EE 109 Unit 2 - USC Bits

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EE 109 Unit 2

Binary Representation Systems

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ANALOG VS. DIGITAL

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Analog vs. Digital

• The analog world is based on continuous events. Observations can take on (real) any value.

• The digital world is based on discrete events. Observations can only take on a finite number of discrete values

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Analog vs. Digital

• Q. Which is better?

• A. Depends on what you are trying to do.

• Some tasks are better handled with analog data, others with digital data.

– Analog means continuous/real valued signals with an infinite number of possible values

– Digital signals are discrete [i.e. 1 of n values]

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Analog vs. Digital

• How much money is in my checking account?

– Analog: Oh, some, but not too much.

– Digital: $243.67

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Analog vs. Digital

• How much do you love me?

– Analog: I love you with all my heart!!!!

– Digital: 3.2 x 103 MegaHearts

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Signal Types • Analog signal

– Continuous time signal where each voltage level has a unique meaning

– Most information types are inherently analog

• Digital signal – Continuous signal where voltage levels are mapped into 2 ranges

meaning 0 or 1

– Possible to convert a single analog signal to a set of digital signals

0

1

0

1

0

vo

lts

vo

lts

time time

Analog Digital

Threshold

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Signals and Meaning

0.0 V

0.8 V

2.0 V

5.0 V

Each voltage value

has unique meaning

0.0 V

5.0 V

Lo

gic

1

Log

ic 0

Il

legal

Analog Digital

Threshold Range

Each voltage maps to '0' or '1'

(There is a small illegal range where meaning is

undefined since threshold can vary based on

temperature, small variations in manufacturing, etc.)

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Analog vs. Digital USC students used to program analog computers!

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Analog vs. Digital

Advantages Disadvantages

Analog

Digital

Can be simpler

for some tasks.

Ex. AM radio

Easier to build

complex system.

Repeatable!

Difficult to build

complex systems.

Not repeatable

More complex

for simple tasks.

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Digital is About Numbers

• In a digital world, numbers are used to represent all the possible discrete events

– Numerical values

– Computer instructions (ADD, SUB, BLE, …)

– Characters ('a', 'b', 'c', …)

– Conditions (on, off, ready, paper jam, …)

• Numbers allow for easy manipulation

– Add, multiply, compare, store, …

• Results are repeatable

– Each time we add the same two number we get the same result

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The Real (Analog) World

• The real world is inherently analog.

• To interface with it, our digital systems need to:

– Convert analog signals to digital values (numbers) at the input.

– Convert digital values to analog signals at the output.

• Analog signals can come in many forms

– Voltage, current, light, color, magnetic fields, pressure, temperature, acceleration, orientation

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DIGITAL REPRESENTATION

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Interpreting Binary Strings

• Given a string of 1’s and 0’s, you need to know the representation system being used, before you can understand the value of those 1’s and 0’s.

• Information (value) = Bits + Context (System)

01000001 = ?

6510 ‘A’ASCII

41BCD

Unsigned

Binary system ASCII

system BCD System

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• Integer Systems – Unsigned

• Unsigned (Normal) binary

– Signed

• Signed Magnitude

• 2’s complement

• Excess-N*

• 1’s complement*

• Floating Point – For very large and small

(fractional) numbers

• Codes – Text

• ASCII / Unicode

– Decimal Codes

• BCD (Binary Coded Decimal) / (8421 Code)

* = Not fully covered in this class

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Number Systems • Number systems consist of

1. A base (radix) r

2. r coefficients [0 to r-1]

• Human System: Decimal (Base 10): 0,1,2,3,4,5,6,7,8,9

• Computer System: Binary (Base 2): 0,1

• Human systems for working with computer systems (shorthand for human to read/write binary)

– Octal (Base 8): 0,1,2,3,4,5,6,7

– Hexadecimal (Base 16): 0-9,A,B,C,D,E,F (A thru F = 10 thru 15)

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Anatomy of a Decimal Number • A number consists of a string of explicit coefficients (digits).

• Each coefficient has an implicit place value which is a power of the base.

