1

Systematic feedback (recursive) encoders

• G’(D) = [1,(1 + D2)/(1 + D + D2) ,(1 + D)/(1 + D + D2) ]

• Infinite impulse response (not polynomial)

• Easier to represent in transform domain

• Sometimes referred to as systematic recursiveconvolutional code (SRCC). But note thatthe feedback is an encoder property

2

Example

11

11

001110

111101

001110

111101

D

DDD

G

)1(

)1(

)1(1

)1(

1)1(

)1)(1(

11

2

2

2

2

22

2

10

01

0

1)1/(1

1)1/(1

1)1/(1

11

)1/(1)1/()1/(1

DD

D

DD

DD

DD

DDD

DDD

DD

DDDDD

DD

D

DDDDDDG

3

Example (Cont.)

1

10

01

)1(

)1(

)1(1

)1(

)1(

)1(1

2

2

2

2

2

2

DD

D

DD

DD

D

DD

H

G

Controller canonical form

1

10

01

)1(

)1(

)1(1

)1(

)1(

)1(1

2

2

2

2

2

2

DD

D

DD

DD

D

DD

H

G

Observer canonical form

4

Some notes on minimality• For rate 1/n codes, a minimal encoder is in controller canonical form

provided common factors are removed from G• Why:

• k = 1, only one row of polynomials• If common factors not removed:

• G(D) = [1 + D2 , 1 + D3 ] =3 but • G’(D) = [1 + D, 1 +D+ D2 ] =2

• For rate (n -1)/n codes, there is a minimal encoder in observer canonical form corresponding to a systematic recursive encoder G’, provided • common factors are removed from H, the systematic rational function parity

check matrix derived from G’, and • all functions in G’ are realizable (delay of denominator exceeds delay of

enumerator)• Why:

• k = n -1, only one row of polynomials in H

• For rate k/n: More difficult

5

Example: Nonsystematic feedback encoder

We need a common denominator in each row for the feedback of the shift registers:

Controller canonical form Observer canonical form

6

Summary

• Any rate 1/n code can be represented by a minimal feedforward encoder G(D), or its rational systematic counterpart

• Any rate (n-1)/n code can be represented by a minimal systematic feedback matrix H(D), or its polynomial counterpart

7

Feedforward Encoder Inverse

• What if we want an encoder inverse; i. e. a way to find an estimate of the information by inverting the received word without decoding?

• Systematic encoders: Easy. Nonsystematic encoders?

• Recall: V(D) = U(D) G(D)

• Then we need an inverse matrix G-1(D) : G(D) G-1(D) = I(D)Dl

• Then V(D) G-1(D) = U(D) G(D) G-1(D) = U(D) I(D)Dl = U(D)Dl

• When does a feedforward inverse exist?

• Rate 1/n code: a feedforward inverse exists iff

• GCD {g (0) (D), g (1)

(D), ..., g (n-1) (D) } = Dl

• Rate k/n code: a feedforward inverse exists iff

• GCD {i(D)} = Dl

• where {i(D)} is the set of all determinants of kk submatrices

8

Example

• G(D) = [1 + D2 + D3, 1 + D + D2 + D3 ]

• GCD {1 + D2 + D3, 1 + D + D2 + D3 } = 1

2

21 1

)(DD

DDDG

9

Example

11

11

D

DDDG

11,1,1 22 DDDDDDDCDG

D

D

1

11

001G

10

Example 11.10 Catastrophic encoders

• G(D) = [1 + D, 1 + D2 ]• GCD {1 + D, 1 + D2 } = 1+D• There is no feedforward inverse

• Example: U(D) = 1/(1+D) = 1 + D + D2 + D3 + ...• will correspond to the codeword

V(D) = U(D) G(D) = 1/(1+D) [1 + D, 1 + D2 ] = (1, 1+D)• infinite input weight, finite output weight• A catastrophic encoder

• Must not be used

• Minimal encoders and systematic encoders are noncatastrophic

11

Structural properties• From now on: Assume a minimal encoder

• The state diagram

• State: Describes the content of the encoder’s memory elements

• Implies a natural state labelling

• State transitions: Describes what happens when k input block arrives when the encoder is in a given state

• Direction: What will be the new state?

• Label: Which n output bits are produced?

12

Example

13

Example

14

Example: Catastrophic encoder• G(D) = [1 + D, 1 + D2 ]

• GCD {1 + D, 1 + D2 } = 1+D

15

Weight Enumerator Functions• A codeword Weight Enumerating Function (WEF) is an expression

that completely describes the weights of all codewords

• The WEF is a code (not encoder specific) property

• The WEF can be derived by inspection of the state diagrams with some enhancements

• Split state S0 into an initial and a terminal state

• Label each branch by Xd, where d is the Hamming weight of the n-bit output block corresponding to the branch

• The path gain of a particular path is the product of the branch labels over the path. Thus the Hamming weight of the path is the power of X in the path gain

16

Modified state diagrams for WEF determination

17

Modified state diagrams for WEF determination

18

Mason’s gain formula• Want to compute the transfer function A(X) = d A d X d

• Definitions:• Forward path: Path going from initial to final state, touching each

state at most once. A finite number of such paths

• Fi : gain of the ith forward path

• Cycle: Path starting at one state and coming back to the same path, touching any other state at most once. A finite number of such cycles. Ci : gain of the ith cycle

= 1 - i Ci + i,j Ci Cj - i,j,l Ci Cj Cl + ... (sum over nonintersecting cycles)

m = 1 - i Ci (m) + i,j Ci

(m) Cj (m)

- i,j,l Ci (m)

Cj (m) Cl(m) + ...

(sum over nonintersecting cycles not touching the mth forward path)

• A(X) = i Fi i /

19

Example 11.13

20

Example (cont.)

21

Other weight enumerators

• Can describe the code parameters in more detail by enhancing the branch labels

• Multiply label by Ww, where w is the branch input weight

• Multiply label by L : denoting that each branch has length 1

• Input-Output weight enumerating function IOWEF

• A(W,X,L) = w,d,l A w,d,l W wX dL l

22

Example

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Suggested exercises

• 11.7-11.16