Chapter10 Convolutional Codes. Dr. Chih-Peng Li ( 李 )

Transcription

1 Chapter Convolutional Codes Dr. Chih-Peng Li ( 李 )

2 Table of Contents. Encoding of Convolutional Codes. tructural Properties of Convolutional Codes. Distance Properties of Convolutional Codes

3 Convolutional Codes Convolutional codes differ from block codes in that the encoder contains memory and the n encoder outputs at any time unit depend not only on the k inputs but also on m previous input blocks. An (n, k, m) convolutional code can be implemented with a k- input, n-output linear sequential circuit with input memory m. Typically, n and k are small integers with k<n, but the memory order m must be made large to achieve low error probabilities. In the important special case when k=, the information sequence is not divided into blocks and can be processed continuously. Convolutional codes were first introduced by Elias in 955 as an alternative to block codes.

4 Convolutional Codes hortly thereafter, Wozencraft proposed sequential decoding as an efficient decoding scheme for convolutional codes, and experimental studies soon began to appear. In 96, Massey proposed a less efficient but simpler-toimplement decoding method called threshold decoding. Then in 967, Viterbi proposed a maximum likelihood decoding scheme that was relatively easy to implement for cods with small memory orders. This scheme, called Viterbi decoding, together with improved versions of sequential decoding, led to the application of convolutional codes to deep-space and satellite communication in early 97s.

5 Convolutional Code A convolutional code is generated by passing the information sequence to be transmitted through a linear finite-state shift register. In general, the shift register consists of K (k-bit) stages and n linear algebraic function generators. 5

6 Convolutional codes Convoltuional Code k = number of bits shifted into the encoder at one time k= is usually used!! n = number of encoder output bits corresponding to the k information bits R c = k/n = code rate K = constraint length, encoder memory. Each encoded bit is a function of the present input bits and their past ones. Note that the definition of constraint length here is the same as that of hu Lin s, while the shift register s representation is different. 6

7 Example : Encoding of Convolutional Code Consider the binary convolutional encoder with constraint length K=, k=, and n=. The generators are: g =[], g =[], and g =[]. The generators are more conveniently given in octal form as (,5,7). 7

8 Example : Encoding of Convolutional Code Consider a rate / convolutional encoder. The generators are: g =[], g =[], and g =[]. In octal form, these generator are (, 5, ). 8

9 Representations of Convolutional Code There are three alternative methods that are often used to describe a convolutional code: Tree diagram Trellis diagram tate disgram 9

10 Representations of Convolutional Code Tree diagram Note that the tree diagram in the right repeats itself after the third stage. This is consistent with the fact that the constraint length K=. The output sequence at each stage is determined by the input bit and the two previous input bits. In other words, we may sat that the -bit output sequence for each input bit is determined by the input bit and the four possible states of the shift register, denoted as a=, b=, c=, and d=. Tree diagram for rate /, K= convolutional code.

11 Representations of Convolutional Code Trellis diagram Tree diagram for rate /, K= convolutional code.

12 Representations of Convolutional Code tate diagram a a a c b a b c c b c d d b d d tate diagram for rate /, K= convolutional code.

13 Representations of Convolutional Code Example: K=, k=, n= convolutional code Tree diagram

14 Representations of Convolutional Code Example: K=, k=, n= convolutional code Trellis diagram

15 Representations of Convolutional Code Example: K=, k=, n= convolutional code tate diagram 5

16 Representations of Convolutional Code In general, we state that a rate k/n, constraint length K, convolutional code is characterized by k branches emanating from each node of the tree diagram. The trellis and the state diagrams each have k(k-) possible states. There are k branches entering each state and k branches leaving each state. 6

17 . Encoding of Convolutional Codes Example: A (,, ) binary convolutional codes: the encoder consists of an m= -stage shift register together with n= modulo- adders and a multiplexer for serializing the encoder outputs. The mod- adders can be implemented as ECLUIVE-OR gates. ince mod- addition is a linear operation, the encoder is a linear feedforward shift register. All convolutional encoders can be implemented using a linear feedforward shift register of this type. 7

18 . Encoding of Convolutional Codes The information sequence u =(u, u, u, ) enters the encoder one bit at a time. ince the encoder is a linear system, the two encoder output sequence v = ( υ, υ, υ, ) and v = ( υ, υ, υ, ) can be obtained as the convolution of the input sequence u with the two encoder impulse response. The impulse responses are obtained by letting u =( ) and observing the two output sequence. ince the encoder has an m-time unit memory, the impulse responses can last at most m time units, and are written as : g g () () = = ( g ( g () (), g, g () () 8,, g,, g () m () m ) )

19 . Encoding of Convolutional Codes The encoder of the binary (,, ) code is () g = ( ) () g = ( ) The impulse response g () and g () are called the generator sequences of the code. The encoding equations can now be written as () v () = u g () v () = u g where * denotes discrete convolution and all operations are mod-. The convolution operation implies that for all l, m ( j) ( j) ( j) ( j) ( j) υl li i l l lm m i= where = u g = u g u g u g, j =,,. u = l i for all l < i. 9

20 . Encoding of Convolutional Codes Hence, for the encoder of the binary (,,) code, υ υ () l () l = u = u l l u l u u l l u u as can easily be verified by direct inspection of the encoding circuit. After encoding, the two output sequences are multiplexed into a signal sequence, called the code word, for transmission over the channel. The code word is given by l l v = () () () () () () ( υ υ, υ υ, υ υ, ).

