binary coded decimal

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Binary Coded Decimal

Presented By Chung Wai Chow

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Introduction:

Although binary data is the most efficient storage scheme; every bit pattern represents a unique, valid value. However, some applications may not be desirable to work with binary data.

For instance, the internal components of digital clocks keep track of the time in binary. The binary value must be converted to decimal before it can be displayed.

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Because a digital clock is preferable to store the value as a series of decimal digits, where each digit is separately represented as its binary equivalent, the most common format used to represent decimal data is called binary coded decimal, or BCD.

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1. The BCD format

2. Algorithms for addition

3. Algorithms for subtraction

4. Algorithms for multiplication

5. Algorithms for division

Explanation of Binary Coded Decimal (BCD):

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1) BCD Numeric Format

Every four bits represent one decimal digit.

Use decimal values

from 0 to 9

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4-bit values above 9 are not used in BCD.


The unused 4-bit values are:

BCD Decimal

1010 10

1011 11

1100 12

1101 13

1110 14

1111 15

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Multi-digit decimal numbers are stored as multiple groups of 4 bits per digit.

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BCD is a signed notation

positive or negative.

For example, +27 as 0(sign) 0010 0111.

-27 as 1(sign) 0010 0111.

BCD does not store negative numbers in two’s complement.

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Values represented

b3b2b1b0 Sign and magnitude 1’s complement 2’s complement

0111 +7 +7 +7

0110 +6 +6 +6

0101 +5 +5 +5

0100 +4 +4 +4

0011 +3 +3 +3

0010 +2 +2 +2

0001 +1 +1 +1

0000 +0 +0 +0

1000 -0 -7 -8

1001 -1 -6 -7

1010 -2 -5 -6

1011 -3 -4 -5

1100 -4 -3 -4

1101 -5 -2 -3

1110 -6 -1 -2

1111 -7 -0 -1

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2) Algorithms for Addition

1100 is not used in BCD.

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Two errors will occurs in a standard binary adder.

1) The result is not a valid BCD digit.

2) A valid BCD digit, but not the correct result.

Solution: You need to add 6 to the result generated by a binary adder.

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A simple example of addition in BCD.

0101

+ 1001

1110

+ 0110

1 0100

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+ 9

Incorrect BCD digit

Add 6

Correct answer

1 4

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2) Algorithms for AdditionA BCD adder

10010101

0001 = 1

0100 = 4

If the result,

S3 S2 S1 S0, is not a valid BCD digit,

the multiplexer causes 6 to be added to the result.

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A simple example of subtraction

3) Algorithms for Subtraction

0111

+ 1101

0100

(+7)

(- 3)

(+4)

0011 is 3, the one’s complement is 1100.

Each of the computations adds 1 to the one’s complement to produce the two’s complement of the number.

1100 + 1 = 1101

The two’s complement of 3 is 1101

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The second change has to do with complements.

The nine’s complement in BCD, generated by subtracting the value to be complemented from another value that has all 9S as its digits. Adding one to this value produces the ten’s complement, the negative of the original value.

e.g, the nine’s complement of 631 is

999 – 631 = 368.

368 + 1 = 369 is the ten’s complement

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The ten’s complement plays the subtraction and negation for BCD numbers.


Hareware generates the nine’s complement of a single BCD digit.

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Conclusion for addition and subtraction

Using a BCD adder and Nine’s complement generation hardware to compute the addition and the subtraction for signed-magnitude binary numbers

The algorithm for adding and subtracting as below:

PM’1: US XS, CU X + Y

PM1: CU X + Y’ + 1, OVERFLOW 0

PM’2: OVERFLOW C

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The algorithm for adding and subtracting

CZ’PM2: US XS

CZPM2: US 0 C’PM2: US X’S, U U’ + 1

2: FINISH 1

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Example of addition of BCD numbers

USU = XSX + YSY

XSX = +33 = 0 0011 0011

YSY = +25 = 0 0010 0101

PM’1: US 0, CU 0 0101 1000

PM’2: OVERFLOW 0

Result: USU = 0 0101 1000 = +58

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Example of subtraction of BCD numbers

USU = XSX + YSY

XSX = +27 = 0 0010 0111

YSY = -13 = 1 0001 0011

PM1: CU 1 0001 0100, OVERFLOW 0

CZ’PM2: US 0

Result: USU = 0 0001 0100 = +14

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4) Algorithms for Multiplication

1101 Multiplicand M

X 1011 Multiplier Q

1101

1101

0000

1101____

10001111 Product P

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Multiplicand

Multiplier

Product

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Required to use the BCD adder and nine’s complement circuitry.

In BCD, each digit of the multiplicand may have any value from 0 to 9; each iteration of the loop may have to perform up to nine additions. We must incorporate an inner loop in the algorithm for these multiple additions.

In addition, use decimal shifts right operation (dshr), which shift one BCD digit, or four bits at a time.

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The BCD multiplication algorithm

1: US XS+YS, VS XS+YS, U 0, i n, Cd 0

ZY0’2: CSU CdU + X, Yd0 Yd0 – 1, GOTO 2

ZY02: i i - 1

3: dshr (CdUV), dshr (Y)

Z’3: GOTO 2

ZT3: US 0, VS 0

Z3: FINISH 1


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Division can be implemented using either a restoring or a non-restoring algorithm. An inner loop to perform multiple subtractions must be incorporated into the algorithm.

5) Algorithms for Division

10

11 ) 1000

11_

10

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A logic circuit arrangement implements the restoring-division technique

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A restoring-division example

Initially 0 0 0 0 0 1 0 0 0

0 0 0 1 1

Shift 0 0 0 0 1 0 0 0

Subtract 1 1 1 0 1

Set q0 1 1 1 1 0

Restore 1 1

0 0 0 0 1 0 0 0 0

Shift 0 0 0 1 0 0 0 0

Subtract 1 1 1 0 1

Set q0 1 1 1 1 1

Restore 1 1

0 0 0 1 0 0 0 0 0

Shift 0 0 1 0 0 0 0 0

Subtract 1 1 1 0 1

Set q0 0 0 0 1 0 0 0 0 1

Shift 0 0 0 1 0 0 0 1

Subtract 1 1 1 0 1

Set q0 1 1 1 1 1

Restore 1 1

0 0 0 1 0 0 0 1 0

remainderQuotient

First cycle

Second cycle

Third cycle

Fourth cycle

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The restoring-division algorithm:

S1: DO n times

Shift A and Q left one binary position.

Subtract M from A, placing the answer back in A.

If the sign of A is 1, set q0 to 0 and add M back to A (restore A); otherwise, set q0 to 1.

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The non-restoring division algorithm:

S1: Do n times

If the sign of A is 0, shift A and Q left one binary position and subtract M from A; otherwise, shift A and Q left and add M to A.

S2: If the sign of A is 1, add M to A.

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References:

Computer Systems Organization & Architecture, Addison Wesley Longman, Inc., 2001

Introduction to Computer Organization 4th Edition. V.Carl hamacher. 1998

http:// www.sfxavier.ac.uk/computing/bcd/bcd1.htm

http:// www.awl.com/carpinelli

Thank you

binary coded decimal

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