Fundamentals of Data Representation, Study notes of Computer Science

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Teach yourself the Fundamentals of Data Representation for AQA GCSE Computer Science (8520) Students Workbook By Nichola Lacey Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 1 By Nichola Lacey Contents Introduction ............................................................................................................................................ 3 How to use this book .......................................................................................................................... 3 Who should use this book? ................................................................................................................. 3 What are number bases? ........................................................................................................................ 4 Decimal (base 10) ................................................................................................................................ 4 Binary (base 2) .................................................................................................................................... 5 What can be represented using binary? ............................................................................................. 7 Hexadecimal (base 16) ........................................................................................................................ 8 Why is hexadecimal used? .................................................................................................................. 8 End of chapter recap ......................................................................................................................... 11 Converting between number bases ...................................................................................................... 12 Convert from binary to decimal ........................................................................................................ 12 Convert from decimal to binary ........................................................................................................ 15 Number base notation ...................................................................................................................... 16 Convert from binary to hexadecimal ................................................................................................ 17 Convert hexadecimal to binary ......................................................................................................... 18 Convert hexadecimal to decimal ...................................................................................................... 19 Convert from decimal to hexadecimal .............................................................................................. 20 End of chapter recap ......................................................................................................................... 21 Units of information .............................................................................................................................. 22 Bits .................................................................................................................................................... 22 Bytes .................................................................................................................................................. 22 Amount of storage space required ................................................................................................... 23 End of chapter recap ......................................................................................................................... 23 Binary Arithmetic .................................................................................................................................. 24 Binary Shifts ...................................................................................................................................... 27 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 4 By Nichola Lacey What are number bases? Objective: Understand the following number bases: decimal (base 10), binary (base 2) and hexadecimal (base 16). Decimal (base 10) Since you first learnt to recognise numbers you have been taught to use a base 10 number system, this is known as a decimal (or denary) number base. It has 10 different digits, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 There is no single digit for the number ten and we use two digits (a 1 and a 0) to represent the place value we know as 10. This stands for "1 ten and 0 ones". To represent any number above 9, we use different values each worth ten times more than the previous column (starting from the right). Take for example the number two thousand, nine hundred and thirty-five (2,935). This can be split into separate columns, each representing a different value. Thousands Hundreds Tens Ones 2 9 3 5 Each of the columns are worth ten times more than the one to the right. Using the example above the 3 is representing the value 30, the 9 is representing the value 900 and the 2 is representing the value 2000. Adding those values together we get the following: 2000 900 30 + 5 2935 A decimal number system uses 10 digits (0 – 9) to represent the value. Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 5 By Nichola Lacey Task 1: Split the following numbers into their correct columns and then write them in the final column as a decimal number. Explanation Thousands Hundreds Tens Ones Decimal number One thousand, four hundred and fifty-six Nineteen ninety-nine Six thousand and twenty Binary (base 2) Computers use electrical pulses and these can either be on or off. As there are only two states a binary number system is used by most computer systems. This on or off state is represented by a 1 for on and 0 for off. Whereas a base 10 number system uses 10 digits 0 – 9, a base 2 number system uses 2 digits 0 – 1. The base 10 number system has columns which were worth ten times the amount of the previous column and a base 2 number system has columns worth twice as much as the previous column. Eight Four Two One 1 0 1 1 Using the example above, the first 1 represents 8, there is nothing in the four-place column, 1 in the two column and 1 in the one column. If we add those column values together we get 11. 8 2 + 1 11 Therefore, the binary number 1011 is the same value as 11 in decimal. A binary number system uses 2 digits (0 and 1) to represent the value. Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 6 By Nichola Lacey Task 2: Place the following numbers into their correct columns and then write them as a binary number by combining them together and finally work out the decimal total by adding together the column headings which contain a 1, for example in the first row the decimal value will be 11 (8 + 2 + 1) to therefore the binary value 1011 is the same as 11 in decimal. Explanation Eight Four Two One Binary number Decimal value 1 x 8, 0 x 4, 1 x 2 and 1 x 1 1 x 8, 1 x 4, 0 x 2 and 0 x 1 1 x 2 and 1 x 1 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 9 By Nichola Lacey For instance, symbols in word processors often show their hexadecimal number and you can see this in some applications. They are also found when referring to colours. Colours are usually made up from three numbers known as the RGB code (which stands for red, green, blue) and these are displayed as hexadecimal numbers. As we have seen, binary is used by computers and one hexadecimal digit can be used to represent 4 binary digits so makes converting between binary and hexadecimal neat and easier than using, say, a base 20 number system. Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 10 By Nichola Lacey Here is a table showing the decimal values from 0 to 15 in both binary and hexadecimal. Decimal Binary Hexadecimal 0 0000 0 1 0001 1 2 0010 2 3 0011 3 4 0100 4 5 0101 5 6 0110 6 7 0111 7 8 1000 8 9 1001 9 10 1010 A 11 1011 B 12 1100 C 13 1101 D 14 1110 E 15 1111 F Here is the same number represented in the three number bases: Decimal 2,890 Binary 1011 0100 1010 Hexadecimal B4A In the next chapter, you will be learning how to convert between these different number bases for yourself but for now you only need to understand the differences between the three number bases. Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 11 By Nichola Lacey End of chapter recap Task 3: Using the table on the previous page, convert these binary numbers into hexadecimal to find the hidden words. Binary number Hidden Word 1011 1110 1110 1111 1011 1110 1101 1101 1110 1010 1111 1010 1101 1101 1010 1100 1110 1011 1110 1010 1101 1111 1110 1110 1101 1011 1010 1101 1100 1010 1011 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 14 By Nichola Lacey Task 5: Convert these binary numbers into decimal. Binary number Decimal value 101 1101 10111 110010 1111100 1001001 10101110 11011001 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 15 By Nichola Lacey Convert from decimal to binary To convert from decimal to binary you need to perform a little more maths. Lets familiarise ourselves with the column headings again. 