QUIZ
What do these bits represent?
1001 0110
1
QUIZ
What do these bits represent?
1101 1110
2
Unsigned integer: …
Signed integer (2’s complement): …
Fraction: …
IBM 437 character: …
Latin-1 character: …
Huffman-compressed text: …
Red-color level: …
QUIZ
What do these bits represent as Huffman-compressed text?
1001 0110
3
QUIZ
What do these bits represent as Huffman-compressed text?
1001 0110 …
4
L D ? Bits from the
next byte
QUIZ: Audio Formats
MP3
– analyzes the frequency spread and discards information that can’t be heard by MOST humans (>16 kHz)
How many MP3 samples are there in a 4-minute song?
After Huffman encoding, the average bits-per-sample has decreased from 8 to 5. What is the total size of the file?
5
Chapter 4
Gates and Circuits (with
some transistors thrown in for good measure)
All hail the HARDWARE!
Abstractions and more abstractions …
7 You are here
Abstractions and more abstractions …
Computers Made of lots of different circuits (CPU, memory, controllers, etc.)
Circuits
Made from gates combined to perform more complicated tasks
Gates
Devices that perform basic logical operations on electrical
signals. They’re built out of transistors
Transistors Very small electronic switches
8
There are 3 layers of computer abstraction that we examine in this
chapter:
9
Circuits
Gates
Transistors
4.2 Gates
There are six basic gates: – NOT
– AND
– OR
– XOR
– NAND
– NOR
Real-life logic diagrams are black and white with gates distinguished only by their shape
We use color for emphasis (and fun) in this text
10
How do we describe the behavior of gates and circuits?
1. Boolean expressions Uses Boolean algebra, a mathematical notation for expressing two-valued logic
2. Logic diagrams A graphical representation of a circuit; each gate has its own symbol
3. Truth tables A table showing all possible input value and the associated output values
11
NOT Gate (a.k.a. inverter)
A NOT gate accepts one input signal (0 or 1) and returns the opposite signal as output
12
Figure 4.1 Various representations of a NOT gate
AND Gate
An AND gate accepts two input signals
If both are 1, the output is 1; otherwise, the output is 0
13
Figure 4.2 Various representations of an AND gate
QUIZ: AND Gate
The inputs to an AND gate have the values A = 1 and B = 0. What is the output?
14
QUIZ: AND Gate
The input B to an AND gate has the value B = 0. What is the output?
15
QUIZ: AND Gate
The input B to an AND gate has the value B = 1. What is the output?
16
Conclusion: the AND gate has the following interesting properties
17
A AND 0 = 0, irrespective of B
A AND 1 = B
OR Gate An OR gate accepts two input signals
If both are 0, the output is 0; otherwise, the output is 1
18
Figure 4.3 Various representations of a OR gate
XOR Gate
19
XOR is called the exclusive OR
Pronunciations: zor, ex-or
If both inputs are the same, the output is 0;
otherwise, the output is 1
QUIZ: recognize the gate and draw its symbol!
20
QUIZ: elementary properties
21
A AND 0 = ?
A AND 1 = ?
1 AND B = ?
A OR 1 = ?
A XOR 0 = ?
A XOR 1 = ?
Quiz
22
For the following circuit diagram:
• Find the Boolean expression
• Find the truth table
X
X =
XOR
NOT
Quiz
23
For the following circuit diagram:
• Find the Boolean expression
• Find the truth table
X
X =
Quiz
24
For the following circuit diagram:
• Find the Boolean expression
• Find the truth table
X
X =
Quiz
25
For the following circuit diagram:
• Find the Boolean expression
• Find the truth table
X
X =
NAND Gate If both inputs are 1, the output is 0;
otherwise, the output is 1
Compare to AND:
Quiz
Can we make a NAND gate out of an AND and an INVERTER?
27
Quiz
Can we make a NAND gate out of an AND and an INVERTER?
28
(AB)’ (AB)’’ = AB
EOL1
QUIZ
Can we build an inverter (NOT) from a NAND gate?
29
QUIZ
Can we build an inverter (NOT) from a NAND gate?
30 Figure 4.7 Various representations of a three-input AND gate
NOR Gate
31
If both inputs are 0, the output is 1;
otherwise the output is 0
Compare to OR:
Quiz
Can we make an OR gate out of a NOR and an INVERTER?
32
Whatever happened with the inverted XOR?
It’s called XNOR, pronounced [ex-nor]
33
Not in text
Review of Gates
A NOT gate inverts its single input
An AND gate produces 1 iff both input values are 1
An OR gate produces 0 iff both input values are 0
A XOR gate produces 0 iff input values are the same
All inverted gates have the opposite outputs
34
If and only if
It’s OK to put the gate representations on your memory-sheet, but
you should be able to remember the word descriptions above!
Quiz
• What are the 3 ways we use to describe gates and circuits?
• Use the 3 ways to describe the NAND gate
– Hint: Describe AND first!
