Performance of Block and Convolutional Codes over Noisy Channel
Almas Uddin Ahmed
Faculty of Engineering, Multimedia University
63100 Cyberjaya, Selangor, almas.mmu@gmail.com
63100 Cyberjaya, Selangor, almas.mmu@gmail.com
Abstract
In this paper, we analyze the performance of block and convolutional codes. Comparison of different values of hamming distance is studied by applying the block coding to binary schematic channel (BSC).Subsequently, we investigate the performance of block code having minimum distance of 3, 6, and 11 as well as convolutional code having rates of 1/2, 2/2, and 3/4 for fixed constraint length in additive white Gaussian noise (AWGN) channel. For the same message length, in comparison to block code, convolutional coding provides coding gain of 5 dB and 7.5 dB utilizing hard and soft decision, respectively.
Keywords: Channel coding, block coding, convolutional coding.
Keywords: Channel coding, block coding, convolutional coding.
1. Introduction
In recent years, there has been growing demand for efficient and reliable digital data transmission system. This demand has been accelerated by the emergence of large-scale, high-speed data transmission for the exchange and processing of digital information in both the government and private spheres. One of the major important issues is that to protect signal strength from the noise in time varying channel. In the past few years, researchers endeavor to establish reliable links of wireless systems to get high performance for different technologies.
Channel coding is one of the most suitable method to obtained good performance in AWGN and Rayleigh fading channels [1,13]. In real wireless transmission systems, channels are affected by large number of scatters namely the superposition of delayed, reflected and scattered (buildings, vehicle and other terrain objects) signals as well as buildings, vehicles and other terrain objects. However, in this work we have considered AWGN and BSC channels only.
Some previous work on channel encoding and decoding designs for wireless communication systems are proposed in [3,9-12]. Other work on convolutional coding [2] can be found in [4] that provide the coding gain and compare the different rates. The main objective of this paper is to demonstrate coding gains of convolutional code in comparison to block code in the terms of same message length. The performance of these coding schemes is also compared with uncoded systems. The importance of Hamming distance is also illustrated in this paper.
The remainder of this paper is organized as follow. In Section 2, background information on channel coding is presented. Overview of block and convolutional codes are discussed in Section 3 and Section 4, respectively. To compare the performance of block and convolutional codes, simulation results are explained in Section 5. Finally, we conclude our paper in Section 6.
2. Background on Channel Coding
In 1948, Shannon published a seminal paper on the proper way of information coding without sacrificing the rate of information [5]. Shannon worked a great deal of effort on information to encode and decode the error control within a noisy channel. It can be seen that it is significant to have high quality digital transmission in modern wireless communication systems and error control has became an integrated part of digital transmission systems.
Involvement of channel coding in data transmission is to protect the signals from the unwanted noise such as interference and fading by increasing the systems performance at the receiver. In this work, block coding is applied to BSC channel with different hamming distances. Subsequently we simulate different rates of trellis and block coding performance for transmission over AWGN channel.
At the encoding stage, it is possible to apply source coding and channel coding. However, in this paper, we only focus on channel coding. Both block code and convolutional code are applied as block coding and trellis systems, respectively. Before transmission, the encoded signals are modulated, whereby differential phase shift keying (DPSK) modulation are considered in our work. Then, the modulated signals are transmitted through the AWGN and BSC channel. As for the decoding system at the receiver end, a Viterbi decoder with free distance is used.
The level of noise and interferences could be high when the signal finally arrived at the receiver. Error detection and correction (EDC) could be employed to enable good communications with lower bit errors at the receiver [7]. The amount of EDC required and its effectiveness depend on the signal to noise ratio (SNR). There are few ways to increase the performance of signal strength and one of these is by increasing the SNR power level. The other way is by decreasing the signals noise.
Assuming that we are free to do anything in the environment, the easiest solution is to increase the signal power but in the wireless communications, the power level of each mobile station is limited thus the transmitted power cannot exceed a certain point. In fact, increasing the power through amplification means both signal and noise are amplified making the system performance even worse. i.e. radio systems.
