Graphics, Sound, Two's Complement & Data Compression

Key terms

You will need to know some key terms before we get started:

1. Most Significant bit (MSB) - this is the bit in the binary number which is on the leftmost side of the bnary number and it carries the highest denary weight

2. Least Significant bit(LSB) - this is the bit on the rightmost side. It carries the lowest denary weight of one

3. Bit 0 is also considered as the LSB

Conversion

You will need to know how to convert from one number system to another:

1. Binary to Denary (vice versa)

2. Denary to Hex (vice versa)

3. Binary to Hex (vice versa)

We will talk about how each on is done briefly and I will also put a video for you to watch at the end of the chapter

Binary to Denary

Let's take a simple binary number

00110011

The binary value has a place value of 2^x

The LSB has a weight of 1 and the 2nd bit has a weight of 2 and so on

So whenever there is a 1 you take that particular place value weight of that digit

Do the same for all other digits which are 1

Then take the sum of all the weights and it will give you the denary value

0  0  1  1  0  0  1  1

-  - 32  16 -  -  2  1

     32+16+2+1 = 51

There is also another method you can follow

00110011

You start with the MSB

First mutliply the MSB with 2 then get the value

After you find the value you add either 1 or 0 depending on the next bit

Then the new value is multiplied by 2 again and the whole process continues

When you receive the last digit(LSB) you just add the bit to the value and this gives you the denary value

0 * 2 = 0
(add 0)
0 * 2 = 0
(add 1)
1 * 2 = 2
(add 1)
3 * 2 = 6
(add 0)
6 * 2 = 12
(add 0)
12* 2 = 24
(add 1)
25* 2 = 50
(Finally just add 1)
50 +1 = 51

This is my opinion a good method to follow

Denary to binary

There are also two methods for this

We will take the denary value 254 as the example

So we keep dividing the denary value by 2. We will sometimes get a remainder of 1 or 0. Then you read the remainders from the bottom to the top which gives us the binary number

2|254
2|127 → 0
2|63  → 1
2|31  → 1
2|15  → 1
2|7   → 1
2|3   → 1
2|1   → 1
0     → 1


Binary number = 111111102

We will also talk about the next method also

These requires you to know the binary weight properly such as 1 , 2 , 4 , 8 , 16 , 32....

This is very useful to convert a decimal number to a binary number (This is an A2 topic so we will discuss that later)

254 - 128 → 1
126 - 64  → 1
62 -  32  → 1
30 -  16  → 1
14 -  8   → 1
6  -  4   → 1
2  -  2   → 1
0  -  1   → 0 // as 0 is not large enough to reduct 1
                      from it

It is better to chose one method or else you would get confused

Binary to Hex

This is the easiest conversion

We group the binary numbers to nibbles starting from the right and we find the denary value of each nibble and this is converted to Hex ( very similar to BCD )

0011011

As we can see, this has 7 bits so how can we group in 4 bits

(0)001   1011

You need to add a 0 to the start to make it 4bits

 
     0001    1011

       1      11

       1      B
           
         = 1B16

To convert from Hex to binary follow the reverse order

   1   B
 
   1   11

 0001  1011

Hex to denary (Vice versa)

There is a method to convert hex to denary and back directly but I won't recommend it to you

Its better to convert denary to binary then Hex ( vice versa ) as this is more simpler and less error prone

Prefixes

Same as physics, we don't like to give storage or other specifications in large numbers, we usually like to give it in 1 d.p

Here are the Decimal/denary prefixes

Prefix	Magnitude	Symbol
kilo	103	KB
Mega	106	MB
Giga	109	GB
Tera	1012	TB

This means that the size of data is grouped in quantities and the prefix tells us the size of data

However, recent questions don't ask this now but, lets see one example

Convert 200000 Bits to MB

First convert the bits to bytes

200000/8 = 25000Bytes

Then to convert to MB just divide by 10⁶

25000/1000000 = 0.025MB

But now we use binary prefixes because it is the correct way of representing quantities

Prefix	Magnitude	Symbol
Kibi	210	KiB
Mebi	220	MiB
Gibi	230	GiB
Tebi	240	TiB

This is a bit complex and requires you to do a lot of workings

Lets take the same example

Convert 200000 Bits to MiB

Note now it MiB

200000/8 = 25000Bytes

25000/1024 = 24.4KiB

24.4/1024 = 0.024MiB

So really they will usually specify which one they want. If they say KB use denary prefixes. If they use KiB use binary prefixes

Also if they say Mib instead of MiB it means they want Mebibits not Mebibytes and so you don't have to divide by 8

Internal Coding Numbers

The computer follows two's complement when storing numbers

There are two types of integer which the computer can store - unsigned and signed integer

Signed integers are integers which can be either positive or negative whereas unsigned integer is always positive

Sign and Magnitude

This is the common form when we display signed integers

The Most significant bit determines the sign ( 1 - negative and 0 - positive ) where as the rest of the bits define the magnitude

For example 10110110

The Most significant bit is 1 and so this is a negative number

If we convert the rest of the bits to denary 54 which is the magnitude

So this byte represents -54 only if this byte is a signed integer

If it is not a signed integer we just use normal conversions like the above examples

So most questions give the byte in a sign and magnitude form ( unless they state the form ) and this must be converted to two's complement

They might ask like this

Convert this signed integer to 2's complement -

                 10011110

Before you know how to convert it we need to first define 2's complement

Two's Complement

This is one's complement plus 1

What is one's complement?

