Error Propagation Significance of Viterbi Decoding of Modal and Non-Modal Ternary Line Codes Khmaies Ouahada, Member, IEEE Department of Electrical and Electronic Engineering Science University of Johannesburg, P.O. Box 524, Auckland Park, 2006, South Africa Email: kouahada@uj.ac.za Abstract The state machine representation of ternary line codes helped making use of the Viterbi decoder (VD), which is considered to be the maximum likelihood decoding algorithm. Viterbi decoding algorithm provides 2 db gain between soft and hard decisions decoding. Since Viterbi decoding algorithm is usually designed for linear convolutional codes, we expect error propagation when applied to ternary line codes. These codes have gain the trellis structure from their state machine representation. To analyze the stability and the accuracy of the Viterbi decoding algorithm, we use the technique of error propagation to assess that stability. We discuss also the reasons behind these error propagation from a code to another and a class to another of ternary line codes as they are presented into modal and nonmodal groups. An objective statement summarizing the observations have been concluded from the conducted simulation study and analysis. Index Terms Ternary line codes, Error propagation, Modal, Non-Modal. I. INTRODUCTION Ternary line codes [1], such as alternative mark inversion and high density bipolar three are nonlinear codes and frequently used in metallic cable systems due to their advantages as the efficient utilization of bandwidth and the DC-free power spectral density function [2]. Although advanced countries have moved to high speed digital transmission technologies as in the case of the optical fiber channel [3] [5], twisted pair cables used in developed countries will function for many years to come. The idea of enhancing the performance and the quality of the digital communications within the twisted pair cable systems is becoming day after day very important and an urgent matter. To study the behavior of the Viterbi decoding algorithm [6] [8], most of the researches and published papers make use in their investigations of linear convolutional codes. The results show the well known 2 db gain for 3-bit quantization at the bit error rate value of 10 6, between soft and hard decisions on wideband Gaussian channels. In this paper we investigate the possibility of having successfully the same results with ternary line codes and the reasons behind the failure. Since the filling patterns is based on the violations used by HDBn and BnZS line codes to overcome consecutive zeros, this will have an impact on the complexity of the sequences generated by this two classes of ternary line codes. We investigate in this paper the differences of gain and the reasons behind such differences when an error propagation test in conducted. The organization of the paper is as follows. In Section II we introduce certain most popular ternary line codes and explain their filling patterns and present their Viterbi decoding results between soft and hard decisions. Section III investigates the Viterbi decoding error propagation for our ternary line codes. The reasons behind the propagation of errors by Viterbi decoding is investigated in Section IV. Finally a conclusion is made to compare between the obtained results. II. VITERBI DECODING TERNARY LINE CODES In the literature [6], it was shown that with soft decisions Viterbi decoding, we have an improvement of 2 db gain over hard decisions for the case of linear binary convolutional codes. As mentioned before, ternary line codes [9], [10] are considered to be nonlinear codes and often are used on channels such as PCM metallic cable systems with transformer decoupling and repeaters. Since these codes were presented in a sate machine [11] form, the trellis presentation and the use of the Viterbi decoding algorithm becomes possible. In the following, we present certain properties of certain useful ternary line codes like AMI, HDB3 and B4ZS and we also investigate the possibility of having the 2 db gain between soft and hard decisions that Viterbi offers with convolutional linear codes. A. Alternative