35
Gates with More Inputs
A three-input AND gate produces an output of 1 iff all input values are 1
36
QUIZ
Draw the gate symbols for:
• 4-input OR
• 5-input NAND
• 3-input NOR
37
QUIZ
Draw the gate symbols for:
• 4-input OR
• 5-input NAND
• 3-input NOR
How many lines does each of the truth tables have?
38
QUIZ
Draw the gate symbols for:
• 4-input OR
• 5-input NAND
• 3-input NOR
How many lines does each of the truth tables have?
Describe in your own words each of the truth tables.
39
Extra-credit QUIZ
40
Remember: There are 3 layers of computer abstraction that we
examine in this chapter:
41
Circuits
Gates
Transistors
4.3 Constructing Gates
Transistor = device that acts either as a wire that conducts electricity or as a resistor that blocks the flow of electricity, depending on the voltage level of an input signal
Made of a semiconductor material • Neither good conductor of electricity nor a good insulator …
• …but with a little help can be either!
Acts like a switch, but w/o moving parts:
• Switch open
• Switch closed
42
How transistors operate as switches
43
High voltage, a.k.a. +
Low voltage, a.k.a. -
When 1 is applied
on the base/gate,
the switch closes
When 0 is applied
on the base/gate,
the switch opens
Base or gate
Transistors
A transistor has three terminals:
– A collector/source, typically connected to the positive terminal of a power source (5 volts, 3.5 volts, etc.)
– An emitter/drain, typically connected to the “ground” (0 volts)
– A base/gate, which controls the flow of current between source and emitter
44
Figure 4.8 The connections of a transistor
The names of transistor terminals -setting the record straight FYI-
45
Bipolar Junction Transistor (BJT)
Collector, Base, Emmiter Field-Effect Transistor (FET)
Drain, Gate, Source
“Mongrel” transistor (our text)
The easiest gates to create are the NOT, NAND, and NOR
46
We can explain their operation for any combination of inputs!
We do this by replacing the transistors with switches!
If there is no path from output to Ground, Vout = 1
If there is a path from output to Ground, Vout = 0
47
QUIZ
The AND gate is obtained as a NAND followed by an inverter. Draw its transistor diagram!
48
QUIZ: Transistors What gate do we have below?
49
QUIZ: Transistors What gate do we have below?
50
To do for next time:
Read pp.97-101 of text
Answer in notebook the end-of-chapter questions: 1 through 11
51 EOL3
QUIZ: Review of Gates
A NOT gate …
An AND gate …
An OR gate …
A XOR gate …
All inverted gates …
52
QUIZ: Based on the switching behavior of transistors, verify the truth table of the OR gate
53
Remember: connection to Ground means
logical zero!
4.4 Intro: From gates to circuits
54
We can go either way between gates and circuits, e.g.
Analysis
Find the truth table for the circuit:
X
From gates to circuits
55
Design
Find the logic diagram of the circuit described by the
following truth table:
Hint: The table is similar to which of the fundamental gates?
1
0 ?
SOLUTION
Having only one 0 in the output column, the circuit most resembles the OR gate!
It is different from the OR gate only in this respect: …
Write the Boolean expression:
56
SOLUTION
Write the Boolean expression: X = A’ + B
Draw the circuit diagram:
57
SOLUTION
Write the Boolean expression: X = A’ + B
Draw the diagram:
58
4.4 Circuits
Combinational circuit
The input values explicitly determine the output
Sequential circuit
The output is a function of the input values and the existing state of the circuit
We describe the circuit operations using Boolean expressions
Logic diagrams
Truth tables
59
Combinational Circuit
Three inputs require 23 = 8 rows to describe all possible input combinations
Boolean expression is:
60
AB+AC
Another Combinational Circuit
61
Another Combinational Circuit
Consider the following Boolean expression A(B + C)
62
Does this truth table look familiar?
Circuit equivalence: Two circuits that produce the same output for identical input are called equivalent
Boolean algebra allows us to apply provable mathematical principles to help design circuits:
A(B + C) = AB + AC (distributive law) so circuits must be equivalent
63
Each property (law) of Boolean Algebra translates directly into a property of equivalent circuits!
64
Null elements A · 0 = 0 A + 1 = 1
Idempotency A · A = A A + A = A
Double complement (A’)’ = A
De Morgan’s Laws
65
Null elements A · 0 = 0 A + 1 = 1
Idempotency A · A = A A + A = A
Double complement (A’)’ = A
DeMorgan’s law applied directly to gates
66
DeMorgan’s law QUIZ
67
Apply DeMorgan’s Law directly on the gate diagram below to obtain equivalent circuits:
DeMorgan’s law QUIZ
68
Apply DeMorgan’s Law directly on the gate diagram below to obtain equivalent circuits:
QUIZ: Write the circuit forms for the first 2 and last 2 properties
69
Null elements A · 0 = 0 A + 1 = 1
Idempotency A · A = A A + A = A
Double complement (A’)’ = A
QUIZ
70
Very Useful Combinational Circuit: the Adder
At the digital logic level, addition is performed in binary
Addition operations are carried out by special circuits called adders
71
Half-Adder truth table
The result of adding two binary digits could produce a carry value
Recall that 1 + 1 = 10 in base two
Half adder
A circuit that computes the sum of two bits and produces the correct carry bit
72
Truth table
Do you recognize
these outputs?
