Convolutional Codes. Lecture 13. Figure 93: Encoder for rate 1/2 constraint length 3 convolutional code.

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1 Convolutional Codes Goals Lecture Be able to encode using a convolutional code Be able to decode a convolutional code received over a binary symmetric channel or an additive white Gaussian channel Convolutional codes are an alternative way of introducing redundancy into the data stream. They are referred to block codes in many situations because of the ease with which soft decision decoding can be erformed. They are called convolutional code because the encoder outut can be written as the convolutional of the encoder inut with a generator sequence. A convolutional code consists of k shift registers of length M or less each. The n oututs are linear combinations of the contents of the shift registers and the inut. A convolutional code is describe by the memory length M (or constraint length KM ), the number of inut bits k the number of outut bits n and the connections between the shift registers and the oututs. XIII- XIII- Examle (K=,M=, rate / code) 0/ 0/ / 0/ / Figure 9: Encoder for rate / constraint length convolutional code. 0/ / / Figure 9: State Diagram for rate / constraint length convolutional code. XIII- XIII-

2 x w We can describe the sequence of states the encoder asses through in time via a trellis diagram. This will be useful for decoding uroses. Figure 95: Trellis Diagram for rate / constraint length convolutional code. Decoding Convolutional Codes Consider a state diagram for a articular convolutional code. Let be the sequence reresenting the state at time m. Let x 0 be the initial state of the rocess x 0. Later on we will denote the states by the integers,,...,n. Since this is a Markov rocess we have that xm x x 0 That is, the state at time m deends only on the state at time m and not any revious state. Let w m corresondence between state sequences and transition sequences. By some mechanism (e.g. a noisy channel) a noisy version z m sequence is observed. Based on this noisy version of wm we wish to estimate the state sequence or the transition sequence w m. Since wm xm be the state transition at time m. There is a one-to-one of the state transition XIII-5 XIII- and contain the same information we have that z z where zz0 x0 channel is memoryless then we have that z zm, x z x M 0 m xm, and w z m w wm w0 wm. If the So given an observation z find the state sequence x for which the a osteriori robability x z is largest. This minimizes the robability that we chose the wrong sequence. Thus the otimum (minimum sequence error robability) decoder chooses x which maximizes x z : i.e ˆx argmaxx x z argmaxx argminx x z log x z XIII- argminx log z Using the memoryless roerty of the channel we obtain z x x x M m0 and using the Markov roerty of the state sequence Define λ x M m0 z k wm wm as follows: λ wm ln Then M ˆx argminx λ m0 This roblem formulation leads to a recursive solution. The recursive solution ln x 0 z m wm wm XIII-8

3 M u is called the Viterbi Algorithm by communication engineers and is a form of Dynamic Programming as studied by control engineers. They are really the same though. xm VITERBI ALGORITHM Let Γ to state at time m. Let ˆx be the shortest ath to state at time m. Let ˆΓ at time m that goes through state at time m. Then the algorithm works as follows. Storage Initialization be the length (otimization criteria) of the shortest (otimum) ath be the length of the ath to state xm m, time index, m0, ˆx Γ xm M o ˆx xm ˆx Γ Γ x 0 x 0 x0 arbitrary, m 0, m 0 x0 XIII-9 XIII- Justification: Recursion ˆΓ Γ xm Let ˆ Γ minxm ˆΓ λ xm argminxm ˆΓ wm for each xm. ˆx ˆx ˆ Basically we are interested in finding the shortest length ath through the trellis. At time m we find the shortest length aths to each of the ossible states at time m by comuting all ossible ways of getting to state ufrom a state at time m. If the shortest ath (denoted by ˆx u ) to get to uat time m goes through state vattime m (i.e. ˆx ˆx v corresonding ath ˆx v to state vmust be the shortest ath to state v at time msince if there was a shorter ath, say x v, to state v at time m then the ath x v time mwould be shorter then what we assumed was the shortest ath). Stated another way if the shortest way of getting to state u at time m is by going through state v at time mthen the ath used to get to state v at time m must be the shortest of all aths to state v at time m. u) then the u to state u at time m that used this shorter ath to state v at XIII- XIII-

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6 XIII- XIII- Error Bounds for Convolutional Codes The erformance of convolutional codes can be uer bounded by l d f ree w l D l Examle Binary Symmetric Channel crossover robability. where w l is the average number of nonzero information bits on aths with Hamming distance l and D is a arameter that deends only on the channel. usually the summation in the uer bound is truncated to some finite number of terms. D XIII- XIII-

7 8D 505D8 Examle Additive White Gaussian Noise channel D EN e 0 Performance Examles Generally hard decisions requires db more signal energy than soft decisions for the same bit error robability. Also soft decisions is only about 0.5dB better than 8 level quantization. XIII-5 XIII- Standard codes: Examle Convolutional Code : Constraint length, memory, state decoder, rate / has the following uer bound. D D 0D There is a chi made by Qualcomm and Stanford Telecommunications that oerates at data rates on the order of Mbits/second that will do encoding and decoding. Examle Convolutional Code : Constraint length 9, memory 8, 5 state decoder, rate / D Examle Convolutional Code : Constraint length 9, memory 8, 5 state decoder, rate / D 8 D0 95D 5D D D 9D XIII- XIII-8

8 D 59D D Examle (K=,M=, rate / code) Examle (K M rate / code) D This code has d f of 5. The weight enumerator olynomial is w D 5 D D5 D D D8 This code has d f of. The weight enumerator olynomial for determining the bit error robability is given by w D D D 0D D 5090D0 D 99D D 8 XIII-9 XIII-0 858D 8 We can uer bound the bit error robability by w P w D w The first bound is the union bound. It is imossible to exactly evaluate this bound because there are an infinite number of terms in the summation. Droing all but the first N terms gives an aroximation. It may no longer be an uer bound though. If the weight enumerator is known we can get arbitrarily close to the union bound and still get a bound as follows. w P N d f w P N d f w P N w D N w P N N d f w d f w P P D D w d f w D The second term is the Union-Bhattacharyya (U-B) bound. The first term is clearly less than zero, so we get something that is tighter than the U-B bound. By choosing N sufficiently large we can sometimes get significant imrovements over the U-B bound. XIII- XIII-

9 0 0 Simulation Uer Bound P e,b Lower Bound E /N (db) b E b /N 0 Figure 9: Error robability of constraint length convolutional codes on an additive white Gaussian noise channel with soft decisions decoding (uerbound, simulation and lower bound). XIII- Figure 9: Error robability of constraint length convolutional codes on an additive white Gaussian noise channel with soft decisions decoding (uerbound, simulation). XIII- 0 U er Bounds on Bit Error Probabilityfor Constraint Length, R ate / Convolutional Code Union Bound - - Simulation Uncoded P e,b - Bit Error Rate Hard Decisions Soft Decisions EbN0(dB) Figure 98: Error robability of constraint length convolutional codes on an additive white Gaussian noise channel with soft decisions decoding (uerbound, simulation). XIII E b /N 0 (db) Figure 99: Error Probability of Constraint Length Convolutional Codes on an Additive White Gaussian Noise Channel (hard and soft decisions). XIII-

10 Bit Error Probability Bit Error Probability (Bound) for Constraint Length 9 Rate / Convolutional Code Convolutional Decoder Hard Decisions Soft Decisions state E b /N 0 (db) Figure 0: Error Probability of Constraint Length 9 Convolutional Codes on an Additive White Gaussian Noise Channel (hard and soft decisions). XIII time XIII-8 Convolutional Decoder 0 Convolutional Decoder state state time time XIII-9 XIII-0