ECE 645: Lecture 4 Number Representation Part 1 Fixed-Point - - PowerPoint PPT Presentation

Number Representation

Part 1 Fixed-Point Representations. Endianness. ECE 645: Lecture 4

Required Reading

• B. Parhami,

Computer Arithmetic: Algorithms and Hardware Design Chapter 1, Numbers and Arithmetic, Sections 1.1-1.6 Chapter 2, Representing Signed Numbers Endianness, from Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Endianness

Historical Representations: Ancient Egyptians Numerals (1)

– ~4000 BC – “Sum of Symbols” = (34)10

Historical Representations: Ancient Egyptians Numerals (2)

What number does this stone carving from Karnak represent?

Historical Representations: Ancient Egyptians Numerals (3)

Historical Representations: Ancient Egyptians Numerals (4)

Symbol for a fraction (part): Special symbols for most commonly used fractions:

– First known positional system – 3100 BC – Radix 60 (sexagesimal) – Two symbols: = 1 = 10

Historical Representations: Ancient Babylonian Numerals (1)

− Integers and fractions were represented identically - a radix point was not written but rather made clear by context

Babylonian Numerals: Example

1 x 602 20 x 601 56 x 600 = (4,856)10

Historical Representations: Ancient Babylonian Numerals (2)

Digits from 1 to 59

Positional Code with Zero

• Zero Represented by Space

– Partial solution – What about trailing zeros?

• Babylonians Introduced New Symbol

– or – 4th to 1st Century BC

• Zero Allows Representation of Fractions

– Fractions started with zero

Mixed System

• Roman Numerals

– Sum of all symbols – I=1 V=5 X=10 L=50 C=100 D=500 M=1000 – Difficult to do arithmetic e.g., MCDXLVII

• IX

?

Hindu-Arabic Numeral System

Brahmi numerals, India, 400 BC-400 AD

Evolution of numerals in early Europe

Positional Code Decimal System

– Documented in the 9th century – Position of coefficient determines its value – Coefficient in position is multiplied by radix (10) raised to the power determined by its position, e.g., ( )

4 10 8 10 5 10 6 10 4 856

3 2 1 10

* * * * , + + + =

Migration of Positional Notation

• ~750 AD

– Zero spread from India to Arabic countries

• ~1250 AD

– Zero spread to Europe

• Importance of Zero

– Ease of arithmetic which leads to improved commerce

Binary Number System

• Binary

– Positional number system – Two symbols, B = { 0, 1 } – Easily implemented using switches – Easy to implement in electronic circuitry – Algebra invented by George Boole (1815-1864) allows easy manipulation of symbols

( ) ( )

10 1 2 3 2

5 2 * 1 2 * 2 * 1 2 * 0101 = + + + =

16

Modern Arithmetic and Number Systems

• Modern number systems used in digital arithmetic can be

broadly classified as:

• Fixed-point number representation systems
• Integers
• Rational numbers of the form x = a/2f, a is an integer, f is a positive integer
• Floating-point number representation systems
• x * bE, where x is a rational number, b the integer base, and E the exponent
• Note that all digital numbers are eventually coded in bits

{0,1} on a computer

17

2 4 6 8 10 12 14 16 −2 −4 −6 −8 −10 −12 −14 −16

Unsigned integers Signed-magnitude 3 + 1 fixed-point, xxx.x Signed fraction, ±.xxx 2’s-compl. fraction, x.xxx 2 + 2 floating-point, s × 2 e in [−2, 1], s in [0, 3] 2 + 2 logarithmic (log = xx.xx)

± ±

Number format

log x s e

e fixed point floating point

Encoding Numbers in 4-Bits

Number system Positional Non-positional Fixed-radix Mixed-radix Conventional Unconventional Signed-digit Non-redundant Redundant Binary Decimal Hexadecimal

Classification of number systems (1)

Classification of number systems (2)

