IFSM 310

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DATA REPRESENTATIONS
IFSM 310
Hardware Software Concepts
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CONTENTS

• Binary, Hex, and Decimal
• N2k
• Excess-8 and Excess-50 notation
• SEEMMMM format
• Example Audio

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Binary, Hex, and Decimal

• Introduction
• At the most basic level, the computer is storing
  (representing) numbers, music, graphics, letters
  etc in electrical states. These states can be
  OFF-ON POSITIVE-NEGATIVE ZERO-ONE etc.

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Binary, Hex, and Decimal
bit
bit
bit
bit
bit
bit
bit
bit

• - - -
• 1 1 0 1 1 0 1 0

5
Binary, Hex, and Decimal
bit
bit
bit
bit
bit
bit
bit
bit

• The states can be created using capacitors
• Each of the capacitors is called a bit.
• 8 Bits is a byte.
• So, each state can be either a 1 or 0. This
  is called a binary state hence computer science
  studies how computers can be used to represent
  the real world in binary ways.
• So, it makes sense to study binary number systems
  and other number systems that relate to binary
  systems like base 8 and base 16.

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Binary, Hex, and Decimal

• Number Systems
• Recall that a number like 34,563 (base 10) can be
  expressed as
• 3x10,000 4x1,000 5x100 6x10 3x1.
• The values 10,000 1,000 100 10 and 1 come from
  the powers of 10.
• Hence the name Base 10 or decimal.

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Binary, Hex, and Decimal

• Consider the Base 2 numbers. They are created the
  same way base 10 numbers are, using the position
  and the power of 2
• The base 2 number 110101
• 1x32 1x16 0x8 1x4 0x2 1x1 53

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Binary, Hex, and Decimal

• Ex 1010011 64162183
• Ex 11111111 128 64 32168421255
• Note
• this is the largest value you can make with 8
  bits
• There are 256 unique representations, because we
  start with 00000000

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Binary, Hex, and Decimal

• You try a few
• Change 1001 to decimal
• Change 1110110 to decimal

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Binary, Hex, and Decimal

• Solution
• 1001 1x8 0x4 0x2 1x1 9 (dec)
• 1110110 1x64 1x32 1x16 0x8 1x4 1x2 0x1
  
• 64 321642
• 96 22 118 (dec)

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Binary, Hex, and Decimal

• How about changing Decimal to Binary?
• First note the powers of 2
• 1, 2, 4, 8,16, 32, 64,128, 256, 512, 1024
• There are a lot of methods to do this, this is
  the one other students have found to be the
  easiest
• Keep subtracting the largest power of 2 until
  there is no remainder, mark the appropriate bit
  with a 1 for the power of two.

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Binary, Hex, and Decimal

• Example Change 23 to binary
• 23-16 7 mark the 16 bit
• 7-4 3 mark the 4 bit
• 3-2 1 mark the 2 bit
• Mark the 1 bit
• Solution 10111

Bit 16 8 4 2 1
mark 1 0 1 1 1
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Binary, Hex, and Decimal

• Example Change 146 to binary
• 146-128 18 mark the 128 bit
• 18-16 2 mark the 16 bit
• 2-2 0 mark the 2 bit
• Solution 1001 0010

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Binary, Hex, and Decimal

• It is easier to see it in the table

Bit 128 64 32 16 8 4 2 1
mark 1 0 0 1 0 0 1 0
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Binary, Hex, and Decimal

• You try a few
• Change 77 to binary
• Change 528 to binary

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Binary, Hex, and Decimal

• Solutions
• Change 77 to binary 100 1101
• Change 528 to binary 10 0001 0000

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Binary, Hex, and Decimal

• HEXADECIMAL or Base 16
• Some interactions with the computer use base 16
  to communicate. We use
• A10
• B11
• C12
• D13
• E14
• F15

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Binary, Hex, and Decimal

• We still use the basic POWERS OF 16
• 1601 16116
• 162256 1634096
• EXAMPLE FA4 Fx256 Ax16 4x1
• Or..15x256 10x16 4x1
• 3840 160 4 4004
• (remember F15, A10)

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Binary, Hex, and Decimal

• What is nice about Hex, is that we can convert to
  binary very easily, by using in groups of 4
• FA4 15 10 4
• 1111 1010 0100
• Because ..
• F 15 (dec) 1111 (bi)
• A 10(dec) 1010 (bi)
• 4 4 (dec) 0100