Mark Inversion (AMI) This type of ternary line codes is considered to be a pseudoternary code [1] and also considered to be the benchmark of all ternary line codes. AMI is designed to give alternating positive and negative pulses for consecutive information data of 1s. For a transmitted digital data, the AMI encoder will consider all the binary zeros as ternary zeros and the alternated marks will be inverted, starting with a + for the first mark. The encoding mechanism can be seen from the state machine as presented in Fig. 1. A simulation set up in Matlab to run the soft and hard decisions for AMI is used and the obtained results are shown
Fig. 1. State Machine of AMI. Fig. 3. Fig. 2. State Machine of HDB3. AMI: Viterbi decoding soft and hard decisions. Fig. 4. HDB3: Viterbi decoding soft and hard decisions. in Fig. 2, where we can see easily the 2 db gain between the soft and hard decisions Viterbi decoding at the BER = 10 6. B. High Density Bipolar 3: HDB3 To overcome the problem of consecutive zeros that AMI can t handle, the maximum number of consecutive zeros is limited to 3 in DB3 codes, where a violation (V) will occur to swap the sings of the consecutive 1s. This will give an advantage to HDB3 over AMI of a better synchronization between the receiver and the transmitter and a low frequency cut-off point provided in power spectral density function. To understand better how the patterns of this ternary line codes are presented we can say that the digital encoded data for HDB3 is represented almost in an identical fashion to AMI except for allowances made to accommodate certain violations. The state machine if AMI is much simpler that the one for HDB3, which is depicted in Fig. 3. The patterns of this kind of codes are described as there is no changes in voltage for a sequence of 0s is solved by changing any incidence of four consecutive 0 bits into a stream containing 000V, where the polarity of the V bit is the same as the previous non-0 voltage (opposite to a 1 bit, which causes a V signal with an alternate voltage according to the previous one). But a new problem arises - because the polarity of the non-zero bits is the same, a non-zero DC level is formed. This is overcome by changing the polarity of the V bit to the opposite of the previous V bit. This changes the bit stream to B00V, where the polarity of the B bit is the same as the polarity of the V bit. The change fools the receiver into thinking a received B bit is a 1 bit, but when it receives the V bit (with the same polarity), it understands the B and the V bits as a 0. In HDB3, the maximum number of consecutive zeros allowed in the substituted string is 3. Using the same simulation setup as previously done with AMI, we found that the difference between soft and hard decisions decoding is a bit less than the 2 db gain at the BER = 10 6 as depicted in Fig. 4. C. Binary Four Zeros Substitution (B4ZS) We make use here of B4ZS ternary line code from the BnZS class of ternary line codes used in North America. It is possible to use the Viterbi decoding algorithm with this code since it has a state machine presentation as shown in Fig. 5. The number of states of this type of codes is depending on the filling patterns that us used to generate this codes. In the case where we make use of the VBVB filling pattern as the filling pattern for B4ZS line code, the state machine will consist of 18 states arranged symmetrically around a horizontal center line as shown in Fig. 5. Every transition from a state in the upper-half, following a data 1, has its destination in the lower half, and vice versa. This feature corresponds to adherence to the bipolar
