what do these bits represent? - tarleton state university · what do these bits represent as...

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




X

X =

Quiz

24




X

X =

Quiz

25




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


• 4-input OR

• 5-input NAND

• 3-input NOR

How many lines does each of the truth tables have?

38

QUIZ


• 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)

http://upload.wikimedia.org/wikipedia/commons/c/cb/BJT_NPN_symbol_%28case%29.svg

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




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




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


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


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

what do these bits represent? - tarleton state university · what do these bits represent as...

Documents