Implementation and performance analysis of convolution error correcting codes with code rate=1/2.

Transcription

1 2016 International Conference on Micro-Electronics and Telecommunication Engineering Implementation and performance analysis of convolution error correcting codes with code rate=1/2. Neha Faculty of engineering and technology Jamia Hamdard New Delhi, India Abstract- Channel coding plays a vital role for the maintenance of quality and reliability in data delivery in a communication system. Practical applications of channel coding includes the area of deep space communication, satellite communication, wireless communication, data transmission etc. Convolution coding is one of large class among them. The main aim of this paper is to elucidate the main features of Convolution coding. This paper explicating the charming reasons which make use of this coding scheme very fluent in long distance communication. Accuracy of data is rationed due to surge in noise level.various features and performance criteria of encoder and decoder has been discussed which give a pertinent ideas to design convolution code. The code is implemented and analysed with varying the Constraint length, number of memory register at code rate=1/2 in AWGN channel. eywords- Channel coding, Convolution encoder, Viterbi decoder. I. Introduction Codes are used for data compression, cryptography, and correction. Codes are first categorized in two forms: a. Source codes: which take data from information source and make it smaller (compressed form).b. Channel Codes: Channel coding refers to the class of signal transformation which improve the performance of communication system. The goal of Channel coding is to find code which transmit quickly, contain many valid code word and can correct or at least detect many errors. Channel coding is very appreciable tool in the design of reliable communication system. Channel coding provides improved error performance by adding redundant information to the input data being transmitted through a channel. Channel coding is having two major form: block coding and convolutional coding [1,9,12]. 1. Data compression (or source coding) 2. Error correction (or channel coding) A functional block diagram illustrating the signal flow and the signal-processing steps through a typical communication system are shown in Figure 2 [1].equations, graphics, and tables are not prescribed, although the various table text styles are provided. The formatter will need to create these components, incorporating the applicable criteria that follow. II. Convolution Codes A convolutional encoder accepts a sequence of message symbols and produces a sequence of code symbols. Its computations depends upon the current set of input symbols as well as the previous input symbols [2]. A block diagram derived from Figure 2, is shown in Figure 3. a. Convolution Codes A convolutional encoder is formed by a fixed number of shift registers. Every input bit get in a shift register and the outcomming of the encoder is perceived by summing the bits in the shift register. The number of outcoming bits depends on the number of modulo 2-adders applied with shift registers. A convolution codes is described by integers m, n, k and. Where, n = number of output bits (Commonly ranges from 1 to 8) k = number of input bits (Commonly ranges from 1 to 8) = constraint length (Commonly ranges from 3 to 9) m= number of memory registers. (Commonly ranges from 2 to ) The quantity k/n is known by the code rate is a measure of the bandwidth efficiency of the code. The code rate from 1/8 to 7/8 other than for deep space implementation where code rates as low as 1/0 (usually below 0.90), or even longer can be employed [3]. The encoder structure of (n=2,k=1=,3) convolution encoder is shown below in figure 4. It is easy to construct a convolutional encoder. We first draw m boxes representing the m memory registers.then we draw n modulo-2 adders (here n=2) representing the n output bits. Finally, we connect the memory registers to the adders using the bits specifying the generator polynomials. The code rate of above encoder is ½. The (2, 1, 3) convolutional encoder has four states, which are: 00, 01,, 11.The encoder operation can be illustrated by any of the three means i.e State diagram representation, Trellis representation, Tree diagram representation. b. Viterbi decoder The Viterbi decoding algorithm was introduced and analysed by Viterbi in 1967.Viterbi decoding was the most effective decoding technique for short constraint length codes by Heller [4,5]. Forney [6] and Omura [ 7].The great advantage of the Viterbi maximum likelihood decoder is that the number of decoder operations performed in decoding N bits is only N2k(-1),which is linear in N. The basic Viterbi decoder consists of five functional units; an input or branch metric calculation section, an ACS arithmetic section, and a path memory and output section. Information can be thought of as passing successively from one section to the next. The branch metric calculation section accepts the input data and /16 $ IEEE DOI.19/ICMETE