One's complement is when the bits in a byte are substracted from 1 ( so making a 1 to 0 and 0 to 1

An example is 00100100 is converted to 11011011 - so every bit is changed

To find two's complement we just need to add 1 to the 11011011

don't use this method as it is quite confusing

To convert a binary number to two's complement directly, follow this steps

Lets take an example

            10011100

Ignore the first 0's on the right side

And also ignore the first 1 on the right

Then invert all other bits - convert 1 to 0 and 0 to 1

So we will get:

 01100100

This is a much easier method and usually used to change negative sign and magnitude binary numbers

Here's the catch!

The computer follows 2's complement and so sign and magnitude numbers must be converted using the above method

However you use this method only when the sign and magnitude is negative, infact this is a very confusing part

If the binary number is positive then the sign and magnitude is the same as the two's complement form

Lets see an example:

Convert 01101111 to 2's complement

The answer is again 01101111 and we will see why later

Convert 10010010 to 2's complement

So for this we need to use the above method and so you will get 01101110

However this is not the answer because the first bit(MSB) is changed from 1 to 0 and so it will become a positive number

For this reason the Two's complement should have the same sign as the sign and magnitude form

The final answer must be 11101110

Why do they follow this

There is a reason why they follow this wierd method and it's because it makes converting from Two's complement to denary much quicker and direct

To convert a 2's complement to the corresponding denary value can be done by reversing the process such as finding the sign and magitude form and then finding the denary value

Or

There is another simpler method

The below binary value is in 2's complement. Find the denary value

               11011010

The most significant bit is 1 so the number is negative

We know that the MSB carries the highest value of 128

The method follows like this:

First find the value of the sign which is negative and then add the positive value of the magnitude(remaining bits)

 -128 + (64 + 16 + 8 +2 ) = -38

As the MSB is 1 then the value would obviously be negative

This is the easiest and fastest method

So this makes sense why the sign and magitude of positive number is the same as the two's complement

The below binary value is in 2's complement. Find the denary value

               01011010

By following the same method above we will get 38

I don't know if textbooks follow this explanation but in my opinion this is the most simplest explanation I could give

Properties

There are some properties of 2's complement and sign and magnitude you need to know

Two's complement has only one representation for 0 where as sign and magnitude have two forms - one for -0 and one for 0. This is actually an advantage of two's complement

The maximum value which can be represented for signed integers using 4 bits is 2ⁿ-1 = 7

The largest negative number formed is -2ⁿ = -8

Binary aritmethics

Lets see an example

      01011101
      01111011+
      ________
      11011000

To do this just follow the normal addition

Binary addition begins from the rightmost side

If 1 + 1 then it gives 0 and 1 is carried

If 1 + 0 then it gives 1 and no carry

If 1 + 1 + 1 then it give 1 and a carry of 1

If 1 + 1 + 1 + 1 then it gives 0 and a carry of 2

There is a pattern if you can see if the summed value is a multiple of 2 then we get 0 if not we get 1

The carry is the number of 2's which can fit the summed value

don't worry I will include a video for all these calculations below

There is also another longer method and that includes converting the binary number to denary and doing the calculation

Lets see an example now to substract binary numbers

      01111101
      01111011-
      ________
      00000010

Binary substraction begins from the right side to the left side

If 1 - 0 then it would be 1

If 1 - 1 then it would be 0

If 0 - 1 then it gives 1 and borrows one from the next digit

Why does it becomes 1? Because when we borrow 1 we get 2

Overflow errors

It is when the result of a calculation is larger than the number of bits used to define the storage for the result

So sometimes when you add 2 binary number the result may have more number of bits

      1111
      1111+
      _____
    (1)1110

Sometimes these overflow can cause the result to become negative when we add two positive numbers

      0111
      0111+
      _____
      1110

Or when two negative numbers are added it can produce a positive number

      10100000
      10100000+
      ________
    (1)01000000

The (1) will not be recorded as it won't fit in the storage, so then computer will read this as a positive number which is wrong

Both of these examples are overflow errors and the processor must be able to identify these - we will see them more in Assembly language chapter

BCD - Binary coded decimal

Denary digits are represented using nibbles or 4 bits

4 bits can be used to represent a single denary digit from 0 to 9

If the nibble contains a value of more than 9 then it is called an invalid BCD

There are 2 types of BCD - packed BCD or one BCD per byte

In packed BCD it means that 2 digits of denary can be represented using 1 byte, whereas in one BCD per byte only one digit is represented(the remaining bits are 0's)