Fig. 5. State Machine of B4ZS. TABLE I PROBABILITY OF n ERRORS Fig. 6. B4ZS: Viterbi decoding soft and hard decisions. Ternary Number of propagation errors n Line Codes 0 1 2 3 4 5 6 7 AMI 0.29 0.02 0.69 0 0 0 0 0 HDB1 0.54 0.03 0.25 0.14 0.04 0 0 0 HDB2 0.49 0.01 0.28 0.08 0.09 0.05 0 0 HDB3 0.22 0.06 0.49 0.09 0.08 0.03 0.02 0.01 CHDB3 0.38 0.01 0.48 0.06 0.02 0.05 0 0 B4ZS 0.34 0 0.66 0 0 0 0 0 B6ZS 0.34 0 0.66 0 0 0 0 0 alternation rule. The pair of states at the left-hand side of the state diagram is occupied whenever the data contains a long string of consecutive data 1s. One of the pair of states at the right-hand side of the diagram is occupied whenever a data 1 is followed by 3 consecutive data 0s. Exiting from them on a data 0 corresponds to commencing the filling sequence. Considering just the upper state of the pair, it is entered only by an arc associated with the previous output +, and on data 0, it begins producing the output sequence + +, that is VBVB. Using the same simulation setup, Fig 6 shows the 2 db gain between hard and soft decisions Viterbi decoding. III. VITERBI DECODING ERROR PROPAGATION Viterbi algorithm is based on the calculation of the distances between the received and the expected transmitted information data in each branch of the trellis diagram designed from the state machine of the convolutional code. Previous published work showed the modeling of line codes to have a state machine presentation, which is important for the use of the Viterbi algorithm. Fig. 7. Error propagation of certain ternary line codes, due to a random single isolated channel error. A simulation experiment was conducted to analyze and prove that the coding gain is predominantly determined by the error propagation when Viterbi decoding fails. The experiment in [12] is simply based on the generation of widely separated single random errors between the levels of the code s symbols, and the observation of the number of errors propagated. The results for certain ternary line codes that we have used in this paper are shown in Table I. These results are also presented differently in Fig. 7. Table II shows the relationship between the achieved gain using 3-bit quantization at BER = 10 6 between soft and hard decisions and the expected number of propagated errors for our line codes and their relations to the modal and nonmodal classes of ternary line codes. It is clear from the previous results that the expected number of error propagation increases when the complexity of the filling pattern increases. As an example, the expected number of error propagation for HDBn line codes, is higher than the expected number of error propagation for BnZS line codes.
TABLE II MODAL AND NON-MODAL CLASSES VS ERROR PROPAGATION Class Ternary Gain using 3-bit Expected number of Maximum number of Line Codes quantization at BER = 10 6 propagated errors propagated errors Non-Modal B4ZS 2.05 1.32 2 B6ZS 2.00 1.34 2 HDB1 1.85 1.40 5 HDB2 1.55 1.42 6 Modal HDB3 1.35 1.48 6 CHDB3 1.45 1.66 7 TABLE III FILLING SEQUENCES FOR B6ZS AND HDB3 Polarity of last violation Preceding pulse Substitution sequence Filling sequence B6ZS (non-modal) + 0+-0-+ 0VB0VB - 0-+0+- 0VB0VB HDB3 (modal) + + -00- B00V - - +00+ + - 000-000V - + 000+ TABLE IV A BINARY STREAM CODED INTO AMI, B6ZS AND HDB3 CODES Binary 1 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 1 1 0 0 0 0 0 0 0 1 1 AMI + - 0 + 0 0 0 0 - + 0 0 0 0 0 0-0 + 0 - + - 0 0 0 0 0 0 0 + - B6ZS + - 0 + 0 0 0 0 - + 0 + - 0 - + - 0 + 0 - + - 0 - + 0 + - 0 + - 0 V B 0 V B 0 V B 0 V B HDB3 + - 0 + 0 0 0 + - + - 0 0-0 0 + 0-0 + - + 0 0 0 + 0 0 0 - + 0 0 0 V B 0 0 V 0 0 0 V IV. REASONS BEHIND THE PROPAGATION OF ERRORS BY VITERBI DECODING The results of the gain using 3-bit quantization at BER = 10 6 in Table II, show that HDB1, HDB2, HDB3 and CHDB3 are those codes in which we could not reach the 2 db gain of the soft decision over the hard decision. But, in the case of B4ZS and B6ZS we could obtain the 2 db gain. We have divided our filled bipolar codes into two groups called non-modal group, which have only one filling sequence, and modal codes, which have more than one filling sequence. We observed that those line codes that did not reach the 2 db gain belong to the modal group. The question is why the VD of modal ternary line codes propagates more errors than others. If we take two ternary line codes from different groups, e.g. HDB3 and B6ZS, where we investigate through their filling sequence how the code which has more than one filling sequence propagates more errors. It is clear from Table II, that the gain difference between the hard and soft decision for B6ZS is 2 db, while that for HDB3 is 1.35 db. The results show that the VD HDB3 propagates more errors than VD B6ZS. From Table III, we can observe the difference in the filling sequence of both codes and we can explain in greater details the reason behind the propagation of errors of each VD. From Table III it can be seen that the filling sequence of HDB3 is much more complicated than that of B6ZS, because in the case of HDB3, we need to know the