2 Figure 1.Classification of codes Figure 2 Block diagram of communication system with source and channel encoding. Figure 3 Encode/decode and modulate/demodulate portions of a communication link. The decision is transmitted to the path memory section and the larger of the two sums becomes a new state metric. One ACS function must be performed for each of the 2-1 states. The path memory section must store about a 4 constraint length history of decisions for each state. The memory requirements are thus nontrivial [8,9,]. Viterbi decoder is further categorized in two parts: hard decision and soft decision. The paper includes the hard decision method. Hard decision is inexpensive and require less memory as compared to soft decision. For calculating the error capability of the code the equation of error correcting capability is written in context of free distance df. Maximum number of guaranteed correctable errors per code word(t)= [(df-1)/2], where df = free distance which is defined as the minimum distance in the set of all arbitrarily long path diverges or remerges [1]. Table 1 and 2 has listed the SNR (Eb/No) in db achieved with MATLAB implementation using Uncoded and coded polynomial with standard as well as tested polynomial. The variation of coding gain with respect to memory size is summarized in figure 12. III. Implementation and Results Convolution code with Viterbi algorithm (hard decision) is implemented and analysed in AWGN channel with,000 bits as an input. And coding gain is calculated for each plot. Coding gain is defined as the reduction in E b/n o that can be realized through the use of code. It is calculated in db. The connection vectors or polynomial generators of a convolutional code are usually selected based on the code s free distance properties. Figure 6 to 12 shows relation between BER and E b/n o of generator polynomials (tested & standard)[,11]. Based on which SNR (in db)as well as coding gain have been calculated and mentioned in tabular form table 1 and table 2. It can be concluded that for a constant BER coding gain increases with increase in memory size. Its value also increases when BER decreases for the same encoder size. As the value of coding gain at encoder size =2 is 0.1 db for BER=-2 and 1.6 db for BER=-4. On calculation of the coding gain for code rate ½, while varying the value of constraint length it is observed that the performance of the system improved in coded curve. The coding gain corresponding to various memory size and polynomial is tabulated below in table 3 and 4 respectively. Figure 4 A (2,1,3) convolution encoder calculates (or looks up) the metric for each distinct branch. For a rate 1/2 code, four branches are possible corresponding to transmission of 00, 01,, and 11, The only section of the decoder that is directly concerned with the number of bits of quantization of the received data, and hence, the only section whose complexity is directly dependent on quantization. The ACS sections perform the basic arithmetic calculation of the decoder. For a rate l / w decoder, an ACS is used to add the state metrics for two states to the appropriate branch metrics, to compare the resulting two sums, and to select the larger

3 Figure 5. Plot of BER vs. Eb/No for a rate 1/2, m=2 encoder Figure 7 Plot of BER vs. Eb/No for a rate ½, m = 4 encoder Figure 6 Plot of BER vs. Eb/No for a rate ½, m = 3 encoder Figure 8 Plot of BER vs. Eb/No for a rate ½, m = 5encoder

4 Figure 9 Plot of BER vs. Eb/No for a rate ½,m=6 encoder Figure11 Plot of BER vs. Eb/No for a rate ½, m=8encoder Figure Plot of BER vs. Eb/No for a rate ½,m=7encoder Figure 12. Plot between Coding Gain Vs Memory size (m) for code rate=1/2. Table 1. SNR (Eb/No) in db for the rate ½ with =3 to 9 (Standard polynomial). BER Unco ded =3 = 4 =5 =6 =7 =8 = Table 2. SNR (Eb/No) in db for the rate ½ convolution encoder for Tested polynomial BER Unco ded

5 Table 3 Coding gain in db for the rate ½ convolution encoder for Tested polynomial. BER Uncoded Table 4 Coding gain in db for the rate ½ convolution encoder for Tested polynomial. B Un E cod = = = = = = =9 R ed iv Conclusion An error correction technique particularly suited for a white Gaussian noise (AWGN) channel, which has been implemented using convolutional encoding with Viterbi decoding. The basic design and implementation of convolution encoder and Viterbi decoder with its essential blocks is discussed here. It is seen, when compared with Figures/Tables that the performance of the convolution code further improves with an encoder having greater number of memory registers or the large constraint length, i.e., a larger value for m or respectively for the same code rate. As was seen from the simulation results obtained, the performance was improved using a convolutional encoder with a larger encoder memory size. Viterbi has derived standard bounds to bit error probability these bounds are particularly tight for the AWGN channel for error probabilities less than about -4. It can be concluded that for a constant BER coding gain increases with increase in memory size. Its value also increases when BER decreases for the same encoder size. This bound has been numerically evaluated over a range of Eb/No. The upper bound supplies performance data at extremely low bit error rates, where simulation results are not available due to excessive computer time required. Finally the results presented here provide the information necessary to evaluate the applicability of Convolution coding & Viterbi decoding to space,satellite or long distance communication systems with a wide range of requirements and constraints. REFERENCES [1] Sklar, Bernard. Digital communications. Vol. 2. NJ: Prentice Hall, [2] umawat, Himmat Lal, and Sandhya Sharma. "An Implementation of a Forward Error Correction Technique using Convolution Encoding with Viterbi Decoding." [3] Jadhao, Mr Vishal G., and Prafulla D. Gawande. "Performance Analysis of Linear Block Code, Convolution code and Concatenated code to Study Their Comparative Effectiveness." [4] Heller, J. A. "Short constraint length convolutional codes." Space Program Summary (1968): [5] Heller, J. A. "Improved performance of short constraint length convolutional codes." Jet Prop. Lab., Space Prog. Summary: [6] Heller, Jerold, and Irwin Jacobs. "Viterbi decoding for satellite and space communication." Communication Technology, IEEE Transactions on 19.5 (1971): [7] Viterbi, Andrew J., and Jim. Omura. Principles of digital communication and coding. Courier Corporation, [8] Heller, Jerold, and Irwin Jacobs. "Viterbi decoding for satellite and space communication." Communication Technology, IEEE Transactions on 19.5 (1971): [9] Shu, Lin, Shu Lin, and Daniel J. Costello. Error control coding. Pearson Education India, [] Jacobs, I. "Practical applications of coding." Information Theory, IEEE Transactions on 20.3 (1974): [11] Sweeney, Peter. Error control coding. Prentice Hall U, [12] Peterson, William Wesley, and Edward J. Weldon. Error-correcting codes. MIT press,