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Binary, Hex, and Decimal

• And Visa Versa
• Convert 1 0111 1101 to Hex
• The first group 1 is just 1 (hex)
• The second group 0111 7 (hex)
• The third group 1101 D (hex)
• So, the answer is 17D
• Note 0001(bi) is the same as 1(bi)

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Binary, Hex, and Decimal

• You try a few
• A) Convert A6 (hex) to decimal
• B) Convert FD (hex) to binary then to decimal
• C) Convert 1101 1110 1010 to Hex
• D) Convert 12ED to decimal and binary

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Binary, Hex, and Decimal

• Solutions
• A) Convert A6 (hex) to decimal
• 166 dec
• B) Convert FD (hex) to binary then to decimal
• 1111 1101 and 253
• C) Convert 1101 1110 1010 to Hex
• DEA
• D) Convert 12ED to decimal and binary
• 4845 and 1 0010 1110 1101

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N2k

• Two basic questions in computer science are
• If I want to represent N things, how many bits do
  I need?
• And
• If I have K bits, how many things can I
  represent?
• The relationship is
• N2k

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N2k

• Lets start with
• If I have K bits, how many things can I
  represent?

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N2k

• Look for the pattern

Bits of things How?
1 2 0,1
2 4 00, 01, 10, 11
3 8 000, 001, 010, 011, 100, 101, 110, 111
4 16 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111
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N2k

• Look for the pattern

Bits of things The relationship is.
1 2 Note 2 21
2 4 Note 4 22
3 8 Note 8 23
4 16 Note 16 24
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N2k

• Generally speaking,

Bits of things The relationship is.
k n n2k
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N2k

• The second question is
• If I want to represent N things, how many bits
  do I need?

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N2k

• Suppose I want to make a new Video Card that has
  14 colors (for the color impaired)
• Then, notice
• 2 bits gives me 4 unique colors TOO SMALL
• 3 bits gives 8 unique colors TOO SMALL
• 4 bits gives 16 unique colors. TOO BIG???
• TOO BIG is ok, there are 2 extra colors. I cant
  avoid this because the number of bits is an
  INTEGER there is no ½ bit

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N2k

• Suppose I want to make a new Video Card that has
  1,400 colors.
• Solution

Bits of things notes
5 2532 way too small, try a larger number
10 2101024 still too small
11 2112048 Got it!
12 2124096 Truly TOO BIG, because 211 works!
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N2k

• Observations
• 1) The ANSWER IS 11 bits.
• 2) 12 bits will work, but since 11 works too, it
  is THE answer.
• 3) 2,048 gt 1,400 so there are a lot of extra
  values that are not used.
• Most designers use POWERS OF 2

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N2k

• Student Questions
• A telephone stores 100 phone numbers, how many
  bits does it use to access each address location?
• My main memory has 64MB address locations, how
  many bits do I need to access each address
  location? M1,048,576 220
• A new TV designer wants to store 512 preset
  stations how many bits are needed to access each
  preset station?

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Excess-50 notation

• Background
• Computers work with numbers of all types Natural
  numbers, Integers, rational and irrational and
  complex numbers.
• What is easy for humans is rather complicated for
  complicated for computers.
• There a many number systems that can represent
  numbers. In this class we will only investigate
  three direct binary representation for natural
  numbers, excess notation for integers, and
  SEEMMMM format for rational numbers.

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Excess-50 notation

• Easier Softer way Excess50.
• There will be NO BINARY representations in
  excess50.
• 55 (excess50) 5 dec
• 52 (excess50) 2 dec
• 45 (excess50) -5 dec
• 99 (excess50) 49 dec
• 39 (excess50) -11 dec

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Excess-50 notation

• 3 (dec) 53 (excess50)
• -3 (dec) 47 (excess50)
• -6 (dec) 44 (excess50)
• 55 (dec) out of range
• -129 (dec) out of range

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SEEMMMM format

• SEEMMMM format
• Background This is the easiest format I have
  seen to represent floating point (rational)
  numbers
• This format is typical of real formats used in
  representing floating point numbers.
• IT ILLUSTRATES the point without getting into too
  much detail!