21 Example.. Encoding of Convolutional Codes Let the information sequence u = ( ). Then the output sequences are () v = ( ) ( () v = ( ) ( and the code word is v = (,,, ) = ( ) = (,,,, ). ) )

22 . Encoding of Convolutional Codes If the generator sequence g () and g () are interlaced and then arranged in the matrix () () () () () () () () g g g g g g g m g m () () () () () () () () g g g g g mg m g m g m G = () () () () () () () g g g mg m g mg m g m g m where the blank areas are all zeros, the encoding equations can be rewritten in matrix form as v = ug. G is called the generator matrix of the code. Note that each row of G is identical to the preceding row but shifted n = places to right, and the G is a semi-infinite matrix, corresponding to the fact that the information sequence u is of arbitrary length. ()

23 . Encoding of Convolutional Codes If u has finite length L, then G has L rows and (ml) columns, and v has length (m L). Example. If u=( ), then v = ug = ( = (, ),,,,,, ), agree with our previous calculation using discrete convolution.

24 . Encoding of Convolutional Codes Consider a (,, ) convolutional codes ince k =, the encoder consists of two m = - stage shift registers together with n = mode- adders and two multiplexers.

25 . Encoding of Convolutional Codes The information sequence enters the encoder k = bits at a time, and can be written as () () () () () () u = ( u u, u u, u u, ) () () () () or as the two input sequences u = ( u, u, u, ) () () () () u = ( u, u, u, ) There are three generator sequences corresponding to each input sequence. ( j) ( j) ( j) ( j) Let gi = ( gi,, gi,,, gi, m ) represent the generator sequence corresponding to input i and output j, the generator sequence of the (,, ) convolutional codes are g g () () = ( = ( ), ), g g () () = ( = ( 5 ), ), g g () () = ( = ( ), ),

26 . Encoding of Convolutional Codes And the encoding equations can be written as v v v () () () = u = u = u () () () The convolution operation implies that υ υ υ () l () l () l = u = = u () () () After multiplexing, the code word is given by () () () () l () l g g g u u () l () l u u u () () () u u u () l () l () l g g g u, () l () () () () () () () () () v = ( υ υ υ, υ υ υ, υ υ υ, ). 6

27 . Encoding of Convolutional Codes Example. If u() = ( ) and u() = ( ), then v v v () () () and = ( = ( = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ( ) = ( ) = ( ) ) ) v = (,,, ). 7

28 . Encoding of Convolutional Codes The generator matrix of a (,, m) code is () () () () () () () () () g, g, g, g, g, g, g, mg, mg, m () () () () () () () () () g,g, g, g,g, g, g, mg, mg, m = () () () () () () () () G g, g, g, g, mg, mg, m g, mg, mg () () () () () () () () g,g, g, g, mg, mg, m g, mg, mg and the encoding equation in matrix are again given by v = ug. Note that each set of k = rows of G is identical to the preceding set of rows but shifted n = places to right. (), m (), m 8

29 Example.. Encoding of Convolutional Codes If u () = ( ) and u () = ( ), then u = (,, ) and v = ug = (,, ) = (,,, ), it agree with our previous calculation using discrete convolution. 9

30 . Encoding of Convolutional Codes In particular, the encoder now contains k shift registers, not all of which must have the same length. If K i is the length of the ith shift register, then the encoder memory order m is defined as m max K An example of a (,, ) convolutional encoder in which the shift register length are,, and. i k i

31 . Encoding of Convolutional Codes The constraint length is defined as n A n(m). ince each information bit remains in the encoder for up to m time units, and during each time unit can affect any of the n encoder outputs, n A can be interpreted as the maximum number of encoder outputs that can be affected by a signal information bit. For example, the constraint length of the (,,), (,,), and (,,) convolutional codes are 8, 6, and, respectively.

