1 / 72

Note 5. Channel Coding: Part 2 : Convolutional Codes

EEE3440-04 Digital Communications: Fall Semester 2012. Note 5. Channel Coding: Part 2 : Convolutional Codes. Sooyong Choi csyong@yonsei.ac.kr School of Electrical and Electronic Engineering Yonsei University. Channel Coding : Part 2. Simple Communication System.

justus
Télécharger la présentation

Note 5. Channel Coding: Part 2 : Convolutional Codes

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. EEE3440-04 Digital Communications: Fall Semester 2012 Note 5. Channel Coding: Part 2 : Convolutional Codes Sooyong Choi csyong@yonsei.ac.kr School of Electrical and Electronic Engineering Yonsei University

  2. Channel Coding : Part 2

  3. Simple Communication System • Encode/decode and modulate/demodulate portions of a communication link Information source Convolutional encode Modulate AWGN channel Information sink Convolutional decode Demodulate

  4. Convolutional Codes • 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.

  5. Convolutional Encoder • Rate = k/n = ½, K = 3 Convolutional Encoder Output branch Word U Input bit m

  6. Convolutional Encoder • Generator polynomial • = Connection vector • = Encoding function • g1 = 1 1 1 • g1(X) = 1+X+X2 • Rate = k/n = ½, K = 3 + u1 : First code symbol Output branch Word U Input bit m u2 : Second code symbol • Generator polynomial • = Connection vector • = Encoding function • g2 = 1 0 1 • g2(X) = 1+X2 + Convolutional encoder

  7. Convolutional Encoder • Example : At time 1 + u1 = 1 1 0 0 0 m = 1 u2 = +

  8. Convolutional Encoder • Example : At time 2 + 0 u1 = 1 0 1 0 m = 1 u2 = +

  9. Convolutional Encoder • Example : At time 3 + 0 u1 = 0 1 1 0 m = 1 u2 = +

  10. Convolutional Encoder • Example : At time 4 + 0 u1 = 1 1 0 1 m = 0 u2 = +

  11. Convolutional Encoder • Example : At time 5 + 0 u1 = 1 0 1 1 m = 1 u2 = +

  12. Convolutional Encoder • Example : At time 6 + 0 u1 = 0 1 1 0 m = 1 u2 = +

  13. Convolutional Encoder • Example : At time 7 + 1 u1 = 0 1 0 1 m = 1 u2 = +

  14. Effective Code Rate • Block code : A fixed word length n • Convolutional code : No particular block size • 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 : No information • Effective code rate : • L is the number of data bits and k=1 is assumed: Encoder data tail codeword

  15. Encoder Representation • Vector representation: • We define n binary vector with K elements (one vector for each modulo-2 adder) • The ith element in each vector, is “1” if the ith stage in the shift register is connected to the corresponding modulo-2 adder, and “0” otherwise. • Example:

  16. Input m Output Modulo-2 sum: Encoder Representation • Impulse response representation: • The response of encoder to a single “one” bit that goes through it. • Example: Branch word Register contents Output = Superposition or the input addition of the time shifted input “impulses” Convolutional codes are linear

  17. Encoder Representation • Polynomial representation: • Define n generator polynomials, one for each modulo-2 adder. • Each polynomial is of degree K-1 or less. • Each polynomial describes the connection of the shift registers to the corresponding modulo-2 adder. • Example: • Lowest order term in the polynomial = Input stage of the register • Output sequence :

  18. State Diagram • Convolutional encoder  A class of devices known as finite-state machines • Finite-state machines have a memory of past signals • 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 • The state provides some knowledge of the past signaling events and the restricted set of possible output in the future • Convolutional encoder :the state represented by the content of the memory • states • A state diagram is a way to represent the encoder • A state diagram contains all the states & all possible transitions between them • Only two transitions initiating from a state • Only two transitions ending up in a state

  19. State Diagram + u1 m u2 0/00 + Output (Branch word) Input 00 1/11 0/11 1/00 10 01 0/10 1/01 0/01 11 Input bit 0 Input bit 1 1/10

  20. Tree Diagram • State Diagram • Cannot represent time history • Tree Diagram • State diagram + dimension of time • Traversing the diagram from left to the right at each successive input bit time. • Input : Zero  Its branch word : Moving to the next rightmost branch in the upward direction • Input : One  Its branch word : Moving to the next rightmost branch in the downward direction Input = 11011

  21. Trellis Diagram • Trellis diagram • Extension of the state diagram that shows the passage of time • Label each node : Four possible states in the shift register a = 00, b = 10, c = 01, and d = 11 • Tree structure : Repeats after K (= constraint length) branching • Sold line : Output generated by an input bit zero • Dashed line : Output generated by an input bit one • All branches emanating from two nodes of the same state generate identical branch word sequences Input bit 0 Input bit 1

