Convolutional encoding Finite State Machine Channel models The Viterbi algorithm Coding and decoding with convolutional codes. The Viterbi Algorithm. J.-M. Brossier 2008 J.-M. Brossier Coding and decoding with convolutional codes. The Viterbi Algorithm.
Convolutional encoding Principles Finite State Machine 1st point of view: infinite length block code Channel models 2nd point of view: convolutions The Viterbi algorithm Some examples Block codes: main ideas Repetition code TX: CODING THEORY RX: C P DING T O EORY No way to recover from transmission errors, we need to add some redundancy at the transmitter side. Repetition of transmitted symbols make detection and correction possible: TX:CCC OOO DDD III NNN GGG TTT HHH EEE OOO RRR YYY RX:CCC OPO DDD III NNN GGD TTT OHO EEE OOO RRR YYY C O D I N G T O E O R Y: 2 corrections - 1 detection. Beyond repetition ... Better codes exist. J.-M. Brossier Coding and decoding with convolutional codes. The Viterbi Algorithm.
Convolutional encoding Principles Finite State Machine 1st point of view: infinite length block code Channel models 2nd point of view: convolutions The Viterbi algorithm Some examples Block codes: main ideas Repetition code TX: CODING THEORY RX: C P DING T O EORY No way to recover from transmission errors, we need to add some redundancy at the transmitter side. Repetition of transmitted symbols make detection and correction possible: TX:CCC OOO DDD III NNN GGG TTT HHH EEE OOO RRR YYY RX:CCC OPO DDD III NNN GGD TTT OHO EEE OOO RRR YYY C O D I N G T O E O R Y: 2 corrections - 1 detection. Beyond repetition ... Better codes exist. J.-M. Brossier Coding and decoding with convolutional codes. The Viterbi Algorithm.
Convolutional encoding Principles Finite State Machine 1st point of view: infinite length block code Channel models 2nd point of view: convolutions The Viterbi algorithm Some examples Block codes: main ideas Repetition code TX: CODING THEORY RX: C P DING T O EORY No way to recover from transmission errors, we need to add some redundancy at the transmitter side. Repetition of transmitted symbols make detection and correction possible: TX:CCC OOO DDD III NNN GGG TTT HHH EEE OOO RRR YYY RX:CCC OPO DDD III NNN GGD TTT OHO EEE OOO RRR YYY C O D I N G T O E O R Y: 2 corrections - 1 detection. Beyond repetition ... Better codes exist. J.-M. Brossier Coding and decoding with convolutional codes. The Viterbi Algorithm.
Convolutional encoding Principles Finite State Machine 1st point of view: infinite length block code Channel models 2nd point of view: convolutions The Viterbi algorithm Some examples Block codes: main ideas Geometric view 0 1 1 1 1 1 k=1 bit of information 0 0 1 1 0 1 2 code words (length n=3): (000) (111) 0 1 0 1 1 0 0 0 0 1 0 0 RX: how can the receiver decide about transmitted words: (001),(010),(100): Detection + correction (000) (110),(101),(011): Detection + correction (111) (000) (111) (Probably) right J.-M. Brossier Coding and decoding with convolutional codes. The Viterbi Algorithm.
Convolutional encoding Principles Finite State Machine 1st point of view: infinite length block code Channel models 2nd point of view: convolutions The Viterbi algorithm Some examples Block codes: main ideas Linear block codes, e.g. Hamming codes. A binary linear block code takes k information bits at its input and calculates n bits. If the 2 k codewords are enough and well spaced in the n -dim space, it is possible to detect or even correct errors. In 1950, Hamming introduced the (7,4) Hamming code. It encodes 4 data bits into 7 bits by adding three parity bits. It can detect and correct single-bit errors but can only detect double-bit errors. The code parity-check matrix is: 1 0 1 0 1 0 1 H = 0 1 1 0 0 1 1 0 0 0 1 1 1 1 J.-M. Brossier Coding and decoding with convolutional codes. The Viterbi Algorithm.
Convolutional encoding Principles Finite State Machine 1st point of view: infinite length block code Channel models 2nd point of view: convolutions The Viterbi algorithm Some examples Convolutional encoding: main ideas In convolutional codes, each block of k bits is mapped into a block of n bits BUT these n bits are not only determined by the present k information bits but also by the previous information bits. This dependence can be captured by a finite state machine. This is achieved using several linear filtering operations: Each convolution imposes a constraint between bits. Several convolutions introduce the redundancy. J.-M. Brossier Coding and decoding with convolutional codes. The Viterbi Algorithm.