32 . Encoding of Convolutional Codes If the general case of an (n, k, m) code, the generator matrix is where each G l is a k n submatrix whose entries are Note that each set of k rows of G is identical to the previous set of rows but shifted n places to the right. = m m m m m m G G G G G G G G G G G G G = ) (, (), (), ) (, (), (), ) (, (), (), n l k l k l k n l l l n l l l l g g g g g g g g g G

33 . Encoding of Convolutional Codes For an information sequence () () ( k ) () () ( k ) u = ( u, u, ) = ( u u u, u u u, ) () () ( n) () () ( n) and the code word v = ( v, v, ) = ( υ υ υ, υ υ υ, is given by v = ug. ince the code word v is a linear combination of rows of the generator matrix G, an (n, k, m) convolutional code is a linear code. )

34 . Encoding of Convolutional Codes A convolutional encoder generates n encoded bits for each k information bits, and R = k/n is called the code rate. For an k L finite length information sequence, the corresponding code word has length n(l m), where the final n m outputs are generated after the last nonzero information block has entered the encoder. Viewing a convolutional code as a linear block code with generator matrix G, the block code rate is given by kl/n(l m), the ratio of the number of information bits to the length of the code word. If L» m, then L/(L m), and the block code rate and convolutional code are approximately equal.

35 . Encoding of Convolutional Codes If L were small, however, the ratio kl/n(l m), which is the effective rate of information transmission, would be reduced below the code rate by a fractional amount k n called the fractional rate loss. k n kl n( L m) To keep the fractional rate loss small, L is always assumed to be much larger than m. Example.5 For a (,,) convolutional codes, L=5 and the fractional rate loss is /8=7.5%. However, if the length of the information sequence is L=, the fractional rate loss is only /=.%. = L m m 5

36 . Encoding of Convolutional Codes In a linear system, time-domain operations involving convolution can be replaced by more convenient transform-domain operations involving polynomial multiplication. ince a convolutional encoder is a linear system, each sequence in the encoding equations can be replaced by corresponding polynomial, and the convolution operation replaced by polynomial multiplication. In the polynomial representation of a binary sequence, the sequence itself is represent by the coefficients of the polynomial. For example, for a (,, m) code, the encoding equations become () () v ( D) = u( D) g ( D) () () v ( D) = u( D) g ( D), where u(d) = u u D u D is the information sequence. 6

37 . Encoding of Convolutional Codes The encoded sequences are () () () () v ( D) = υ υ D υ D () () () () v ( D) = υ υ D υ D The generator polynomials of the code are () () () g ( D) = g g D g () () () g ( D) = g g D g and all operations are modulo-. After multiplexing, the code word become () () v ( D) = v ( D ) Dv ( D ) the indeterminate D can be interpreted as a delay operator, and the power of D denoting the number of time units a bit is delayed with respect to the initial bit. 7 () m () m m D D m

38 Example.6. Encoding of Convolutional Codes For the previous (,, ) convolutional code, the generator polynomials are g () (D) = D D and g () (D) = DD D. For the information sequence u(d) = D D D, the encoding equation are () 7 v ( D) = ( D D D )( D D ) = D () v ( D) = ( D D D )( D D D ) 5 7 = D D D D D, and the code word is () () ( D) = D D D = D D D D D D D v v v. Note that the result is the same as previously computed using convolution and matrix multiplication. 8

39 . Encoding of Convolutional Codes The generator polynomials of an encoder can be determined directly from its circuit diagram. ince the shift register stage represents a one-time-unit delay, the sequence of connection (a representing a connection and a no connection) from a shift register to an output is the sequence of coefficients in the corresponding generator polynomial. ince the last stage of the shift register in an (n, ) code must be connected to at least one output, at least one of the generator polynomials must have degree equal to the shift register length m, that is j m = max deg g D j n [ ] 9

40 . Encoding of Convolutional Codes In an (n, k) code where k>, there are n generator polynomials for each of the k inputs. Each set of n generators represents the connections from one of the shift registers to the n outputs. The length K i of the ith shift register is given by K i = [ ] j g D, i k, max deg j n i j where g i D is the generator polynomial relating the ith input to the jth output, and the encoder memory order m is m = max K i i k = max deg j n i k [ ] j g i.

41 . Encoding of Convolutional Codes ince the encoder is a linear system, and u (i) (D) is the ith input sequence and v (j) (D)is the jth output sequence, the generator polynomial ( j g ) i D can be interpreted as the encoder transfer function relating input i to output j. As with k-input, n-output linear system, there are a total of k n transfer functions. These can be represented by the k n transfer function matrix G D () ( n ) g D g D g D () n g D g D g D = () ( ) ( n ) k D k D k D g g g

42 . Encoding of Convolutional Codes Using the transfer function matrix, the encoding equation for an (n, k, m) code can be expressed as ( D) ( D) G( D) V = U where () () ( k ) U D u ( D), u ( D),, u ( D) is the k-tuple of () () ( n) input sequences and V( D) is v ( D), v ( D),, v ( D) the n-tuple of output sequences. After multiplexing, the code word becomes v ( ) ( n ) ( ) ( n ) n ( n ) ( n D = v D Dv D D v D ).