  22. Trellis Diagram • Nodes of the trellis • Characterize the encoder states • First row nodes : State a = 00 • Second & subsequent rows : States b = 10, c = 01, and d = 11. • Each unit of time : 2K-1 nodes to represent the 2K-1 possible encoder states • After depth K : Fixed structure • Branches of the trellis • Code symbol sequence • Characterized by N branches (representing N data bits) : N intervals of time Input bit 0 Input bit 1

  23. Trellis Diagram • Encoder trellis diagram with rate k/n = ½, K = 3 State t1 t2 a = 00 b = 10 c = 01 d = 11 + u1 = 0 0 0 0 0 m = 0 u2 = +

  24. Trellis Diagram • Encoder trellis diagram with rate k/n = ½, K = 3 State t1 t2 00 a = 00 b = 10 c = 01 d = 11 + u1 = 0 0 0 0 m = 0 u2 = +

  25. Trellis Diagram • Encoder trellis diagram with rate k/n = ½, K = 3 State t1 t2 00 a = 00 b = 10 c = 01 d = 11 + u1 = 1 1 0 0 0 m = 1 u2 = +

  26. Trellis Diagram • Encoder trellis diagram with rate k/n = ½, K = 3 State t1 t2 00 a = 00 b = 10 c = 01 d = 11 + 11 u1 = 1 0 1 0 m = 1 u2 = +

  27. Trellis Diagram • Encoder trellis diagram with rate k/n = ½, K = 3 State t1 t2 t3 00 00 a = 00 b = 10 c = 01 d = 11 + 11 11 u1 = 1 0 0 1 X m = 0 u2 = +

  28. Trellis Diagram • Encoder trellis diagram with rate k/n = ½, K = 3 State t1 t2 t3 00 00 a = 00 b = 10 c = 01 d = 11 + 11 11 u1 = 1 1 0 0 m = 0 u2 = 10 +

  29. Trellis Diagram • Encoder trellis diagram with rate k/n = ½, K = 3 State t1 t2 t3 00 00 a = 00 b = 10 c = 01 d = 11 + 11 11 u1 = 0 1 0 1 X m = 1 u2 = 10 +

  30. Trellis Diagram • Encoder trellis diagram with rate k/n = ½, K = 3 State t1 t2 t3 00 00 a = 00 b = 10 c = 01 d = 11 + 11 11 u1 = 0 1 1 0 m = 1 u2 = 10 + 01

  31. + Trellis Diagram u1 m u2 • Encoder trellis diagram with rate k/n = ½, K = 3 + State t1 t2 t3 t4 t5 t6 00 00 00 00 00 a = 00 b = 10 c = 01 d = 11 11 11 11 11 11 11 11 11 00 00 00 10 10 10 10 01 01 01 01 01 01 01 10 10 10

  32. Convolutional Decoding • We talked about: • Another class of linear codes, known as Convolutional codes. • We studied the structure of the encoder and different ways for representing it. • State Diagram • Tree diagram • Trellis Diagram • From now on, we are going to talk about: • How the decoding is performed for Convolutional codes? • What is a Maximum likelihood decoder? • What are the soft decisions and hard decisions? • How does the Viterbi algorithm work?

  33. Block Diagram of the DCS Information source Rate 1/n Conv. encoder Modulator Channel Information sink Rate 1/n Conv. decoder Demodulator

  34. Formulation of the Convolutional Decoding Problem • Maximum Likelihood Decoding • All input message sequences are equally likely, • Decoder : Minimum probability of error • Compares the conditional probabilities called likelihood functions • : Received sequence • : One of possible transmitted sequences • Chooses the maximum • Maximum likelihood concept • Binary demodulation: Only two equally likely possible signals

  35. Maximum Likelihood Decoding • Convolutional code • Has memory • Received sequence = Superposition of currents and prior bits • Applying ML = Choosing the most likely sequence • Multitude of possible codeword transmitted • Binary code • Sequence of L branch words = Member of a set of 2L possible sequences • ML decoder minimizes the error probability • Maximum likelihood • The decoder chooses a particular as the transmitted sequence if the likelihood is greater than the likelihoods of all other possible transmitted sequemces

  36. Maximum Likelihood Decoding • Assumption • Additive white Gaussian noise with zero mean • Memoryless channel • Noise affects each code symbol independently of all the other symbols • Convolutional code of rate 1/n • Likelihood : • Zi : i th branch of the received sequence Z • : i th branch of a particular codeword sequence • zji : j th code symbol of Zi • : j th code symbol of • Each branch = n code symbols • The decoder problem : Choose a path through the trellis

  37. Path metric Branch metric Bit metric Maximum Likelihood Decoding • Log-likelihood function • The decoder problem ⇒ Choose a path through the tree such that is maximized • Binary code • Sequence of L branch words = Member of a set of 2L possible sequences • Not practical to consider maximum likelihood decoding with a tree structure • The decoded path is chosen from some reduced set of surviving paths • Viterbi decoding algorithm • Performs maximum likelihood decoding • Optimal decoding

  38. Channel Models: Soft & Hard Decisions • Hard decision • Demodulator • Makes a firm or hard decision whether one or zero is transmitted • Provides no other information for the decoder such that how reliable the decision is • Output of the demodulator : Only zero or one (the output is quantized only to two level) which are called “hard-bits” • Decoding based on hard-bits is called the “hard-decision decoding”.

