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Lecture 1. Number System and Logic Gates
COSC3330 Computer Architecture
Instructor: Weidong Shi (Larry), PhDComputer Science Department
University of Houston
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Today
• Numbers
• Logic gates
2
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1000 Apples in 10 Boxes
3
• How to arrange 1000apples in 10 boxes sothat any number ofapples can be pickedin terms of boxes?
117 Apples
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A Computer System
• What are there inside a computer?
4
CPU
NorthBridge
SouthBridge
MainMemory(DDR2)
FSB(Front-Side Bus)
DMI(Direct Media I/F)
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Bottom Layer of a Computer
• Each component inside a computer is basicallymade based on analog and digital circuits
Analog• Continuous signal
Digital• Only knows 1 and 0
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What Do You Mean by 0 or 1 in Digital Circuit?
• In fact, everything in this world is analogFor example, sound, light, electric signals are all analog since theyare continuous in time
Actually, digital circuit is a special case of analog circuit
• Power supply provides power to the computer system• Power supply has several outlets (such as 3.3V, 5V, and 12V)
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What Do You Mean by 0 or 1 in Digital Circuit?
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Digital circuit treats a signal above a certain level as “1” and asignal below a certain level as “0”
Different components in a computer have different voltagerequirements ( Voltage requirements change as the technologyadvances)
• CPU (Core 2 Duo): 1.325 V• Chipsets: 1.45 V• Peripheral devices: 3.3V, 1.5V
0V
1.325V
time
“1”
“0”
Not determined
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Logic Levels
• Define a range of voltages to represent 1 and 0• Define different ranges for outputs and inputs toallow for noise in the system
Noise is anything that degrades the signal
• For example, a gate (driver) could output a 5 voltsignal but, because of losses in the wire and othernoise, the signal could arrive at the receiver with adegraded value, for example, 4.5 volts
8
Driver Receiver Noise
5 V 4.5 V
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Logic Levels
9
Driver Receiver
ForbiddenZone
NM L
NM H
Input CharacteristicsOutput Characteristics
V O H
V DD
V O
L
GND
V IH
V IL
Logic HighInput Range
Logic LowInput Range
Logic HighOutput Range
Logic LowOutput Range
Noise Margin
NMH = V OH – V IHNML = V IL – V OL
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Digital vs. Analog
10
Digital Analog music
video
wireless signal
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DSP – Digital Signal Processing
a bit loudAnalog Computer
Digital Computer
ADC
DSP
DAC OUTPUT
1010 1001
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Number Systems
• Analog information (video, sound etc) isconverted to a digital format for processing• Computer processes information in digital• Since digital knows “1” and “0”, we use different
number systems in computerBinary and Hexadecimal numbers
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Number Systems - Decimal
• Decimal numbersMost natural to human because we have ten fingers(?) and/or because we are used to it (?)Each column of a decimal number has 10x the weight
of the previous column• Decimal number has 10 as its base
ex) 5374 10 = 5 x 10 3 + 3 x 10 2 + 7 x 10 1 + 4 x 10 0
N-digit number represents one of 10 N possibilities
ex) 3-digit number represents one of 1000 possibilities: 0 ~ 999
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Number Systems - Binary
• Binary numbersBit represents one of 2 values: 0 or 1Each column of a binary number has 2x the weight ofthe previous column
• Binary number has 2 as its baseex) 10110 2 = 1 x 2 4 + 0 x 2 3 + 1 x 2 2 + 1 x 2 1 + 0 x 2 0 = 22 10
N-bit binary number represents one of 2 N possibilitiesex) 3-bit binary number represents one of 8 possibilities: 0 ~ 7
14
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Counting Binary Numbers
• 0• 1• 2
• 3• 4• 5• 6• 7• 8…
01
10
11
100
101
110
111
1000
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Power of 2
• 20 =• 21 =• 22 =
• 23
=• 24 =• 25 =• 26 =• 27 =
16
• 28
=• 29 =• 210 =• 211 =• 212 =• 213 =• 214 =• 215 =
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Power of 2
• 20 = 1• 21 = 2• 22 = 4
• 23
= 8• 24 = 16• 25 = 32• 26 = 64• 27 = 128
* Handy to memorize up to 2 9
17
• 28
= 256• 29 = 512• 210 = 1024• 211 = 2048• 212 = 4096• 213 = 8192• 214 = 16384• 215 = 32768
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Number Systems - Hexadecimal
