Contents.History Convolutional codes were introduced in 1955. It was thought that convolutional codes could be decoded with arbitrary quality at the expense of computation and delay. In 1967 determined that convolutional codes could be maximum-likelihood decoded with reasonable complexity using time invariant trellis based decoders — the. Other trellis-based decoder algorithms were later developed, including the decoding algorithm.Recursive systematic convolutional codes were invented by around 1991.
These codes proved especially useful for iterative processing including the processing of concatenated codes such as.Using the 'convolutional' terminology, a classic convolutional code might be considered a (FIR) filter, while a recursive convolutional code might be considered an (IIR) filter.Where convolutional codes are used. Stages of channel coding in GSM. Block encoder and Parity check - error detection part. Convolutional encoder and Viterbi decoder - error correction part. And Deinterleaving - code words separation increasing in time domain and to avoid bursty distortions.Convolutional codes are used extensively to achieve reliable data transfer in numerous applications, such as, radio, (e.g., in GSM, GPRS, EDGE and 3G networks (until 3GPP Release 7) ). These codes are often implemented in with a hard-decision code, particularly.
Prior to such constructions were the most efficient, coming closest to the.Convolutional encoding To convolutionally encode data, start with k, each holding one input bit. Unless otherwise specified, all memory registers start with a value of 0.
The encoder has n modulo-2 (a modulo 2 adder can be implemented with a single, where the logic is: 0+0 = 0, 0+1 = 1, 1+0 = 1, 1+1 = 0), and n — one for each adder (see figure below). An input bit m 1 is fed into the leftmost register. Using the generator polynomials and the existing values in the remaining registers, the encoder outputs n symbols. These symbols may be transmitted or punctured depending on the desired code rate. Now all register values to the right ( m 1 moves to m 0, m 0 moves to m −1) and wait for the next input bit. If there are no remaining input bits, the encoder continues shifting until all registers have returned to the zero state (flush bit termination). Rate 1/3 non-recursive, non-systematic convolutional encoder with constraint length 3The figure below is a rate 1⁄ 3 ( m⁄ n) encoder with constraint length ( k) of 3.
Generator polynomials are G 1 = (1,1,1), G 2 = (0,1,1), and G 3 = (1,0,1). Therefore, output bits are calculated (modulo 2) as follows:n 1 = m 1 + m 0 + m −1 n 2 = m 0 + m −1 n 3 = m 1 + m −1.Convolutional codes can be systematic and non-systematic:. systematic repeats the structure of the message before encoding. non-systematic changes the initial structureNon-systematic convolutional codes are more popular due to better noise immunity. It relates to the free distance of the convolutional code. Rate 1/2 8-state recursive systematic convolutional encoder. Used as constituent code in 3GPP 25.212 Turbo Code.The example encoder is because the input data is also used in the output symbols (Output 2).
Codes with output symbols that do not include the input data are called non-systematic.Recursive codes are typically systematic and, conversely, non-recursive codes are typically non-systematic. It isn't a strict requirement, but a common practice.The example encoder in Img. Is an 8-state encoder because the 3 registers will create 8 possible encoder states (2 3). A corresponding decoder trellis will typically use 8 states as well.Recursive systematic convolutional (RSC) codes have become more popular due to their use in Turbo Codes. Recursive systematic codes are also referred to as pseudo-systematic codes.Other RSC codes and example applications include. See also:A convolutional encoder is a.
An encoder with n binary cells will have 2 n states.Imagine that the encoder (shown on Img.1, above) has '1' in the left memory cell ( m 0), and '0' in the right one ( m −1). ( m 1 is not really a memory cell because it represents a current value). We will designate such a state as '10'. According to an input bit the encoder at the next turn can convert either to the '01' state or the '11' state. One can see that not all transitions are possible for (e.g., a decoder can't convert from '10' state to '00' or even stay in '10' state).All possible transitions can be shown as below.
A trellis diagram for the encoder on Img.1. A path through the trellis is shown as a red line. The solid lines indicate transitions where a '0' is input and the dashed lines where a '1' is input.An actual encoded sequence can be represented as a path on this graph. One valid path is shown in red as an example.This diagram gives us an idea about decoding: if a received sequence doesn't fit this graph, then it was received with errors, and we must choose the nearest correct (fitting the graph) sequence. The real decoding algorithms exploit this idea.Free distance and error distribution.
