Sequential decoding - Revision history

Wbm1058: Bayes' theorem (via WP:JWB)

2025-04-10T10:17:56Z

← Previous revision		Revision as of 10:17, 10 April 2025
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	Given a partially explored tree (represented by a set of nodes which are limit of exploration), we would like to know the best node from which to explore further. The Fano metric (named after [[Robert Fano]]) allows one to calculate from which is the best node to explore further. This metric is optimal given no other constraints (e.g. memory).		Given a partially explored tree (represented by a set of nodes which are limit of exploration), we would like to know the best node from which to explore further. The Fano metric (named after [[Robert Fano]]) allows one to calculate from which is the best node to explore further. This metric is optimal given no other constraints (e.g. memory).

	For a [[binary symmetric channel]] (with error probability <math>p</math>) the Fano metric can be derived via [[Bayes theorem]]. We are interested in following the most likely path <math>P_i</math> given an explored state of the tree <math>X</math> and a received sequence <math>{\mathbf r}</math>. Using the language of [[probability]] and [[Bayes theorem]] we want to choose the maximum over <math>i</math> of:		For a [[binary symmetric channel]] (with error probability <math>p</math>) the Fano metric can be derived via [[Bayes' theorem]]. We are interested in following the most likely path <math>P_i</math> given an explored state of the tree <math>X</math> and a received sequence <math>{\mathbf r}</math>. Using the language of [[probability]] and [[Bayes' theorem]] we want to choose the maximum over <math>i</math> of:
	:<math>\Pr(P_i\|X,{\mathbf r}) \propto \Pr({\mathbf r}\|P_i,X)\Pr(P_i\|X)</math>		:<math>\Pr(P_i\|X,{\mathbf r}) \propto \Pr({\mathbf r}\|P_i,X)\Pr(P_i\|X)</math>

Karn: /* Fano algorithm */

2023-11-17T01:33:11Z

Fano algorithm

← Previous revision		Revision as of 01:33, 17 November 2023
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	# At each decoding stage, the Fano algorithm retains the information regarding three paths: the current path, its immediate predecessor path, and one of its successor paths.		# At each decoding stage, the Fano algorithm retains the information regarding three paths: the current path, its immediate predecessor path, and one of its successor paths.
	# Based on this information, the Fano algorithm can move from the current path to either its immediate predecessor path or the selected successor path; hence, no stack is required for queuing all examined paths.		# Based on this information, the Fano algorithm can move from the current path to either its immediate predecessor path or the selected successor path; hence, no stack is required for queuing all examined paths.
	# The movement of the Fano algorithm is guided by a dynamic threshold {{var\|T}} that is an integer multiple of a fixed step size ¢.		# The movement of the Fano algorithm is guided by a dynamic threshold {{var\|T}} that is an integer multiple of a fixed step size Δ.
	# Only the path whose path metric is no less than {{var\|T}} can be next visited. According to the algorithm, the process of codeword search continues to move forward along a code path, as long as the Fano metric along the code path remains non-decreasing.		# Only the path whose path metric is no less than {{var\|T}} can be next visited. According to the algorithm, the process of codeword search continues to move forward along a code path, as long as the Fano metric along the code path remains non-decreasing.
	# Once all the successor path metrics are smaller than {{var\|T}}, the algorithm moves backward to the predecessor path if the predecessor path metric beats {{var\|T}}; thereafter, threshold examination will be subsequently performed on another successor path of this revisited predecessor.		# Once all the successor path metrics are smaller than {{var\|T}}, the algorithm moves backward to the predecessor path if the predecessor path metric beats {{var\|T}}; thereafter, threshold examination will be subsequently performed on another successor path of this revisited predecessor.

