Chapter 4 Number Representations

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1 Chapter 4 Number Representations SKEE2263 Digital Systems Mun im/ismahani/izam {munim@utm.my,e-izam@utm.my,ismahani@fke.utm.my} February 9, 2016

2 Table of Contents 1 Fundamentals 2 Signed Numbers 3 Fixed-Point Numbers 4 Floating Point

3 Taxonomy of Number Systems Numbers Integers Reals Unsigned Signed Fixed Point Floating Point Signed- Magnitude Ones' Complement Two's Complement

4 Integers Number of bits Number of values Machine 4 16 Intel , PDP11, 8086, , VAX11, IEEE single Unisys Cray, IEEE double

5 Integers Value for integer bit pattern: Example : V unsigned = N 1 i=0 b i 2 i = = 22 10

6 Signed-Magnitude Value for N-bit signed-magnitude pattern is: Example 1010 SM : N 2 V SM = ( 1) b N 1 b i 2 i i=0 V SM = ( 1) b 3 [b b b ] = ( 1) 1 [0(4) + 1(2) + 0(1)] = 1 2 = 2

7 Signed-Magnitude Signed Integer Signed-Magnitude

8 Ones Complement Value for N-bit ones complement pattern is: Example C : N 2 V 1C = b N 1 2 N 1 + b i 2 i + b N 1 i=0 V 1C = b b b b b 3 = 1(8) + 0(4) + 1(2) + 0(1) + 1 = = 5

9 Ones Complement Signed Integer Ones Complement

10 Two s Complement Value for N-bit Two s complement pattern is: Example C : N 2 V 2C = b N 1 2 N 1 + b i 2 i i=0 V 2C = b b b b = 1(8) + 0(4) + 1(2) + 0(1) = = 6

11 Twos Complement Signed Integer Twos Complement

12 Sign Extension convert a number to a larger format. just copy the sign bit to fill the new high order bits in 8-bit two s-complement binary in 16-bit two s-complement binary in 8-bit two s-complement binary in 16-bit two s-complement binary

13 Offset binary a.k.a. biased-k representation Variation of two s complement Uses a value K as biasing value Applications: Exponent of floating-point number (biased-127 or biased-1023) Analog interfacing Excess-3 code (actual value = binary - 3)

14 Comparing Number Systems Decimal Signed- One s Two s Offset Magnitude Complement Complement Binary

15 Signed Systems Compared Unsigned Signed- Ones Two s Magnitude Complement Complement Smallest 0 (2 n 1 1) (2 n 1 1) 2 n 1 Largest 2 n 1 +(2 n 1 1) +(2 n 1 1) +(2 n 1 1)

16 Real Numbers Number System Format Characteristics Fixed-point ±i.f Low-precision Rational ±p/q Difficult to work with Floating-point ±m b e Most common way to handle reals

17 Fixed-Point Numbers The general expression for an N-bit fixed point 2 s complement where: N = total #bits x = b N 12 N 1 + N 2 i=0 b i 2 i 2 f f = #bits in fraction (0 f N 1) N-1 0 S int frac imaginary binary point

18 Expression for Two s Comp. Value of N-bit two s complement integer, f = 0 x = b N 12 N 1 + N 2 i=0 b i 2 i 2 0 = b N 1 2 N 1 + b N 2 2 N b b N-1 0 S int imaginary binary point

19 Expression for Q-Format Value of N-bit fixed point, f = N 1 x = b N 12 N 1 + N 2 i=0 b i 2 i 2 N 1 = b 0 + b b b (N 1) 2 (N 1) N-1 0 S frac imaginary binary point

20 N = 8, f = 4 Weights Bit value = = = OR Weights Bit value x = ( ) 2 4 = ( ) 16 =

21 N = 8, f = 7 Q7format Weights Bit value max = = ( ) 2 7 = 127/128 = Weights Bit value min = 2 0 = 1

22 Multiplying Q15 Numbers Q S x Q15 S Q30 31 S S Q30 31 S S r rounding by addition a '1' here

23 What is Floating Point? = = = = Binary point floats to a pre-defined position Process is called normalization

24 Floating Point Parts x Exponent Sign of mantissa Location of decimal point Mantissa Sign of exponent Radix ±X = m b e where m = mantissa, b = number base and e = exponent.

25 IEEE Single-Precision Format S 8-bit exp 23-bit frac ±X = ( 1) s 1.m 2 e 127 Sign field: 0 for positive numbers (-1 0 = +1) 1 for negative numbers (-1 1 = -1). Exponent field: Unsigned 8 bit, biased-127. Mantissa field: Bits to the right of normalized binary number.

26 IEEE Single-Precision Format X = ( 1) = = 7

27 IEEE Double-Precision Format Double precision is more common S 11-bit exp 52-bit frac ±X = ( 1) s 1.m 2 e 1023 public class TryFP { public static void main(string[ ] args) { double d = 1/3.; // Java likes double-prec more float f = 1f/3f; // Must force use of single-prec System.out.println("Value of d="+d); System.out.println("Value of f="+f); } } Value of d= Value of f=

28 FX vs FP Fixed Point Arithmetic Simple circuit Small area and faster Less accurate (the result is truncated if it exceeds the size) Smaller range of values can be handled Floating-Point Arithmetic Complex circuit (due to rounding and normalization) Large area and slower More accurate (high precision) Wider range of values can be handled

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