This is when coding comes into action where instead of increasing the power level, techniques of coding are able to reduce the noise level in different communication channels. Coding has different terminology and characteristics. It is true that the noise level reduction technique through coding is an excellent scheme for any communication systems.
3. Block Coding
The block coding is often referred to as (n,k). A block of k information bits are coded and become a block of n encoded bits. However the error correction of the system needs Hamming distance. In general, the maximum number of error is given by [6]
In block coding, the generator matrix G is an binary matrix and c is the codeword. These are described as follows with generator matrix of , and . Notation represents an identity matrix representing the binary symbol codeword, while P is an matrix representing the parity check symbols and u denotes an matrix whose rows are all binary sequences of length k.
Consider a channel as BSC, then the error probability of a linear block coding with minimum distance in hard-decision decoding is given by [8].For soft-decision decoding, the message error probability is where is the number of codeword and is the minimum Euclidean distance.
4. Convolutional Coding
Besides block coding, convolution coding is the other major class of channel coding that can be applied for error correction [6], whereby convolution coding technique has three parameters namely n, k and m. One important feature of convolutional coding is the constraint length generator polynomial. This acts as trellis structure that provides coding gain, which does not available in other coding schemes. This constraint length is given as where k is the number of input bits, m is the number of shift register and the quantity k/n is called the code rate.
In recent years, there has been growing demand for efficient and reliable digital data transmission system. This demand has been accelerated by the emergence of large-scale, high-speed data transmission for the exchange and processing of digital information in both the government and private spheres. One of the major important issues is that to protect signal strength from the noise in time varying channel. In the past few years, researchers endeavor to establish reliable links of wireless systems to get high performance for different technologies.
Channel coding is one of the most suitable method to obtained good performance in AWGN and Rayleigh fading channels [1,13]. In real wireless transmission systems, channels are affected by large number of scatters namely the superposition of delayed, reflected and scattered (buildings, vehicle and other terrain objects) signals as well as buildings, vehicles and other terrain objects. However, in this work we have considered AWGN and BSC channels only.
Some previous work on channel encoding and decoding designs for wireless communication systems are proposed in [3,9-12]. Other work on convolutional coding [2] can be found in [4] that provide the coding gain and compare the different rates. The main objective of this paper is to demonstrate coding gains of convolutional code in comparison to block code in the terms of same message length. The performance of these coding schemes is also compared with uncoded systems. The importance of Hamming distance is also illustrated in this paper.
The remainder of this paper is organized as follow. In Section 2, background information on channel coding is presented. Overview of block and convolutional codes are discussed in Section 3 and Section 4, respectively. To compare the performance of block and convolutional codes, simulation results are explained in Section 5. Finally, we conclude our paper in Section 6.
2. Background on Channel Coding
In 1948, Shannon published a seminal paper on the proper way of information coding without sacrificing the rate of information [5]. Shannon worked a great deal of effort on information to encode and decode the error control within a noisy channel. It can be seen that it is significant to have high quality digital transmission in modern wireless communication systems and error control has became an integrated part of digital transmission systems.
Involvement of channel coding in data transmission is to protect the signals from the unwanted noise such as interference and fading by increasing the systems performance at the receiver. In this work, block coding is applied to BSC channel with different hamming distances. Subsequently we simulate different rates of trellis and block coding performance for transmission over AWGN channel.
At the encoding stage, it is possible to apply source coding and channel coding. However, in this paper, we only focus on channel coding. Both block code and convolutional code are applied as block coding and trellis systems, respectively. Before transmission, the encoded signals are modulated, whereby differential phase shift keying (DPSK) modulation are considered in our work. Then, the modulated signals are transmitted through the AWGN and BSC channel. As for the decoding system at the receiver end, a Viterbi decoder with free distance is used.