Karn: /* Computational cutoff rate */

2023-11-17T01:25:01Z

Computational cutoff rate

← Previous revision		Revision as of 01:25, 17 November 2023
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	==Computational cutoff rate==		==Computational cutoff rate==

	For sequential decoding to a good choice of decoding algorithm, the number of states explored ~~wants to~~ remain small (otherwise an algorithm which deliberately explores all states, e.g. the [[Viterbi algorithm]], may be more suitable). For a particular noise level there is a maximum coding rate <math>R_0</math> called the computational cutoff rate where there is a finite backtracking limit. For the binary symmetric channel:		For sequential decoding to be a good choice of decoding algorithm, the number of states explored should remain small (otherwise an algorithm which deliberately explores all states, e.g. the [[Viterbi algorithm]], may be more suitable). For a particular noise level there is a maximum coding rate <math>R_0</math> called the computational cutoff rate where there is a finite backtracking limit. For the binary symmetric channel:
	:<math>R_0 = 1-\log_2(1+2\sqrt{p(1-p)})</math>		:<math>R_0 = 1-\log_2(1+2\sqrt{p(1-p)})</math>

PrimeBOT: Task 24: remove a maintenance template following a TFD

2022-01-18T21:40:00Z

Task 24: remove a maintenance template following a TFD

← Previous revision		Revision as of 21:40, 18 January 2022
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	{{Expert-subject\|Mathematics\|date=December 2009}}
	Recognised by [[John Wozencraft]], '''sequential decoding''' is a limited memory technique for decoding [[tree codes]]. Sequential decoding is mainly used as an approximate decoding algorithm for long constraint-length [[convolutional code]]s. This approach may not be as accurate as the [[Viterbi algorithm]] but can save a substantial amount of computer memory. It was used to decode a convolutional code in 1968 [[Pioneer 9]] mission.		Recognised by [[John Wozencraft]], '''sequential decoding''' is a limited memory technique for decoding [[tree codes]]. Sequential decoding is mainly used as an approximate decoding algorithm for long constraint-length [[convolutional code]]s. This approach may not be as accurate as the [[Viterbi algorithm]] but can save a substantial amount of computer memory. It was used to decode a convolutional code in 1968 [[Pioneer 9]] mission.

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	*<math>n_i</math> to be the length of <math>P_i</math> in branches.		*<math>n_i</math> to be the length of <math>P_i</math> in branches.

	We express the [[likelihood]] <math>\Pr({\mathbf r}\|P_i,X)</math> as <math>p^{d_i} (1-p)^{n_ib-d_i} 2^{-(N-n_i)b}</math> (by using the binary symmetric channel likelihood for the first <math>n_ib</math> bits followed by a uniform prior over the remaining bits).		We express the [[likelihood]] <math>\Pr({\mathbf r}\|P_i,X)</math> as <math>p^{d_i} (1-p)^{n_ib-d_i} 2^{-(N-n_i)b}</math> (by using the binary symmetric channel likelihood for the first <math>n_ib</math> bits followed by a uniform prior over the remaining bits).

	We express the [[prior probability\|prior]] <math>\Pr(P_i\|X)</math> in terms of the number of branch choices one has made, <math>n_i</math>, and the number of branches from each node, <math>2^{Rb}</math>.		We express the [[prior probability\|prior]] <math>\Pr(P_i\|X)</math> in terms of the number of branch choices one has made, <math>n_i</math>, and the number of branches from each node, <math>2^{Rb}</math>.

	Therefore:		Therefore:

