Digital Logic Circuits, Digital Component and Data Representation Course: MCA-I Subject: Computer Organization And Architecture Unit-1 1
Jul 16, 2015
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