The level of noise and interferences could be high when the signal finally arrived at the receiver. Error detection and correction (EDC) could be employed to enable good communications with lower bit errors at the receiver [7]. The amount of EDC required and its effectiveness depend on the signal to noise ratio (SNR). There are few ways to increase the performance of signal strength and one of these is by increasing the SNR power level. The other way is by decreasing the signals noise.
Assuming that we are free to do anything in the environment, the easiest solution is to increase the signal power but in the wireless communications, the power level of each mobile station is limited thus the transmitted power cannot exceed a certain point. In fact, increasing the power through amplification means both signal and noise are amplified making the system performance even worse. i.e. radio systems.
This is when coding comes into action where instead of increasing the power level, techniques of coding are able to reduce the noise level in different communication channels. Coding has different terminology and characteristics. It is true that the noise level reduction technique through coding is an excellent scheme for any communication systems.
3. Block Coding
The block coding is often referred to as (n,k). A block of k information bits are coded and become a block of n encoded bits. However the error correction of the system needs Hamming distance. In general, the maximum number of error is given by [6]
In block coding, the generator matrix G is an binary matrix and c is the codeword. These are described as follows with generator matrix of , and . Notation represents an identity matrix representing the binary symbol codeword, while P is an matrix representing the parity check symbols and u denotes an matrix whose rows are all binary sequences of length k.
Consider a channel as BSC, then the error probability of a linear block coding with minimum distance in hard-decision decoding is given by [8].For soft-decision decoding, the message error probability is where is the number of codeword and is the minimum Euclidean distance.
4. Convolutional Coding
Besides block coding, convolution coding is the other major class of channel coding that can be applied for error correction [6], whereby convolution coding technique has three parameters namely n, k and m. One important feature of convolutional coding is the constraint length generator polynomial. This acts as trellis structure that provides coding gain, which does not available in other coding schemes. This constraint length is given as where k is the number of input bits, m is the number of shift register and the quantity k/n is called the code rate.
The constraint length and the free distance are fixed where the code rates of 1/2, 2/2 and 3/4 are applied. The free distance of convolutional coding provides the approximation of the coding bit error rate where probability of error is [4] where d and df symbolize distance and free distance respectively. Notation is the sum of bit errors and is the pairwise probability of error is given by [4] where R is the code rate, is the per bit energy at the receiver, is the two sided spectral density of the noise process, and Q(X) is given by the following:
5. Simulation Results Analysis
we study the effect of Hamming distance (Hd) for block code. Fig. 2 shows the bit error rate (BER) performance of block coding for transmission over BSC channel with Hd of 1, 3, and 5. Simulation result indicates that the error detection and correction ability increases proportionally with increasing Hd.
Subsequently, we investigate both block coding with free distance and convolutional coding using trellis structure for transmission over AWGN channel. In Fig. 3, generally it is observed that convolutional coding with trellis structure resulted in excellent performance. On the other hand, block coding with high value of free distance can obtain good BER results.
From our simulation, the constraint length is found to be 7, 9 for the convolutional code. The constraint length is applied to convolutional encoder to the compare of block coding in the case of hard and soft decision Viterbi decoder.
Fig. 3 show the BER against Eb/N0 for block coding with free distance of 3, 6 and 11, and convolutional coding as trellis structure with rates of TR1H = 1/2, TR2H= 2/2 and TR3H = 3/4 consider as hard decision.
Fig. 2. BER performance of block code with deferent value of Hamming distance Hd simulated over BSC channel.
Fig. 3. BER performance comparison of block (B) and convolutional codes (T) with hard decision simulated over AWGN channel.
Fig. 4. BER performance comparison of block (B) and convolutional codes (T) with soft decision simulated over AWGN channel.