Frap: /* Fano algorithm */ use var template

2020-06-04T11:45:50Z

Fano algorithm: use var template

← Previous revision		Revision as of 11:45, 4 June 2020
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	# At each decoding stage, the Fano algorithm retains the information regarding three paths: the current path, its immediate predecessor path, and one of its successor paths.		# At each decoding stage, the Fano algorithm retains the information regarding three paths: the current path, its immediate predecessor path, and one of its successor paths.
	# Based on this information, the Fano algorithm can move from the current path to either its immediate predecessor path or the selected successor path; hence, no stack is required for queuing all examined paths.		# Based on this information, the Fano algorithm can move from the current path to either its immediate predecessor path or the selected successor path; hence, no stack is required for queuing all examined paths.
	# The movement of the Fano algorithm is guided by a dynamic threshold T that is an integer multiple of a fixed step size ¢.		# The movement of the Fano algorithm is guided by a dynamic threshold {{var\|T}} that is an integer multiple of a fixed step size ¢.
	# Only the path whose path metric is no less than T can be next visited. According to the algorithm, the process of codeword search continues to move forward along a code path, as long as the Fano metric along the code path remains non-decreasing.		# Only the path whose path metric is no less than {{var\|T}} can be next visited. According to the algorithm, the process of codeword search continues to move forward along a code path, as long as the Fano metric along the code path remains non-decreasing.
	# Once all the successor path metrics are smaller than T, the algorithm moves backward to the predecessor path if the predecessor path metric beats T; thereafter, threshold examination will be subsequently performed on another successor path of this revisited predecessor.		# Once all the successor path metrics are smaller than {{var\|T}}, the algorithm moves backward to the predecessor path if the predecessor path metric beats {{var\|T}}; thereafter, threshold examination will be subsequently performed on another successor path of this revisited predecessor.
	# In case the predecessor path metric is also less than T, the threshold T is one-step lowered so that the algorithm is not trapped on the current path.		# In case the predecessor path metric is also less than {{var\|T}}, the threshold {{var\|T}} is one-step lowered so that the algorithm is not trapped on the current path.
	# For the Fano algorithm, if a path is revisited, the presently examined dynamic threshold is always lower than the momentary dynamic threshold at the previous visit, guaranteeing that looping in the algorithm does not occur, and that the algorithm can ultimately reach a terminal node of the code tree, and stop.		# For the Fano algorithm, if a path is revisited, the presently examined dynamic threshold is always lower than the momentary dynamic threshold at the previous visit, guaranteeing that looping in the algorithm does not occur, and that the algorithm can ultimately reach a terminal node of the code tree, and stop.

Claude.E.Shannon at 17:07, 13 May 2019

2019-05-13T17:07:15Z

← Previous revision		Revision as of 17:07, 13 May 2019
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	{{Expert-subject\|Mathematics\|date=December 2009}}		{{Expert-subject\|Mathematics\|date=December 2009}}
	Recognised by [[John Wozencraft]], '''~~Sequential~~ decoding''' is a limited memory technique for decoding [[tree codes]]. Sequential decoding is mainly used as an approximate decoding algorithm for long constraint-length [[convolutional code]]s. This approach may not be as accurate as the [[Viterbi algorithm]] but can save a substantial amount of computer memory. It was used to decode a convolutional code in 1968 [[Pioneer 9]] mission.		Recognised by [[John Wozencraft]], '''sequential decoding''' is a limited memory technique for decoding [[tree codes]]. Sequential decoding is mainly used as an approximate decoding algorithm for long constraint-length [[convolutional code]]s. This approach may not be as accurate as the [[Viterbi algorithm]] but can save a substantial amount of computer memory. It was used to decode a convolutional code in 1968 [[Pioneer 9]] mission.

	Sequential decoding explores the tree code in such a way to try to minimise the computational cost and memory requirements to store the tree.		Sequential decoding explores the tree code in such a way to try to minimise the computational cost and memory requirements to store the tree.

Claude.E.Shannon at 17:06, 13 May 2019

2019-05-13T17:06:49Z

← Previous revision		Revision as of 17:06, 13 May 2019
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	{{Expert-subject\|Mathematics\|date=December 2009}}		{{Expert-subject\|Mathematics\|date=December 2009}}
	Recognised by [[John Wozencraft]], '''Sequential decoding''' is a limited memory technique for decoding [[tree codes]]. Sequential decoding is mainly used as an approximate decoding algorithm for long constraint-length [[convolutional code]]s. This approach may not be as accurate as the [[Viterbi algorithm]] but can save a substantial amount of computer memory. It was used to decode a convolutional code in 1968 [[Pioneer 9 ~~mission~~]].		Recognised by [[John Wozencraft]], '''Sequential decoding''' is a limited memory technique for decoding [[tree codes]]. Sequential decoding is mainly used as an approximate decoding algorithm for long constraint-length [[convolutional code]]s. This approach may not be as accurate as the [[Viterbi algorithm]] but can save a substantial amount of computer memory. It was used to decode a convolutional code in 1968 [[Pioneer 9]] mission.

