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Digital Representation and Encoding: Bits, Bytes, and Analog vs. Digital, Study Guides, Projects, Research of Information Technology

Saint Peter's UniversityInformation Technology

The concept of digital representation, focusing on the conversion of real-world values into numbers, the differences between analog and digital systems, and the use of transducers for conversion. Topics include encoding sound, ASCII, interpretation of bits, and binary arithmetic.

Typology: Study Guides, Projects, Research

2021/2022

Uploaded on 09/12/2022

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Bits, bytes, and representation of information
• digital representation means that everything is represented by
numbers only
• the usual sequence:
– something (sound, pictures, text, instructions, ...) is converted into
numbers by some mechanism
– the numbers can be stored, retrieved, processed, copied, transmitted
– the numbers might be reconstituted into a version of the original
• for sound, pictures, other real-world values
– make accurate measurements
– convert them to numeric values
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Download Digital Representation and Encoding: Bits, Bytes, and Analog vs. Digital and more Study Guides, Projects, Research Information Technology in PDF only on Docsity!

Bits, bytes, and representation of information

  • digital representation means that everything is represented by

numbers only

  • the usual sequence:
    • something (sound, pictures, text, instructions, ...) is converted into numbers by some mechanism
    • the numbers can be stored, retrieved, processed, copied, transmitted
    • the numbers might be reconstituted into a version of the original
  • for sound, pictures, other real-world values
    • make accurate measurements
    • convert them to numeric values

Analog versus Digital

  • analog: "analogous" or "the analog of"
    • smoothly or continuously varying values
    • volume control, dimmer, faucet, steering wheel
    • value varies smoothly with something else no discrete steps or changes in values small change in one implies small change in another infinite number of possible values
    • the world we perceive is largely analog
  • digital: discrete values
    • only a finite number of different values
    • a change in something results in sudden change from one discrete value to another digital speedometer, digital watch, push-button radio tuner, …
    • values are represented as numbers

Encoding sound

  • need to measure intensity/loudness often enough and accurately

enough that we can reconstruct it well enough

  • higher frequency = higher pitch
  • human ear can hear ~ 20 Hz to 20 KHz
    • taking samples at twice the highest frequency is good enough (Nyquist)
  • CD audio usually uses
    • 44,100 samples / second
    • accuracy of 1 in 65,536 (= 2^16) distinct levels
    • two samples at each time for stereo
    • data rate is 44,100 x 2 x 16 bits/sample = 1,411,200 bits/sec = 176,400 bytes/sec ~ 10.6 MB/minute
  • MP3 audio compresses by clever encoding and removal of sounds

that won't really be heard

  • data rate is ~ 1 MB/minute

ASCII: American Standard Code for Information Interchange

  • an arbitrary but agreed-upon representation for USA
  • widely used everywhere del

A review of how decimal numbers work

  • how many digits?
    • we use 10 digits for counting: "decimal" numbers are natural for us
    • other schemes show up in some areas clocks use 12, 24, 60; calendars use 7, 12 other cultures use other schemes (quatre-vingts)
  • what if we want to count to more than 10?
    • 0 1 2 3 4 5 6 7 8 9 1 decimal digit represents 1 choice from 10; counts 10 things; 10 distinct values
    • 00 01 02 … 10 11 12 … 20 21 22 … 98 99 2 decimal digits represents 1 choice from 100; 100 distinct values we usually elide zeros at the front
    • 000 001 … 099 100 101 … 998 999 3 decimal digits …
  • decimal numbers are shorthands for sums of powers of 10
    • 1492 = 1 x 1000 + 4 x 100 + 9 x 10 + 2 x 1
    • = 1 x 10^3 + 4 x 10^2 + 9 x 10^1 + 2 x 10^0
  • counting in "base 10", using powers of 10

Binary numbers: using bits to represent numbers

  • just like decimal except there are only two digits: 0 and 1
  • everything is based on powers of 2 (1, 2, 4, 8, 16, 32, …)
    • instead of powers of 10 (1, 10, 100, 1000, …)
  • counting in binary or base 2: 0 1 1 binary digit represents 1 choice from 2; counts 2 things; 2 distinct values 00 01 10 11 2 binary digits represents 1 choice from 4; 4 distinct values 000 001 010 011 100 101 110 111 3 binary digits …
  • binary numbers are shorthands for sums of powers of 2 11011 = 1 x 16 + 1 x 8 + 0 x 4 + 1 x 2 + 1 x 1 = 1 x 2^4 + 1 x 2^3 + 0 x 2^2 + 1 x 2^1 + 1 x 2^0
  • counting in "base 2", using powers of 2

Bytes

  • "byte" = group of 8 bits
    • on modern machines, the fundamental unit of processing and memory addressing
    • can encode any of 2 8 = 256 different values, e.g., numbers 0 .. 255 or a single letter like A or digit like 7 or punctuation like $ ASCII character set defines values for letters, digits, punctuation, etc.
  • group 2 bytes together to hold larger entities
    • two bytes (16 bits) holds 2 16 = 65536 values
    • a bigger integer, a character in a larger character set Unicode character set defines values for almost all characters anywhere
  • group 4 bytes together to hold even larger entities
    • four bytes (32 bits) holds 2 32 = 4,294,967,296 values
    • an even bigger integer, a number with a fractional part (floating point), a memory address
  • etc.
    • recent machines use 64-bit integers and addresses (8 bytes) 2 64 = 18,446,744,073,709,551,

Interpretation of bits depends on context

  • meaning of a group of bits depends on how they are interpreted
  • 1 byte could be
    • 1 bit in use, 7 wasted bits (e.g., M/F in a database)
    • 8 bits storing a number between 0 and 255
    • an alphabetic character like W or + or 7
    • part of a character in another alphabet or writing system (2 bytes)
    • part of a larger number (2 or 4 or 8 bytes, usually)
    • part of a picture or sound
    • part of an instruction for a computer to execute instructions are just bits, stored in the same memory as data different kinds of computers use different bit patterns for their instructions laptop, cellphone, game machine, etc., all potentially different
    • part of the location or address of something in memory
    • ...
  • one program's instructions are another program's data
    • when you download a new program from the net, it's data
    • when you run it, it's instructions

Converting binary to decimal

from right to left:

if bit is 1 add corresponding power of 2

i.e. 2

0

1

2

3

(rightmost power is zero)

! 1101 = 1 x 2

0

+ 0 x 2

1

+ 1 x 2

2

+ 1 x 2

3

= 1 x 1 + 0 x 2 + 1 x 4 + 1 x 8!

Converting decimal to binary

repeat while the number is > 0:

divide the number by 2

write the remainder (0 or 1)

use the quotient as the number and repeat

the answer is the resulting sequence

in reverse (right to left) order

divide 13 by 2, write "1", number is 6

divide 6 by 2, write "0", number is 3

divide 3 by 2, write "1", number is 1

divide 1 by 2, write "1", number is 0

answer is 1101

ASCII again

Color

  • TV & computer screens use Red-Green-Blue (RGB) model
  • each color is a combination of red, green, blue components
    • R+G = yellow, R+B = magenta, B+G = cyan, R+G+B = white
  • for computers, color of a pixel is usually specified by three

numbers giving amount of each color, on a scale of 0 to 255

  • this is often expressed in hexadecimal so the three components

can be specified separately (in effect, as bit patterns)

  • 000000 is black, FFFFFF is white
  • printers, etc., use cyan-magenta-yellow[-black] (CMY[K])