43 Example.7 V. Encoding of Convolutional Codes For the previous (,, ) convolutional code D D D G( D) = D For the input sequences u () (D) = D and u () (D)=D, the encoding equations are [ ] [ ] D D D D = v D, v D, v D = D, D D = D, D, D D () and the code word is v [ ] ( D) = ( ) ( ) ( ) D D D D D D = D D D D D 8 9

44 . Encoding of Convolutional Codes Then, we can find a means of representing the code word v(d) directly in terms of the input sequences. A little algebraic manipulation yields where k () i ( n D = D ) g ( D) v u i= () ( n ) ( ) ( n ) n ( n ) ( n ) g D g D Dg D D g D, i k, i i i i is a composite generator polynomial relating the ith input sequence to v(d). i

45 Example.8. Encoding of Convolutional Codes For the previous (,, ) convolutional codes, the composite generator polynomial is g () ( ) ( ) ( ) D = g D Dg D = D D D D D D and for u(d)=d D D, the code word is ( D = D ) ( D) v u g ( 6 8) ( D D D D D D D D D ) = = D D D D D D D again agreeing with previous calculations. 5

46 . tructural Properties of Convolutional Codes ince a convolutional encoder is a sequential circuit, its operation can be described by a state diagram. The state of the encoder is defined as its shift register contents. For an (n, k, m) code with k >, the ith shift register contains K i previous information bits. k Defined K as the total encoder memory, the encoder i = K i state at time unit l is the binary K-tuple of inputs ( ( k ) ( k ) ( k ) u u u u u u u u u ) l l l K l l l K l l and there are a total K different possible states. l K k 6

47 . tructural Properties of Convolutional Codes For a (n,, m) code, K = K = m and the encoder state at time unit l is simply ( u ). l ul ul m Each new block of k inputs causes a transition to a new state. There are k branches leaving each state, one corresponding to each different input block. Note that for an (n,, m) code, there are only two branches leaving each state. Each branch is labeled with the k inputs causing the transition () ( k ) () ( n) u u and n corresponding outputs υ υ υ. ( u ) l l l The states are labeled,,, K-, where by convention i represents the state whose binary K-tuple representation b,b,,b K- is equivalent to the integer i b b b K = K 7 l l l

48 . tructural Properties of Convolutional Codes Assuming that the encoder is initially in state (all-zero state), the code word corresponding to any given information sequence can be obtained by following the path through the state diagram and noting the corresponding outputs on the branch labels. Following the last nonzero information block, the encoder is return to state by a sequence of m all-zero blocks appended to the information sequence. 8

49 . tructural Properties of Convolutional Codes Encoder state diagram of a (,, ) code If u = ( ), the code word v = (,,,,,,, ) 9

50 . tructural Properties of Convolutional Codes Encoder state diagram of a (,, ) code 5

51 . tructural Properties of Convolutional Codes The state diagram can be modified to provide a complete description of the Hamming weights of all nonzero code words (i.e. a weight distribution function for the code). tate is split into an initial state and a final state, the selfloop around state is deleted, and each branch is labeled with a branch gain i,where i is the weight of the n encoded bits on that branch. Each path connecting the initial state to the final state represents a nonzero code word that diverge from and remerge with state exactly once. The path gain is the product of the branch gains along a path, and the weight of the associated code word is the power of in the path gain. 5

52 . tructural Properties of Convolutional Codes Modified encoder state diagram of a (,, ) code. The path representing the sate sequence has path gain =. 5

53 . tructural Properties of Convolutional Codes Modified encoder state diagram of a (,, ) code. The path representing the sate sequence has path gain =. 5

54 . tructural Properties of Convolutional Codes The weight distribution function of a code can be determined by considering the modified state diagram as a signal flow graph and applying Mason s gain formula to compute its generating function T i ( ) = A i, where A i is the number of code words of weight i. i In a signal flow graph, a path connecting the initial state to the final state which does not go through any state twice is called a forward path. A closed path starting at any state and returning to that state without going through any other state twice is called a loop. 5

55 . tructural Properties of Convolutional Codes Let C i be the gain of the ith loop. A set of loops is nontouching if no state belongs to more than one loop in the set. Let {i} be the set of all loops, {i, j } be the set of all pairs of nontouching loops, {i, j, l } be the set of all triples of nontouching loops, and so on. Then define = C C C C C C, i ' ' '' '' '' i j i j l ' ' '' '' '' i i, j where C is the sum of the loop gains, C C i ' ' i jis the product of i ' ' i, j the loop gains of two nontouching loops summed over all pairs of nontouching loops, C '' C '' C '' i j l is the product of the loop gains of '' '' '' i, j, l three nontouching loops summed over all nontouching loops. i, j, l 55

56 . tructural Properties of Convolutional Codes And i is defined exactly like, but only for that portion of the graph not touching the ith forward path; that is, all states along the ith forward path, together with all branches connected to these states, are removed from the graph when computing i. Mason s formula for computing the generating function T() of a graph can now be states as T F i =, where the sum in the numerator is over all forward paths and F i is the gain of the ith forward path. i i 56

57 57. tructural Properties of Convolutional Codes C Loop C Loop C Loop C Loop C Loop C Loop C Loop C Loop C Loop C Loop C Loop = = = = = = = = = = = : : 9 : 8 : 7 : 6 : 5 : : : : : Example (,,) Code: There are loops in the modified encoder state diagram.

