. 2

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

distance conserving mapping codes from different constructions.

B. Distance Preserving Mappings Construction

C. Simulation Results

The technique of distance-preserving mappings, which has the purpose of using the existing error correcting capability of the base convolutional code to the permutation codes, needs an algorithm that help choosing the best combination of permutation sequences for the mapping. This algorithm is called construction. Many contributions in this field basically on the search for the optimum mapping or construction that gives better error correction results. We now provide a brief synopsis of some of the previous work. Ferreira and Vinck [2] proposed the prefix construction, which they applied for 4 ≤ M ≤ 8, and it was later generalized by Chang et al. [9] to all values of M . Since then several research papers proposing different constructions were presented in [9]–[10] and [11]–[12]. All the work used the Hamming distance as the distance metric for the permutations codes. Initially, Chang [10], [13] studied distance-increasing mappings (DIMs) and later on, Lee [12] presented new constructions for DPMs of odd lengths and investigated the distance increase achieved by the mappings. He proved that the distance increase in his constructions is more than what Chang achieved with his mappings. Lee [11], [12] then generalized Chang’s work to all values of M and was the first to introduce the graph concept in the mapping to illustrate and explain the functioning of his algorithms. Ouahada and Ferreira [8] have used graph theory to design a construction based on the properties of k-cube graph. The new designed cube graph multilevel construction is considered to be robustly optimum on the sum of distances since it reached the upper bound on the sum of the Hamming distances for any value of M = 2k . It also achieved better results for other values of M than previous constructions for most values of M . Table II compares the sum on the Hamming distances for

To see the advantage of the technique of distance preserving mappings, we have taken a look at the DPM code rate in its binary form and compare it to different other similar-rate convolutional codes. With trial and error search, we could have found different convolutional codes (BCC1 → BCC4 in 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 we need to be sure that our transmitted signals are transmitted with all M frequencies. Unfortunately this is not guaranteed when we use similar rate and constrain length codes since the free distance is becoming the criteria. If we need a good free distance then we have to increase the generator function entries and this will minimize the chance to cover all the frequencies since the outputs of the designed convolutional code will have similar codewords and this also can lead to a catastrophic code. Daut and Lee [14], [15] in their papers and with a better and well structured search for codes with better error correction capabilities, have found very good error correcting convolutional codes (BCC5 → BCC8 in Table III), for different rate and different constraint lengths that can give very good free distance. Here the trade off is with the increase of the constrain length and therefore the number of states which will make the design and the implementation of the Viterbi decoding algorithm very complicated. Based on previous researches, we can see the advantages and the benefits of using distance preserving mappings techniques that keeps the number of state as it was with the base code and increase the free distance and cover all frequencies with 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 codebooks generated from two different constructions, the cube construction represented by the codebook Q(4, 4, 0) and the prefix

TABLE III C OMPARISON OF DISTANCES FOR DISTANCE - CONSERVING MAPPINGS Convolutional Code

Binary Rate

df ree

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 we can see the δ = 0 which means that we are in the case of distance-conserving mappings. The codebooks and their corresponding sums on the Hamming distances are presented below.

where |EQ(4,4,0) | > |EP(4,4,0) |. The higher sum on the Hamming distances, the better error correction capability. III. M-FSK M ODULATION S CHEME A. Description

  1243, 1342, 4213, 4312,      1234, 1432, 3214, 3412, Q(4, 4, 0) = , |EQ(4,4,0) | = 768 2143, 2341, 4123, 4321,      2134, 2431, 3124, 3421   1234, 1243, 1324, 1342,      1423, 1432, 2134, 2143, P(4, 4, 0) = , |EP(4,4,0) | = 732 3214, 3241, 2314, 2341,      3421, 3412, 3124, 3142

2

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

The discussion in this section revolves around a few of the major properties of the M -ary frequency shift keying (FSK) modulation scheme [17]–[18] that has been used in the telecommunication systems, especially in power-line communications. Combined with other codes like permutation codes, M -ary FSK has shown robustness against permanent frequency disturbances and impulse noise [1]. The modulation scheme with its constant envelope signal is in agreement with the European Committee for Electrotechnical Standardization Norms (CENELEC) [19]. The M -ary FSK is considered to be an orthogonal frequency modulation scheme, the same as OFDM modulation. In this communication 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 vector r is given by r = [r1 , . . . , rM ], with (√ Es ejφ + ni,c + jni,s , for ri = ni,c + jni,s , for  p(r|sk ) =