Convolutional encoding Principles Finite State Machine 1st point of view: infinite length block code Channel models 2nd point of view: convolutions The Viterbi algorithm Some examples Infinite generator matrix A convolutional code can be described by an “infinite matrix”: G 0 G 1 · · · G M 0 k × n · · · 0 k × n G 0 · · · G M − 1 G M . . . ... ... ... . . . . . . . . . 0 k × n G 0 G 1 G = . ... ... . . G 0 . ... . . 0 k × n ... This matrix depends on K = M + 1 k × n sub-matrices { G i } i =0 .. M . K is known as the constraint length of the code. J.-M. Brossier Coding and decoding with convolutional codes. The Viterbi Algorithm.
Convolutional encoding Principles Finite State Machine 1st point of view: infinite length block code Channel models 2nd point of view: convolutions The Viterbi algorithm Some examples Infinite generator matrix A convolutional code can be described by an “infinite matrix”: · · · · · · G 0 G 1 G M 0 k × n 0 k × n G 0 · · · G M − 1 G M . . . ... ... ... . . . . . . . . . 0 k × n G 0 G 1 ( C 0 , C 1 · · · ) = ( I 0 , I 1 · · · ) . ... ... . . G 0 . ... . . 0 k × n ... It looks like a block coding: C = IG J.-M. Brossier Coding and decoding with convolutional codes. The Viterbi Algorithm.
Convolutional encoding Principles Finite State Machine 1st point of view: infinite length block code Channel models 2nd point of view: convolutions The Viterbi algorithm Some examples Infinite generator matrix Denoting by: I j = ( I j 1 · · · I jk ) the j th block of k informative bits, C j = ( C j 1 · · · C jn ) a block of n coded bits at the output. Coding an infinite sequence of blocks (length k ) I = ( I 0 I 1 · · · ) produces an infinite sequence C = ( C 0 C 1 · · · ) of coded blocks (length n ). C 0 = I 0 G 0 C 1 = I 0 G 1 + I 1 G 0 Block form of the coding . . . scheme: it looks like a block C M = I 0 G M + I 1 G M − 1 + · · · + I M G 0 coding: . . . C = IG C j = I j − M G M + · · · + I j G 0 for j ≥ M . . . J.-M. Brossier Coding and decoding with convolutional codes. The Viterbi Algorithm.
Convolutional encoding Principles Finite State Machine 1st point of view: infinite length block code Channel models 2nd point of view: convolutions The Viterbi algorithm Some examples Infinite generator matrix performs a convolution Using the convention I i = 0 for i < 0, the encoding structure C = IG is clearly a convolution : M � C j = I j − l G l . l =0 For an informative bits sequence I whose length is finite,only L < + ∞ blocks of k bits are different from zero at the input of the coder: I = ( I 0 · · · I L − 1 ). The sequence C = ( C 0 · · · C L − 1+ M ) at the coder output is finite too. This truncated coded sequence is generated by a linear block code whose generator matrix is a size kL × n ( L + M ) sub-matrix of G J.-M. Brossier Coding and decoding with convolutional codes. The Viterbi Algorithm.
Convolutional encoding Principles Finite State Machine 1st point of view: infinite length block code Channel models 2nd point of view: convolutions The Viterbi algorithm Some examples Shift registers based realization Let us write g ( l ) αβ elements of matrix G l . We now expand the convolution C j = � M l =0 I j − l G l to explicit the n components C j 1 , · · · , C jn of each output block C j : � M k M k � I j − l ,α g ( l ) I j − l ,α g ( l ) � � � � C j = [ C j 1 , · · · , C jn ] = α 1 , · · · , α n l =0 α =1 l =0 α =1 If the length of the shift register is L , there are M L different internal configurations. The behavior of the convolutional coder can be captured by a M L states machine. J.-M. Brossier Coding and decoding with convolutional codes. The Viterbi Algorithm.
Convolutional encoding Principles Finite State Machine 1st point of view: infinite length block code Channel models 2nd point of view: convolutions The Viterbi algorithm Some examples Shift registers based realization k M � � I j − l ,α g ( l ) C j β = αβ α =1 l =0 depends on: the present input I j M previous input blocks I j − 1 , · · · , I j − M . C j β can be calculated by memorizing M input values in shift registers One shift register α ∈ 1 · · · k for each k bits of the input. For register α , only memories for which g ( l ) αβ = 1 are connected to adder β ∈ 1 · · · n . J.-M. Brossier Coding and decoding with convolutional codes. The Viterbi Algorithm.
Recommend
More recommend