• The value of a decimal number (a string of decimal coefficients) is the sum of each coefficient times it place value

Explicit coefficients Implicit place values

radix

(base)

(934)10 = 9*102 + 3*101 + 4*100 = 934

(3.52)10 = 3*100 + 5*10-1 + 2*10-2 = 3.52

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Positional Number Systems (Unsigned)

• A number in base r has place values/weights that are the powers of the base

• Denote the coefficients as: ai

r -1 r -2 r 1 r 0

.

r 2 r 3

Left-most digit =

Most Significant

Digit (MSD)

Right-most digit =

Least Significant

Digit (LSD)

... ... a -1 a -2 a 1 a 0 a 2 a 3

Nr = Σi(ai*ri) = D10

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Examples

(746)8 = 7*82 + 4*81 + 6*80 = 448 + 32 + 16 = 48610

(1A5)16 = 1*162 + 10*161 + 5*160 = 256 + 160 + 5 = 42110

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Anatomy of a Binary Number

• Same as decimal but now the coefficients are 1 and 0 and the place values are the powers of 2

(1011)2 = 1*23 + 0*22 + 1*21 + 1*20

Least Significant

Bit (LSB)

Most Significant

Digit (MSB)

coefficients place values

= powers of 2

radix

(base)

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Binary Examples

(1001.1)2 = 8 + 1 + 0.5 = 9.510 .5 1 2 4 8

(10110001)2 = 128 + 32 + 16 + 1 = 17710 16 32 128 1

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Powers of 2

20 = 1 21 = 2 22 = 4 23 = 8

24 = 16 25 = 32 26 = 64

27 = 128 28 = 256 29 = 512

210 = 1024

512 256 128 64 32 16 8 4 2 1 1024

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Practice On Your Own

(11010)2 = 1*24 + 1*23 + 1*21

= 16 + 8 + 2 = (26)10

(6523)8 = 6*83 + 5*82 + 2*81 + 3*80 = 3072 + 320 + 16 + 3 = (3411)10

(AD2)16 = 10*162 + 13*161 + 2*160

= 2560 + 208 + 2 = (2770)10

• Decimal equivalent is…

… the sum of each coefficient multiplied by its

implicit place value (power of the base)

= Σi(ai * ri) [ai = coefficient, r = base]

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Unique Combinations

• Given n digits of base r, how many unique numbers can be formed? rn

– What is the range? [0 to rn-1]

Main Point: Given n digits of base r, rn unique numbers can

be made with the range [0 - (rn-1)]

2-digit, decimal numbers (r=10, n=2)

3-digit, decimal numbers (r=10, n=3)

4-bit, binary numbers (r=2, n=4)

6-bit, binary numbers

(r=2, n=6)

0-9 0-9

100 combinations:

00-99

0-1 0-1 0-1 0-1

1000 combinations:

000-999

16 combinations:

0000-1111

64 combinations:

000000-111111

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Approximating Large Powers of 2

• Often need to find decimal approximation of a large powers of 2 like 216, 232, etc.

• Use following approximations: – 210 ≈ 103 (1 thousand) = 1 Kilo-

– 220 ≈ 106 (1 million) = 1 Mega-

– 230 ≈ 109 (1 billion) = 1 Giga-

– 240 ≈ 1012 (1 trillion) = 1 Tera-

• For other powers of 2, decompose into product of 210 or 220 or 230 and a power of 2 that is less than 210 – 16-bit half word: 64K numbers

– 32-bit word: 4G numbers

– 64-bit dword: 16 million trillion numbers

216 = 26 * 210

≈ 64 * 103 = 64,000

224 = 24 * 220

≈ 16 * 106 = 16,000,000

228 = 28 * 220

≈ 256 * 106 = 256,000,000

232 = 22 * 230

≈ 4 * 109 = 4,000,000,000

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Decimal to Unsigned Binary

• To convert a decimal number, x, to binary:

– Only coefficients of 1 or 0. So simply find place values that add up to the desired values, starting with larger place values and proceeding to smaller values and place a 1 in those place values and 0 in all others

16 8 4 2 1

2510 = 1 1 1

32

For 2510 the place value 32 is too large to include so we include

16. Including 16 means we have to make 9 left over. Include 8

and 1.

