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Path register savings in a syndrome decoder of a binary rate

$(n-1)/n$ convolutional code

Citation for published version (APA):

Post, K. A. (1976). Path register savings in a syndrome decoder of a binary rate $(n-1)/n$ convolutional code. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7604). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1976

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

Memorandum 1976-04 February 1976

n - I Path register savings in a syndrome decoder of a binary rate

----n University of Technology Department of Mathematics PO Box 513, Eindhoven The Netherlands convolutional code by K.A. Post

(3)

n-l Path register savings ~n a syndrome decoder of a binary rate n

convolutional code by

K.A. Post

1. Introduction. Physical states. Abstract states. Let nJ k and l be integers satisfying

(1 ) 2 s: n s: k; I s: £ s: k - n + 1.

The class r of n-tuples of binary polynomials (A,B,C, ••• ,D) is n,k,t

defined as follows:

Suppose n polynomials A,B,C, ••• ,D are given by A := akX k + ~_lX k-] + ••• + a2X 2 + a1X + 8 0 B := b Xk + b Xk-1 k k-I + ... + b X2 2 + btX + bO (2) C := ~X k + ~_lX k-) + ••• + c2X 2 + c1X + cO'

...

'" D •'

=

d Xk k + d k-I Xk-1 + • • . + d x2 2 + d X I + d 0

Then (A,B,C, ••• ,D) E

r

if and only if

n,k,Q, (i) (ii) (iii) a. + b. 0 (l s: ~ s: Q, - 1 ) 1. 1. (iv) a. + b.

=

0 (k - Q, + s: i s: k - I) 1. 1.

(4)

2 -(3) (vi) rank

...

n dkd k_1

...

d1

I

l

(vii) gcd (A,B,C, ••• ,D)

=

I The condition (1) implies that

r

:f

0.

n,k,t

Condition (3 vii) reflects the fact that, as a consequence of the invariant factor theorem every n-tuple (A,B,C, ... ,D) E r is a complete set of

n,k'£n_l

syndrome polynomials for some non-catastrophic rate convolutional code.

n

Obviously we have

(4)

r

:J

r

n,k, I n,k,2

r

k 3 :J

.0.

n, ,

Because of (3 i) f (3 v) the matrix of all coefficients of our polynomials

A,B,C, .•• ,D has the form

E R,----'>lJ' A 1

·

· .

I

·

·

~ 0

·

·

0 B (5 ) C

·

·

·

1

·

· ·

I

· · · .

1

n

.

·

·

.

0

·

·

a

D L

1

I L • t

---____ :_1

+---

k + I

where bits in the same solid rectangular box should have the same value. Let the elements of this matrix represent the tap connections of a multiple shift register, which are supposed to be connected with one single common binary adder.

(5)

3

--

-; xI X -1 Xo y-I YO Yl

.

,

Z -I 20 Zl

'

.

t -1

t~J

t)

1.8 shifted from right to left in this mUltiple shift register, and corres-pondingly a s~quence of syndrome bits

is formed by the adder

The physioaZ state of this system is defined to be the nk-dimensional binary vector representing the contents of all A,B,e, •.• ,D-shift register cells numbered k - 1, k - 2, ••• ,1, O. Every noise vector that enters the sys tem causes a transition of its physical state and gives rise to a syndrome bit. Looking at the sequence of syndrome bits, however, one is notable to

distinguish between certain physical states.

Two physical states are said to be equivaZent, if the syndrome sequence of length k, obtained by introducing k successive zero noise-vectors into the system is the same for both physical states. The corresponding equivalence classes are called abstract states. Physical states belonging to the same abstract state are syndrome-indistinguishable.

The zero noise syndrome sequence~ i.e. the syndrome sequence (w

1,w2, ••• ,wk) obtained by introducing k successive zero noise vectors into the system with initial state

(6)

4

-is given by the formula

w. a k ...• al bk· ••. b I .••• dk · ••• d I xI 1

,

"

.

"

,

=

O\"k

\ (6)

0'\

0'\ ·

w k 'd k

x

k £:1

Since, by (3 ii) the multitriangular matrix in (6) has rank k it follows that every binary vector of length k occurs as zero noise syndrome sequence for some physical state. Combining the arguments of this section we may state that there is a one-to-one correspondence between abstract states and k-dimensional vectors. For reasons of simplicity we shall use the word

state for abstract state and denote states by their zero noise syndrome

sequence.

