Logische Funktionen - uni-due.de...010 0110 & 010 0111 ' 010 1000 (010 1001 ) 010 1010 * 010 1011 + 010 1100 , 010 1101 - ... From Decimal to Binary. Grundlagen der Technische Informatik
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Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
1
Logische FunktionenGrundlagen der technischen InformatikWintersemester 2018/19
Folien basierend auf Material von F. Vahid und S. Werner
Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
2Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
• Look “under the hood” of computers– Solid understanding --> confidence,
insight, even better programmer when aware of hardware resource issues
• Electronic devices becoming digital– Enabled by shrinking and more capable chips– Enables:
• Better devices: Sound recorders, cameras, cars, cell phones, medical devices,...
• New devices: Video games, smartphones, ...
– Known as “embedded systems”• Thousands of new devices every year• Designers needed: Potential career
direction
Review: Why Study Digital Design?
Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
3Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
• ASCII*: 7- (or 8-) bit encoding of each letter, number, or symbol
• Unicode: popular 16-bit encoding– Encodes
characters from various world languages
Question: What does this ASCII bit sequence represent?
1010010 1000101 1010011 1010100
R E S T
Sample ASCII encodingsSymbolEncoding
010 0000 <space>010 0001 !010 0010 "010 0011 #010 0100 $010 0101 %010 0110 &010 0111 '010 1000 (010 1001 )010 1010 *010 1011 +010 1100 ,010 1101 -010 1110 .010 1111 /
SymbolEncoding100 1110 N100 1111 O101 0000 P101 0001 Q101 0010 R101 0011 S101 0100 T101 0101 U101 0110 V101 0111 W101 1000 X101 1001 Y101 1010 Z
100 0001 A100 0010 B100 0011 C100 0100 D100 0101 E100 0110 F100 0111 G100 1000 H100 1001 I100 1010 J100 1011 K100 1100 L100 1101 M
SymbolEncoding
SymbolEncoding
011 0000 0011 0001 1011 0010 2011 0011 3011 0100 4011 0101 5011 0110 6011 0111 7011 1000 8011 1001 9
110 0001 a110 0010 b ...111 1001 y111 1010 z
Review: ASCII, Unicode
*American Standard Code for Information Interchange
Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
4Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
• Put 1 in leftmost place without sum exceeding number
• Track sum
16 18 4 20
16 18 4 28
1
16 18 4 212 1 1
16 18 4 2done 1 1 00
016 > 12, too big;Put 0 in 16’s place
8 <= 12, so put1 in 8’s place,current sum is 8
8+4=12 <= 12, so put1 in 4’s place,current sum is 12
Reached desired 12,so put 0s in remainingplaces
Desired decimalnumber: 12
Currentsum
Binarynumber
(a)
(d)
(c)
(b)0
0
0
a
Review: From Decimal to Binary
Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
5Grundlagen der Technische InformatikWintersemester 2018/19
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• Microprocessors a common choice to implement a digital system– Easy to program– Cheap (as low as
1 €)– Readily available
I3I4I5I6I7
I2I1I0
P3P4P5P6P7
P2P1P0 void main()
while (1)
P0 = I0 && !I1; // F = a and !b,
0F
b
a
10101
6:00 7:05 7:06 9:00 9:01 time
Desired motion-at-night detector Programmedmicroprocessor
Custom designeddigital circuit
1.3Review: µProzessor vs. Digital System
Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
6Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
• Let’s learn to design digital circuits, starting with a simple form of circuit:– Combinational circuit
• Outputs depend solely on the present combination of the circuit inputs’ values
• vs. sequential circuit: Has “memory” that impacts outputs too
DigitalSystem
b=0 F=0
DigitalSystem
if b=0, then F=0if b=1, then F=1
b=1 F=1
(a)
Motionsensor Digital
System
Lightsensor
Lamp
a
b
F
if a=0 and b=0, then F=0if a=0 and b=1, then F=0if a=1 and b=0, then F=1if a=1 and b=1, then F=0
(b) Cannot determine value of F solely from present value
(c)
Introduction
DigitalSystem
b=0 F=0
DigitalSystem
b=1 F=1
DigitalSystem
b=0 F=1
Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
7Grundlagen der Technische InformatikWintersemester 2018/19
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• Electronic switches are the basis of binary digital circuits– Electrical terminology
• Voltage: Difference in electric potential between two points (volts, V)
– Analogous to water pressure• Current: Flow of charged particles (amps, A)
– Analogous to water flow• Resistance: Tendency of wire to resist current
flow (ohms, Ω)– Analogous to water pipe diameter
• R = V/I (Ohm’s Law)R = 9 V / 4.5 AR = 2 ohms
4.5 A
4.5
A
4.5 A
2 ohms
9 V
0 V 9 V
+–
If a 9V potential difference is appliedacross a 2 ohm resistor, then 4.5 A of current will flow.