0 0 0

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Decimal to Unsigned Binary

7310=

128 64 32 16 8 4 2 1

.5 .25 .125 .0625 .03125

0 1 0 0 1 0 0 1

0 1 0 1 0 1 1 1

1 0 0 1 0 0 0 1

1 0 1 0 0

8710=

14510=

0.62510=

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Decimal to Another Base

• To convert a decimal number, x, to base r:

– Use the place values of base r (powers of r). Starting with largest place values, fill in coefficients that sum up to desired decimal value without going over.

16 1

7510 = 4 B

256

0 hex

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Timeout-Liars & Truth Tellers • You're walking on an island with a volcano and natives who are either truth tellers or liars. You meet a

native, and you want to know which kind of person he is. So you ask him

– “Are you a truth-teller? ”

– When he is answering, the volcano makes a loud noise and you cannot hear the answer. So you ask him again

– “Excuse me, I couldn't hear what you said, did you say you were a truth-teller?” - and he answers

– “No, I didn't say that, I said I was a liar. “

• Is the native a liar or a truth-teller? (Hint: Think about what the native's could have answered to the first question, first assuming he's a truth-teller and then assuming he's a liar.)

http://math.berkeley.edu/~antonio/MEC/liars.html

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SIGNED SYSTEMS

Signed Magnitude

2’s Complement System

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– Signed




• Excess-N*



• Codes – Text

• ASCII / Unicode

– Decimal Codes


* = Not covered in this class

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Unsigned and Signed

• Normal (unsigned) binary can only represent positive numbers

– All place values are positive

• To represent negative numbers we must use a modified binary representation that takes into account sign (pos. or neg.)

– We call these signed representations

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Signed Number Representation

• 2 Primary Systems

– Signed Magnitude

– Two’s Complement (most widely used for integer representation)

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Signed numbers

• All systems used to represent negative numbers split the possible binary combinations in half (half for positive numbers / half for negative numbers)

• In both signed magnitude and 2’s complement, positive and negative numbers are separated using the MSB

– MSB=1 means negative

– MSB=0 means positive

0000 0001

0010

0011

0100

0101

0110

0111

1000

1111

1110

1101

1100

1011

1010

1001

+ -

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Signed Magnitude System

• Use binary place values but now MSB represents the sign (1 if negative, 0 if positive)

1 2 4 8

4-bit

Unsigned

4-bit Signed

Magnitude

0 to 15

Bit

0

Bit

1

Bit

2

Bit

3

1 2 4 +/-

-7 to +7

Bit

0

Bit

1

Bit

2

Bit

3

8-bit Signed

Magnitude 16 32 64 +/-

-127 to +127

Bit

4

Bit

5

Bit

6

Bit

7

1 2 4 8

Bit

0

Bit

1

Bit

2

Bit

3

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Signed Magnitude Examples

4-bit Signed

Magnitude

1 2 4 +/-

= -5

8-bit Signed

Magnitude

1 0 1 1

1 2 4 +/-

= +3 1 1 0 0

16 32 64 +/- 1 2 4 8

Notice that +3 in signed

magnitude is the same

as in the unsigned

system

1 2 4 +/-

= -7 1 1 1 1

1 0 0 1 1 1 0 0 = -19

16 32 64 +/- 1 2 4 8

1 0 0 0 1 0 0 1 = +25

Important: Positive numbers have the same representation in signed

magnitude as in normal unsigned binary

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Signed Magnitude Range

• Given n bits…

– MSB is sign

– Other n-1 bits = normal unsigned place values

• Range with n-1 unsigned bits = [0 to 2n-1-1]

Range with n-bits of Signed Magnitude

[ -2n-1 –1 to +2n-1–1]

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Disadvantages of Signed Magnitude

1. Wastes a combination to represent -0

0000 = 1000 = 010

2. Addition and subtraction algorithms for signed magnitude are different than unsigned binary (we’d like them to be the same to use same HW)

4

- 6

Swap &

make res.

