Digital Logic Circuits, Digital Component and Data

Representation

Course: MCA-ISubject: Computer Organization

And Architecture Unit-1

1

Fixed point numbers

• Fast and inexpensive implementation

• Limited in the range of numbers

• Susceptible to problems of overflow

•In a fixed-point processor, numbers are represented in integer format.

• Fixed-point numbers and their data types are

characterized by their -

word size in bits binary point and whether they are signed or

unsigned

• The dynamic range of an N-bit number based on 2’s-complement representation is between -(2N-1) & (2 N-1 - 1), or between -32,768 and 32,767 for a 16-bit system.

• By normalizing the dynamic range between -1 and 1, the range will have 2N sections, 2 -(N-1) -size of each section starting at -1 up to 1 – 2 -(N-1).

• For a 4-bit system, there would be 16 sections, each of size 1/8, from -1 to 7/8 .

• In unsigned integer

the stored number can take on any integer value from 0 to 65,535.

• signed integer

uses two's complement

allows negative numbers

it ranges from -32,768 to 32,767

• With unsigned fraction notation

65,536 levels spread uniformly between 0 and 1

• the signed fraction format

allows negative numbers, equally spaced between -1 and 1

15+1=0 6+(-2)=4

• The 4-bit unsigned numbers represent a modulo (mod) 16 system.

• If 1 is added to the largest number (15), the operation wraps around to give 0 as the answer.

• A number wheel graphically demonstrates the addition properties of a finite bit system.

• Addition procedure – 1Find the first number x on the wheel.

– 2. Step off y units in the clockwise direction, which brings you to the answer.

• Carry applies to unsigned numbers — when adding or subtracting, result is incorrect.

• Overflow applies to signed numbers — when adding or subtracting, result is incorrect.

Carry and Overflow

01111 + 100+

00111 111

-------- -------------

10110 1011

Overflow Carry

Sign bitCarry

Examples:

Sign bit

Fractional Fixed Point Rep

• Rather than using the integer values just discussed, a fractional fixed-point number that has values between +0.99 . . . and -1 can be used.

Data types1.Short:

it is of size 16 bits represented as 2’s complement with a range from -215 to (215 -1)

2.Int or signed int: it is of size 32 bits represented as 2’s complement with a

range from -231 to ( 231-1)3.Float: it is of size 32 bits represented as IEEE 32 bit with a range

from 2-126(1.175494x10-38) to 2+128 (3.40282346x1038)4.Double: it is of size 64 bits represented as IEEE 64 bit with a range

from 2-1022(2.22507385x10-308) to 2 1024(1.79769313x10308)

Floating-point representation

•The advantage over fixed-point representation is that

it can support a much wider range of values.

• The floating-point format needs slightly more storage

• The speed of floating-point operations is measured in

FLOPS.

General format of floating point number :

X= M. be

where M is the value of the significand (mantissa), b is the base e is the exponent.Mantissa determines the accuracy of the numberExponent determines the range of numbers that can be

represented

Floating point numbers can be represented as:

Single precision : • called "float" in the C language family

• it is a binary format that occupies 32 bits • its significand has a precision of 24 bits

Double precision :• called "double" in the C language family

• it is a binary format that occupies 64 bits • its significand has a precision of 53 bits

Single Precision (SP):

Bit 31 represents sign bit

Bits 23 to 30 represents exponent bits

Bits 0 to 22 represents fractional bits

Numbers as small as 10-38 and as large as 10 38 can be represented

S e f

022233031

Double precision (DP) :• since 64 bits, more exponent and fractional bits are available • a pair of registers are used

Bits 0 to 31 of first register represents fractional bitsBits 0 to 19 second register also represents fractional bitsBits 20 to 30 represents exponent bitsBits 31 is the sign bit

Numbers as small as 10 -308 and as large as 10 +308 can be represented

ffes

031019203031

• Instructions ending in SP or DP represents single and double precision

• Some Floating point instructions have more latencies than fixed point instructions

Eg: MPY requires one delay

MPYSP has three delays

MPYDP requires nine delays

• Single precision floating point value can be loaded into a single register where as Double precision values need a pair of registers

A1:A0, A3:A2 ,…….. B1:B0, B3:B2 ,……………

• C6711 processor has a single precision reciprocal instruction RCPSP for performing division

Reference

Reference Book

• Computer Organization & Architecture 7e By Stallings

• Computer System Architecture By Mano

• Digital Logic & Computer Design By Mano