Switches
Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
8Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
• A switch has three parts– Source input, and output
• Current tries to flow from source input to output
– Control input• Voltage that controls whether
that current can flow • The amazing shrinking switch
– 1930s: Relays– 1940s: Vacuum tubes– 1950s: Discrete transistor– 1960s: Integrated circuits (ICs)
•Initially just a few transistors on IC•Then tens, hundreds, thousands...
“off”
“on”
outputsourceinput
outputsourceinput
controlinput
controlinput
relay vacuum tube
discrete transistor
IC
quarter(to see the relative size)
Switches
Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
9Grundlagen der Technische InformatikWintersemester 2018/19
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Moore‘s Law
• Prediction announced by Gordon Moore in 1965
• Semiconductor Technology to double effectiveness every 18 Months
16151413121110987654321
Log2
ofth
eN
umbe
rof
Com
pone
nts
Per
Inte
grat
edFu
nctio
n
Year
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
Year Source: Moore, 1965
Log 2
Anza
hl d
er T
rans
istor
en fü
r jed
e in
tegr
iert
e Fu
nktio
n
Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
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Moore‘s Law
Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
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Moore‘s Law
0
300
600
900
1200
1500
1800
2100
2002 2004 2006 2008 2010 2012 2014/15
# Tr
ansi
stor
s (M
ill.)
Year
Sandy Bridge995 Mill.
Northwood55 Mill.
Prescott125 Mill.
Yonah151 Mill.
Wolfdale410 Mill.
Ivy Bridge1860 Mill.
Broadwell1900 Mill.
130 nm
90 nm
65 nm
45 nm32 nm
22 nm14 nm
0
20
40
60
80
100
120
140
0
300
600
900
1200
1500
1800
2100
2002 2004 2006 2008 2010 2012 2014/15
Tech
nolo
gy
# Tr
ansi
stor
s (M
ill.)
Year
Sandy Bridge995 Mill.
Northwood55 Mill.
Prescott125 Mill.
Yonah151 Mill.
Wolfdale410 Mill.
Ivy Bridge1860 Mill.
Broadwell1900 Mill.
Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
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Moore‘s Law
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Moore‘s Law
1 m10 cm1 cm1 mm100 µm10 µm100 nm
„22 nm“-Transistorsource: Intel
Source: „Spektrum der Wissenschaften“
Grundlagen der Technische InformatikWintersemester 2018/19
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• CMOS transistor– Basic switch in modern ICs
Silicon -- not quite a conductor or insulator:Semiconductor
gate
source drainoxide
A positive voltage here...
IC package
IC
...attracts electrons here, turning the channel between the source and drain into a conductor
The CMOS Transistor
Grundlagen der Technische InformatikWintersemester 2018/19
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does notconduct
0
conducts
1gate
nMOS
does notconduct
1gate
pMOS
conducts
0
The CMOS Transistor
Grundlagen der Technische InformatikWintersemester 2018/19
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gatesource
draindrain
sourcegate
CMOS Transistor Analogy
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• “Logic gates” are better digital circuit building blocks than switches (transistors)– Why?...
