Simulation Results for Permutation Trellis Codes using M-ary FSK

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1 Simulation Results or Permutation Trellis Codes using M-ary FSK T.G. Swart, I. de Beer, H.C. Ferreira Department o Electrical and Electronic Engineering University o Johannesburg Auckland Park, South Arica {ts,idb,hc}@ing.rau.ac.za A.J.H. Vinck Institute or Experimental Mathematics University o Essen Essen, Germany vinck@exp-math.uni-essen.de Abstract It has previously been shown that using the combination o permutation codes and M-ary requency shit keying has special properties and error correcting capabilities that are suitable or the noise types encountered in power line communications. Furthermore, the permutation codes are used to map onto the outputs o a binary convolutional code to orm permutation trellis codes. We investigate and compare the perormance o dierent permutation trellis codes when used with M-FSK or power line communications. Keywords-channel coding; convolutional codes; intererence suppression I. INTRODUCTION Previous research [1,2] has shown how combining M-ary requency shit keying (M-FSK) with permutation codes can be used to correct broad-, narrowband and background noise. A quick overview o this is given in Section II. It has also been shown [] how distance mappings can be used to map permutation codes onto the outputs o a binary convolutional code to orm permutation trellis codes. This is covered in a brie overview in Section III. Section IV discusses the simulations and the results obtained or various mappings o permutation trellis codes. II. PERMUTATION CODES AND M-ARY FSK The deinition or a permutation code is as ollows: Deinition 1: A permutation code C consists o C code words o length M, where every code word contains the M dierent integers 1,2,,M as symbols. Every symbol corresponds uniquely to a requency rom an M-FSK modulator. The M-ary symbols are transmitted in time as the corresponding requencies, thus the transmitted signal has a constant envelope. The demodulator consists o a modiied envelope detector or each requency, that outputs a 1 i the signal envelope is above a certain threshold and outputs a 0 otherwise. Thus or each symbol transmitted, M outputs are obtained rom the demodulator. These result in an M M binary matrix that is used or decoding, where the rows represent the requencies used and the columns represent the position or time in the code word. When combined with convolutional codes (as described in the next section) this binary matrix is used in the Viterbi decoder. Example 1: The M=4 permutation code word (1,4,2,) is sent. I received correctly, the output o the demodulator would be t t t t where i represents the output or the detector at requency i and t j represents the time interval j in which it occurs, or 1 i,j 4. Channel noise causes errors in the matrix, which can be represented by the ollowing: background noise insertion or deletion o ones impulse noise a complete column is set to ones narrowband noise a complete row is set to ones /05/$20.00 c2005 IEEE. 17

2 III. PERMUTATION TRELLIS CODES Using convolutional codes it is possible to easily decode permutation codes using a trellis and the Viterbi algorithm. The outputs o a binary convolutional encoder are mapped onto the code words rom a permutation code. A mapping consists o choosing an ordered subset o 2 n M-tuples, rom the ull set o permutation M-tuples, to map onto the corresponding convolutional base code s n-tuples. The subset is chosen such that the Hamming distance between any two permutation M- tuples is at least as large as the distance between the corresponding convolutional code s output n-tuples which are mapped onto them. This property was previously called distance preserving in [4], since the Hamming distance o the base code is at least conserved, and may sometimes even be increased in the resulting trellis code. Example 2: Using the standard R=1/2, ν=2, d ree =5 convolutional code, we map n=2 M= by applying the ollowing mapping: {00,01,10,11} {21,21,12,12}. This results in the state systems as shown in Fig (0) 11 (0) 11 (1) 01 (0) 0 00 (1) (0) 10 (1) 01 (1) 21 (0) 12 (0) 12 (1) 21 (0) (a) (b) Figure 1. State systems or (a) convolutional base code and (b) permutation trellis code It can easily be veriied that the distance between any permutation code words and the corresponding binary outputs increases by one. The shortest remerging paths in the trellis, which or this code determines the ree distance, have a length o three steps. Thus or each step there is an increase in distance o one, resulting in a ree distance o d' ree =8 or the permutation trellis code. Three dierent mappings can be obtained, depending on how the Hamming distance is preserved: distance conservative mapping (DCM) guarantees conservation o the base code s ree distance distance increasing mapping (DIM) guarantees that the resulting trellis code s distance will always have some increase above the base code s ree distance distance reducing mapping (DRM) has a distance loss which is guaranteed to be not more than a ixed amount per step between any two unremerged paths in the trellis diagram o the resulting trellis code (1) (0) 12 (1) 21 (1) IV. SIMULATION RESULTS A simple error model, which generates errors in the received matrix according to certain error parameters, was used to evaluate the dierent mappings. The error parameters were assumed to be equal or all requency sub-bands. The dierent types o noise are generated as ollows (similar to Example 1): background noise each element in the received matrix has a probability, p b, o being in error impulse noise each column in the received matrix has a probability, p i, o resulting in an impulse noise permanent requency disturbance (or narrowband noise) all received matrices will have 1 s in the row that corresponds to the requency error The ollowing mappings were used: (1) DIM, mapping n=4 onto M=5 while increasing the distance by 1, ound by trail and error, (2) DCM 1, mapping n=5 onto M=5 while conserving the distance, obtained by applying the preix construction method rom [], () DCM 2, mapping n=5 onto M=5 while conserving the distance, ound by exhaustive search, and (4) DRM, mapping n=6 onto M=5 while decreasing the distance by at most 1, ound by trail and error. The exact mappings used can be ound in Table I. TABLE I. Description n=4 M=5 DIM n=5 M=5 DCM 1 n=5 M=5 DCM 2 n=6 M=5 DRM DISTANCE MAPPINGS USED TO OBTAIN RESULTS Mapping a 1245,1452,1452,1524,2514,2514, 2145,2451,1542,2154,4215,5421, 5241,5241,5124, ,5124,5124,5142,5142,5142, 5214,5214,5214,5241,5214,5241, 5421,5412,5124,5142,4125,4125, 4125,4152,4152,4152,4215,4215, 4215,4251,4215,4251,4521,4512, 4125, ,2145,1254,2415,2154,1245, 2514,2145,1524,5142,2514,5241, 4125,2154,1524,5142,1425,1245, 4251,5412,4251,2415,4512,4152, 5421,5241,4521,5421,5214,4512, 4521, ,1254,1254,1254,1245,1254, 1542,1524,1425,1452,1452,1452, 1524,1542,1542,1542,2145,2154, 2154,2154,2145,2154,2541,2514, 2415,2451,2451,2451,2514,2541, 2541,2541,4215,4215,4251,4251, 4125,4125,4152,4152,4512,4512, 4521,4521,4251,4215,4521,4512, 2154,2145,2514,2541,1254,1245, 1524,1542,5241,5214,5421,5412, 4251,4215,4521,4512 a. Binary code words have been omitted. Use normal lexicographical ordering. 18