The BER performance block and convolutional codes applying soft decision is given in Fig. 4. For convolutional code with rates TR1S = 1/2, TR2S= 2/2 and TR3S = 3/4 (for the soft decision) was simulated. It is noted that a coding gain of 5 dB and 7.5dB is achieved for convolutional coding as compared to the same message length for block coding. Also, it is observed that convolutional code can obtain about 2.5 dB improvement in comparison to the hard decision method. Hence, we can conclude that the soft decoding decision is better than hard decision method as the later employs make binary decision concerning the probability of a binary one and a binary zero [14].
6. Conclusion
This paper reviews block and convolutional coded as these are the two major classification of channel coding methods. In general, the BER performance of convolutional code is better than block coding scheme. Convolutional coding can provide excellent coding gain as compared to the block coding with high free distance for the same message length. However, as a drawback, for the convolutional code, the complexity for the decoding of trellis structure is higher than that of block code. If the complexity is not an issue, then convolution code with trellis structure is preferred due to its attainable coding gain. To obtain lower the BER performance, considerable amount of research are directed to investigate multiple input multiple output (MIMO) systems. Significant efforts of trellis and block coding are also being conducted in different multi antenna systems.
References
[1] S. Haykin and M. Moher, Modern wireless communications, Prentice Hall, 2005.
[2] K. J. Larsen, “Short convolutional codes with maximal free distance for rates 1/2, 1/3, and 1/4,” IEEE Trans. Inform. Theory, vol. IT-19, pp. 371–372, May 1973.
[3] B. Sklar and J. F. Harris, “The ABCs of linear block codes”, IEEE Signal Processing Magazine, vol. 21, pp.14 – 35, Jul 2004.
[4] P. Frenger, P. Orten and T. Ottosson, “Convolutional codes with optimum distance spectrum”, IEEE Trans. Inform. Theory, vol. 3, pp.317–319, Nov 1999.
[5] C.E. Shannon, “A mathematical theory of communication,” Bell System Technical Journal, vol. 27, pp. 379–423, 623–657, 1948.
[6] M. Bossert, Channel Coding for Telecommunications, John Wiley & Sons, New York, 1999.
[7] V. Pless, Introduction to the Theory of Error-Correcting Codes, John Wiley & Sons, New York, 1998.
[8] G. J. Proakis and M. Salehi, Contemporaray Communication Systems, Brooks/Cole, 2000.
[9] B. Sklar, Digital Communications: Fundamentals and Applications, Englewood Cliffs, NJ: Prentice Hall, 2001.
[10] T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inform. Theory, vol. 47, pp. 619–637, Feb. 2001.
[11] E. Yeo and V. Anantharam, “Iterative decoder architectures,” IEEE Commun. Magazine, vol. 41, no. 8, pp. 132–140, Aug. 2003.
[12] P. Frenger, P. Orten, and T. Ottosson, "Comments and additions to recent papers on new convolutional codes," IEEE Transactions on Information Theory, vol. 47, no. 3, March 2001, pp. 1199-1201.
[13] J. G. Proakis, Digital Communications, 4th ed., New York, McGraw-Hill, 2001.
[14] L. Hanzo, T. H. Liew and B. L. Yeap., Turbo Coding, Turbo Equalization and Space-Time Coding for Transmission over Fading Channels, John Wiley & Sons, West Sussex 2002.
5. Simulation Results Analysis
we study the effect of Hamming distance (Hd) for block code. Fig. 2 shows the bit error rate (BER) performance of block coding for transmission over BSC channel with Hd of 1, 3, and 5. Simulation result indicates that the error detection and correction ability increases proportionally with increasing Hd.
Subsequently, we investigate both block coding with free distance and convolutional coding using trellis structure for transmission over AWGN channel. In Fig. 3, generally it is observed that convolutional coding with trellis structure resulted in excellent performance. On the other hand, block coding with high value of free distance can obtain good BER results.
From our simulation, the constraint length is found to be 7, 9 for the convolutional code. The constraint length is applied to convolutional encoder to the compare of block coding in the case of hard and soft decision Viterbi decoder.