1 2πσv2

N e

(|r|2 +Es2 )/2σv2

i = k, i 6= k, p

I0

Es |rk |2 σv2

! ,

where, Es represents the energy per symbol, φ, the phase shift of the signal and I0 is the modified Bessel function of order zero. At the demodulator the optimum detector computes the magnitude of the different coordinates of the received vector and 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 improve the performance of the code when combined with M -FSK since we are using the measure of the bit error rate (BER). On the other hand, Fig. 3(b) shows that the increase of the value of M decreases the performance of the code when combined with 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 noncoherent demodulation will have great importance in the correction of the narrow band interferences in a power-line channel using different detectors as it will be presented in the following section.

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

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

B. advantage The combination of M-FSK modulation and coding for a constant envelope modulation signal has the advantage of having a frequency spreading that can help us avoiding some bad parts of the frequency spectrum to facilitate correction of frequency disturbances and impulse noise simultaneously. M-FSK has the advantage of a constant envelope signal modulation and a demodulation in a coherent as also in a non-coherent way. as it is known, in a M-FSK modulation scheme, symbols are modulated as one of the sinusoidal waves described by the following description [1], r 2Es cos(2πfi t) ; 0 ≤ t ≤ Ts (3) si (t) = Ts taking 1 ≤ i ≤ M and Es as the signal energy per modulation symbol and fi = f0 + i−1 Ts , 1 ≤ i ≤ M . The correspondence between symbols generated by the distance preserving mappings and the frequencies from the M-FSK modulation make this combination very practical in a environment like the powerline communication channel.

Finally we can say that the modulation scheme may play an important role in the error correction process. As we can say that the M-FSK is the best candidate for power line communications channel. IV. A PPLICATION : C OMBINED P ERMUTATION T RELLIS C ODES AND M- ARY FSK FOR THE NARROWBAND C HANNEL In this section, we make use of different detectors that M-FSK modulation can use and see the performance of our permutation trellis codes when narrowband noise [20] is present. We can describe the narrowband noise as a continuous disturbing source that effects one of the transmitted frequency signals. This source can have a probability of presence denoted by pn for a duration of Tn symbols, where Tn ∈ [0, ∞). In this example of application, we assume and model the narrowband noise signal as a signal with an energy Ens = 4Es , which is chosen to reflect the fact that it is much higher than the energy

and the sum on the Hamming distance when using the optimal construction. The distance-preserving mappings technique also kept the number of state the same as the base codes which makes Viterbi decoding algorithm simple and more applicable. The M-FSK modulation scheme has shown also its robustness in a hostile environment as the power line channel in the presence of permanent frequency disturbances which are also called narrow band interferences. All these advantages make of the combined permutation trellis codes to M-FSK modulation scheme to be the best choice for power-line communications channel. R EFERENCES

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

of the transmitted symbols. This will cause a total saturation of the signal at the corresponding frequency. The corresponding frequency of the effected signal by the narrowband noise is denoted by fns = fi , where fi corresponds to one of the transmission frequencies (in our simulation we just add the value two to the symbols’ energy). Example 3 As discussed previously, when combining PTC codes with an M-FSK modulator, every symbol, 1, 2, . . . , M corresponds uniquely to one of the M frequencies. The M-ary symbols are transmitted in time as the corresponding frequencies, thus the transmitted signal has a constant envelope. In this example, we make use of the case of M = 8 and also make use of the cube construction to design our permutation trellis 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 the channel. We make use here of distance-increasing mappings to design our permutation trellis codes. We run our simulation for uncoded data (UC) and our PTC codes for different detectors as the envelope detector (ED), the threshold detector (TD) for different values of its threshold τ . And also for the Viterbi’s ratio threshold detector (VRTT) for different values of λ. Although TD and VRTT detector are designed for erasure channels, we make use of them of their soft decision properties and consider our channel as not an erasure channel. The obtained results for the ED detector, the TD detector and the VRTT detector are respectively presented in Figures 4, 5 and 6. 2 V. C ONCLUSION We have shown from the investigation conducted in this paper and the obtained results, that the technique of distancepreserving mappings to design permutation trellis codes is a good choice for power line communication when combined with M-FSK modulation. This technique helped allocate symbols to corresponding frequencies when combined with MFSK modulation. And also helped increasing the free distance

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