Explicit evaluation of Viterbi's union bounds for the first event error probability and the bit error probability of a binary convolutional code on a binary symmetric channel

Explicit evaluation of Viterbi's union bounds for the first event

error probability and the bit error probability of a binary

convolutional code on a binary symmetric channel

Citation for published version (APA):

Post, K. A. (1976). Explicit evaluation of Viterbi's union bounds for the first event error probability and the bit error probability of a binary convolutional code on a binary symmetric channel. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7607). Technische Hogeschool Eindhoven.

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

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TECHNISCHE HOGESCHOOL EINDHOVEN

Onderafdeling der Wiskunde

Memorandum 1976-07 mei 1976

EXPLICIT EVALUATION OF VITERBI'S UNION BOUNDS FOR THE FIRST EVENT ERROR PROBABILITY AND THE BIT ERROR PROBABILITY OF A BINARY CONVOLUTIONAL CODE

ON A BINARY SYMMETRIC CHANNEL

Technische Hogeschool Onderafdeling der Wiskunde PO Box 513, Eindhoven Nederland

by

EXPLICIT EVALUATION OF VITERBI'S UNION BOUNDS FOR THE FIRST EVENT ERROR PROBABILITY AND THE BIT ERROR PROBABILITY OF A BINARY CONVOLUTIONAL CODE

ON A BINARY SYMMETRIC CHANNEL

by

K.A. Post

Abstract. An explicit method is given to evaluate Viterbi's union bounds [IJ on both the first event error probability and the bit error probability of binary convolutional codes on a BSC. These bounds are explicitly given f or t e rate h I ' 2 code w1th generators 1 + D + D and 2 2 . 1 + D . Compar1son is made with bounds and experimental results of Van de Meeberg [2J.

I. Let p and q be positive num~ers, p <

!,

p + q = 1. Following Viterbi [IJ we'define a sequence

by the formulas ( ) ) k-) \ 2k-) 2k-l-j j /. ( J')p q j=O (k = 1,2,3, ... ).

Using the well-known addition property 1n Pascal's triangle we find (p + q)P_2k -1 = P2k, so that P2k = P2k-1 (k ::: 1,2,3, .•. ) and furthermore so that (p + q)[P _ I

e

2k ) k k_J 2k :2 k P q 2k k k ::: P 2k+ I - P ( k ) P q , P _{2k+ 1 =} P _2k _ (1 _ _{2 P} )(2k) k k _{k P q} (k = 1, 2 • 3 , ... ) •

- 2

-Combining these results we obtain

I

o -

0 2 P3 P₄ = :1

-

p)[<O) + ( 1 ) pq] I

o -

0 2

_<i)p2q2]

P

s

P6 = :1

-

p)[ (0) + (j)pq + P = P == 1 _

o -

_p)[(O) 0 _{+ (1)} 2 _{pq +} (4) 2 2

(~)p\3J

2 P q + 7 8 2

...

"

...

"

..

It is a well-known fact that for complex

z, Izi

<

!

(2) (I - 4z)

(0)

°

₊

(2)

_{I Z +}

(4) 2

_{2 Z +}

(6) 3

_{3 z +}

so that the generating function F for the sequence P

1,P2,P3"" must read as follows (3) 00 F(z) :=

I

Pkzk k=) z 2 _1 - - [~ - (~ - p) (1 - 4 pq z ) 2] • - Z _1

The Taylor-series in this formula converges for complex z,

Izl

< (4pq) 2 since z

=

I is a removable singularity of the function F.

n(z) 2. Let G be a

rational

function of the complex variable z, G(z) = d(z) ,

where nand d are polynomials having gcd one, and nCO)

=

0.

G has a Taylor-expansion around z

=

0;

(4)

G(z)

For our purpose we are interested 1n finding an explicit form for the expression

3

-Recall that the polynomial d can be uniquely factorized

(5)

where C is a complex constant and a. are the distinct zeros of d(z) with

J mu 1 tip 1 i ci tie sm. (j

=

1, ••• , r) .

J

Then G(z) can be decomposed uniquely into partial fractions

(6) G(z)

m. r J

q(z) +

I I

j=1 £=1

where q ~s a polynomial and A

j£ are complex constants, q(O) + fj,£A j £ O. Hence we obtain

(7) (k 1,2,3, ... ),

where qk ~s the k-th term of q(z) and hj£k ~s the k-th Taylor-coefficient

z -£ of (I - - ) .

a.

J

Let us consider this Taylor-coefficient separately. By the binomial ser~es­

expansion we have

00

(8)

f

k=O

and hence, by repeated application of the addition property ~n Pascal's

triangle

(9) (1 __ _a z)-£

Therefore, the k-th Taylor-coefficient of (I - ~)-£ ~s equal to

a.

(10) _a. -k

J

4 -(k) -k+t k (t) _(zk) (t) Recall that _{t o.} j =:

tT

(z ) z = : -1 • where Z

=

a. ex • J J

denotes the value of the t-th derivative of the function 2 k taken at 2

a..

Hence, the contribution of the term A.

(1-J9,

Z -9,

to the value of the J

ro a..

J

desired expression equal to

9,-1

£-1 _{F(t) (_I )}

A.

_I

_{( t )}

Jil,

t=O t!a. t a..

J J

1_11

_1

provided that < (4pq) 2 (cf .

(3» .

a.. J

3. We now apply our results to the well-known convolutional code over GF(2)

' h 1 ' 1 2 2

w~t generator po ynom~a s I + D + D and I + D • The function G = G

E to be taken for the construction of the union bound

for the first event error prob ab i l i ty P

E ~n [ 1 ] ~s equal to 5 1 I I 2 I 3 I 4 1 G E (z) Z :=

=-- -

_T6z

-8

2 - - z - - z + 32 1 - 2z 32 4 2

So we find for the union bound for P E

_1 1 3

(l - 2p)(1 - 16pq) 2 -

3'2-

p

For the construction of the union bound for the bit error probability P

B in [I] we must take G

=

G B 1 z5 I 3 1 2 1 3 GB(z) := ----2:0- ==

'8

+ T6 z + 4"z +

4"Z -

5 ~~- + - - ---:;,-I - 2z 32 (1 _ 2z) (l - 2z)

Hence we find as union bound for P B PB < ]36PI+-t-P2+-t-P3 - }2 F(2) + 1 [F(2) + 2F'(2)] = 5 7 3 2 1 3 I 32 + T6P+ 4"P - Zp -

32

(1-2p) -96pq)(I-16pq) 3 2

5

-1

These bounds hold for pq <

16

4. The first attempt to find an upper bound for the un10n bounds for P

E and P

B was made by Viterbi [IJ. Van de Meeberg [2J improved upper

bound using only the fact that P

Zk

=

P2k- 1 (k

=

1,2,3, ... ). Experimen-tal measurements also have been made on PH for the special code with

2 2

generators 1 + D + D and 1 + D • It turns out numerically that the union bound on PH for this code is closer to Van de Meeberg's bound than to the experimental results. So it seems to be useful now to study the philosophy behind Viterbi's union bound in order to obtain better bounds.

[IJ A.J. Viterbi. Convolutional codes and their performance 1n

[2J

communications systems.

IEEE Trans. Comm. Techn. COM-19 (1971), 751-772.

L. van de Meeberg. A ghtened upper bound on the error probability

of binary convolutional codes with Viterbi decoding. IEEE Trans. Inf. Theory IT-20 (1974), 389-391.