EE 109 Unit 3 - USC Bits
Transcript of EE 109 Unit 3 - USC Bits
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EE 109 Unit 3
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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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
systemBCD System
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Binary Representation Systems
• 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 coefficientsImplicit 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 -2r 1 r 0
.
r 2r 3
Left-most digit =
Most Significant
Digit (MSD)
Right-most digit =
Least Significant
Digit (LSD)
......a -1 a -2a 1 a 0a 2a 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)
coefficientsplace values
= powers of 2
radix
(base)
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Binary Examples
(1001.1)2 = 8 + 1 + 0.5 = 9.510.51248
(10110001)2 = 128 + 32 + 16 + 1 = 177101632128 1
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Powers of 2
20 = 121 = 222 = 423 = 8
24 = 1625 = 3226 = 64
27 = 12828 = 25629 = 512
210 = 1024
512 256 128 64 32 16 8 4 2 11024
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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)
0-90-9
100 combinations:
00-99
0-10-10-10-1
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 00
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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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Binary Representation Systems
• Integer Systems– Unsigned
• Unsigned (Normal) binary
– Signed
• Signed Magnitude
• 2’s complement
• 1’s complement*
• Excess-N*
• Floating Point – For very large and small
(fractional) numbers
• Codes– Text
• ASCII / Unicode
– Decimal Codes
• BCD (Binary Coded Decimal) / (8421 Code)
* = 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
00000001
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)
1248
4-bit
Unsigned
4-bit Signed
Magnitude
0 to 15
Bit
0
Bit
1
Bit
2
Bit
3
124+/-
-7 to +7
Bit
0
Bit
1
Bit
2
Bit
3
8-bit Signed
Magnitude163264+/-
-127 to +127
Bit
4
Bit
5
Bit
6
Bit
7
1248
Bit
0
Bit
1
Bit
2
Bit
3
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Signed Magnitude Examples
4-bit Signed
Magnitude
124+/-
= -5
8-bit Signed
Magnitude
1011
124+/-
= +31100
163264+/- 1248
Notice that +3 in signed
magnitude is the same
as in the unsigned
system
124+/-
= -71111
1001 1100 = -19
163264+/- 1248
1000 1001 = +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)
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- 6Swap &
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
1248
4-bit
Unsigned
4-bit
2’s complement
0 to 15
Bit
0
Bit
1
Bit
2
Bit
3
124-8
-8 to +7
Bit
0
Bit
1
Bit
2
Bit
3
8-bit
2’s complement163264-128
-128 to +127
Bit
4
Bit
5
Bit
6
Bit
7
1248
Bit
0
Bit
1
Bit
2
Bit
3
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2’s Complement Examples
4-bit
2’s complement
124-8
= -5
8-bit
2’s complement
1101
124-8
= +31100
163264-128 1248
Notice that +3 in 2’s
comp. is the same as
in the unsigned system
124-8
= -11111
0001 1000 = -127
163264-128 1248
1000 1001 = +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
101001110.11101001110.11 0 00 000
• 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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Binary Representation Systems
• Integer Systems– Unsigned
• Unsigned (Normal) binary
– Signed
• Signed Magnitude
• 2’s complement
• 1’s complement*
• Excess-N*
• Floating Point – For very large and small
(fractional) numbers
• Codes– Text
• ASCII / Unicode
– Decimal Codes
• BCD (Binary Coded Decimal) / (8421 Code)
* = 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 1001BCD 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 TableLSD/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)