• Hexadecimal numbersWriting long binary numbers is tedious and error-proneWe group 4 bits to form a hexadecimal (hex)
• A hex represents one of 16 values0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F
Each column of a hex number has 16x the weight of theprevious column
• Hexadecimal number has 16 as its base
ex) 2ED 16 = 2 x 16 2 + E (14) x 16 1 + D (13) x 16 0 = 749 10
N-hexadigit number represents one of 16 N possibilitiesex) 2-hexadigit number represents one of 16 2 possibilities: 0 ~ 255
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Number Systems
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Hex Number Decimal Equivalent Binary Equivalent
0 0 00001 1 0001
2 2 0010
3 3 0011
4 4 0100
5 5 0101
6 6 0110
7 7 0111
8 8 1000
9 9 1001
A 10 1010
B 11 1011
C 12 1100
D 13 1101
E 14 1110
F 15 1111
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Number Conversions
75237
18
9421
222
22
02
1
101001
Convert 75 10 to binary number
7510
= 100 10112
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Bits, Bytes, Nibbles
• Bits (b)
• Bytes & NibblesByte (B) = 8 bits
• Used everyday
Nibble (N) = 4 bits• Not commonly used
22
10010110nibble
byte
CEBF9AD7least
significantbyte
mostsignificant
byte
10010110leastsignificant
bit
mostsignificant
bit
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Quick Checks
• 222 =?2 2 × 2 20 = 4 Mega
• How many values can a 32-bit variable
represent?2 2 × 2 30 = 4 Giga
• Suppose that you have 2GB main memory in
your computer. How many bits you need toaddress (cover) 2GB?2 1 × 2 30 = 2 GB, so 31 bits
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Addition
25
37345168+
8902
carries11
10110011+1110
11 carries
• Decimal
• Binary
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Signed Binary Numbers
• How does the computer represent positive andnegative integer numbers?• There are 2 ways
Sign/Magnitude Numbers
Two’s Complement Numbers
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Sign/Magnitude Numbers
• 1 sign bit, N -1 magnitude bits• Sign bit is the most significant (left-most) bitNegative number: sign bit = 1Positive number: sign bit = 0
• Example: 4 -bit representations of ± 5:+5 = 0101 2 - 5 = 1101 2
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Sign/Magnitude Numbers
SignedDecimal
Sign/MagnitudeNumbers
UnsignedDecimal
0 000 0
1 001 1
2 010 2
3 011 3
-0 100 4
-1 101 5
-2 110 6
-3 111 7
• Range of anN
-bitsign/magnitudenumber:[- (2 N -1-1), 2 N -1-1]
• Example N=3,[-(4-1), 4-1] = [-3, 3]
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Sign/Magnitude Numbers
• Problems Addition doesn’t work naturally Example: 5 + (-5)
0101+ 1101
10010Two representations of 0 ( ± 0)
0000 (+0)1000 (-0)
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Two’s Complement Numbers
• Ok, so what’s a solution to these problems? 2’s complement numbers!
• Don’t have same problems as sign/magnitude
numbers Addition works fineSingle representation for 0
• So, hardware designers like it and uses 2’scomplement number system whendesigning CPU
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How to Make 2’s Complement Numbers?
• Reversing the sign of a two’s complement
numberMethod:1. Flip (Invert) the bits2. Add 1
Example
-7 : 2’s complement number of +7
0111 (+7) 1000 (flip all the bits)+ 1 (add 1)
1001 (-7)
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Two’s Complement Numbers
• The most significant bitstill indicates the sign
If MSB == 1, a negativenumberIf MSB == 0, a positivenumber
• Range of an N -bit two’scomplement number[-2N -1, 2 N -1-1]
• Example N=3,[-(4), 4-1] = [-4, 3]
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Signed
Decimal
2’s complement Unsigned
Decimal
0 000 0
1 001 1
2 010 2
3 011 3
-4 100 4
-3 101 5
-2 110 6
-1 111 7
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Two’s Complement Examples
• Take the two’s complement of 01102
1001 (flip all the bits)+ 1 (add 1)
1010
• Take the two’s complement of 1101 2
0010 (flip all the bits)+ 1 (add 1)
0011
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Two’s Complement Addition
• Add 6 + (- 6) using two’s complement numbers
• Add -2 + 3 using two’s complement numbers
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+01101010
10000
111
+11100011
10001
111
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Sign-Extension
• Sign bit is copied into most significant bits.Number value remains the same
• Examples4-bit representation of 3 = 00118-bit sign-extended value:
4-bit representation of -5 = 10118-bit sign-extended value:
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1111 1011
0000 0011
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Zero-Extension
• Zeros are copied into most significant bits.Number value may change.