Positional

wi - weight of the digit xi

Fixed-radix

i k l i i w

x X ⋅ = ∑

− − = 1

i k l i i r

x X ⋅ = ∑

− − = 1

r - radix of the number system

Conventional fixed-radix

i k l i i r

x X ⋅ = ∑

− − = 1

r integer, r > 0 xi ∈ {0, 1, …, r-1}

Classification of number systems (3)

Unconventional fixed-radix

i k l i i r

x X ⋅ = ∑

− − = 1

xi ∈ {-α, …, β } Non-redundant number of digits = α + β + 1 ≤ r Redundant number of digits = α + β + 1 > r Signed-digit α>0 ⇒ negative digits

Example of a mixed-radix positional system?

What is it?

What is it?

What is it?

Samrat Yantra in Jantar Mantar - Jaipur, India

• The largest sundial in the world
• 90 feet (27 m) tall
• Conceived by Maharaja Sawai Raja Jai Singh II
• Built of local stone and marble 1728-1734
• The style's horizontal angle is equal the sundial's

geographical latitude 27° north

• The time is read by observing where the shadow

is sharpest at the time

• Accuracy of 2 seconds
• Reconstructed in 1901

Jantar Mantar - Jaipur, India Original Units of Time

• 1 day = 60 ghadis (1 ghadi = 24 minutes)
• 1 ghadi = 60 pals (1 pal = 24 seconds)
• 1 pal = 4 breaths (1 breath = 6 seconds)

Original weights: 60*60*4 60*4 4 breaths day ghadi pal Current weights: 24*60*60 60*60 60 seconds day hour minute

Integral and fractional part X = xk-1 xk-2 … x1 x0 . x-1 x-2 … x-l

Integral part Fractional part Radix point

• NOT stored in the register
• understood to be in a fixed position

Fixed-point representation

Fixed-Radix Conventional (Unsigned) Representations

29

Fixed Point Number system Positional Non-positional Fixed-radix Mixed-radix Conventional (unsigned) Unconventional (signed) Signed-digit Non-redundant Redundant Binary Decimal Hexadecimal

Fixed-Radix Conventional Number Systems

Range of numbers

Decimal X = (xk-1 xk-2 … x1 x0.x-1 … x-l)10 Xmin Xmax 10k - 10-l Binary Number system X = (xk-1 xk-2 … x1 x0.x-1 … x-l)2 2k - 2-l Conventional fixed-radix X = (xk-1 xk-2 … x1 x0.x-1 … x-l)r rk - r-l ulp = r-l Notation:

unit in the least significant position unit in the last position

Number of digits

Number system Number of digits in the integer part necessary to cover the range 0..Xmax Decimal Binary Conventional fixed-radix

⎣ ⎦ ⎡ ⎤

) 1 ( log 1 log

max 10 max 10

+ = = + = X X k

⎣ ⎦ ⎡ ⎤

) 1 ( log 1 log

max 2 max 2

+ = = + = X X k

⎣ ⎦ ⎡ ⎤

) 1 ( log 1 log

max max

+ = = + = X X k

r r

32

Radix Conversion

Option 1) Radix conversion, using arithmetic in the old radix r Convenient when converting from r = 10 or familiar radix

u = w . v = ( xk–1xk–2 . . . x1x0 . x–1x–2 . . . x–l )r Old = ( XK–1XK–2 . . . X1X0 . X–1X–2 . . . X–L )R New

Option 2) Radix conversion, using arithmetic in the new radix R Convenient when converting to R = 10 or familiar radix

Whole part Fractional part Example: (31)eight = (25)ten

From: Parhami, Computer Arithmetic: Algorithms and Hardware Design

Two methods:

33

Option 1: Arithmetic in old radix r

Converting whole part w: (105)ten = (?)five Repeatedly divide by five Quotient Remainder 105 21 1 4 4 Therefore, (105)ten = (410)five Converting fractional part v: (105.486)ten = (410.?)five Repeatedly multiply by five Whole Part Fraction .486 2 .430 2 .150 .750 3 .750 3 .750 Therefore, (105.486)ten ≅ (410.22033)five