2. Some notations

It useful to denote a state by a greek letter with a subscript, e.g.

and so on.

Occasionally, i.e. if sufficiently many terminating components sk' sk-l"" vanish, we also write the right shifts, e.g.

(7)

5

-Bij (3 ii), (3 iv) and (3 v) these definitions imply that

and also

3. R k .-equivalence.

n,

,N

For (A,B,C, •.. ,D) E

r

k n an t-singZeton state is defined to be a state

n,

,N

the last Q. components of which are zero. Linear combinations and left shifts

of 9,-singleton states are 9,-singleton states, too. For any state ~I the shifts ~.(i ~ 9, + I) are 9,-singleton states.

~

We have the following lemma, the proof of which is left to the reader (use (3 ii»:

Lemma I. For every state cr

l there exists a unique i-singleton and a unique index set I c {1,2, ••. ,i} such that °1

=

~9,+1 +

state Cfl 9, +.1

I

a 1 •

i€I

we now define the class of the state cp 1 +

I

a. to be the set

9,+ iEI ~

(7) cp H 1 +

L

a i :

=

~ ~

9, + I +

I

(r . a. + (I - r . ) 13 . )

1

r. E { 0 , I} f or all

i}

iEI l iEI ~ ~ ~ ~ ~

Since, by (3 iii) and (3 vi), the set {(a + (3)

I"r'

(a + (3) 9,} ~s linearly independent the cardinality of such a class is

21

I.

Moreover, the classes are disjoint. In fact, these

equivalence ~elation R k o' n, ,N

classes are the equivalence classes of some For every 9,' ~ 9, the R k .-equivalence

n, ,N

classes are unions of R k o ,-equivalence classes.

n,

,N

classes exactly contain one 9,-singleton state. The classes ~s ( 8) N n,k,9, k-29, 2 The one-element R k o-n,

,N

total number of R n,k,9,

(8)

6

-4. State transitions and syndrome bits

T

From sections I and 2 it is clear that a noise vector [x,y,z, ••• ,t] cause~ a state transition with syndrome bit w according to the formula

(9) T [x,y,z, .•• ,t]

1

\ '"2 + x'l +

ya

l + zy I + ... + tOI w

=

sl + xa O + ybO + zcO + ••• + tdO

Since, by (3 vi), the set {aj,SI'Y1""'c

1} is linearly independent, every

state 01 has exactly 2n images under the state transition mappings due to the various (= 2n) possible noise vectors, and these images form in the k-dimensional state space a coset of the n-dimensional subspace

<a1,SI,yl, ••• ,Ot>· It ~s easily verified that different states 01 and

°

1'

have either identical or disjoint image sets under state transition. Now let us define 8} := [1,O,O, ••

;,OJ~

Then, as a consequence of (3 ii) and (3 vi) the set {8

1,(a + 13)0' YO, .. "oO} is also linearly independent,

so that <£I,(a + 13)0' Yo, ••• ,oO> is an n-dimensional subspace of the state space, too. It is again easily seen that two distinct states in the same coset of <E1,(a + 13)0' YO"",oO have a common state transition image (under dif;£erent noise vectors). Finally, every state CPI can be a state .transition image: If the final component of CPI vanishes then

. [ ~T f h'

rpO +!PI due to the n01.se vector O,O,O, ••• ,OJ; I t 1.S last component has the value 1 then (use (3 ii» (rp + a)o +!P, due to the noise vector

[1,O,O, •••

,oj.

So apparently the state transition pre-images of a state form in the k-dimensional state space a coset of the n-dimensional subspace <E:

1,(a + S)O'Yo, ... ,oO>' Different states CPI and CPI' have either

identical or disjoint pre-image sets under state transition. Combining these arguments we have the following theorem

THEOREM J. For every system

(A,B,e, .•.

,D) E

r

k" there exists a one-to-one n,

,x.

correspondence between the cosets 8i of < £1' (a + S)O,Yo""'oO> and the cosets Ti of <a

1,131,yl, ... ,Oj> in such a way that every state transition

is an affine bijection from B. to T. for every i.

