coding5 - Coding for Communications Lecture 5 Convolutional...

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1 Coding for Communications Lecture 5 Convolutional codes
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2 Convolutional Codes ± Another class of linear codes ± We study the structure of the encoder. ± We study different ways for representing the encoder. Questions on convolutional codes ± How the decoding is performed for Convolutional codes? ± What are the state diagram and trellis representation of the code?
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3 Convolutional codes ± Convolutional codes offer an approach to error control coding substantially different from that of block codes. ± A convolutional encoder: ± encodes the entire data stream, into a single codeword. ± does not need to segment the data stream into blocks of fixed size ( Convolutional codes are often forced to block structure by periodic truncation ). ± is a machine with memory. ± This fundamental difference in approach imparts a different nature to the design and evaluation of the code. ± Block codes are based on algebraic/combinatorial techniques. ± Convolutional codes are based on construction techniques. Convolutional codes-cont’d ± A Convolutional code is specified by three parameters or where ± is the coding rate , determining the number of data bits per coded bit. ± In practice, usually k=1 is chosen and we assume that from now on. ± K is the constraint length of the encoder a where the encoder has K-1 memory elements. ± There is different definitions in literatures for constraint length. ) , , ( K k n ) , / ( K n k n k R c / =
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4 Block diagram of the DCS Information source Rate 1/n Conv. encoder Information sink Rate 1/n Conv. decoder Channel 4 43 4 42 1 sequence Input 1 0 ,...) ,..., , ( i x x x = x 4 4 4 4 1 4 4 4 4 1 bits) coded ( word Branch ) 1 ( ) 1 ( ) 0 ( sequence Codeword 2 1 0 ,...) ,..., , , ( n n i i i i i ,...,y ...,y y Y Y Y Y Y = = = G(x) Y ,...) ' ,..., ' , ' ( ' 1 0 i y y y = y { 4 4 4 4 1 4 4 4 4 1 word per Branch outputs ) 1 ( ) 1 ( ) 0 ( word for Branch outputs Channel sequence received 3 2 1 ,...) ,..., , , ( n n i i i i i i ,...,r ,...,r r R R R R R = = R A Rate ½ Convolutional encoder ± Convolutional encoder (rate ½, K=3) ± 3 shift-registers where the first one takes the incoming data bit and the rest, form the memory of the encoder. Input data bits Output coded bits m 0 y 1 y First coded bit Second coded bit 1 0 , y y (Branch word)
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5 A Rate ½ Convolutional encoder 1 0 0 1 t 0 y 1 y 1 1 1 0 y y 0 1 0 2 t 1 u 2 u 0 1 1 0 y y 1 0 1 3 t 1 u 2 u 0 0 1 0 y y 0 1 0 4 t 1 u 2 u 0 1 1 0 y y ) 101 ( = x Time Output Output Time Message sequence: (Branch word) (Branch word) A Rate ½ Convolutional encoder Encoder ) 101 ( = x ) 11 10 00 10 11 ( = Y 0 0 1 5 t 0 y 1 y 1 1 1 0 y y 0 0 0 6 t 0 y 1 y 0 0 1 0 y y Time Output Time Output (Branch word) (Branch word)
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6 Effective code rate ± Initialize the memory before encoding the first bit (all- zero) ± Clear out the memory after encoding the last bit (all- zero) ± Hence, a tail of zero-bits is appended to data bits. ± Effective code rate (with fractional rate loss): ± L is the number of data bits and k=1 is assumed: data Encoder codeword tail c eff R K L n L R < + = ) 1 ( Encoder representation ± Vector representation: ± We define n binary vector with K elements (one vector for each modulo-2 adder). The i:th element in each vector, is “1” if the i:th stage in the shift
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This note was uploaded on 04/24/2011 for the course EE 4421 taught by Professor Mondin during the Spring '11 term at California State University Los Angeles .

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coding5 - Coding for Communications Lecture 5 Convolutional...

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