Theoretical bit-error rate curves for uncoded and coded QPSK, additive white Gaussian noise channel. Hard decision means that the decoder expects binary symbols (0's and 1's); Soft decision means that the decoder expects.Several exist for decoding convolutional codes. For relatively small values of k, the is universally used as it provides performance and is highly parallelizable. Viterbi decoders are thus easy to implement in hardware and in software on CPUs with instruction sets.Longer constraint length codes are more practically decoded with any of several algorithms, of which the algorithm is the best known. Unlike Viterbi decoding, sequential decoding is not maximum likelihood but its complexity increases only slightly with constraint length, allowing the use of strong, long-constraint-length codes. Such codes were used in the of the early 1970s to Jupiter and Saturn, but gave way to shorter, Viterbi-decoded codes, usually concatenated with large codes that steepen the overall bit-error-rate curve and produce extremely low residual undetected error rates.Both Viterbi and sequential decoding algorithms return hard decisions: the bits that form the most likely codeword.
An approximate confidence measure can be added to each bit by use of the. (MAP) soft decisions for each bit can be obtained by use of the.Popular convolutional codes. Theoretical bit-error rate curves of encoded QPSK (soft decision), additive white Gaussian noise channel. Longer constraint lengths produce more powerful codes, but the of the Viterbi algorithm with constraint lengths, limiting these more powerful codes to deep space missions where the extra performance is easily worth the increased decoder complexity.In fact, predefined convolutional codes structures obtained during scientific researches are used in the industry. This relates to the possibility to select catastrophic convolutional codes (causes larger number of errors).An especially popular Viterbi-decoded convolutional code, used at least since the has a constraint length k of 7 and a rate r of 1/2., and the to Saturn use a k of 15 and a rate of 1/6; this code performs about 2 dB better than the simpler k=7 code at a cost of 256× in decoding complexity (compared to Voyager mission codes).The convolutional code with a constraint length of 2 and a rate of 1/2 is used in as an error correction technique. Punctured convolutional codes.
Convolutional codes with 1/2 and 3/4 code rates (and constraint length 7, Soft decision, 4-QAM / QPSK / OQPSK).Convolutional code with any code rate can be designed based on polynom selection, however, in practice, puncturing procedure is used to achieve the required code rate. Is a technique used to make a m/ n rate code from a 'basic' low-rate (e.g., 1/ n) code. It is reached by deletion of some bits in the encoder output.
Bits are deleted according to a puncturing matrix. The following puncturing matrices are the most frequently used:Code ratePuncturing matrixFree distance (for NASA standard K=7 convolutional code)1/2(No perf.)1103For example, if we want to make a code with rate 2/3 using the appropriate matrix from the above table, we should take a basic encoder output and transmit every first bit from the first branch and every bit from the second one. The specific order of transmission is defined by the respective communication standard.Punctured convolutional codes are widely used in the, for example, in systems and.Punctured convolutional codes are also called 'perforated'.Turbo codes: replacing convolutional codes. A turbo code with component codes 13, 15. Turbo codes get their name because the decoder uses feedback, like a turbo engine.
Permutation means the same as the interleaving. C1 and C2 are recursive convolutional codes. Recursive and non-recursive convolutional codes are not so much different in BER performance, however, recursive type of is implemented in Turbo convolutional codes due to better interleaving properties.Simple Viterbi-decoded convolutional codes are now giving way to, a new class of iterated short convolutional codes that closely approach the theoretical limits imposed by with much less decoding complexity than the Viterbi algorithm on the long convolutional codes that would be required for the same performance. With an outer algebraic code (e.g., ) addresses the issue of inherent to turbo code designs.See also.References. This article incorporates from the document. Publications. Francis, Michael.
'Viterbi Decoder Block Decoding-Trellis Termination and Tail Biting.' Xilinx XAPP551 v2.
0, DD (2005): 1-21. Chen, Qingchun, Wai Ho Mow, and Pingzhi Fan. 'Some new results on recursive convolutional codes and their applications.' Information Theory Workshop, 2006. ITW'06 Chengdu.
IEEE, 2006. Fiebig, U-C., and Patrick Robertson.
'Soft-decision and erasure decoding in fast frequency-hopping systems with convolutional, turbo, and Reed-Solomon codes.' IEEE Transactions on Communications 47.11 (1999): 1646-1654. Bhaskar, Vidhyacharan, and Laurie L. 'Performance of punctured convolutional codes in asynchronous CDMA communications under perfect phase-tracking conditions.' Computers & Electrical Engineering 30.8 (2004): 573-592.
Modestino, J., and Shou Mui. 'Convolutional code performance in the Rician fading channel.'
IEEE Transactions on Communications 24.6 (1976): 592-606. Chen, Yuh-Long, and Che-Ho Wei. 'Performance evaluation of convolutional codes with MPSK on Rician fading channels.' IEE Proceedings F-Communications, Radar and Signal Processing.
