Iterative Arrays for Code Conversions EUR 4908

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4908 e

ITERATIVE ARRAYS FOR CODE CONVERSIONS by M. COMBET

Commission of the European Communities

Joint Nuclear Research Centre - Ispra Establishment (Italy) Technology

Luxembourg, November 1972 - 22 Pages - 15 Figures - B.Fr. 40.—

This paper describes a division matrix which can be used for the conversion of numbers written in base a in their equivalent in base b.

The description of the elementary cell and the matrix is made with their mathematical expressions.

Application to the conversion of codes is made with examples for the Iinary to decimal conversion and vice-versa.

4908 e

ITERATIVE ARRAYS FOR CODE CONVERSIONS by M. COMBET

Commission of the European Communities

Joint Nuclear Research Centre - Ispra Establishment (Italy) Technology

Luxembourg, November 1972 - 22 Pages - 15 Figures - B.Fr. 40.—

This paper describes a division matrix which can be used for the conversion of numbers written in base a in their equivalent in base b.

The description of the elementary cell and the matrix is made with their mathematical expressions.

EUR 4908 e

COMMISSION OF THE EUROPEAN COMMUNITIES

ITERATIVE ARRAYS FOR CODE CONVERSIONS

by

M. COMBET

1972

Joint Nuclear Research Centre Ispra Establishment - Italy

ABSTRACT

This paper describes a division matrix which can be used for the conversion of numbers written in base a in their equivalent in base b.

The description of the elementary cell and the matrix is made with their mathematical expressions.

Application to the conversion of codes is made with examples for the Iinary to decimal «onversion and vice-versa.

KEYWORDS

ITERATIVE METHODS MATHEMATICAL OPERATORS MATHEMATICS DIGITAL COMPUTERS

s

-TABLE OF CONTENTS

1. Introduction 5

2. Division Matrix 6 2.1 Elementary cell 6

2.2 Division Matrix 7

2. 3 Example 9

3. Code conversion 11

3.1 Integers 11 3.2 Fractionnai numbers 11

3.3 Examples 12

4. Example for BCD Code 17

4.1 Minimisation of the functions 18 4.2 Cell and matrix in I 2 4,Β code 18

5

-I. INTRODUCTION *)

Iterative arrays offer lot of new possibilities for the logical design.

Such circuits, which will be realised in a next future are very usefull for parallel arithmetical operations in the computers.

This paper describes a division matrix which can be U3ed for the conversion of numbers written in base a in their equivalent in base b.

The description of the elementary cell and the matrix is made with their mathematical expressions.

Application to the conversion of codes is made with examples for the binary to decimal conversion and vice-versa.

­ 6 ­

2. DIVISION MATRIX

The division matrix and its elementary cell are n o w described.

2.1 Elementary Cell

Elementary cell is a mathematical operator which com­ putes quotient and remainder of a division .

Bi.

A;

_­*

_[au.b]

nu, ·

Q U O T

Rt h

Fig; I. Elementary Cell

This cell has 2 inputs: an horizontal one on which it receives digit Au in base a and a vertical one. Bt

in base b .

Such a cell computes the division of A­u + a.B¿ by b

and gives the quotient on A;.+1output ( in base a ) and

the remainder on B^v1 output ( in base b ) .

Mathematical expressions of such a division may be done by using modulo operations.

The remainder can be expressed in the following way

and the quotient A. ­ —

b

Λ

­t­ à. · B i j r*\o¿ b

A, ■* a.. B L ­ ( Au 4. a.B.

Γ A·,.

+ a.

Bi·

­ Β

L+^J

m o d Ι b

­ 7 ­

see in the following paragraphs.

Inputs and outputs are actually made of several wires

when number is expressed with bits I and 0 ( for ins­

tance 4 wires at least will be necessary for represen­

ting decimal digits ). But relations between inputs

A;.and B;_ and the outputs A^+, and B^i will be always

the same.

2.2 Division Matrix

A matrix can be now buildt w^ith m rows of n

identical cells and connected as indicated in Fig. 2

5.

B;

e,

1

A,

r

1 '

r

Γι ι 1

[a., oj

I

­ \*M * /»:

[â,bj

Λ

A [a .b]

Β :

[ÌM

ι

- ß o - . y

.J

Β,

Fig. 2 ­ Division Matrix

A

■ m i d

Now, let us consider the first comumn of the matrix

( Fig. 3 ) . It is obvious that this column will divide

by b a number of which each digit is entered in the

A;,inputs successively, most sigificant digit in the

top cell.

If B"0 = 0, most significant digit is divided by b

in the first cell where it sends the remainder to the

second cell . The remainder is added to the second digit

after multiplication by a and divided by b, etc..