58 . tructural Properties of Convolutional Codes Example (,,) Code: (cont.) There are pairs of nontouching loops : Loop pair Loop pair Loop pair Loop pair Loop pair Loop pair Loop pair Loop pair Loop pair Loop pair : : : : 5 : 6 : 7 : 8 : 9 : : ( loop, loop 8) ( CC8 = ) ( loop, loop ) ( CC 8 = ) ( loop, loop 8) ( CC8 = ) ( loop, loop ) ( CC = ) ( loop 6, loop ) ( C6C 9 = ) ( loop 7, loop 9) ( C7C9 8 = ) ( loop 7, loop ) ( CC8 7 = ) ( loop, loop 8) ( C7C = ) ( loop 7, loop ) ( C7C = ) ( loop 8, loop ) ( C C = ) 58 8

59 . tructural Properties of Convolutional Codes Example (,,) Code : (cont.) There are two triples of nontouching loops : Loop triple Loop triple : : ( loop, loop 8, loop C ) C8C = ( 8 loop 7, loop, loop C C C = ) 7 There are no other sets of nontouching loops. Therefore, = ( ) ( ) ( 8 ) = 59

60 6. tructural Properties of Convolutional Codes Example (,,) Code : (cont.) There are seven forward paths in this state diagram :. path 7 : Foward path 6 : Foward path 5 : Foward path : Foward path : Foward path : Foward path : Foward F F F F F F F = = = = = = =

61 . tructural Properties of Convolutional Codes Example (,,) Code : (cont.) Forward paths and 5 touch all states in the graph, and hence the sub graph not touching these paths contains no states. Therefore, = 5 =. The subgraph not touching forward paths and 6: = 6 = - 6

62 . tructural Properties of Convolutional Codes Example (,,) Code : (cont.) The subgraph not touching forward path : = - The subgraph not touching forward path : = ( ) ( ) = 6

63 . tructural Properties of Convolutional Codes Example (,,) Code : (cont.) The subgraph not touching forward path 7: 7 = ( 5 ) ( 5 ) = 6

64 6. tructural Properties of Convolutional Codes Example (,,) Code : (cont.) The generating function for this graph is then given by T() provides a complete description of the weight distribution of all nonzero code words that diverge from and remerge with state exactly once. In this case, there is one such code word of weight 6, three of weight 7, five of weight 8, and so on. = = = T

65 65. tructural Properties of Convolutional Codes Example (,,) Code : (cont.) There are eight loops, six pairs of nontouching loops, and one triple of nontouching loops in the graph of previous modified encoder state diagrams : (,, ) code, and = =

66 66. tructural Properties of Convolutional Codes F i i i = = Example (,,) Code : (cont.) There are 5 forward path in this graph, and Hence, the generating function is This code contains two nonzero code word of weight, five of weight, fifteen of weight 5, and so on. = = T

67 . tructural Properties of Convolutional Codes Additional information about the structure of a code can be obtained using the same procedure. If the modified state diagram is augmented by labeling each branch corresponding to a nonzero information block with Y j, where j is the weight of the k information bits on the branch, and labeling every branch with Z, the generating function is given by T i j, Y, Z = A Y Z. i, j, l The coefficient A i,j,l denotes the number of code words with weight i, whose associated information sequence has weight j, and whose length is l branches. i, j, l l 67

68 . tructural Properties of Convolutional Codes The augment state diagram for the (,, ) codes. 68

69 69. tructural Properties of Convolutional Codes Example (,,) Code: For the graph of the augment state diagram for the (,, ) codes, we have: ) ( ) ( Z Y Z Y Z Y Z Z Y Z Y YZ Z Z Y Z Z Y Z Y Z Y Z Y Z Y Z Y Z Y Z Y Z Y Z Y Z Y Z Y Z Y YZ Z Y Z Y YZ YZ Z Y Z Y Z Y Z Y Z Y Z Y = =

70 7. tructural Properties of Convolutional Codes Example (,,) Code: (cont.) Hence the generating function is ) ( Z Y YZ Z Y Z Y YZ YZ YZ Z Y Z Y Z Y Z Z Y Z Y YZ Z Y YZ Z Y Z Y F i i i = = = = ,, Z Y Z Y Z Y Z Y Z Y Z Y YZ Z Y Z Y YZ Z Y Z Y T