Boolean Logic Gates -Building Blocks for Digital Circuits
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• To understand the benefits of “logic gates” vs. switches, we should first understand Boolean algebra
• “Traditional” algebra– Variables represent real numbers (x, y)– Operators operate on variables, return real numbers (2.5*x + y - 3)
• Boolean Algebra– Variables represent 0 or 1 only– Operators return 0 or 1 only– Basic operators
• AND: a AND b returns 1 only when both a=1 and b=1• OR: a OR b returns 1 if either (or both) a=1 or b=1• NOT: NOT a returns the opposite of a (1 if a=0, 0 if a=1)
Boolean Algebra and its Relation to Digital Circuits
a0011
b0101
AND0001
a0011
b0101
OR0111
a01
NOT10
Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
19Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
George Boole
George Boole(November 2, 1815 –December 8, 1864) was a British mathematicianand philosopher
Grundlagen der Technische InformatikWintersemester 2018/19
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• Developed mid-1800’s by George Boole to formalize human thought– Ex: “I’ll go to lunch if Mary goes OR John goes, AND Sally does not
go.”• Let F represent my going to lunch (1 means I go, 0 I don’t go)• Likewise, m for Mary going, j for John, and s for Sally• Then F = (m OR j) AND NOT(s)
– Nice features• Formally evaluate
– m=1, j=0, s=1 --> F = (1 OR 0) AND NOT(1) = 1 AND 0 = 0• Formally transform
– F = (m and NOT(s)) OR (j and NOT(s)) » Looks different, but same function» We’ll show transformation techniques soon
• Formally prove– Prove that if Sally goes to lunch (s=1), then I don’t go (F=0)– F = (m OR j) AND NOT(1) = (m OR j) AND 0 = 0
a0011
b0101
AND0001
a0011
b0101
OR0111
a01
NOT10
Boolean Algebra and its Relation to Digital Circuits
Grundlagen der Technische InformatikWintersemester 2018/19
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21
• Evaluate the Boolean equation F = (a AND b) OR (c AND d)for the given values of variables a, b, c, and d:– Q1: a=1, b=1, c=1, d=0.
•Answer: F = (1 AND 1) OR (1 AND 0) = 1 OR 0 = 1.
– Q2: a=0, b=1, c=0, d=1. •Answer: F = (0 AND 1) OR (0 AND 1) = 0 OR 0 = 0.
– Q3: a=1, b=1, c=1, d=1. •Answer: F = (1 AND 1) OR (1 AND 1) = 1 OR 1 = 1.
Evaluating Boolean Equations
a0011
b0101
AND0001
a0011
b0101
OR0111
a01
NOT10
Grundlagen der Technische InformatikWintersemester 2018/19
Vorlesender:Dr. Ing. Frank Sill Torres
22Grundlagen der Technische InformatikWintersemester 2018/19
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• Convert the following English statements to a Boolean equation– Q1. a is 1 and b is 1.
• Answer: F = a AND b– Q2. either of a or b is 1.
• Answer: F = a OR b– Q3. a is 1 and b is 0.
• Answer: F = a AND NOT(b) – Q4. a is not 0.
• Answer:–(a) Option 1: F = NOT(NOT(a))–(b) Option 2: F = a
Converting to Boolean Equations
Grundlagen der Technische InformatikWintersemester 2018/19
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23Grundlagen der Technische InformatikWintersemester 2018/19
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• Q1. A fire sprinkler system should spray water if high heat is sensed and the system is set to enabled.– Answer: Let Boolean variable h represent “high heat is sensed,” e
represent “enabled,” and F represent “spraying water.” Then an equation is: F = h AND e.
• Q2. A car alarm should sound if the alarm is enabled, and either the car is shaken or the door is opened. – Answer: Let a represent “alarm is enabled,” s represent “car is
shaken,” d represent “door is opened,” and F represent “alarm sounds.” Then an equation is: F = a AND (s OR d).
– (a) Alternatively, assuming that our door sensor d represents “door is closed” instead of open (meaning d=1 when the door is closed, 0 when open), we obtain the following equation: F = a AND (s OR NOT(d)).