This method is identical to that we use when we divide

[i. b J

8

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_?A.

- [a.b] -

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3*

Fig. 3 - Column division.

When Bo fO, the top cell divides A0 + a.Bc and we must

correct the preceding results by adding a.B0 to A0

But A,, = AD.a + A0 .a + .... A0 .a

v/here A has a coefficient a ; then, we must add

a .a.B0 = a .B0to Ae and say that the first column

rvi Λ

divides A0 + a ,B0 by b, the quotient is A^ and the

remainder B? .

B^ = ( Ao + &η"·Βΐ )mod b

A, = -( A0+ a ,BD- B „J

When two columns of cells are involved, we have two successives division, but term B enters only in the second division.

We can write for every division :

A0 + a~.B¿ = A, .b + Bl

A. a .B, = A,.b Eleminating A0 , it yields :

B Ì

­ 9 ­

When the n columns of the complete matrix are

involved, we can write the expressions for every

division and replace succesively every quotient in the

following expression; finally, we get :

A0 + a.B0 + a";B0b + . . . + a B0b =

=An.b + B^ + B^b + + B„ b

Ap + a ( B0b° + B ' b1+ + B0bn1) = A„. b +(B^b + B^b + + B^b J

Now, t h e f i r s t p a r e n t h e s i s i s B0, the second B ^

A0 + a " . B0 = An. b " + B „

An i s the q u o t i e n t and B^the r e m a i n d e r of the d i v i s i o n

of A0 + a ^ . B . by b n

B», A , = = o ( 1 b A0 « < + KYI a

A0 +

■ Β σ

ru

a .

)mo

B0

d

-bn

Bw )

2 . 3 Example

An example i s now g i v e n when a = 10 and b = 2;

A;.are d e c i m a l d i g i t s 0 , 1 , 2 , . . , 9 an9s B¡.is 0 o r I .

Twenty c a s e s a r e p o s s i b l e of which we g i v e the l i s t

w i t h the c o r r e s p o n d i n g o u t p u t s . Tor L k ce-LL·

A; 0 I 2 3 4 5 6 7 8 9 0 I 2 3 4 B, 0 0 0 0 0 0 0 0 0 0 I I I I I

AL Bi

10

-Ai*-i B i*

5 6 7 8 9 I I I I I 7 8 8 9 9 I 0 I 0 I

Indeed, the cell [l0,2] divides AL + 10. BLby 2

AC41is the quotient and B^ + 1the remainder.

Using tie matrix Fig. 4, on which we put

k0 = 237 base 10 B0= H O base 2 ( = 6 )

we can write the various values on the different points of the matrix, using the preceding table, and get the quotient 779 and the remainder 101 ( = 5τ η ).

This matrix has divided 237 + 10 .6 = 6237 by 2^ = 8 the quotient is 779, and the remainder 5.

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-3. CODE CONVERSION

A division matrix may be very usefull for the conversion of a number written in code a into its equivalent in

code b. Integers and number less than I have to be considered separately.

3.1 Integers

For this purpose, we put the number to be converted at input A,, and we take B0 = 0.

We can notice that the successive quotients A^, A , will be more and more small, and if the number of

columns is large enough, the last quotients will be 0. Making Bö= 0 and A„= 0, in the expression giving A„,

it yields :

0 = 4-„U

0

+a\0 - B

m

)

A0 = B„

Bmhas the same value as Ac, but written in code b

The number n of columns necessary wil be given by noticing in your preceding example, that the largest decimal number with 3 digits may be 999; this number divided by 2 = 1024 will give 0, as a quotient, and 10 columns are necessary.

In the general case, we must have

am- I>/bn

n >L° ^ " - ^

Lou b 3.2 Fractionnai numbers

The same matrix may be used for converting numbers less than I, written in 'base b in their equivalent in base a

If the matrix has an adequate number of rows, we feed the number B to be converted in the inputs B0, and

12

-A . m .

Α. = Τ^Γ ( ° +_ο a -Bo" Ο )

. ' . Α = —-Β

a~ η b* °

— A„is a number of which the point is just before the

a"

n

most significant digit.

In the case of fractionnai numbers, it is not always possible to have an adequate number· of rows, because the numbers of digit may be infinite, if m rows are involved, the preceding formula becomes :

Λ _Δ ή (

ρ

luS

~ãr n

~

b" v

°

ÔTJ -g

m must be large enough to give the error — — less

a

than the required precision.

Therefore, an [a,b]matrix converts integers from base a into base b and fractionnai numbers from base b into base a.