polarity of the last violation in conjunction with the preceding pulse, which is not the case with B6ZS, where we need only to know the preceding pulse. This is why for HDB3 code we have four possibilities of filling sequence or substitution sequence as defined in the table. Whereas with B6ZS code we have only two. The complexity of the filling sequence of a Non-Modal group compared to a Modal group can be seen clearly in Table IV. We can notice from the above that the filling sequence of B6ZS affects only the consecutive zeros. This means the binary stream codes into B6ZS will differ to AMI only in the area of zeros and the rest will be the same as AMI. The situation with HDB3 is totally different because for modal codes, we have, besides the filling sequences- which is a solution for the consecutive of zeros- a change of data also in between the filling sequence. Thus, the complexity of Modal codes is very clear from the previous example and this causes the VD to propagate more errors than in the non-modal codes. [8] K. Ouahada and H. C. Ferreira, Simulation study of the performance of ternary line codes under Viterbi decoding, in IEE Proc-Commun., vol. 151, no. 5, pp. 409 414, 2004. [9] Buchner J. B. Ternary Line Codes, Philips Telecommunications Review, vol. 34, no. 2, pp. 72 86, June 1976. [10] H. C. Ferreira, J. F. Hope, and A. L. Nel, On Ternary Error Correcting Line Codes, IEEE Trans. Commun. vol. 37 no. 5, pp. 510 515, May 1989. [11] D.B. Keogh, Finite-State Machine Descriptions of Filled Bipolar Codes, A.T.R. vol. 18, no. 2, pp. 3-12, June 1984. [12] N. Q. Duc, Line coding techniques for baseband digital transmission, in Australian Telecommunications Research, vol. 9, no. 1, pp. 351 357, 1977. V. CONCLUSION From the obtained results it was clear that the error propagation Viterbi decoding increases with the complexity of the pattern of the code as the case between the HDBn codes and BnZS line codes. It can be seen clearly that the pulse B always ends the filling sequence of B6ZS line code and this will have no effect on the coming data of the message as presented in boxes in Table IV, where we can see that the output - is similar to the output of AMI at the same position. Whereas for HDB3 line codes, we always have the violation V at the end of the filling sequence, and this has an affect on the coming data of the message, where we can see that the corresponding output at the same position changed to +. This is why we have to choose the number of pulse B as odd in between two consecutive V to correct the data. From here it is clear that for modal class of ternary line codes, we have higher complexity in coding data and this creates difficulties for the Viterbi decoding algorithm to correct the information data easily compared to the Non-Modal class of ternary line codes. REFERENCES [1] A. Croisier, Introduction to Pseudo-ternary Transmission Codes, IBM J. Res. Develop., vol. 14, pp. 354 367, Jul. 1970. [2] G. L. Pierobon, Codes for zero spectral density at zero frequency, IEEE Trans. Inf. Theory, vol. 30, pp. 435 439, Mar. 1984. [3] R. M. Brooks and A. Jessop, Line coding for optical fibre systems, in International Journal of Electronics, vol. 55, no. 1, pp. 81 120, 1983. [4] S. D. Personick, Optical Fiber Transmission Systems,New York: Plenum, 1981. [5] C. Matrakidis and J. J. O Reilly, A Block decodable line code for high speed optical communication, in Proceedings of the International Symposium on Information Theory, pp. 221, Ulm, Germany, June 29 Jul 4, 1997. [6] A. Viterbi and J. Omura, Principles of Digital Communication and Coding, McGraw-Hill Kogakusha LTD, Tokyo Japan, 1979. [7] G. David Forney, JR., The Viterbi Algorithm, Proceedings of the IEEE, vol. 61, no. 3, pp. 268 278, Mar. 1973.