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Data Representation and Gates
These slides are taken from your textbook Dale & Lewis
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Data and Computers
Computers store, present, and help us modify
• Numbers• Text• Audio• Images and graphics• Video
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Data and Computers
Data compression Reduction in the amount of space needed to store a piece of dataCompression ratio The size of the compressed data divided by the size of the original dataA data compression techniques can be
lossless, which means the data can be retrieved without any loss of the original information
lossy, which means some information may be lost in the process of compaction
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Analog and Digital Information
Information can be represented in one of two ways: analog or digital
Analog data
A continuous representation, analogous to the actual information it represents
Digital data
A discrete representation, breaking the information up into separate elements
Analog and Digital Information
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Analog and Digital Information
Computers cannot work well with analog data, so we digitize the data
Digitize
Breaking data into pieces and representing those pieces separately
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Electronic Signals
Important facts about electronic signals• An analog signal continually fluctuates in
voltage up and down • A digital signal has only a high or low state,
corresponding to the two binary digits
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Electronic Signals (Cont’d)
Figure 3.2 An analog and a digital signal
Figure 3.3 Degradation of analog and digital signals
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Binary RepresentationsCounting withbinary bits
Figure 3.4
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Representing Negative Values
Signed-magnitude number representation
The sign represents the ordering, and the digits represent the magnitude of the number
Signed binary• signed-magnitude
• The most significant bit (MSB) is set aside to indicate the sign, zero for positive and 1 for negative.
• One’s-complement• The MSB of a negative number will begin with a
one and a positive number with a zero. 0011 will represent +3 and 1100 (complement of the positive 3) will represent -3.
• Two’s-complement. • a negative number is represented by complementing
the bits of the positive number and adding a 1. a –3 is represented as 1101 (complement of 0011 is 1100, and add a 1).
Two’s complement• a binary number’s MSB is a zero then it is a
positive and no further action is necessary. • If the MSB is 1, it is a negative number. • Suppose the number is 10000001, we know
it is a negative number. Complement the bits and add a 1, we get 01111110 +1 = 01111111 which is 127.
• The sign is already determined to be negative it is –127.
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Number Overflow
What happen if the computed value won't fit?
Overflow
If each value is stored using eight bits, adding 127 to 3 overflows
1111111+ 0000011 10000010
Problems occur when mapping an infinite world onto a finite machine!
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Representing Text
What must be provided to represent text?
There are finite number of characters to represent, so list them all and assign each a binary string
Character set A list of characters and the codes used to represent each one Computer manufacturers agreed to standardize
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The ASCII Character Set
ASCII stands for American Standard Code for Information Interchange
ASCII originally used seven bits to represent each character, allowing for 128 unique characters
Later extended ASCII evolved so that all eight bits were used
How many characters could be represented?
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ASCII Character Set Mapping
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The ASCII Character Set
The first 32 characters in the ASCII character chart do not have a simple character representation to print to the screen
What do you think they are used for?
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The Unicode Character Set
Extended ASCII is not enough for international use
One Unicode mapping uses 16 bits per character
How many characters can this mapping represent?
Unicode is a superset of ASCII
The first 256 characters correspond exactly to the extended ASCII character set
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The Unicode Character Set
Figure 3.6 A few characters in the Unicode character set
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Text Compression
Assigning 16 bits to each character in a document uses too much file space
We need ways to store and transmit text efficiently
Text compression techniqueskeyword encodingrun-length encodingHuffman encoding
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Keyword Encoding
Replace frequently used words with a single character
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Keyword Encoding
Given the following paragraph,We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. ム That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, ム That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness.
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Keyword Encoding
The encoded paragraph isWe hold # truths to be self-evident, $ all men are created equal, $ ~y are endowed by ~ir Creator with certain unalienable Rights, $ among # are Life, Liberty + ~ pursuit of Happiness. — $ to secure # rights, Governments are instituted among Men, deriving ~ir just powers from ~ consent of ~ governed, — $ whenever any Form of Government becomes destructive of # ends, it is ~ Right of ~ People to alter or to abolish it, + to institute new Government, laying its foundation on such principles + organizing its powers in such form, ^ to ~m shall seem most likely to effect ~ir Safety + Happiness.
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Keyword EncodingWhat did we save?
Original paragraph 656 characters
Encoded paragraph 596 characters
Characters saved60 characters
Compression ratio596/656 = 0.9085
Could we use this substitution chart for all text?
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Run-Length Encoding
A single character may be repeated over and over again in a long sequenceReplace a repeated sequence with
– a flag character – repeated character– number of repetitions
*n8– * is the flag character– n is the repeated character– 8 is the number of times n is repeated
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Run-Length Encoding
Original textbbbbbbbbjjjkllqqqqqq+++++
Encoded text*b8jjjkll*q6*+5 (Why isn't l encoded? J?)
The compression ratio is 15/25 or .6
Encoded text*x4*p4l*k7
Original textxxxxpppplkkkkkkk
This type of repetition doesn’t occur in English text; can you think of a situation where it might occur?