• Examples4-bit value = 00118-bit zero-extended value:
4-bit value = 10118-bit zero-extended value:
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0000 1011
0000 0011
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Online Tools
• Cisco’s binary teaching game Set and reset bits to display a binary representationof specific decimal numbershttp://forums.cisco.com/CertCom/game/binary_game_page.htm
• Number converter And online number converter from binary to hex anddecimal and back.
http://www.free-test-online.com/binary/signed_converter.html
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Zoom-in a System Component
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Logic Gates
• Logic gates perform logic functions such as NOT(inversion), AND, OR, NAND, NOR, etc.Single-input logic gates
• NOT gate, buffer
Two-input logic gates• AND, OR, XOR, NAND, NOR, XNOR etc
Multiple-input logic gates• AND, OR, XOR, NAND, NOR, XNOR etc
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Logic Function and Truth Table
F ABC
Y
Y
10
101001
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Single-Input Logic Gates
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NOT
Y = A
A Y 0 1
1 0
A Y
BUF
Y = A
A Y 0 0
1 1
A Y
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More Two-Input Logic Gates
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XNOR
Y = A + B
A B Y 0 0
0 1
1 0
1 1
AB
Y
XOR NAND NOR
Y = A + B Y = AB Y = A + B
A B Y 0 0 0
0 1 1
1 0 1
1 1 0
A B Y 0 0 1
0 1 1
1 0 1
1 1 0
A B Y 0 0 1
0 1 0
1 0 0
1 1 0
AB
Y AB
Y AB
Y
• 2 input XOR (Exclusive OR) is “true” if either A or B (not both) is true
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Transistor
• Transistor is a three-ported voltage-controlled switch
• Electronic switch, a path exists when the Switch Control isclosed• If (Open) OUTPUT = unknown ; Switch is open ( OFF )• Else OUTPUT = INPUT ; Switch is closed ( ON )
• Analogy — water through a pipe: the gate acts like a valve,allowing/preventing a flow between the source and drain
49
Hmmm, what is it really built from?
INPUT OUTPUT
Switch Control
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Th A l f A T i
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The Analogy of A Transistor
Cross Section
An N-Channel Metal-Oxide Semiconductor Field Effect Transistor (MOSFET)
Source Drain
Switch Control (Gate)
OS Si l f
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MOS Signal Transfer Property
Gate Path
0 volts Conduct
2.9 volts Open
Gate
Drain
Source
Gate
Source
Drain
Gate Path0 volts Open
2.9 volts Conduct
pMOS
nMOS
• Transmits 1 well• Transmits 0 poorly
• Transmits 0 well• Transmits 1 poorly
CMOS
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CMOS
V DD
In Out
GND
p-type
n-type
• CMOS is used to build the vast majority of all transistorsfabricated today
pMOS transistors pass good 1’s, so connect source to V DD nMOS transistors pass good 0’s, so connect source to GND
= 2.9 volts
In Out
0 volts 2.9 volts
2.9 volts 0 volts
I i I (NOT G )
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It is an Inverter (NOT Gate)
V DD
A Y
GND
N1
P1
NOT
Y = A
A Y 0 1
1 0
A Y
A P1 N1 Y0 ON OFF 1
1 OFF ON 0
NAND G t
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NAND Gate
A
B
Y
N2
N1
P2 P1NAND
Y = AB
A B Y 0 0 1
0 1 1
1 0 1
1 1 0
AB
Y
A B P1 P2 N1 N2 Y
0 0 ON ON OFF OFF 1
0 1 ON OFF OFF ON 1
1 0 OFF ON ON OFF 1
1 1 OFF OFF ON ON 0
E t C dit Q ti
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Extra Credit Question
• ProblemDesign and build a mechanical NOR gateFree to use any mechanical parts
• tubes, rods, wheels, valves, pipes
No magnets and electronic components allowed
• Competition ruleSend your design by Wednesday midnightFirst five working designs will get 3 points on the finalgrades3 additional points if the winner can build it anddemonstrate it in class
56
A l ti l E i
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Analytical Engine
• Charles Babbage’s Analytical Engine • Mechanical decimal general purpose computer• Steam engine, punchcards, gears• CPU complete by death in 1871• The first complete Babbage Engine was completed in London in
2002, 153 years after it was designed.
Another age must be the judge
Z1
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Z1
• 1938 Konrad Zuse: the Z1• First binary programmable computer, completely mechanical• Punchcard input, processing implemented with metal plates
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