negative

6

- 4

-

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2’s Complement System

• Normal binary place values except MSB has negative weight – MSB of 1 = -2n-1

1 2 4 8

4-bit

Unsigned

4-bit

2’s complement

0 to 15

Bit

0

Bit

1

Bit

2

Bit

3

1 2 4 -8

-8 to +7

Bit

0

Bit

1

Bit

2

Bit

3

8-bit

2’s complement 16 32 64 -128

-128 to +127

Bit

4

Bit

5

Bit

6

Bit

7

1 2 4 8

Bit

0

Bit

1

Bit

2

Bit

3

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2’s Complement Examples

4-bit

2’s complement

1 2 4 -8

= -5

8-bit

2’s complement

1 1 0 1

1 2 4 -8

= +3 1 1 0 0

16 32 64 -128 1 2 4 8

Notice that +3 in 2’s

comp. is the same as

in the unsigned system

1 2 4 -8

= -1 1 1 1 1

0 0 0 1 1 0 0 0 = -127

16 32 64 -128 1 2 4 8

1 0 0 0 1 0 0 1 = +25

Important: Positive numbers have the same representation in 2’s complement

as in normal unsigned binary

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2’s Complement Range

• Given n bits…

– Max positive value = 011…11

• Includes all n-1 positive place values

– Max negative value = 100…00

• Includes only the negative MSB place value

Range with n-bits of 2’s complement

[ -2n-1 to +2n-1–1]

– Side note – What decimal value is 111…11?

• -110

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Comparison of Systems

0000

0001

0010

0011

0100

0101

0110

0111

1000

1111

1110

1101

1100

1011

1010

1001

0 +1

+2

+3

+4

+5

+6 +7 -0

-1

-2

-3

-4

-5

-6 -7 Signed

Mag.

2’s comp.

0 +1 +2

+3

+4

+5 +6

+7 -8

-7 -6

-5

-4

-3 -2

-1

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Unsigned and Signed Variables

• In C, unsigned variables use unsigned binary (normal power-of-2 place values) to represent numbers

• In C, signed variables use the 2’s complement system (Neg. MSB weight) to represent numbers

128 64 32 16 8 4 2 1

1 0 0 1 0 0 1 1 = +147

-128 64 32 16 8 4 2 1

1 0 0 1 0 0 1 1 = -109

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IMPORTANT NOTE

• All computer systems use the 2's complement system to represent signed integers!

• So from now on, if we say an integer is signed, we are actually saying it uses the 2's complement system unless otherwise specified

– We will not use "signed magnitude" unless explicitly indicated

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Zero and Sign Extension

2’s complement = Sign Extension (Replicate sign bit):

Unsigned = Zero Extension (Always add leading 0’s):

111011 = 00111011

011010 = 00011010

110011 = 11110011

pos.

neg.

Increase a 6-bit number to 8-bit

number by zero extending

Sign bit is just repeated as

many times as necessary

• Extension is the process of increasing the number of bits used to represent a number without changing its value

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Zero and Sign Truncation

• Truncation is the process of decreasing the number of bits used to represent a number without changing its value

2’s complement = Sign Truncation (Remove copies of sign bit):

Unsigned = Zero Truncation (Remove leading 0’s):

00111011 = 111011

00011010 = 011010

11110011 = 10011

pos.

neg.

Decrease an 8-bit number to 6-bit

number by truncating 0’s. Can’t

remove a ‘1’ because value is changed

Any copies of the MSB can be

removed without changing the

numbers value. Be careful not to

change the sign by cutting off

ALL the sign bits.

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Data Representation

• In C/C++ variables can be of different types and sizes

– Integer Types (signed and unsigned)

– Floating Point Types

C Type Bytes Bits MIPS Name ATmega328

[unsigned] char 1 8 byte byte

[unsigned] short [int] 2 16 half-word word

[unsigned] long [int] 4 32 word -1

[unsigned] long long [int] 8 64 double-word -1

C Type Bytes Bits MIPS Name ATmega328

float 4 32 single N/A

double 8 64 double N/A

1Can emulate but has no single-instruction support

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SHORTHAND FOR BINARY Hexadecimal and Octal

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Binary, Octal, and Hexadecimal

• Octal (base 8 = 23)

• 1 Octal digit ( _ )8 can represent: 0 – 7

• 3 bits of binary (_ _ _)2 can represent: 000-111 = 0 – 7

• Conclusion… 1 Octal digit = 3 bits

• Hex (base 16=24)

• 1 Hex digit ( _ )16 can represent: 0-F (0-15)