3 These M=5 mappings were used as ollows: (1) DIM was mapped onto a rate R=/4, ν=2, d ree =, punctured convolutional base code, (2) DCM 1 and () DCM 2 were mapped onto a rate R=/5, ν=2, d ree =4, punctured convolutional base code, and (4) DRM was mapped onto a rate R=/6, ν=2, d ree =5, punctured convolutional base code. These rates ensured that the overall system rate was the same in all our cases. Also, in all cases the guaranteed ree distance o the permutation trellis code, d' ree, is equal to 4. The mappings only guarantee a minimum increase or maximum loss o distance. Thus it is generally able to exceed these, with even the DRM showing an increase o distance on average. For the mappings used, we have an average distance increase as ollows: (1) DIM with 2.10, (2) DCM 1 with 1.06, () DCM 2 with 1.54, and (4) DRM with This would suggest that the DCM 2 should perorm better than the DCM 1. Fig. 2 compares the mappings when only one permanent requency disturbance occurs in the presence o background noise. This was done with the disturbance present on a dierent requency position each time. The DIM is perorming the best, with the perormance irrespective o which requency the disturbance is on. The DCM 1 is the worst with perormance varying, depending on the position o the disturbance. Fig. compares the mappings when zero, one (in position 1), two (in positions 1 and 2) and three (in positions 1, 2 and ) requency disturbances respectively occur at the same time in the presence o background noise. Again, the DIM is perorming the best, with the DCM 2 and DRM very close to each other. While the DCM 1 was close to the others or no disturbances, its perormance degrades signiicantly as the number o disturbances increases. The previous result has shown that this mapping s perormance depends on the disturbance positions, thus certain combinations o disturbances can render it useless (as in the case o three disturbances on positions 1, 2 and ). Other combinations might result in slightly better perormance. V. CONCLUSION Various simulation results have been presented or dierent mappings to be used with permutation trellis codes combined with M-FSK. The eect o permanent requency disturbances in the presence o background noise was investigated, taking into account the position and number o disturbances. The results gave a clear indication o how the various mappings perormed under these conditions. REFERENCES [1] A.J.H. Vinck, Coded Modulation or Powerline Communications, AEU International Journal o Electronics and Communications, vol. 54, no. 1, pp , [2] A.J.H. Vinck, A. Hasbi, H.C. Ferreira and T.G. Swart, On Coded M-ary Frequency Shit Keying, Proceedings o the International Symposium on Power Line Communications 2004, Zaragoza, Spain, pp , March 1-April 2, [] H.C. Ferreira and A.J.H. Vinck, Intererence Cancellation with Permutation Trellis Codes, Proceedings o the IEEE Vehicular Technology Conerence Fall 2000, Boston, MA., U.S.A., pp , September 24-28, [4] H.C. Ferreira, D.A. Wright and A. L. Nel, Hamming Distance Preserving Mappings and Trellis Codes with Constrained Binary Symbols, IEEE Transactions on Inormation Theory, vol. 5, no. 5, pp , September

4 M=5 DIM M=5 DCM 1 M=5 DCM 2 M=5 DRM PFD in pos 1 PFD in pos 2 PFD in pos PFD in pos 4 PFD in pos 5 Figure 2. Single permanent requency disturbance (PDF) in dierent positions 20

5 0 PFD 1 PFD 2 PFD PFD M=5 DIM M=5 DCM 1 M=5 DCM 2 M=5 DRM Figure. Comparison o various number o permanent requency disturbances (PFD) 21

THIS LETTER reports the results of a study on the construction

THIS LETTER reports the results of a study on the construction 1782 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 11, NOVEMBER 2005 Permutation Trellis Codes Hendrik C. Ferreira, Member, IEEE, A. J. Han Vinck, Fellow, IEEE, Theo G. Swart, and Ian de Beer Abstract