Contents.History Convolutional codes were introduced in 1955. It was thought that convolutional codes could be decoded with arbitrary quality at the expense of computation and delay.
In 1967 determined that convolutional codes could be maximum-likelihood decoded with reasonable complexity using time invariant trellis based decoders — the. Other trellis-based decoder algorithms were later developed, including the decoding algorithm.Recursive systematic convolutional codes were invented by around 1991. These codes proved especially useful for iterative processing including the processing of concatenated codes such as.Using the 'convolutional' terminology, a classic convolutional code might be considered a (FIR) filter, while a recursive convolutional code might be considered an (IIR) filter.Where convolutional codes are used. Stages of channel coding in GSM. Block encoder and Parity check - error detection part. Convolutional encoder and Viterbi decoder - error correction part. And Deinterleaving - code words separation increasing in time domain and to avoid bursty distortions.Convolutional codes are used extensively to achieve reliable data transfer in numerous applications, such as, radio, (e.g., in GSM, GPRS, EDGE and 3G networks (until 3GPP Release 7) ).
These codes are often implemented in with a hard-decision code, particularly. Prior to such constructions were the most efficient, coming closest to the.Convolutional encoding To convolutionally encode data, start with k, each holding one input bit. Unless otherwise specified, all memory registers start with a value of 0. The encoder has n modulo-2 (a modulo 2 adder can be implemented with a single, where the logic is: 0+0 = 0, 0+1 = 1, 1+0 = 1, 1+1 = 0), and n — one for each adder (see figure below). An input bit m 1 is fed into the leftmost register. Using the generator polynomials and the existing values in the remaining registers, the encoder outputs n symbols.
Binary Convolutional Code
These symbols may be transmitted or punctured depending on the desired code rate. Now all register values to the right ( m 1 moves to m 0, m 0 moves to m −1) and wait for the next input bit. If there are no remaining input bits, the encoder continues shifting until all registers have returned to the zero state (flush bit termination). Rate 1/3 non-recursive, non-systematic convolutional encoder with constraint length 3The figure below is a rate 1⁄ 3 ( m⁄ n) encoder with constraint length ( k) of 3. Generator polynomials are G 1 = (1,1,1), G 2 = (0,1,1), and G 3 = (1,0,1). Therefore, output bits are calculated (modulo 2) as follows:n 1 = m 1 + m 0 + m −1 n 2 = m 0 + m −1 n 3 = m 1 + m −1.Convolutional codes can be systematic and non-systematic:.
systematic repeats the structure of the message before encoding. non-systematic changes the initial structureNon-systematic convolutional codes are more popular due to better noise immunity. It relates to the free distance of the convolutional code. Rate 1/2 8-state recursive systematic convolutional encoder.
Used as constituent code in 3GPP 25.212 Turbo Code.The example encoder is because the input data is also used in the output symbols (Output 2). Codes with output symbols that do not include the input data are called non-systematic.Recursive codes are typically systematic and, conversely, non-recursive codes are typically non-systematic. It isn't a strict requirement, but a common practice.The example encoder in Img. Is an 8-state encoder because the 3 registers will create 8 possible encoder states (2 3).
A corresponding decoder trellis will typically use 8 states as well.Recursive systematic convolutional (RSC) codes have become more popular due to their use in Turbo Codes. Recursive systematic codes are also referred to as pseudo-systematic codes.Other RSC codes and example applications include. See also:A convolutional encoder is a. An encoder with n binary cells will have 2 n states.Imagine that the encoder (shown on Img.1, above) has '1' in the left memory cell ( m 0), and '0' in the right one ( m −1). ( m 1 is not really a memory cell because it represents a current value).
We will designate such a state as '10'. According to an input bit the encoder at the next turn can convert either to the '01' state or the '11' state.
One can see that not all transitions are possible for (e.g., a decoder can't convert from '10' state to '00' or even stay in '10' state).All possible transitions can be shown as below. A trellis diagram for the encoder on Img.1.
A path through the trellis is shown as a red line. The solid lines indicate transitions where a '0' is input and the dashed lines where a '1' is input.An actual encoded sequence can be represented as a path on this graph. One valid path is shown in red as an example.This diagram gives us an idea about decoding: if a received sequence doesn't fit this graph, then it was received with errors, and we must choose the nearest correct (fitting the graph) sequence.
The real decoding algorithms exploit this idea.Free distance and error distribution. Theoretical bit-error rate curves for uncoded and coded QPSK, additive white Gaussian noise channel. Hard decision means that the decoder expects binary symbols (0's and 1's); Soft decision means that the decoder expects.Several exist for decoding convolutional codes. For relatively small values of k, the is universally used as it provides performance and is highly parallelizable.