From: Parhami, Computer Arithmetic: Algorithms and Hardware Design

Radix Conversion of the Integral Part

XI = (xk-1 xk-2 … x1 x0)R = =

R - destination radix

i k i i R

x ⋅

∑

− = 1

= ((...((xk-1 R + xk-2) R + xk-3 ) R + … + x2 ) R + x1 ) R + x0 Quotient Remainder (...((xk-1 R + xk-2) R + xk-3 ) R + … + x2 ) R + x1 x0 ...((xk-1 R + xk-2) R + xk-3 ) R + … + x2 x1 xk-1 xk-2 xk-1

………. ……….

xk-1 R + xk-2 xk-3

Radix Conversion of the Fractional Part

XF = (. x-1 x-2 … x-l+1 x-l)R = =

R - destination radix

i l i i R

x ⋅

∑

− − = 1

= R-1 (x-1 + R-1 (x-2 + R-1 (…. + R-1 ( x-l+1 + R-1 x-l )….))) Integer part Fractional part x-1 R-1 (x-2 + R-1 (….. + R-1 ( x-l+1 + R-1 x-l)….)) R-1 (….. + R-1 ( x-l+1 + R-1 x-l)….) x-2 x-l+1 R-1 x-l x-l

………. ……….

x-l+2 R-1 ( x-l+1 + R-1 x-l)

36

Option 2: Arithmetic in new radix R

Converting (22033)five = (?)ten

((((2 × 5) + 2) × 5 + 0) × 5 + 3) × 5 + 3 |-----| : : : : 10 : : : : |-----------| : : : 12 : : : |---------------------| : : 60 : : |-------------------------------| : 303 : |-----------------------------------------|

• 1518

Converting fractional part v: (410.22033)five = (105.?)ten (0.22033)five × 55 = (22033)five = (1518)ten 1518 / 55 = 1518 / 3125 = 0.48576 Therefore, (410.22033)five = (105.48576)ten

Horner’s rule or formula

From: Parhami, Computer Arithmetic: Algorithms and Hardware Design

37

Option 2 cont'd: Horner's rule for fractions

Converting fractional part v: (0.22033)five = (?)ten

• (((((3 / 5) + 3) / 5 + 0) / 5 + 2) / 5 + 2) / 5

|-----| : : : : 0.6 : : : : |-----------| : : : 3.6 : : : |---------------------| : : 0.72 : : |-------------------------------| : 2.144 : |-----------------------------------------| 2.4288 |-----------------------------------------------| 0.48576

Horner’s rule or formula

From: Parhami, Computer Arithmetic: Algorithms and Hardware Design

Horner’s rule is also applicable: Proceed from right to left and use division instead of multiplication

38

r=bg → b → R=bG 4=22 → 2 → 8=23 (2301.302)4 = (10 110 001. 110 010)2 = (261.62)8

Radix Conversion Shortcut for r=bg, R=bG

• Trick here is to first convert to a number in radix b, then to R
• Cluster in groups of 3 (because 23 = 8) moving away from binary point

Signed Number Representations

Representations of signed numbers

Signed-magnitude Biased Complement

Radix-complement Diminished-radix complement (Digit complement) Two’s complement One’s complement r=2 r=2

7 0111 1111 0111 0111 6 0110 1110 0110 0110 5 0101 1101 0101 0101 4 0100 1100 0100 0100 3 0011 1011 0011 0011 2 0010 1010 0010 0010 1 0001 1001 0001 0001 0000 1000 0000 0000