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Huffman Encoding
Why should the character “X" and "z" take up the same number of bits as "e" or " "?
Huffman codes use variable-length bit
strings to represent each character
More frequently used letters have shorter
strings to represent them
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Huffman Encodingballboard would be1010001001001010110001111011
compression ratio
28/56
Encode roadbed
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Representing Audio Information
We perceive sound when a series of air compressions vibrate a
membrane in our ear, which sends signals to our brain
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Representing Audio Information
A stereo sends an electrical signal to a speaker to produce sound This signal is an analog representation of the sound wave The voltage in the signal varies in direct proportion to the sound wave
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Representing Audio Information
Digitize the signal by sampling – periodically measure the voltage – record the numeric value
How often should we sample?
A sampling rate of about 40,000 times per second is enough to create a reasonable sound reproduction
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Representing Audio Information
Figure 3.8 Sampling an audio signal
Some datais lost, but areasonablesound is reproduced
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Representing Audio Information
CDs store audio information digitally
On the surface of the CD are microscopic
pits that represent binary digits
A low intensity laser is pointed as the disc
The laser light reflects
strongly if the surface is smooth and poorly if the surface is pitted
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Representing Audio Information
Figure 3.9 A CD player reading binary information
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Audio Formats
Audio Formats– WAV, AU, AIFF, VQF, and MP3
MP3 (MPEG-2, audio layer 3 file) is dominant – analyzes the frequency spread and discards
information that can’t be heard by humans – bit stream is compressed using a form of Huffman
encoding to achieve additional compression
Is this a lossy or lossless compression (or both)?
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Representing Images and Graphics
Color
Perception of the frequencies of light that reach the retinas of our eyes
Retinas have three types of color
photoreceptor cone cells that correspond to
the colors of red, green, and blue
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Representing Images and Graphics
Color is expressed as an RGB (red-green-blue) value--three numbers that indicate the relative contribution of each of these three primary colors
An RGB value of (255, 255, 0) maximizes the contribution of red and green, and minimizes the contribution of blue, which results in a bright yellow
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Representing Images and Graphics
Figure 3.10 Three-dimensional color space
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Digitized Images and Graphics
Digitizing a picture
Representing it as a collection of individual dots called pixels
Resolution
The number of pixels used to represent a picture
Raster Graphics
Storage of data on a pixel-by-pixel basis
Bitmap (BMP), GIF, JPEG, and PNG are raster-
grahics formats
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Digitized Images and GraphicsBitmap formatContains the pixel color values of the image from left to right and from top to bottomGIF format (indexed color)Each image is made up of only 256 colorsJPEG formatAverages color hues over short distancesPNG formatLike GIF but achieves greater compression with wider range of color depths
Which is better for line drawings? Pictures?
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Digitized Images and Graphics
Figure 3.12 A digitized picture composed of many individual pixels
Wholepicture
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Digitized Images and Graphics
Figure 3.12 A digitized picture composed of many individual pixels
Magnified portionof the picture
See the pixels?
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Representing Video
Video codec COmpressor/DECompressor Methods used to shrink the size of a movie to allow it to be played on a computer or over a network
Almost all video codecs use lossy compressions to minimize the huge amounts of data associated with video
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Representing Video
Temporal compression
A technique based on differences between consecutive frames: If most of an image in two frames hasn’t changed, why should we waste space to duplicate all of the similar information?
Spatial compression
A technique based on removing redundant information within a frame: This problem is essentially the same as that faced when compressing still images
Computers and Electricity
Gate A device that performs a basic operation onelectrical signals
Circuits Gates combined to perform morecomplicated tasks
Computers and Electricity
How do we describe the behavior of gates and circuits?Boolean expressionsUses Boolean algebra, a mathematical notation for expressing two-valued logic Logic diagramsA graphical representation of a circuit; each gate has itsown symbolTruth tablesA table showing all possible input value and the associatedoutput values
Gates
Six types of gates– NOT– AND– OR– XOR– NAND– NOR
Typically, logic diagrams are black and white with gates distinguished only by their shape
We use color for emphasis (and fun)
NOT Gate
A NOT gate accepts one input signal (0 or 1) and returns the opposite signal as output
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
Figure 4.2 Various representations of an AND gate
OR GateAn OR gate accepts two input signals
If both are 0, the output is 0; otherwise,the output is 1
Figure 4.3 Various representations of a OR gate
XOR Gate
Figure 4.4 Various representations of an XOR gate
An XOR gate accepts two input signals
If both are the same, the output is 0; otherwise,the output is 1
XOR Gate
Note the difference between the XOR gate and the OR gate; they differ only in one input situation
When both input signals are 1, the OR gate produces a 1 and the XOR produces a 0
XOR is called the exclusive OR
NAND Gate
The NAND gate accepts two input signalsIf both are 1, the output is 0; otherwise,the output is 1
Figure 4.5 Various representations of a NAND gate
NOR Gate
Figure 4.6 Various representations of a NOR gate
The NOR gate accepts two input signals
If both are 0, the output is 1; otherwise, the output is 0
Review of Gate Processing
A NOT gate inverts its single input
An AND gate produces 1 if both input values are 1
An OR gate produces 0 if both input values are 0
An XOR gate produces 0 if input values are the same
A NAND gate produces 0 if both inputs are 1
A NOR gate produces a 1 if both inputs are 0
Gates with More Inputs
Gates can be designed to accept three or more input values
A three-input AND gate, for example, produces an output of 1 only if all input values are 1
Figure 4.7 Various representations of a three-input AND gate
Constructing Gates
Transistor A 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
A transistor has no moving parts, yet acts like a switch
It is made of a semiconductor material, which is neither a particularly good conductor of electricity nor a particularly good insulator
Constructing Gates
A transistor has three terminals– A source– A base– An emitter, typically
connected to a ground wire
If the electrical signal is grounded, it is allowed to flow through an alternative route to the ground (literally) where it can do no harm
Figure 4.8 The connections of a transistor
Constructing Gates
The easiest gates to create are the NOT, NAND, and NOR gates
Figure 4.9 Constructing gates using transistors
Circuits
Combinational circuit The input values explicitly determine the outputSequential circuit The output is a function of the input values and the
existing state of the circuitWe describe the circuit operations using
Boolean expressionsLogic diagramsTruth tables
Are you surprised?