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SEEMMMM format

• Pick a number, any number.
• 45.7909
• I can always place a number into this format
• .457909 x E2The 457909 is the mantissa, 2 is
  the exponent the sign is positive

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SEEMMMM format

• So, all floating point numbers have
• SIGN
• MANTISSA
• EXPONENT

This is all I need to know about a number to
represent it (or recreate it!)
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SEEMMMM format

• SEEMMMM has the following
• S sign, 5 for negative 0 for Positive
• EE exponent, excess50 notation
• MMMM mantissa regular decimal digits
• If my mantissa has more than 4 digits, well
  round to the nearest value

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SEEMMMM format

• EXAMPLE
• How would 45.7909 look in SEEMMMM format?
• STEP 1 get into .ZMMM format with Z nonzero
  (note the exponent)
• 45.7909 .4579 (rounded) E2

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SEEMMMM format

• STEP 2 Convert the exponent into excess50
  notation
• 2 (dec) 52 (excess50)
• STEP 3 construct the SEEMMMM including the sign
• 0524579answer

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SEEMMMM format

• Example Convert .00078 to SEEMMMM format
• .00078 .7800 E-3
• -3 (dec) 47 (excess50)
• 0477800 .answer

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SEEMMMM format

• Example Convert 92,821,376 to SEEMMMM format
• 92,821,376 .9282 E8
• 8 (dec) 58 excess50
• 5589282 answer (note the sign!)

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SEEMMMM format

• Example Construct the original floating point
  from 5568376 (SEEMMMM)
• 5 ? negative value
• 56 ? exponent is 6
• - .8376 E6 - 837,600 .answer

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SEEMMMM format

• Example Construct the original floating point
  from 0477869
• 0 ? positive
• 47 ? exponent -3
• .7869 E-3 .0007869 .answer

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SEEMMMM format

• Student problems
• Convert 3.56008 to SEEMMMM format
• Convert .009876 to SEEMMMM format
• Convert 5509999 (SEEMMMM) to decimal
• Convert 0458001 (SEEMMMM) to decimal

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DATA REPRESENTAION
EXAMPLE Audio
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INTRODUCTION

• Real World -- analog/continuous data
• Vs
• Computer World -- digital/discrete data
• This is the problem!

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AUDIO DATA

• Audio data is an analog source
• Can be viewed as a waveform

The digital waveform approximates the analog wave
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AUDIO

• Changing the analog waveform to digital format is
  called digitizing. Another term is Pulse Code
  Modulation or PCM
• What information needs to be collected?
• method used to digitize the sound (like MP3)
• sampling rate in samples per sec (50KHz)
• Data transfer rate
• Number of bits per sample (4,8,16, 32,)
• Recorded in mono or stereo

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AUDIO

• Example
• CD quality, you sample at 44.1 kHz, stereo, with
  16 bits per sample (or 2bytes/sample).
• Question How much space will a one-minute
  CD-quality song require?
• Ans 44,100samples/sec x 2bytes/sample x 2
  (stereo) x 60 sec/min 10,584,000 bytes
• Convert to MB you divide by M1048576 Gives
  you 10.09MB

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AUDIO

• Example continued
• How long will this take to download from the
  internet via a ISDN connection at 64kbps?
• Ans 10,584,000 bytes x 8bits/byte / (64(1024)
  bits/sec) x 1min/60sec 21.5 minutes! Formula
  is SIZE/RATE TIME
• remember K1024

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AUDIO

• Based on the previous example, the need for
  compression or coding music is very important
• Lets look at MP3 audio format (MP3The
  compression standard for MPEG-1 Layer 3 audio)
• It is a lossy data compression standard (as
  opposed to lossless data compression)
• This means that DATA IS LOST DURING THE
  COMPRESSION PHASE

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AUDIO

• MP3 audio format
• They say MP3 Coding to mean the process of
  changing the digital music into a compressed
  format (encoding) and then converting the MP3
  file from a series of 0s and 1s into audible
  sound (decoding)

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AUDIO

• How it works
• The human auditory system 2kHz -20KHz
• More sensitive between 2.5-5KHz less sensitive
  over 16KHz
• audio signal a masking If another tone lies
  below this masking threshold, it will be masked
  by the louder tone and remains inaudible.

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AUDIO
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AUDIO

• How it works
• So, eliminate or use less encoding for
• Below 2.5 KHz and above 20KHz (or 16KHz)
• Tones hidden by other tones
• And Sample more between 2.5 and 5KHz

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Background Image

• World map of major Internet routers pathways