1000 1111

• 1

1001 0111 1111 1110

• 2 1010

0110 1110 1101

• 3

1011 0101 1101 1100

• 4

1100 0100 1100 1011

• 5

1101 0011 1011 1010

• 6

1110 0010 1010 1001

• 7

1111 0001 1001 1000

• 8 0000

1000

Signed- magnitude Biased Two’s complement One’s complement

Signed-magnitude representation of signed numbers

Advantages: Disadvantages:

• conceptual simplicity
• symmetric range: -(2k-1-1) .. -(2k-1-1)
• simple negation
• addition of numbers with the same sign and with

a different sign handled differently

k-2 k-1

sign magnitude

Biased (excess-B) representation of signed integers

• 8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

B = 2k-1, k=4 R(X) = X + B

• 2k-1 ≤ X ≤ 2k-1-1

X R(X) R

Signed number X Unsigned Representation R(X) Bit vector (xk-1xk-2...x0.x-1...x-l) Binary mapping Representation mapping

i k l i i

x X R 2 ) (

1

⋅ ∑ =

− − =

Biased representation with radix 2

Complement Signed Number Representations

Signed number X Unsigned Representation R(X) Bit vector (xk-1xk-2...x0.x-1...x-l) Binary mapping Representation mapping

i k l i i

x X R 2 ) (

1

⋅ ∑ =

− − =

Complement representations with radix 2

1 – xi = xi 1 – xi xi xi 1 1 1

Useful dependencies

|X| = X when X ≥ 0

• X when X < 0

One’s complement transformation OC(A) = A = 2k – 2-l - A For

i k l i i

A A 2

1

⋅ ∑ =

− − =

≥ 0

0 ≤ OC(A) ≤ 2k – 2-l OC(OC(A)) = A

def

k-1 k-2 ... 0 -1 -2 ... -l

1 1 ... 1 . 1 1 ... 1 – Ak-1 Ak-2 … A0 . A-1 A-2 ... A-l Ak-1 Ak-2 … A0 . A-1 A-2 ... A-l Properties:

One’s Complement Representation

• f Signed Numbers

R(X) = X for X > 0 0 or OC(0) for X = 0 OC(|X|) for X < 0 For –(2k-1 – 2-l) ≤ X ≤ 2k-1 – 2-l 0 ≤ R(X) ≤ 2k – 2-l

def

One’s complement representation of signed integers

• 8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

X k=4 X>0 X<0 X+2k-1 = 2k-1 - |X| 0, 2k-1

One’s complement representation of signed numbers

+0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15

Two’s complement transformation (1) A + 2-l = 2k – A for A > 0 For

i k l i i

A A 2

1

⋅ ∑ =

− − =

≥ 0

def

Properties: TC(A) = 0 for A = 0 0 ≤ TC(A) ≤ 2k – 2-l TC(TC(A)) = A 2k – A = 2k – A – 2-l + 2-l = = (2k – 2-l – A)+2-l = A + 2-l

Two’s complement transformation (2) For

i k l i i

A A 2

1

⋅ ∑ =

− − =

≥ 0 A + 2-l mod 2k = 2k – A mod 2k

def

TC(A) =

Two’s Complement Representation

• f Signed Numbers

R(X) = X for X ≥ 0 TC(|X|) for X < 0 For –2k-1 ≤ X ≤ 2k-1 – 2-l 0 ≤ R(X) ≤ 2k – 2-l

def

Two’s complement representation of signed integers

• 8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

X k=4 X>0 X<0 X+2k = 2k - |X|

Two’s complement representation of signed integers

+0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15

Signed-magnitude representation of signed numbers

• 8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

X k=4 X>0 X<0 | X|+2k-1 = -X+2k-1 0, 2k-1

Signed-magnitude representation of signed numbers

+0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15

Biased representation of signed numbers

• 8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

X+B B = 2k-1, k=4 X>0 X<0 X+B B

Biased representation of signed numbers

+0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15

Arithmetic Operations in Signed Number Representations

Unsigned addition vs. signed addition

Machine Programmer

0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0

1 1 1 Unsigned mind Signed mind

128 64 32 16 8 4 2 1

weight carry

X Y S

+ = FA x0 y0 s0 c1 FA x1 y1 s1 c2 FA x2 y2 s2 c3 FA x3 y3 s3 c4 FA x4 y4 s4 c5 FA x5 y5 s5 c6 FA x6 y6 s6 c7 FA x7 y7 s7 c8