Combinational Circuits
Gates are combined into circuits by using the output of one gate as the input for another
Combinational Circuits
Three inputs require eight rows to describe all possible input combinations
This same circuit using a Boolean expression is (AB + AC)
Combinational Circuits
Consider the following Boolean expression A(B + C)
Does this truth table look familiar?
Compare it with previous table
Combinational Circuits
Circuit equivalenceTwo circuits that produce the same output for
identical input
Boolean algebra allows us to apply provable mathematical principles to help design circuits
A(B + C) = AB + BC (distributive law) so circuits must be equivalent
Properties of Boolean Algebra
Adders
At the digital logic level, addition is performed in binary
Addition operations are carried out by special circuits called, appropriately, adders
Adders
The result of adding two binary digits could produce a carry value
Recall that 1 + 1 = 10 in base two
Half adderA circuit that computes
the sum of two bits and produces the correct carry bit
Truth table
Half Adder#include <iostream.h> void main() { short x,y, sum, carry; bool a,b,c,d; cout <<"Enter two bits to add seperated by a space ";cin >> x>>y; a=x==1; b=y==1; //actually c++ assigns 0 for false and 1 for true, we
could have read these directly; c=(a||b) && !(a&&b); d=(a&&b); 9sum=c?1:0; //if c is true then sum gets 1 else sum gets 0 carry=d?1:0; cout <<x <<"+"<<y <<"="<<carry<<sum<<endl; if (a = true) cout <<"it is true"; if (a) cout << "it is ture"; }
Adders
Circuit diagram representing a half adder
Boolean expressions
sum = A Bcarry = AB
Adders
Full adder
A circuit that takes the carry-in value into account
Figure 4.10 A full adder
Multiplexers
Multiplexer
A circuit that uses a few input control signals to determine which of several output data lines is routed to its output
Multiplexers
The control lines S0, S1, and S2 determine which of eight other input lines
(D0 … D7)
are routed to the output (F)
Figure 4.11 A block diagram of a multiplexer with three select control lines
Circuits as Memory
Digital circuits can be used to store information
These circuits form a sequential circuit, because the output of the circuit is also used as input to the circuit
Circuits as Memory
An S-R latch stores a single binary digit (1 or 0)
There are several ways an S-R latch circuit can be designed using various kinds of gates
Figure 4.12 An S-R latch
Circuits as Memory
The design of this circuit guarantees that the two outputs X and Y are always complements of each other
The value of X at any point in time is considered to be the current state of the circuit
Therefore, if X is 1, the circuit is storing a 1; if X is 0, the circuit is storing a 0
Figure 4.12 An S-R latch
Integrated Circuits
Integrated circuit (also called a 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
Integrated Circuits
Integrated circuits (IC) are classified by the number of gates contained in them
Integrated Circuits
Figure 4.13 An SSI chip contains independent NAND gates
CPU Chips
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 essentially all communication in a computer system occurs
University of Texas Pan Am
Dr. John P. Abraham
Computer Programming Languages
• Computers perform operations such as moving data and data manipulation by activating switches and gates.
• Instructions to do that also must be given in the form of 1s and 0s or on and off.
• When we program computers using just ones and zeros we are using the “machine language”.
• When a program is written using human readable code line for line of machine code, that language is called Assembly language. Human readable code is called Mnemonics.
• When a programmer can use mathematical symbols and familiar English words such as write (a+b) that language is called a High Level language. Each of the program line will be converted to several lines of assembly or machine code.
• More about it later