Viterbi decoders are thus easy to implement in hardware and in software on CPUs with instruction sets.Longer constraint length codes are more practically decoded with any of several algorithms, of which the algorithm is the best known. Unlike Viterbi decoding, sequential decoding is not maximum likelihood but its complexity increases only slightly with constraint length, allowing the use of strong, long-constraint-length codes.
Such codes were used in the of the early 1970s to Jupiter and Saturn, but gave way to shorter, Viterbi-decoded codes, usually concatenated with large codes that steepen the overall bit-error-rate curve and produce extremely low residual undetected error rates.Both Viterbi and sequential decoding algorithms return hard decisions: the bits that form the most likely codeword. An approximate confidence measure can be added to each bit by use of the. (MAP) soft decisions for each bit can be obtained by use of the.Popular convolutional codes. Theoretical bit-error rate curves of encoded QPSK (soft decision), additive white Gaussian noise channel.
Longer constraint lengths produce more powerful codes, but the of the Viterbi algorithm with constraint lengths, limiting these more powerful codes to deep space missions where the extra performance is easily worth the increased decoder complexity.In fact, predefined convolutional codes structures obtained during scientific researches are used in the industry. This relates to the possibility to select catastrophic convolutional codes (causes larger number of errors).An especially popular Viterbi-decoded convolutional code, used at least since the has a constraint length k of 7 and a rate r of 1/2., and the to Saturn use a k of 15 and a rate of 1/6; this code performs about 2 dB better than the simpler k=7 code at a cost of 256× in decoding complexity (compared to Voyager mission codes).The convolutional code with a constraint length of 2 and a rate of 1/2 is used in as an error correction technique. Punctured convolutional codes. Convolutional codes with 1/2 and 3/4 code rates (and constraint length 7, Soft decision, 4-QAM / QPSK / OQPSK).Convolutional code with any code rate can be designed based on polynom selection, however, in practice, puncturing procedure is used to achieve the required code rate. Is a technique used to make a m/ n rate code from a 'basic' low-rate (e.g., 1/ n) code.
It is reached by deletion of some bits in the encoder output. Bits are deleted according to a puncturing matrix. The following puncturing matrices are the most frequently used:Code ratePuncturing matrixFree distance (for NASA standard K=7 convolutional code)1/2(No perf.)1103For example, if we want to make a code with rate 2/3 using the appropriate matrix from the above table, we should take a basic encoder output and transmit every first bit from the first branch and every bit from the second one. The specific order of transmission is defined by the respective communication standard.Punctured convolutional codes are widely used in the, for example, in systems and.Punctured convolutional codes are also called 'perforated'.Turbo codes: replacing convolutional codes. A turbo code with component codes 13, 15.
Turbo codes get their name because the decoder uses feedback, like a turbo engine. Permutation means the same as the interleaving. C1 and C2 are recursive convolutional codes.
Recursive and non-recursive convolutional codes are not so much different in BER performance, however, recursive type of is implemented in Turbo convolutional codes due to better interleaving properties.Simple Viterbi-decoded convolutional codes are now giving way to, a new class of iterated short convolutional codes that closely approach the theoretical limits imposed by with much less decoding complexity than the Viterbi algorithm on the long convolutional codes that would be required for the same performance. With an outer algebraic code (e.g., ) addresses the issue of inherent to turbo code designs.See also.References. This article incorporates from the document.
Publications. Francis, Michael. 'Viterbi Decoder Block Decoding-Trellis Termination and Tail Biting.'
Xilinx XAPP551 v2. 0, DD (2005): 1-21.
Chen, Qingchun, Wai Ho Mow, and Pingzhi Fan. 'Some new results on recursive convolutional codes and their applications.' Information Theory Workshop, 2006. ITW'06 Chengdu. IEEE, 2006. Fiebig, U-C., and Patrick Robertson. 'Soft-decision and erasure decoding in fast frequency-hopping systems with convolutional, turbo, and Reed-Solomon codes.'
Convolutional Encoder Matlab
IEEE Transactions on Communications 47.11 (1999): 1646-1654. Bhaskar, Vidhyacharan, and Laurie L. 'Performance of punctured convolutional codes in asynchronous CDMA communications under perfect phase-tracking conditions.' Computers & Electrical Engineering 30.8 (2004): 573-592. Modestino, J., and Shou Mui. 'Convolutional code performance in the Rician fading channel.' IEEE Transactions on Communications 24.6 (1976): 592-606.
C Program For Convolutional Code Ber Free
Chen, Yuh-Long, and Che-Ho Wei. 'Performance evaluation of convolutional codes with MPSK on Rician fading channels.' IEE Proceedings F-Communications, Radar and Signal Processing.