154 Part 2 Lecture 5 and 6

154 Part 2 Lecture 5 and 6 - Convolutional Codes...

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Coding 1 1/1/2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Convolutional Codes
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Coding 2 1/1/2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Error Control Coding Convolutional codes n Convolutional codes are used in many wireless systems including IS-95, and 3GPP wireless cellular communication systems for forward error correction n Unlike block codes in which encoding is done on a block by block basis , encoding is continuous for convolutional codes. n When a new information bit arrives it is shifted into a shift register of length M bits, and the encoder generates n output bits using linear combinations ( moduo 2 sums) of the M bits in the shift register; M is called the constraint or memory length n The decoder performs error correction operation using a moderately complex signal processing algorithm called the Viterbi algorithm that is very conducive to VLSI implementation ( Single chip Viterbi decoders have been available for a while) n Viterbi algorithm operates on the quantized values of the matched filter output. n Binary quantization decoding is called hard decision decoding and multi-level quantization decoding schemes are called soft decision decoding. n Digital cellular systems use rate 1/2, 1/3 1nd 1/4 convolutional codes with constraint lengths from 5 to 10 and coding gains ranging from 3 to 5 dB
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Coding 3 1/1/2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Example of a Convolutional code n Consider the rate 1/3 binary convolutional code whose encoder block diagram is shown below n State transitions along with the corresponding input bits and output bits are conveniently depicted a 0 State diagram or a 0 Trellis diagram Current Bit b k Prev. Bit b k-1 Prev Bit b k-2 Output bits Input bits n =3 , k =1 Rate = 1/3; Total Memory M= 3 State of the Encoder : Right most two bits 00, 01, 10, 11
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Coding 4 1/1/2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Convolutional Code Example: Rate 2/3 Encoder Split Merge r b r b /2 r b /2 r b (3/2) Shift in two bits at a time into the encoder, one bit to the top and one bit to the bo Shift out three bits at a time out of the encoder k = 2; n = 3 M =2 Constraint length = 2 symbols or 6 bits
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Coding 5 1/1/2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Examples of Generator Polynomials and Bound on Aymptotic Coding Gain (soft decision) 1653 37 g 4 (x) 1633 33 1633 10101 g 3 (x) 27 16 20 12 8 5 d f: Free distance 8.28 1365 1117 10 1/4 6.01 27 25 5 1/4 8.23 1365 1117(Oct) 10 1/3 6.01 11011 11111 5 1/3 6.01 110101 101111 6 1/2 3.97 101 111 3 1/2 Max Coding gain(db) g 2 (x) g 1 (x) Memory Code Rate R Bound on asymptotic coding gain (soft decision) < 10 log 10 (Rd f ) [ 10 log10 (Rd f /2 ) for hard decision] Note: Increasing memory and 1/R will increase the free distance and the coding gain; Increasing 1/R will reduce E c and hence will offset the effect of increased free distance Example: R =1/2; M =3; Gen Polynomials = 5,7 Octal = 101, 111 111 101
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Coding 6 1/1/2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Convolutional Code: State Diagram Input bit = 1 Input bit = 0 Current Bit b k Prev.
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154 Part 2 Lecture 5 and 6 - Convolutional Codes...

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