71 . tructural Properties of Convolutional Codes Example (,,) Code: (cont.) This implies that the code word of weight 6 has length 5 branches and an information sequence of weight, one code word of weight 7 has length branches and information sequence weight, another has length 6 branches and information sequence weight, the third has length 7 branches and information sequence weight, and so on. 7

72 The Transfer Function of a Convolutional Code The state diagram can be used to obtain the distance property of a convolutional code. Without loss of generality, we assume that the all-zero code sequence is the input to the encoder. 7

73 The Transfer Function of a Convolutional Code First, we label the branches of the state diagram as either D =, D, D, or D, where the exponent of D denotes the Hamming distance between the sequence of output bits corresponding to each branch and the sequence of output bits corresponding to the all-zero branch. The self-loop at node a can be eliminated, since it contributes nothing to the distance properties of a code sequence relative to the all-zero code sequence. Furthermore, node a is split into two nodes, one of which represents the input and the other the output of the state diagram. 7

74 The Transfer Function of a Convolutional Code Use the modified state diagram, we can obtain four state equations: c = Da Db = D D b c d = D D d c d e = D b The transfer function for the code is defined as T(D)= e / a. By solving the state equations, we obtain: 6 D 6 8 d T D = = D D D 8D = a d D D a d = ( d 6) 7 ( even d) ( odd d) d = 6

75 The Transfer Function of a Convolutional Code The transfer function can be used to provide more detailed information than just the distance of the various paths. uppose we introduce a factor N into all branch transitions caused by the input bit. Furthermore, we introduce a factor of J into each branch of the state diagram so that the exponent of J will serve as a counting variable to indicate the number of branches in any given path from node a to node e. 75

76 The Transfer Function of a Convolutional Code The state equations for the state diagram are: c = JND a JND b = JD JD b c d = JND JND d c d e = JD b Upon solving these equations for the ratio e / a, we obtain the transfer function: 6 J ND T( D, N, J) = JND J 76 = J ND J N D J N D J N D J ND JND

77 The Transfer Function of a Convolutional Code The exponent of the factor J indicates the length of the path that merges with the all-zero path for the first time. The exponent of the factor N indicates the number of s in the information sequence for that path. The exponent of D indicates the distance of the sequence of encoded bits for that path from the all-zero sequence. Reference: John G. Proakis, Digital Communications, Fourth Edition, pp. 77 8, McGraw-Hill,. 77

78 . tructural Properties of Convolutional Codes An important subclass of convolutional codes is the class of systematic codes. In a systematic code, the first k output sequences are exact replicas of the k input sequences, i.e., v (i) = u (i), i =,,, k, and the generator sequences satisfy ( j ) if j = i gi =, i =,,...k, if j i 78

79 79. tructural Properties of Convolutional Codes The generator matrix is given by where I is the k k identity matrix, is the k k all-zero matrix, and P l is the k (n k) matrix = m m m m m m P P P P I P P P P I P P P P I G,,,,,,,,,, = n l k k l k k l k n l k l k l n l k l k l l g g g g g g g g g P

80 . tructural Properties of Convolutional Codes And the transfer matrix becomes ( k ) ( n g ) D g D ( k ) ( n g ) D g D G D = k n g k D g k D ince the first k output sequences equal the input sequences, they are called information sequences and the last n k output sequences are called parity sequences. Note that whereas in general k n generator sequences must be specified to define an (n, k, m) convolutional code, only k (n k) sequences must be specified to define a systematic code. ystematic codes represent a subclass of the set of all possible codes. 8

81 8. tructural Properties of Convolutional Codes Example (,,) Code: Consider the (,, ) systematic code. The generator sequences are g () =( ) and g () =( ). The generator matrix is = G The (,, ) systematic code.

82 . tructural Properties of Convolutional Codes Example (,,) Code: The transfer function matrix is g()=( ) and g()=( ). G(D) = [ D D ]. For an input sequence u(d) = D D, the information sequence is v () ( ) ( ) D = u D g D = D D = D D and the parity sequence is v ( ) ( ) ( )( D = u D g D = D D D D ) = D D 8 D D D 5 D 6

83 . tructural Properties of Convolutional Codes One advantage of systematic codes is that encoding is somewhat simpler than for nonsystematic codes because less hardware is required. For an (n, k, m) systematic code with k > n k, there exits a modified encoding circuit which normally requires fewer than K shift register states. The (,, ) systematic code requires only one modulo- adder with three inputs. 8

84 . tructural Properties of Convolutional Codes Example (,,) ystematic Code: Consider the (,, ) systematic code with transfer function matrix D D G( D) = D The straightforward realization of the encoder requires a total of K = K K = shift registers. 8

85 . tructural Properties of Convolutional Codes Example (,,) ystematic Code: ince the information sequences are given by v () (D) = u () (D) and v () (D) = u () (D), and the parity sequence is given by ( ) ( ) ( ) ( ) ( D = u D g D u D g ) ( D), v The (,, ) systematic encoder, and it requires only two stages of encoder memory rather than. 85