Fig. 3 show the BER against Eb/N0 for block coding with free distance of 3, 6 and 11, and convolutional coding as trellis structure with rates of TR1H = 1/2, TR2H= 2/2 and TR3H = 3/4 consider as hard decision.
Fig. 2. BER performance of block code with deferent value of Hamming distance Hd simulated over BSC channel.
Fig. 3. BER performance comparison of block (B) and convolutional codes (T) with hard decision simulated over AWGN channel.
Fig. 4. BER performance comparison of block (B) and convolutional codes (T) with soft decision simulated over AWGN channel.
The BER performance block and convolutional codes applying soft decision is given in Fig. 4. For convolutional code with rates TR1S = 1/2, TR2S= 2/2 and TR3S = 3/4 (for the soft decision) was simulated. It is noted that a coding gain of 5 dB and 7.5dB is achieved for convolutional coding as compared to the same message length for block coding. Also, it is observed that convolutional code can obtain about 2.5 dB improvement in comparison to the hard decision method. Hence, we can conclude that the soft decoding decision is better than hard decision method as the later employs make binary decision concerning the probability of a binary one and a binary zero [14].
6. Conclusion
This paper reviews block and convolutional coded as these are the two major classification of channel coding methods. In general, the BER performance of convolutional code is better than block coding scheme. Convolutional coding can provide excellent coding gain as compared to the block coding with high free distance for the same message length. However, as a drawback, for the convolutional code, the complexity for the decoding of trellis structure is higher than that of block code. If the complexity is not an issue, then convolution code with trellis structure is preferred due to its attainable coding gain. To obtain lower the BER performance, considerable amount of research are directed to investigate multiple input multiple output (MIMO) systems. Significant efforts of trellis and block coding are also being conducted in different multi antenna systems.
References
[1] S. Haykin and M. Moher, Modern wireless communications, Prentice Hall, 2005.
[2] K. J. Larsen, “Short convolutional codes with maximal free distance for rates 1/2, 1/3, and 1/4,” IEEE Trans. Inform. Theory, vol. IT-19, pp. 371–372, May 1973.
[3] B. Sklar and J. F. Harris, “The ABCs of linear block codes”, IEEE Signal Processing Magazine, vol. 21, pp.14 – 35, Jul 2004.
[4] P. Frenger, P. Orten and T. Ottosson, “Convolutional codes with optimum distance spectrum”, IEEE Trans. Inform. Theory, vol. 3, pp.317–319, Nov 1999.
[5] C.E. Shannon, “A mathematical theory of communication,” Bell System Technical Journal, vol. 27, pp. 379–423, 623–657, 1948.
[6] M. Bossert, Channel Coding for Telecommunications, John Wiley & Sons, New York, 1999.
[7] V. Pless, Introduction to the Theory of Error-Correcting Codes, John Wiley & Sons, New York, 1998.
[8] G. J. Proakis and M. Salehi, Contemporaray Communication Systems, Brooks/Cole, 2000.
[9] B. Sklar, Digital Communications: Fundamentals and Applications, Englewood Cliffs, NJ: Prentice Hall, 2001.
[10] T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inform. Theory, vol. 47, pp. 619–637, Feb. 2001.
[11] E. Yeo and V. Anantharam, “Iterative decoder architectures,” IEEE Commun. Magazine, vol. 41, no. 8, pp. 132–140, Aug. 2003.
[12] P. Frenger, P. Orten, and T. Ottosson, "Comments and additions to recent papers on new convolutional codes," IEEE Transactions on Information Theory, vol. 47, no. 3, March 2001, pp. 1199-1201.
[13] J. G. Proakis, Digital Communications, 4th ed., New York, McGraw-Hill, 2001.
[14] L. Hanzo, T. H. Liew and B. L. Yeap., Turbo Coding, Turbo Equalization and Space-Time Coding for Transmission over Fading Channels, John Wiley & Sons, West Sussex 2002.
Author:©Almas Uddin Ahmed, 2007All rights reserved