Example, Packed BCD - 11011001 and One BCD per byte - 11010000

So how do you convert it to denary digits

For example 00100111 is a packed BCD

We can break it to two nibbles

 0010     1001

Each nibble could be converted to the corresponding denary value

0010	1001
2	9

This gives a value of 29

If the nibble had a value of more than 9 then it is invalid

BCD is used to display denary digits in calculators and watches

You also need to know how to perform calculations and addition with BCD

When we do addition with BCD we isolate each nibble such as the example below

29 + 45

   0010   1001     (29)
   0100   0101     (45)
   ____   ____
   0110   1110
          0110
   0110 (1)0100 
   
   0111   0100     (74)
   

So to correct this we need to always add 0110 or 6

We can see after adding 0110 we get a carry of 1 which taken to the next BCD and added

If the next BCD also was invalid then we had to add 0110 and 1 = 0111 and there will be a carry to the next BCD

Internal coding of character

There are two ways which text and characters can be represented

• ASCII

American standard code for information interchange

Originally ASCII uses 7 bits to store each character(this is known as the ANSI ASCII)

The bits required to store each ASCII character are known as character codes

This could be used to represent 128 different characters which is enough to represent the english set and numbers and other command keys

The new ASCII is called the Extended ASCII as it is used to represent modified alphabets such as the ISO - Latin characters

This uses 8 bits or 1 byte to store each character and this could represent 256 characters

This is still not enough to represent other characters from different languages



• Unicode

The standard uses 2 bytes to store each character allowing 65536 possible characters

However most of the time its usually less as some bits are required to describe the encoding used

UTF - 8 and UTF - 16 bits means that in UTF - 8 the system handles each 8 bits at a time.

Unicode use the term code point instead of character code

Unicode is able to represent all the characters in the world and many other language so it used more commonly than ASCII

Unicode code point are usually represented using Hexadecimals - for example U - FFFF

Notes in ASCII

All you need to know is ASCII is enough to represent the english character set and numbers and some commands

Characters which are in sequence are changed by the ASCII value of 1 - Ex A and B are only different by one value

The difference between the uppercase and lowercase ASCII characters is in bit 5 - Ex (A=65) where as (a=97) so difference in 32

Representation of graphics

There are 2 types of graphics:

• Vector Graphics

Graphics made out of drawing objects and their associated properties

So these drawing objects are shapes

A drawing object is component which is defined using geometric formulae

This follows a very simple idea, the image is made up of shapes or components which are defined using a drawing list

A drawing list is a list which contains one set values that contains many attributes for each drawing objects

So the drawing list contains a list of attributes which defines a property of the drawing object

A property is an aspect of the appearance of the drawing object

An example of the property includes the radius or the color or the border

Vector graphics are stored in the SVG format and requires a graphic plotter to output them directly

Vector graphics are relative to the canvas and so it is scalable without geting distorted and pixelated

Also when image is scaled both the dimensions are scaled so the image doesn't get squashed or stretched

Another important point is that vector graphics uses a lot of processing power however less storage compared to BMP graphics and it is also used to represent simple images such as shapes

If an vector graphics must be displayed on a screen or printed then it must be be converted to bitmap form



• Bitmap Graphics

An image which is made out of elements or components called pixels which stores a color and its position in the BMP matrix

The pixels are displayed in a 2D matrix where the color of each pixel and its position in the 2D matrix is stored

Pixel is the smallest component which is identified using its color and position in the 2D matrix

So each pixel can represent a color and the number of bits required to store each pixel is known as color depth

Color depth is the number of bits required to store each pixel

The color depth of the pixel determines how many possible colors the pixel can represent

For example 24 bit color depth means that 2²⁴ colors can be represented

Some questions use the word bit depth for images and this means the number of bits required to store each primary color

For example if you have a bit depth of 8 bits it means it requires 8 bits to store each primary color of RGB and so the color depth will be 24 bits

Bit depth is the number of its required to store each primary color




So back to BMP images it is made of many tiny pixels in a 2D matrix. To find the number of pixels in a image we need to find the product of the pixel high and pixel wide of an image, this is known as the image resolution

Image resolution is the product of the number of pixels high and number of pixels wide an image is

In a simpler terms, it means how many pixels are there in an image

The same concept could be applied for display screen

The screen resolution is the product of the number of pixels high and number of pixels wide the screen can display

So whatever we see through the screen, it is limited by the screen resolutuion - for example if the screen is 2K but the image is 4K we will only see a 2K image




There are some points on Bitmap graphics - it uses the file extension .BMP(raw file) where as the compressed versions are .PNG and .JPG

Also the bitmap images can be enlargened but, when we enlarge the image even the pixels are enlarged and so the image seems pixelated and distorted

Bitmap images are good for storing photographs and pictures but uses way alot of storage




Image file size

They will ask us to calculate the file size of an image

We find the number of pixels(using the image resolution) and multiply it by the color depth

File size = Resolution * Color depth

Even bit depth could be used but we need to then multiply by 3

File size = Resolution * Bit depth * 3

Depending on what you took for the color depth ( if we took it in bits ) then the file size will be in bits

In my opinion it's better to use bytes as you may have to convert it to MiB

Let's take an example

What is the image size if the pixel height is 500 and the pixels wide 1000 and the color depth is 24 bits

File size = Resolution * Bit depth * 3

Resolution = Pixels high * pixels wide

File size = 500 * 1000 * 3