86 . tructural Properties of Convolutional Codes A complete discussion of the minimal encoder memory required to realize a convolutional code is given by Forney. In most cases the straightforward realization requiring K states of shift register memory is most efficient. In the case of an (n,k,m) systematic code with k>n-k, a simpler realization usually exists. Another advantage of systematic codes is that no inverting circuit is needed for recovering the information sequence from the code word. 86

87 . tructural Properties of Convolutional Codes Nonsystematic codes, on the other hand, require an inverter to recover the information sequence; that is, an n k matrix G - (D) must exit such that G(D)G - (D) = ID l for some l, where I is the k k identity matrix. ince V(D) = U(D)G(D), we can obtain V(D)G - (D) = U(D)G(D)G - (D) = U(D)D l, and the information sequence can be recovered with an l-timeunit delay from the code word by letting V(D) be the input to the n-input, k-output linear sequential circuit whose transfer function matrix is G - (D). 87

88 . tructural Properties of Convolutional Codes For an (n,, m) code, a transfer function matrix G(D) has a feedforward inverse G - (D) of delay l if and only if GCD[g () (D), g () (D),, g (n) (D)] = D l for some l, where GCD denotes the greatest common divisor. n For an (n, k, m) code with k >, let i ( D), i =,,,, k be the determinants of the n distinct k k submatrices of the k transfer function matrix G(D). A feedforward inverse of delay l exits if and only if for some l. GCD i ( D ) : i =,,, k n = D l 88

89 . tructural Properties of Convolutional Codes Example (,,) Code: For the (,, ) code, GCD D D, D D D = = D and the transfer function matrix D D G ( D) = D D provides the required inverse of delay [i.e., G(D)G (D) = ]. The implementation of the inverse is shown below 89

90 . tructural Properties of Convolutional Codes Example (,,) Code: For the (,, ) code, the submatrices of G(D) yield determinants D D, D, and. ince GCD D D, D, = there exists a feedforward inverse of delay. The required transfer function matrix is given by: G ( D) = D D 9

91 . tructural Properties of Convolutional Codes To understand what happens when a feedforward inverse does not exist, it is best to consider an example. For the (,, ) code with g () (D) = D and g () (D) = D, GCD D, D = D, and a feedforward inverse does not exist. If the information sequence is u(d) = /( D) = D D, the output sequences are v () (D) = and v () (D) = D; that is, the code word contains only three nonzero bits even though the information sequence has infinite weight. If this code word is transmitted over a BC, and the three nonzero bits are changed to zeros by the channel noise, the received sequence will be all zeros. 9

92 . tructural Properties of Convolutional Codes A MLD will then produce the all-zero code word as its estimated, since this is a valid code word and it agrees exactly with the received sequence. The estimated information sequence will be implying an infinite number of decoding errors u ( D caused ) =, by a finite number (only three in this case) of channel errors. Clearly, this is a very undesirable circumstance, and the code is said to be subject to catastrophic error propagation, and is called a catastrophic code. () ( ) ( n ) l GCD D, D,, ( D) and g g g = D GCD i ( D) l : i =,,..., n D are necessary and sufficient conditions k = for a code to be noncatastrophic. 9

93 . tructural Properties of Convolutional Codes Any code for which a feedforward inverse exists is noncatastrophic. Another advantage of systematic codes is that they are always noncatastrophic. A code is catastrophic if and only if the state diagram contains a loop of zero weight other than the self-loop around the state. Note that the self-loop around the state has zero weight. tate diagram of a (,, ) catastrophic code. 9

94 . tructural Properties of Convolutional Codes In choosing nonsystematic codes for use in a communication system, it is important to avoid the selection of catastrophic codes. Only a fraction /( n ) of (n,, m) nonsystematic codes are catastrophic. A similar result for (n, k, m) codes with k > is still lacking. 9

95 . Distance Properties of Convolutional Codes The performance of a convolutional code depends on the decoding algorithm employed and the distance properties of the code. The most important distance measure for convolutional codes is the minimum free distance d free, defined as { v v u u } dfree min d, :, where v and v are the code words corresponding to the information sequences u and u, respectively. In equation above, it is assumed that if u and u are of different lengths, zeros are added to the shorter sequence so that their corresponding code words have equal lengths. 95

96 . Distance Properties of Convolutional Codes d free is the minimum distance between any two code words in the code. ince a convolutional code is a linear code, ( v) u ( ug) u { w v v u u } dfree = min : { w } = min : { w } = min :, where v is the code word corresponding to the information sequence u. d free is the minimum-weight code word of any length produced by a nonzero information sequence. 96