Converting to Boolean Equations
Grundlagen der Technische InformatikWintersemester 2018/19
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24Grundlagen der Technische InformatikWintersemester 2018/19
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• Implement Boolean operators using transistors– Call those implementations logic gates. – Lets us build circuits by doing math -- powerful concept
Boolean algebra(mid-1800s)
Boole’s intent: formalizehuman thought
Switches(1930s)
Shannon (1938)
Digital design
Showed application of Boolean algebra to design of switch-
based circuits
For telephone switching and other electronic uses
Relating Boolean Algebra to Digital Design
Grundlagen der Technische InformatikWintersemester 2018/19
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25Grundlagen der Technische InformatikWintersemester 2018/19
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Logic Gates
x0011
y0101
F0001
x0011
y0101
F0111
x01
F10
Fxx
yF
ORN O T
Fx
y
AND
1
0
Fx
Symbol
TruthTable
Transistorcircuit
F
1
0
x
x
y
y
F
1
0
x y
x
y
1.8 V
1.2 V
0.6 V
0V“0”
“1”
1 and 0 each actually
corresponds to a voltage range
Grundlagen der Technische InformatikWintersemester 2018/19
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26Grundlagen der Technische InformatikWintersemester 2018/19
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0
1
1
0
time
F
x
x
0
1
F
1
0
1
0
F1x 0
(a)
1
0
F0x1
(b)
When the input is 0 When the input is 1
NOT gate
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27Grundlagen der Technische InformatikWintersemester 2018/19
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x
0
0
1
1
y
0
1
0
1
F
0
1
1
1
1
0x
y
F1
1
0
0
timeWhen one input is 1 When both inputs are 0
OR gate
1
0
11
0
1
1
1
10
0
1
1
0
0
0
0
0
1
0
Grundlagen der Technische InformatikWintersemester 2018/19
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28Grundlagen der Technische InformatikWintersemester 2018/19
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x
0
0
1
1
y
0
1
0
1
F
0
0
0
1
1
0x
y
F1
1
0
0
timeWhen both inputs are 1 When an input is 0
AND gate
10
1
0
0 1
0
1
11
1
0
1 1
1
1
0
Grundlagen der Technische InformatikWintersemester 2018/19
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29Grundlagen der Technische InformatikWintersemester 2018/19
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More Gates
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30Grundlagen der Technische InformatikWintersemester 2018/19
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• Recall Chapter 1 motion-in-dark example– Turn on lamp (F=1) when motion sensed (a=1) and no light (b=0)– F = a AND NOT(b)– Build using logic gates, AND and NOT, as shown– We just built our first digital circuit!
Building Circuits Using Gates
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31Grundlagen der Technische InformatikWintersemester 2018/19
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• Q: Convert the following equation to logic gates: F = a AND NOT( b OR NOT(c) )
ab
cF
Start from the output, work back towards the inputs
Example: Converting a Boolean Equation to a Circuit of Logic Gates
Grundlagen der Technische InformatikWintersemester 2018/19
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32Grundlagen der Technische InformatikWintersemester 2018/19
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Start from the output, work back towards the inputs
as
dF
(a)
F = a AND (s OR d)1 2 a
b
c
F
(b)
F = (a AND NOT(b)) OR (b AND NOT(c))2 1 3
More examples
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33Grundlagen der Technische InformatikWintersemester 2018/19
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(a) (b)
F = a AND b AND c
ab
FF
abc
c
Can think of as AND(a,b,c)
Using gates with more than 2 inputs
Grundlagen der Technische InformatikWintersemester 2018/19
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34Grundlagen der Technische InformatikWintersemester 2018/19
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• Design circuit for warning light• Sensors
– s=1: seat belt fastened– k=1: key inserted
• Capture Boolean equation– seat belt not fastened, and key
inserted• Convert equation to circuit
• Timing diagram illustrates circuit behavior– We set inputs to any values– Output set according to circuit
w = NOT(s) AND k
k
s
w
BeltWarn
Seatbelt
time
Inputs
Outputs
1
1
1
0
0
0
k
s
w
Example: Seat Belt Warning Light System
Grundlagen der Technische InformatikWintersemester 2018/19
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35Grundlagen der Technische InformatikWintersemester 2018/19
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Notice • Boolean algebra enables easy capture as equation
and conversion to circuit• How design with switches?
• Of course, logic gates are built from switches, but we think at level of logic gates, not switches
w = NOT(s) AND k
k
s
w
BeltWarn
Seatbelt
BeltWarn
w
1
0
0
1
s
k
Gates vs. switches
Grundlagen der Technische InformatikWintersemester 2018/19
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36Grundlagen der Technische InformatikWintersemester 2018/19
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• Only illuminate warning light if person is in the seat (p=1), and seat belt not fastened and key inserted
• w = p AND NOT(s) AND k
k
p
s
w
Belt W a r n
• Given t=1 for 5 seconds after key inserted. Turn on warning light when t=1 (to check that warning lights are working)
• w = (p AND NOT(s) AND k) OR t
k
wp
s
t
BeltWarn
More examples: Seat belt warning light extensions
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37Grundlagen der Technische InformatikWintersemester 2018/19
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no yes
not ok
ok
xy
F
no yes
Some Gate-Based Circuit Drawing Conventions
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38Grundlagen der Technische InformatikWintersemester 2018/19
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• Unterschied kombinatorische und sequentielle Schaltungen
• Schalter (Transistoren) sind Grundelemente
• Logikgatter bestehen aus Transistoren
• Logikgatter ermöglichen Umsetzung Boolescher Logik zu digitalen Schaltungen
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