• 4 bits of binary (_ _ _ _)2 can represent: 0000-1111= 0-15

• Conclusion… 1 Hex digit = 4 bits

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Binary to Octal or Hex

• Make groups of 3 bits starting from radix point and working outward

• Add 0’s where necessary

• Convert each group of 3 to an octal digit

101001110.11 101001110.11 0 0 0 0 0 0

• Make groups of 4 bits starting from radix point and working outward

• Add 0’s where necessary

• Convert each group of 4 to an octal digit

516.68 14E.C16

5 1 6 6 1 4 E C

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Octal or Hex to Binary

• Expand each octal digit to a group of 3 bits

• Expand each hex digit to a group of 4 bits

317.28 D93.816

011001111.0102 110110010011.10002

11001111.012 110110010011.12

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Hexadecimal Representation

• Since values in modern computers are many bits, we use hexadecimal as a shorthand notation (4 bits = 1 hex digit) – 11010010 = D2 hex or 0xD2 if you write it in C/C++

– 0111011011001011 = 76CB hex or 0x76CB if you write it in C/C++

• Important Point: To interpret the value of a hex number, you must know what underlying binary system is assumed (unsigned, 2’s comp. etc.) – D2 hex (what if underlying binary system is unsigned?)

– D2 hex (what if underlying binary system is signed?)

– D2 hex ( = 'Ò' in Arial Unicode font)

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BINARY CODES ASCII & Unicode

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– Signed




• Excess-N*



• Codes – Text

• ASCII / Unicode

– Decimal Codes


* = Not covered in this class

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Binary Codes

• Using binary we can represent any kind of information by coming up with a code

• Using n bits we can represent 2n distinct items

Colors of the rainbow:

•Red = 000

•Orange = 001

•Yellow = 010

•Green = 100

•Blue = 101

•Purple = 111

Letters:

•‘A’ = 00000

•‘B’ = 00001

•‘C’ = 00010

.

.

.

•‘Z’ = 11001

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BCD • Rather than convert a decimal number to binary which may lose

some precision (i.e. 0.110 = infinite binary fraction), BCD represents each decimal digit as a separate group of bits (exact decimal precision) – Each digits is represented as a separate 4-bit number (using place values 8,4,2,1

for each dec. digit)

– Often used in financial and other applications where decimal precision is needed

(439)10

0100 0011 1001 BCD Representation:

This is the Binary

Coded Decimal (BCD)

representation of 439

Important: Some processors have specific

instructions to operate on #’s represented in BCD

Unsigned Binary Rep.: 1101101112

This is the binary

representation of 439

(i.e. using power of 2

place values)

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ASCII Code • Used for representing text characters

• Originally 7-bits but usually stored as 8-bits = 1- byte in a computer

• Example: – printf(“Hello\n”);

– Each character is converted to ASCII equivalent • ‘H’ = 0x48, ‘e’ = 0x65, …

• \n = newline character is represented by either one or two ASCII character

– LF (0x0A) = line feed (moves cursor down a line)

– CR (0x0D) = carriage return character (moves cursor to start of current line)

– Newline for Unix / Mac = LF only

– Newline for Windows = CR + LF

printf() is the method to print text to the screen in C

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ASCII Table LSD/MSD 0 1 2 3 4 5 6 7

0 NULL DLW SPACE 0 @ P ` p

1 SOH DC1 ! 1 A Q a q

2 STX DC2 “ 2 B R b r

3 ETX DC3 # 3 C S c s

4 EOT DC4 $ 4 D T d t

5 ENQ NAK % 5 E U e u

6 ACK SYN & 6 F V f v

7 BEL ETB ‘ 7 G W g w

8 BS CAN ( 8 H X h x

9 TAB EM ) 9 I Y i y

A LF SUB * : J Z j z

B VT ESC + ; K [ k {

C FF FS , < L \ l |

D CR GS - = M ] m }

E SO RS . > N ^ n ~

F SI US / ? O _ o DEL

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UniCode

• ASCII can represent only the English alphabet, decimal digits, and punctuation

– 7-bit code => 27 = 128 characters

– It would be nice to have one code that represented more alphabets/characters for common languages used around the world

• Unicode

– 16-bit Code => 65,536 characters

– Represents many languages alphabets and characters

– Used by Java as standard character code Unicode hex value

(i.e. FB52 => 1111101101010010)

EE 109 Unit 2 - USC Bits

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