(9)

7

-5. Metric functions. Metric equations.

A metric function is defined as a nonnegative integer-valued function on the states.

With every state transition we now associate the (Hamming) weight of its noise vector. This leads to the formulation of the following problem:

Problem. Given a metric function f and a syndrome bit w every state has several state transition pre-{mages with syndrome bit w. Find a metric function g, which 1S statewise minimal, and for every state is consistent with at least one of thp values of f on its w-pre-images increased by the weight of its corresponding state transition.

The solution of this problem expresses g in terms of f and w, and can be formulated, by theorem I, in the sense that the values of g on T, are

1 completely determined by the values of f on S. and the syndrome bit w.

1

The equations that express g in terms of f and ware called metric equations.

We can also repeat our construction, starting with a metric function fa, given a syndrome sequence w

1,w2,w3, ••• as to form a sequence of metric functions f

l ,f2,f3, ••. iteratively by means of the metric equations

(l0)

. ..

..

We shall prove the following theorem:

THEOREM 2. Assume that (A,B,C, ••• ,D) E

r

k • n, , R,

Let fa be any starting metric function and w

l ,w2,w3, ••• be any syndrome sequence. Then every iterate f 1S constant on the R k -equivalence

u n, ,u

classes of (A,B,C, .•• ,D) (1 ~ u s ~).

Proof. By induction.

(10)

I

8

-Now consider two Rn,k,J-equivalent states ~2 + a1 and ~2 + 131,

We shall list the pre-images, corresponding noise vectors and syndrome bits for both states:

Pre-image q>Z+a l ~2+f31

noise; syndrome noise; syndrome

q>

I + zyO + ••• + too [1,O,z, ... ,t]; w [O,l,z, ••• ,t]; w

q> I + (a + (3)

°

+ zyo + ••• + t oo [O,l,z, ... ,t]; w [1,O,z, ••• ,tJ; w

-

-(jll + £ 1 + zYO + ••• + too [l,O,z, •.•• tJ; w [O,I,z, ••• ,t]; w

-

-(jll +£1 +(a+S)O+zYO+···+to

o

[O,I,z, ••• ,t]; w [1,O,z, ... ,t]; w

We see that on every line, i.e. for every pre-image, the syndromes and the weights of the state-transitions to (jl2 + a

l and (jl2 + 131 are identical. Hence f

1(q>2 + at) = fl«(jl2 + 131) for all syndrome bits. This proves the

assertion for u = 1.

Now let us assume that the assertion is proved for a fixed u, I ~ u ~ ~ - 1. Let fO be any starting metric function and let wl,w

Z, ... be any syndrome sequence. Then f is constant on the R k -equivalence classes.

*

u n, ,u

Let Xl and Xl be any pair of R k -equivalent states. Then there is a n, ,U

state ~u+l and a fixed index set I c {1,2, ••• ,u} such that

XI

=

~ u+ I + .

I

a. ~ ~EI

*

Xl

=

J/lu+l +

I

I (r.a. + (1 -r.)S.) ~ ~ ~ ~ Xl and

*

consider the cosets Sand S of <£J,(a + S)O'Yo, ••• ,oO>' to which

x;

belong, respectively and compare them elementwise. We now

Xl + E] and X; + E:l are obviously Rn,k,u-equivalent since the last k -I ~ u components of £1 vanish.

*

(11)

'> '

9

-Finally, Xl + r y

o

+ ••• + soO and Xl + ryO + ••• + soO are Rn,k,u-equivalent because of (3 v).

This implies that the states

and

are R k -equivalent for all p,q,r, ••• ,s E {O,l}.

n, ,u

Hence, by induction

f or all p, q, r , ••• ,s E

{o,

I }.

It should also be remarked, in view of the syndrome forming, that the first

*

components of Xl and of Xl are the same, since, by (3 iii)

2

iEI a ..

=

1. iE I

2

r.a. 1. 1. + (1 -r.)b. 1 1.

Thes~ arguments together, however, imply that the values of f 1 on the

u+

corresponding state-transition-images of Xl and

x7

are equal, and hence is constant on the R k 1 equivalence classes.

n, ,u+

f

u+1

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