97 Also, it is the minimum weight of all paths in the state diagram that diverge from and remerge with the all-zero state, and it is the lowest power of in the code-generating function T(). For example, d free = 6 for the (,, ) code of example.(a), and d free = for the (,, ) code of Example.(b). Another important distance measure for convolutional codes is the column distance function (CDF). Letting. Distance Properties of Convolutional Codes i i i [ ] ( n) ( n) ( n) v = v v v, v v v,, i v v v denote the ith truncation of the code word v, and i i i [ ] ( k) ( k) ( k) u = u u u, u u u,, i u u u denote the ith truncation of the code word u. 97

98 . Distance Properties of Convolutional Codes The column distance function of order i, d i, is defined as d { d( [ v ] [ v ] ) [ u ] [ u ] } min, :, i i i { w[ v] [ u] } = min : i where v is the code word corresponding to the information sequence u. d i is the minimum-weight code word over the first (i ) time units whose initial information block is nonzero. In terms of the generator matrix of the code, [ v] = [ u] [ G] i i i 98

99 [G] i is a k(i ) n(i ) submatrix of G with the form or [ G] i. Distance Properties of Convolutional Codes [ G] G G Gi = G Gi, i m, i G G G Gm Gm G Gm Gm Gm G m G Gm = G G G G m m, i > m. G G Gm G G G G G 99

100 Then d { ([ ] [ ] ) [ ] } i = min w u G : u i i is seen to depend only on the first n(i ) columns of G and this accounts for the name column distance function. The definition implies that d i cannot decrease with increasing i (i.e., it is a monotonically nondecreasing function of i). The complete CDF of the (,, 6) code with G ( D) = D D D D D D D. Distance Properties of Convolutional Codes, D D D D D D D D D D is shown in the figure of the following page.

101 . Distance Properties of Convolutional Codes Column distance function of a (,,6) code

102 . Distance Properties of Convolutional Codes Two cases are of specific interest: i = m and i. For i = m, d m is called the minimum distance of a convolutional code and will also be denoted d min. From the definition d i = min{w([u] i [G] i ):[u] }, we see that d min represents the minimum-weight code word over the first constraint length whose initial information block is nonzero. For the (,,6) convolutional code, d min = d 6 = 8. For i, lim i d i is the minimum-weight code word of any length whose first information block is nonzero. Comparing the definitions of lim i d i and d free, it can be shown that for noncatastrophic codes lim d i i = d free

103 . Distance Properties of Convolutional Codes d i eventually reaches d free and then increases no more. This usually happens within three to four constraint lengths (i.e., when i reaches m or m). Above equation is not necessarily true for catastrophic codes. Take as an example the (,,) catastrophic code whose state diagram is shown in Figure.. For this code, d = and d = d = = lim i d i =, since the truncated information sequence [u] i =(,,,, ) always produces the truncated code word [v] i =(,,,,, ), even in the limit as i. Note that all paths in the state diagram that diverge from and remerge with the all-zero state have weight at least, and d free =.

104 . Distance Properties of Convolutional Codes Hence, we have a situation in which lim i d i = d free =. It is characteristic of catastrophic codes that an infinite-weight information sequence produces a finite-weight code word. In some cases, as in the example above, this code word can have weight less than the free distance of the code. This is due to the zero weight loop in the state diagram. In other words, an information sequence that cycles around this zero weight loop forever will itself pick up infinite weight without adding to the weight of the code word.

105 . Distance Properties of Convolutional Codes In a noncatastrophic code, which contains no zero weight loop other than the self-loop around the all-zero state, all infiniteweight information sequences must generate infinite-weight code words, and the minimum-weight code word always has finite length. Unfortunately, the information sequence that produces the minimum-weight code word may be quite long in some cases, thereby causing the calculation of d free to be a rather formidable task. The best achievable d free for a convolutional code with a given rate and encoder memory has not been determined exactly. Upper and lower bound on d free for the best code have been obtained using a random coding approach. 5

106 . Distance Properties of Convolutional Codes A comparison of the bounds for nonsystematic codes with the bounds for systematic codes implies that more free distance is available with nonsystematic codes of a given rate and encoder memory than with systematic codes. This observation is verified by the code construction results presented in succeeding chapters, and has important consequences when a code with large d free must be selected for use with either Viterbi or sequential decoding. Heller (968) derived the upper bound on the minimum free distance of a rate /n convolutional code: l d free min K l n l l 6

107 Maximum Free Distance Codes Rate ½ 7

108 Maximum Free Distance Codes Rate / 8

109 Maximum Free Distance Codes Rate / 9

110 Maximum Free Distance Codes Rate /5

111 Maximum Free Distance Codes Rate /6

112 Maximum Free Distance Codes Rate /7

113 Maximum Free Distance Codes Rate /8

114 Rate / Maximum Free Distance Codes Rate k/5

115 Rate / and /8 Maximum Free Distance Codes Rate k/7 5