If, now, we need convert integers from base b into base a and fractionnai numbers from base a into base b,we

have to design a new matrix of which th elementary cell will have parameters permutated.

3.3 Examples

We will designed the two matrices for the conversion of numbers, integers and fractionnais, from binary into decimal and vice-versa.

Fig. 5 shows the cell[l0,2] , its table (Fig. 6 ) and the matrix [IO, 2] converting I3 (base IO) and 0,011 (base 2 ) (Fig. 7)

Fig. 8 shows the cell[2,I0] , its table (Fig. 9) and the matrix [2, IO] converting H O I (base 2) and 0,375

13

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A if\ Β ι +■

0 0 I I 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 0 I 0 I 0 I 0 I 0 I 0 I 0 I 0 I 0 I 0 I

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4. EXAMPLE FOR BCD CODE

Now, we must show how to realise the elementary cell

with electronic components.

As we must use a representation of numbers with bits,

inputs and outputs of the cell will have more than I

wire when numbers have a base more than 2.

As an example, we shal compute the circuitry of a

[l0,2]cell when 1,2,4,8 BCD code is used.

We shalL write the table of this cell in this code :

't 0 0 0 0 0 0 0 0 I Î 0 0 0 0 0 0 0 0 I I ζ 0 0 0 0 I I I I 0 0 0 0 0 0 I I I I 0 0 AL y 0 0 I I 0 0 I I 0 0 0 0 I I 0 Ü I

τ

0 0 X 0 I 0 I 0 I 0 Τ 0 I 0 I 0 I 0 I 0 I 0 I B : 0 0 0 0 0 0 0 0 0 0 I I I I I I I I I I Τ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I I I I A ζ 0 0 0 0 0 0 0 0 I I I I I I Τ I 0 0 0 0

i + 1 Y 0 0 0 0 I I I I 0 0 0 0 I I I I 0 0 0 0

t

0 0 I I 0 0 T I 0 0 I I 0 0 I T 0 0 I I B 0 I 0 I 0 I 0 I 0 I 0 T 0 I 0 I 0 I 0 I

Fig. II ­ 10,2 cell table in 1,2,4,8 code

We can consider Χ,Υ,Ζ,Τ and B¿t1 as "tooleean functions

18

-4»I Minimisation of the functions

Before beginning the minimisation, we will notice that Χ,Υ,Ζ and Τ don't depend of variable χ because when

considering the preceding table by group of 2 rows we see they are identical but x; thus, Χ,Υ,Ζ and Τ

are function of y,z,t and Β'u only.

We see also immediately, that:

BL*') = x

Now, Χ,Υ,Ζ and Τ are minimised with Karnaugh's tables

b

r

o o h '\

(1—ΤΊ o o

UJ-

_{o .}

_p-η

-ψ

βι χ

ο ο ο Ο f

ι) oj o Q

- U

ß:

>

>

τ

7

BL

Ζ-ι

ο

ο

0

0

ο

ο

-ο -ο[

Τ| ο

I I "~)

-

ν

- ί

Su

Fig. 12 - Minimisation of Χ,Υ,Ζ and T.

The Boolean expressions of these functions are

X = BL.y + Bi.y

Y = BL.ζ + B^.y.z + y.ζ

Ζ = BL.t + BL.ζ.t + BL.y.t

T = Bl.t + Bo.y.z

BLT1= x

4.2 - Cell and matrix in 1,2,4,8 code

f oc

>-Il

y-y

• t

>4—I-L

- 19

fri

3>

L>

T

L>

v

3ν.Λ

X

r*y

ζ

TJ

Fig. 13 - Cell [io,2] in 1,2,4,8 code

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3 ;

I

ì

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/ ι . Ζ Λ , ΐ )

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Τ

"δ­

ι

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o

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^

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Ιί ζ

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Fig. 14 - Example of Fig.4 in 1,2,4,8 code

5. ADVANTAGES OF DIVISION BEATRICES

The principal advantages of division matrices are now summarised

5.1 Conversion of integers and fractionnai numbers

Generally,converters can only convert integer or fractionnais numbers, but non both.

5.2 Unique circuit for conversion and reverse conversion Two matrices are needed for converting from base a in base b, and the same two matrices may be used for the reverse conversion

5.3 Various codes possible'

Cells may be easily designed for various representation of digits A and Β with bits (1,2,4,8 - 1,2,4,2 - excess 3 2 out of 5 - etc.. )

5.4 Expansibility

21

-Fig. 15 - Expansibility of matrices

ACKNOWLEDGMENTS

The author wishes to thank Mr Weinmiller for fruitful suggestions and Mr Boldrin for the constructive comments during the preparation of this manuscript.

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