	Sequential decoding explores the tree code in such a way to try to minimise the computational cost and memory requirements to store the tree.		Sequential decoding explores the tree code in such a way to try to minimise the computational cost and memory requirements to store the tree.

Claude.E.Shannon at 17:05, 13 May 2019

2019-05-13T17:05:22Z

← Previous revision		Revision as of 17:05, 13 May 2019
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	{{Expert-subject\|Mathematics\|date=December 2009}}		{{Expert-subject\|Mathematics\|date=December 2009}}
	Recognised by [[~~Jack~~ Wozencraft]], '''Sequential decoding''' is a limited memory technique for decoding [[tree codes]]. Sequential decoding is mainly used as an approximate decoding algorithm for long constraint-length [[convolutional code]]s. This approach may not be as accurate as the [[Viterbi algorithm]] but can save a substantial amount of computer memory. It was used to decode a convolutional code in 1968 [[Pioneer 9 mission]].		Recognised by [[John Wozencraft]], '''Sequential decoding''' is a limited memory technique for decoding [[tree codes]]. Sequential decoding is mainly used as an approximate decoding algorithm for long constraint-length [[convolutional code]]s. This approach may not be as accurate as the [[Viterbi algorithm]] but can save a substantial amount of computer memory. It was used to decode a convolutional code in 1968 [[Pioneer 9 mission]].

	Sequential decoding explores the tree code in such a way to try to minimise the computational cost and memory requirements to store the tree.		Sequential decoding explores the tree code in such a way to try to minimise the computational cost and memory requirements to store the tree.

Claude.E.Shannon at 17:04, 13 May 2019

2019-05-13T17:04:07Z

← Previous revision		Revision as of 17:04, 13 May 2019
Line 1:		Line 1:
	{{Expert-subject\|Mathematics\|date=December 2009}}		{{Expert-subject\|Mathematics\|date=December 2009}}
	'''Sequential decoding''' is a limited memory technique for decoding [[tree codes]]. Sequential decoding is mainly used as an approximate decoding algorithm for long constraint-length [[convolutional code]]s. This approach may not be as accurate as the [[Viterbi algorithm]] but can save a substantial amount of computer memory.		Recognised by [[Jack Wozencraft]], '''Sequential decoding''' is a limited memory technique for decoding [[tree codes]]. Sequential decoding is mainly used as an approximate decoding algorithm for long constraint-length [[convolutional code]]s. This approach may not be as accurate as the [[Viterbi algorithm]] but can save a substantial amount of computer memory. It was used to decode a convolutional code in 1968 [[Pioneer 9 mission]].

	Sequential decoding explores the tree code in such a way to try to minimise the computational cost and memory requirements to store the tree.		Sequential decoding explores the tree code in such a way to try to minimise the computational cost and memory requirements to store the tree.

Claude.E.Shannon at 16:55, 13 May 2019

2019-05-13T16:55:17Z

← Previous revision		Revision as of 16:55, 13 May 2019
Line 1:		Line 1:
	{{Expert-subject\|Mathematics\|date=December 2009}}		{{Expert-subject\|Mathematics\|date=December 2009}}
	'''Sequential decoding''' is a limited memory technique for decoding [[tree codes]]. Sequential decoding is mainly used is as an approximate decoding algorithm for long constraint-length [[convolutional code]]s. This approach may not be as accurate as the [[Viterbi algorithm]] but can save a substantial amount of computer memory.		'''Sequential decoding''' is a limited memory technique for decoding [[tree codes]]. Sequential decoding is mainly used as an approximate decoding algorithm for long constraint-length [[convolutional code]]s. This approach may not be as accurate as the [[Viterbi algorithm]] but can save a substantial amount of computer memory.

	Sequential decoding explores the tree code in such a way to try to minimise the computational cost and memory requirements to store the tree.		Sequential decoding explores the tree code in such a way to try to minimise the computational cost and memory requirements to store the tree.

	There is a range of sequential decoding approaches based on choice of metric and algorithm. Metrics include:		There is a range of sequential decoding approaches based on the choice of metric and algorithm. Metrics include:
	*Fano metric		*Fano metric
	*Zigangirov metric		*Zigangirov metric