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Modulation and coding course- lecture 10

Another class of linear codes, known as Convolutional codes. -We study the structure of the encoder. -We study different ways for representing the encoder. Digital Communications I: Modulation and Coding Course Period 3 - 2007 Catharina Logothetis Lecture 10 Last time, we talked about: Channel coding Linear block codes The error detection and correction capability Encoding and decoding Hamming codes Cyclic codes Lecture 10 2 Today, we are going to talk about: Another class of linear codes, known

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  1. Digital Communications I: Modulation and Coding Course Period 3 - 2007 Catharina Logothetis Lecture 10
  2. Last time, we talked about: Channel coding Linear block codes The error detection and correction capability Encoding and decoding Hamming codes Cyclic codes Lecture 10 2
  3. Today, we are going to talk about: Another class of linear codes, known as Convolutional codes. We study the structure of the encoder. We study different ways for representing the encoder. Lecture 10 3
  4. 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. Lecture 10 4
  5. Convolutional codes-cont’d A Convolutional code is specified by three parameters (n, k , K ) or (k / n, K ) where Rc = k / n 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. Lecture 10 5
  6. Block diagram of the DCS Information Rate 1/n Modulator source Conv. encoder m = (m1 , m2 ,..., mi ,...) U = G(m) 144 44 2 3 = (U1 , U 2 , U 3 ,..., U i ,...) Channel Input sequence 144 2444 4 3 Codeword sequence U i = u1i ,...,u ji ,...,u ni 14 2444 3 Branch word ( n coded bits) Information Rate 1/n Demodulator sink Conv. decoder m = (m1 , m2 ,..., mi ,...) ˆ ˆ ˆ ˆ Z = ( Z1 , Z 2 , Z 3 ,..., Z i ,...) 144 2444 4 3 received sequence Zi = z1i ,...,z ji ,...,z ni { 14 2444 3 Demodulator outputs n outputs per Branch word for Branch word i Lecture 10 6
  7. 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. u1 First coded bit (Branch word) Input data bits Output coded bits m u1 ,u2 u2 Second coded bit Lecture 10 7
  8. A Rate ½ Convolutional encoder Message sequence: m = (101) Time Output Time Output (Branch word) (Branch word) u1 u1 u1 u 2 u1 u 2 t1 1 0 0 t2 0 1 0 1 1 1 0 u2 u2 u1 u1 u1 u 2 u1 u 2 t3 1 0 1 t4 0 1 0 0 0 1 0 u2 u2 Lecture 10 8
  9. A Rate ½ Convolutional encoder Time Output Time Output (Branch word) (Branch word) u1 u1 u1 u 2 u1 u 2 t5 0 0 1 t6 0 0 0 1 1 0 0 u2 u2 m = (101) Encoder U = (11 10 00 10 11) Lecture 10 9
  10. 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. data tail Encoder codeword Effective code rate : L is the number of data bits and k=1 is assumed: L Reff = < Rc n( L + K − 1) Lecture 10 10
  11. 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 register is connected to the corresponding modulo- 2 adder, and “0” otherwise. Example: u1 g1 = (111) m u1 u 2 g 2 = (101) u2 Lecture 10 11
  12. Encoder representation – cont’d Impulse response representaiton: The response of encoder to a single “one” bit that goes through it. Example: Branch word Register contents u1 u2 100 1 1 Input sequence : 1 0 0 010 1 0 Output sequence : 11 10 11 001 1 1 Input m Output 1 11 10 11 0 00 00 00 1 11 10 11 Modulo-2 sum: 11 10 00 10 11 Lecture 10 12
  13. Encoder representation – cont’d Polynomial representation: We define n generator polynomials, one for each modulo-2 adder. Each polynomial is of degree K-1 or less and describes the connection of the shift registers to the corresponding modulo-2 adder. Example: g1 ( X ) = g 01) + g1(1) . X + g 21) . X 2 = 1 + X + X 2 ( ( g 2 ( X ) = g 02 ) + g1( 2 ) . X + g 22 ) . X 2 = 1 + X 2 ( ( The output sequence is found as follows: U( X ) = m( X )g1 ( X ) interlaced with m( X )g 2 ( X ) Lecture 10 13
  14. Encoder representation –cont’d In more details: m( X )g1 ( X ) = (1 + X 2 )(1 + X + X 2 ) = 1 + X + X 3 + X 4 m( X )g 2 ( X ) = (1 + X 2 )(1 + X 2 ) = 1 + X 4 m( X )g1 ( X ) = 1 + X + 0. X 2 + X 3 + X 4 m( X )g 2 ( X ) = 1 + 0. X + 0. X 2 + 0. X 3 + X 4 U( X ) = (1,1) + (1,0) X + (0,0) X 2 + (1,0) X 3 + (1,1) X 4 U = 11 10 00 10 11 Lecture 10 14
  15. State diagram A finite-state machine only encounters a finite number of states. State of a machine: the smallest amount of information that, together with a current input to the machine, can predict the output of the machine. In a Convolutional encoder, the state is represented by the content of the memory. Hence, there are 2 states. K −1 Lecture 10 15
  16. State diagram – cont’d A state diagram is a way to represent the encoder. A state diagram contains all the states and all possible transitions between them. Only two transitions initiating from a state Only two transitions ending up in a state Lecture 10 16
  17. State diagram – cont’d Current input Next output 0/00 Output state state (Branch word) S0 Input S0 0 S0 00 1/11 00 0/11 00 1 S2 11 1/00 S1 0 S0 11 S2 S1 10 01 01 1 S2 00 0/10 S2 0 S1 10 1/01 S3 0/01 10 1 S3 01 0 01 11 S3 S1 1/10 11 1 S3 10 Lecture 10 17
  18. Trellis – cont’d Trellis diagram is an extension of the state diagram that shows the passage of time. Example of a section of trellis for the rate ½ code State S 0 = 00 0/00 1/11 S 2 = 10 0/11 1/00 S1 = 01 1/01 0/10 0/01 S3 = 11 1/10 ti ti +1 Time Lecture 10 18
  19. Trellis –cont’d A trellis diagram for the example code Input bits Tail bits 1 0 1 0 0 Output bits 11 10 00 10 11 0/00 0/00 0/00 0/00 0/00 1/11 1/11 1/11 1/11 1/11 0/11 0/11 0/11 0/11 0/11 1/00 1/00 1/00 1/00 1/00 0/10 0/10 0/10 0/10 0/10 1/01 1/01 1/01 1/01 1/01 0/01 0/01 0/01 0/01 0/01 t1 t 2 t 3 t 4 t 5 t 6 Lecture 10 19
  20. Trellis – cont’d Input bits Tail bits 1 0 1 0 0 Output bits 11 10 00 10 11 0/00 0/00 0/00 0/00 0/00 1/11 1/11 1/11 0/11 0/11 0/11 0/10 1/00 0/10 0/10 1/01 1/01 0/01 0/01 t1 t 2 t 3 t 4 t 5 t 6 Lecture 10 20
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