Out of range flags

C = 1 if result > MAX_UNSIGNED or result < 0 0 otherwise

where MAX_UNSIGNED = 28-1 for 8-bit operands 216-1 for 16-bit operands

V = 1 if result > MAX_SIGNED or result < MIN_SIGNED 0 otherwise

where MAX_SIGNED = 27-1 for 8-bit operands 215-1 for 16-bit operands MIN_SIGNED = -27 for 8-bit operands

• 215 for 16-bit operands

Carry flag - C Overflow flag - V

• ut-of-range for unsigned numbers
• ut-of-range for signed numbers

Overflow for signed numbers

Indication of overflow Positive + Positive = Negative Negative + Negative = Positive Formulas Overflow2’s complement = xk-1 yk-1 sk-1 + xk-1 yk-1 sk-1 = = ck ⊕ ck-1

Two’s complement representation of signed integers

+0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15

Addition of Signed and Unsigned Numbers

C=1 and V=0 1 1 0 1 -3 1 0 1 1 -5

• 8 4 2 1

1 1 0 0 0 -8 0 0 1 1 3 0 1 1 0 6

• 8 4 2 1

1 0 0 1 9≠-7 1 1 0 1 13 1 0 1 1 11

8 4 2 1

1 1 0 0 0 24≠8 C=0 and V=1 0 0 1 1 3 0 1 1 0 6

8 4 2 1

1 0 0 1 9

Addition of Signed and Unsigned Numbers

C=0 and V=0 0 1 0 1 5 1 0 0 1 -7

• 8 4 2 1

1 1 1 0 -2 1 1 0 0 -4 1 0 1 1 -5

• 8 4 2 1

1 0 1 1 1 -9≠-7 0 1 0 1 5 1 0 0 1 9

8 4 2 1

1 1 1 0 14 C=1 and V=1 1 1 0 0 12 1 0 1 1 11

8 4 2 1

1 0 1 1 1 23≠7

Can replace this with k xor gates

Two's Complement Adder/Subtractor Architecture

Arithmetic Shift

Two’s complement Sh.L {001012 = +5} = 010102 = +10 Sh.L {110112 = -5} = 101102 = -10 Sh.L {010102 = +10} = 101002 = - 12

• verflow

Sh.R {001012 = +5} = 000102 = +2 rem 1 Sh.R {110112 = -5} = 111012 = -3 rem 1

Shift left may cause overflow Shift right requires sign extension

Addition and subtraction

One’s complement

Numbers of the same sign Numbers of the opposite sign 0 0 1 0 2 1 1 1 0 -1

8 4 2 1

1 0 0 0 0 1 0 1 0 -5 1 1 0 1 -2

8 4 2 1

1 0 1 1 1 + 1 1 0 0 0

• 7

end-arround carry + 0 0 0 1 1 1

Disadvantage: Need another adder after the addition is complete!

One’s complement representation of signed numbers

+0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15

Addition and subtraction

Signed-magnitude

Numbers of the same sign Numbers of the opposite sign 0 1 0 1 1 11 0 0 1 1 0 6 sign bit magnitude + 0 1 0 0 0 1 17 carry = overflow 1 1 0 1 1 -11 0 0 1 1 0 6 sign bit magnitude + 11 > 6 1 0 1 1 11 0 1 1 0 6 – 0 1 0 1 5 1