Half Adder
Circuit diagram
Boolean expressions
Sum = A B
Carry = A·B
73
How many transistors are here?
Full Adder
This adder takes the Carry-in value into account!
74
Do you recognize
these circuits?
Adding multiple bits - “Ripple Carry” Adder -
75
SKIP Multiplexers
76 EOL4
There are 2 basic types of circuits
Combinational circuit
The input values explicitly determine the output
Sequential circuit
The output is a function of the input values and the existing state of the circuit
77
4.5 Circuits as Memory a.k.a. Sequential Circuits
A sequential circuit is one whose output depends not only on the current values of its inputs, but also on the past sequence of those inputs (history).
It can be used to store information, i.e. as memory.
78
The S – R latch
There are several ways to build S – R latches using various kinds of gates, but there’s always feedback.
79
Figure 4.12 An S-R latch
How many transistors are here?
S – R latch
If X is 1, we say that the circuit is storing a 1; if X is 0, the circuit is storing a 0
As long as S = R = 1, an S-R latch stores a single binary digit,1 or 0.
The design guarantees that the two outputs X and Y are (almost always) complements of each other.
80
Figure 4.12 An S-R latch
The value of X at
any point in time is
considered to be the
state of the circuit
Set means “make 1”, Reset means “make 0”
81
To make X = 1,
make S = 0 (while
keeping R = 1).
To make X = 0,
make R = 0 (while
keeping S = 1).
QUIZ: S – R latch
What happens if both S and R are made 0 at the same time?
82
Integrated Circuit (a.k.a. IC or chip) = A piece of silicon on which multiple gates have been embedded
Silicon pieces are mounted on a plastic or ceramic package with pins along the edges that can be soldered onto circuit boards or inserted into appropriate sockets
83
Integrated Circuits
Integrated circuits (IC) are classified by the number of gates contained in them
84
VVLSI (?) more than 1B
3rd gen.
4th gen.
5th gen.
Integrated Circuits
85
Figure 4.13 An SSI chip containing NAND gates
VLSI chip: AMD Phenom II CPU
contains 768 million transistors
How many transistors are here?
86
As of 2014, the highest transistor count in a commercially available CPU is over 4.3 billion transistors, in Intel's 15-core Xeon IvyBridge-EX.
On August 7, 2014, IBM announced their second generation SyNAPTIC chip, which contains the most transistors in a Neurosynaptic chip to date: 5.4 billion.
Xilinx currently holds the "world-record" for a FPGA containing more than 20 billion transistors.
Source: http://en.wikipedia.org/wiki/Transistor_count
Pricing ranges from $1200 to
$6800 per chip and with up to 8 per
server, that can get costly but is
only part of the puzzle.
Each chip can address up to 1.5TB
of memory meaning that in an 8
processor configuration one could
get up to 12TB of RAM.
Source: http://www.servethehome.com/skip-
tock-intel-xeon-e7-v2-released-ivy-bridge-ex/
The most important integrated circuit in any computer is the Central Processing Unit, or CPU
Each CPU chip has a large number of pins through which communication takes place in a computer system
87
motherboard CPU
Each CPU chip has a large number of pins (Pin Grid Array = PGA) through which communication takes place
88 Image source: http://www.digitale-fotografien.com/freie-galerie/sonstiges/prozessor/
Wait a second: Didn’t we say that there are Billions of transistors on a chip? This means many hundreds of millions of gates!
89 Image source: http://www.digitale-fotografien.com/freie-galerie/sonstiges/prozessor/
Answer: Like in the case of the F.A., the gates are integrated into more complex circuits, with relatively few outputs.
90
Latest packaging technology:
Ball Grid Array = BGA
91
Image source: http://www.electronicsweekly.com/news/design/test-and-
measurement/boundary-scan-testing-grows-as-bgas-proliferate-2006-01/
Image source: http://www.etech-web.com/bga-reballing.htm
Chapter review questions
• Identify the basic gates and describe the behavior of each – Includes the inverted gates!
• Describe how gates are implemented using transistors
• Combine basic gates into circuits – Analysis vs. design
• Describe the behavior of a gate or circuit using Boolean expressions, truth tables, and logic diagrams
92
Chapter review questions
• Write deMorgan’s laws in Boolean form and in circuit form
• Compare and contrast a half adder and a full adder
• Explain how an S-R latch operates
• Describe the characteristics of the four (five?) generations of integrated circuits
• What is the CPU?
93
Homework due Friday, Oct.9
End-of-chapter 37, 48, 49, 50, 62, 66, 70, 72, 73
Thought question #4
94