  39. Soft and Hard Decisions • Soft decision • Demodulator • Provides the decoder with some side information together with the decision • The side information provides the decoder with a measure of confidence for the decision • The demodulator outputs which are called soft-bits, are quantized to more than two levels • Decoding based on soft-bits, is called the “soft-decision decoding” • On AWGN channels, 2 dB and on fading channels 6 dB gain are obtained by using soft-decoding over hard-decoding

  40. AWGN Channels • BPSK modulation • Transmitted sequence corresponding to the codeword is denoted by • Log-likelihood function • Maximizing the correlation = Minimizing the Euclidean distance Inner product or correlation between Z and S • ML decoding rule: Choose the path which with minimum Euclidean distance to the received sequence

  41. The Viterbi Algorithm • Viterbi algorithm • Maximum likelihood decoding • Complexity : Not a function of the number of symbols in the codeword sequence • Calculating a measure of similarity or distance between the received signal at time ti and all the trellis paths entering each state at time ti • Find a path through trellis with the largest metric (maximum correlation or minimum distance). • Process the demodulator outputs in an iterative manner. • Compares the metric of all paths entering each state • Keeps only the path with the largest metric, called the survivor (surviving path), together with its metric. • Eliminate the least likely paths. • Goal of selecting the optimum path • Choosing the codeword with the maximum likelihood metric, or with the minimum distance metric

  42. Viterbi Algorithm • Do the following set up: • Form the trellis for a data block of L bits • The trellis has L+K-1 sections or levels and starts at time t1 and ends up at time tL+K. • Label all the branches in the trellis with their corresponding branch metric • Define a parameter (S(ti), ti) for each state in the trellis at the time ti which is denoted by S(ti)  {0, 1, 2,…, 2K-1} • Then, do the following: • Set (0, t1) = 0 and i = 2. • Compute the partial path metrics for all the paths entering each state at time ti • Set (S(ti), ti) equal to the best partial path metric entering each state at time ti Keep the survivor path and delete the dead paths from the trellis • If i < L+K, increase i by 1 and return to step 2. • Start at state zero at time tL+K • Follow the surviving branches backwards through the trellis.  The path thus defined is unique and corresponds to the ML codeword

  43. Example : Viterbi Decoding • Input data m: 1 1 0 1 1 • Codeword U: 11 01 01 00 01 Received seq. Z: 11 01 01 10 01 t1 t2 2 a = 00 b = 10 0

  44. Example : Viterbi Decoding • Input data m: 1 1 0 1 1 • Codeword U: 11 01 01 00 01 Received seq. Z: 11 01 t1 t2 t3 1 2 a = 00 b = 10 0 1 2 0 c = 01 d = 11

  45. Example : Viterbi Decoding • Input data m: 1 1 0 1 1 • Codeword U: 11 01 01 00 01 Received seq. Z: 11 01 01 t1 t2 t3 t4 1 1 2 a = 00 b = 10 0 1 1 1 1 2 2 0 c = 01 d = 11 0 0 2

  46. Example : Viterbi Decoding • Input data m: 1 1 0 1 1 • Codeword U: 11 01 01 00 01 Received seq. Z: 11 01 01 t1 t2 t3 t4 1 1 2 Path metric a = 00 b = 10 0 1 1 4 1 1 2 3 2 0 c = 01 d = 11 0 0 2

  47. Example : Viterbi Decoding • Input data m: 1 1 0 1 1 • Codeword U: 11 01 01 00 01 Received seq. Z: 11 01 01 t1 t2 t3 t4 1 1 2 Path metric a = 00 b = 10 0 1 1 4 1 1 2 3 2 0 c = 01 d = 11 0 0 2

  48. Example : Viterbi Decoding • Input data m: 1 1 0 1 1 • Codeword U: 11 01 01 00 01 Received seq. Z: 11 01 01 t1 t2 t3 t4 1 1 2 a = 00 b = 10 0 1 1 Path metric 1 1 2 4 2 0 c = 01 d = 11 0 0 0 2

  49. Example : Viterbi Decoding • Input data m: 1 1 0 1 1 • Codeword U: 11 01 01 00 01 Received seq. Z: 11 01 01 t1 t2 t3 t4 1 1 2 a = 00 b = 10 0 1 1 Path metric 1 1 2 2 0 c = 01 d = 11 0 3 2 0 2

  50. Example : Viterbi Decoding • Input data m: 1 1 0 1 1 • Codeword U: 11 01 01 00 01 Received seq. Z: 11 01 01 01 t1 t2 t3 t4 t5 1 a = 00 b = 10 0 1 1 1 1 1 2 0 0 c = 01 d = 11 2 2 0 2 0

More Related