Signed Number Representations

Summary

Representing k-bit signed binary numbers

Representation for X>0 Representation for X<0 Representation for 0 Representation Signed- magnitude X 0, 2k-1 2k-1+|X| Biased X+B B X+B Complement X M-|X|=M+X 0, M mod 2k Two’s complement One’s complement X X 2k-|X|= 2k-ulp-|X|= 0, 2k-ulp

typical B=2k-1 or 2k-1-ulp

ulp X +

X

Value of a number in the signed representations

Representation Value of (xk-1 xk-2 … x1 x0.x-1 … x-l)

Signed- magnitude Biased Two’s complement One’s complement

i k l i i x

x X

k

2 ) 1 (

2

1

⋅ − =

∑

− − =

−

B x X

i k l i i

− ⋅ = ∑

− − =

2

1 i k l i i k k

x x X 2 2

2 1 1

⋅ + − =

∑

− − = − − i k l i i k k

x ulp x X 2 ) 2 (

2 1 1

⋅ + − − =

∑

− − = − −

Extending the number of bits of a signed number

xk-1 xk-2 … x1 x0 . x-1 x-2 … x-l yk’-1 yk’-2 … yk yk-1 yk-2 … y1 y0 . y-1 y-2 … y-l y-(l+1) … y-l’ X Y signed-magnitude xk-1 0 0 0 0 0 0 0 xk-2 … x1 x0 . x-1 x-2 … x-l 0 0 0 0 0 0 two’s complement xk-1 xk-1 xk-1 . . .xk-1 xk-2 … x1 x0 . x-1 x-2 … x-l 0 0 0 0 0 0

• ne’s complement

xk-1 xk-1 xk-1 . . .xk-1 xk-2 … x1 x0 . x-1 x-2 … x-l xk-1 . . . .xk-1 biased xk-2 … x1 x0 . x-1 x-2 … x-l 0 0 0 0 0 0

1 − k

x

. . . xk-1

1 − k

x

Generalized Complement Representation

Generalized complement representation

• f signed integers

M-N > P M ≥ N+P+1

Generalized complement representation

• f signed integers

Little-Endian vs. Big-Endian Representation of Integers

Little-Endian vs. Big-Endian Representation

A0 B1 C2 D3 E4 F5 67 8916 LSB MSB

MSB = A0 B1 C2 D3 E4 F5 67 LSB = 89

Big-Endian Little-Endian

LSB = 89

MAX

67 F5 E4 D3 C2 B1 MSB = A0 address

Little-Endian vs. Big-Endian Camps

Big-Endian Little-Endian

MAX

address MSB LSB . . . LSB MSB . . . Motorola 68xx, 680x0 Intel IBM Hewlett-Packard DEC Alpha Internet TCP/IP Sun SuperSPARC

Bi-Endian

Motorola Power PC Silicon Graphics MIPS RS 232 AMD Xilinx MicroBlaze Altera Nios II ARM v.3 and above IA-64

Origin of the terms

Little-Endian vs. Big-Endian

Jonathan Swift, Gulliver’s Travels

• A law requiring all citizens of Lilliput to break their soft-eggs

at the little ends only

• A civil war breaking between the Little Endians and

the Big-Endians, resulting in the Big Endians taking refuge on a nearby island, the kingdom of Blefuscu

• Satire over holy wars between Protestant Church of England

and the Catholic Church of France

Little-Endian vs. Big-Endian Big-Endian Little-Endian

• easier to determine a sign of

the number

• easier to compare two numbers
• easier to divide two numbers
• easier to print
• easier addition and multiplication
• f multiprecision numbers

Advantages and Disadvantages

Pointers (1)

89 67 F5 E4 D3 C2 B1 A0

Big-Endian Little-Endian

MAX

address int * iptr; (* iptr) = 8967; (* iptr) = 6789; iptr+1

Pointers (2)

89 67 F5 E4 D3 C2 B1 A0

Big-Endian Little-Endian

MAX

address long int * lptr; (* lptr) = 8967F5E4; (* lptr) = E4F56789; lptr + 1