128 64 32 16 8 4 2 1 Step 1: Decide on the column to start with. This should be lower than or equal to the value you are looking for so if we wanted to convert 50 to binary we would start with the column 32. Enter a 1 in that column. Step 2: Find out the remainder (50 – 32 = 18) Step 3: Repeat steps 1 and 2 until there is no more remainder (in this case we would also put a 1 in the 16 and the 2 columns. Step 4: Fill in the other columns with 0’s. Please note: you do not need to add 0s before your first 1 as these are unnecessary. Using the example of 50 our binary number would be 110010 (32 + 16 + 2). Task 6: Convert decimal values into binary numbers. Decimal value 128 64 32 16 8 4 2 1 Binary number 27 33 52 63 85 207 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 16 By Nichola Lacey Number base notation As you may have worked out, the number “10” could mean ten in decimal or it could be two in binary. To make it clear as to which number base is being used it is common to include the number base as a subscript after the number. For instance, 1002 would be a binary number (in this case it is equivalent to four in decimal) as opposed to 10010 which is one hundred in decimal. Task 7: Convert the following numbers. If the number is currently a binary number convert it to a decimal number and if it is currently a decimal number, convert it to a binary number. Original number Converted number 11102 1010 10110 110011102 11010 110000112 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 19 By Nichola Lacey Convert hexadecimal to decimal There are several ways to do this, all of which involve complicated maths involving your 16 times table. Not many of us know our 16 times table (unless you are that delightful woman on Countdown of course) so the best advice I can give is to follow these much simpler steps, which, although may be more long winded, are less likely to make your brain hurt and make a mistake in your calculations: Step 1: Convert the hexadecimal number to binary (see page 18) Step 2: Convert the binary number to decimal (see page 12) Make sure you are using the whole binary number and working out the column headings as shown below, rather than working with the blocks of 4 individually. Task 9: Convert the following hexadecimal numbers into decimal by first converting it to binary and then converting the whole binary number to decimal. Hexadecimal Binary Decimal 1A16 B816 F916 4E16 BC16 5216 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 20 By Nichola Lacey Convert from decimal to hexadecimal To reverse the process and convert from decimal to hexadecimal you need to do the following: Step 1: Convert the decimal number to binary (see page 15) Step 2: Convert from binary to hexadecimal (see page 17). Task 10: Convert the following decimal numbers into hexadecimal by first converting it to binary and then converting the binary number to hexadecimal. Decimal Binary Hexadecimal 1210 4910 7810 9510 11610 25510 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 21 By Nichola Lacey End of chapter recap Task 11: Fill in the blanks on this conversion table. Decimal Binary Hexadecimal 510 100010012 5B16 101011012 2710 E716 111000112 16310 110100012 6610 7C16 10110002 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 24 By Nichola Lacey Binary Arithmetic Objectives: To be able to add up to three binary numbers. Before we look at adding with binary let’s have a quick recap on the first few digits with binary using the first three digits Binary Decimal equivalent 000 0 001 1 010 2 011 3 100 4 101 5 110 6 111 7 Now let’s have a look at some very simple binary maths Binary arithmetic Binary Answer Decimal equivalent 0 + 0 0 0 + 0 = 0 0 + 1 1 0 + 1 = 1 1 + 0 1 1 + 0 = 1 1 + 1 10 1 + 1 = 2 1 + 1 + 1 11 1 + 1 + 1 = 3 1 + 1 + 1 + 1 100 1 + 1 + 1 + 1 = 4 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 25 By Nichola Lacey When you are adding two binary numbers together, for instance 1010 and 111, lay them out as follows: 1 0 1 0 + 1 1 1 Description Example Take the first column on the right and add together the individual 1’s. In this case 0 + 1 = 1 Next take the next column and add together the 1’s. In this case 1 + 1 = 10. Don’t forget we are adding in binary so the answer should also be in binary. Put the 0 in the same column and carry the 1. This is usually shown below the bottom line. In the next column don’t forget to include the 1 that has been carried forward. In this example the calculation is 0 + 1 + 1 which is 10. Again, put the 0 in the column and carry the 1. In the final column add together the digits, including the carried digit. In this case 1 + 1 = 10. As there are no other columns instead of putting the 1 that is being carried below the line, move it up to the main answer row. The answer to the sum 1010 + 111 = 10001. You can check this is correct by converting it to decimal. 1010 (ten) + 111 (seven) = 10001 (seventeen). Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 26 By Nichola Lacey Task 13: Complete the following binary addition 1 1 0 0 + 0 1 1 0 1 0 0 1 + 1 1 1 1 1 0 1 1 1 0 + 1 1 0 1 1 1 0 0 0 + 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 + 1 1 1 0 0 1 1 0 0 1 1 0 + 0 1 1 1 11110000 + 10011 = 1101011 + 10111 + 1101001 = 1111 + 111111 + 11111111 = Use this space to convert the binary numbers into decimal to check if your answers are correct. Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 29 By Nichola Lacey Character encoding Objective: Understand what a character set is and be able to describe the following character encoding methods; 7-bit ASCII, Unicode. 7-bit ASCII Look at the following table, known as a character set: Binary Dec Hex Glyph Binary Dec Hex Glyph Binary Dec Hex Glyph 010 0000 32 20 ? 100 0000 64 40 @ 110 0000 96 60 ` 010 0001 33 21 ! 100 0001 65 41 A 110 0001 97 61 a 010 0010 34 22 " 100 0010 66 42 B 110 0010 98 62 b 010 0011 35 23 # 100 0011 67 43 C 110 0011 99 63 c 010 0100 36 24 $ 100 0100 68 44 D 110 0100 100 64 d 010 0101 37 25 % 100 0101 69 45 E 110 0101 101 65 e 010 0110 38 26 & 100 0110 70 46 F 110 0110 102 66 f 010 0111 39 27 ' 100 0111 71 47 G 110 0111 103 67 g 010 1000 40 28 ( 100 1000 72 48 H 110 1000 104 68 h 010 1001 41 29 ) 100 1001 73 49 I 110 1001 105 69 i 010 1010 42 2A * 100 1010 74 4A J 110 1010 106 6A j 010 1011 43 2B + 100 1011 75 4B K 110 1011 107 6B k 010 1100 44 2C , 100 1100 76 4C L 110 1100 108 6C l 010 1101 45 2D - 100 1101 77 4D M 110 1101 109 6D m 010 1110 46 2E . 100 1110 78 4E N 110 1110 110 6E n 010 1111 47 2F / 100 1111 79 4F O 110 1111 111 6F o 011 0000 48 30 0 101 0000 80 50 P 111 0000 112 70 p 011 0001 49 31 1 101 0001 81 51 Q 111 0001 113 71 q 011 0010 50 32 2 101 0010 82 52 R 111 0010 114 72 r 011 0011 51 33 3 101 0011 83 53 S 111 0011 115 73 s 011 0100 52 34 4 101 0100 84 54 T 111 0100 116 74 t 011 0101 53 35 5 101 0101 85 55 U 111 0101 117 75 u 011 0110 54 36 6 101 0110 86 56 V 111 0110 118 76 v 011 0111 55 37 7 101 0111 87 57 W 111 0111 119 77 w 011 1000 56 38 8 101 1000 88 58 X 111 1000 120 78 x 011 1001 57 39 9 101 1001 89 59 Y 111 1001 121 79 y 011 1010 58 3A : 101 1010 90 5A Z 111 1010 122 7A z 011 1011 59 3B ; 101 1011 91 5B [ 111 1011 123 7B { 011 1100 60 3C < 101 1100 92 5C \ 111 1100 124 7C | 011 1101 61 3D = 101 1101 93 5D ] 111 1101 125 7D } 011 1110 62 3E > 101 1110 94 5E ^ 111 1110 126 7E ~ 011 1111 63 3F ? 101 1111 95 5F _ This is known as the 7-bit ASCII table as each character is represented by 7 binary bits. They are displayed in sequences for instance you can see, a = 97, b = 98, c = 99. This means that if we are told what value a character is we can easily work out the value of subsequent or prior characters. Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 30 By Nichola Lacey There are some additional characters known as invisible characters: Binary Dec Hex Abbr. Description 000 0000 0 00 [NUL] Null character 000 0001 1 01 [SOH] Start of Header 000 0010 2 02 [STX] Start of Text 000 0011 3 03 [ETX] End of Text 000 0100 4 04 [EOT] End of Transcript 000 0101 5 05 [ENQ] Enquiry 000 0110 6 06 [ACK] Acknowledgement 000 0111 7 07 [BEL] Bell 000 1000 8 08 [BS] Backspace 000 1001 9 09 [HT] Horizontal Tab 000 1010 10 0A [LF] Line feed 000 1011 11 0B [VT] Vertical Tab 000 1100 12 0C [FF] Form feed 000 1101 13 0D [CR] Carriage return 000 1110 14 0E [SO] Shift Out 000 1111 15 0F [SI] Shift In 001 0000 16 10 [DLE] Data link escape 001 0001 17 11 [DC1] Device control 1 001 0010 18 12 [DC2] Device control 2 001 0011 19 13 [DC3] Device control 3 001 0100 20 14 [DC4] Device control 4 001 0101 21 15 [NAK] Negative acknowledgement 001 0110 22 16 [SYN] Synchronous idle 001 0111 23 17 [ETB] End of trans. block 001 1000 24 18 [CAN] Cancel 001 1001 25 19 [EM] End of medium 001 1010 26 1A [SUB] Substitute 001 1011 27 1B [ESC] Escape 001 1100 28 1C [FS] File separator 001 1101 29 1D [GS] Group separator 001 1110 30 1E [RS] Record separator 001 1111 31 1F [US] Unit separator 111 1111 127 7F [DEL] Delete In the following message Hello world, Computers are fun! You may assume there are only 29 characters (including the spaces) but in fact there are a couple of invisible characters involved too. [STX] Hello world,[CR] Computers are fun! [ETX] Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 31 By Nichola Lacey Task 15: Convert the following messages into binary. Make sure you include the invisible characters, for the first example the invisible characters have been included for you. Original message Converted into binary [STX]LOL[ETX] 1 2 3 Computers! Objectives: Understand that character codes are commonly grouped and run in sequence within encoding tables. As we have seen, the characters are in sequence, so you can work out the characters if you know the number of one of them in the sequence. Using the following code work out the decimal numbers for the following: A = 65 i = 105 Task 16: Work out the decimal values of the following characters WITHOUT looking at the table on page 29. Character Decimal value Character Decimal value J e T m Characters of a similar type (upper case, lower case or numeric digits) run in sequence within a character set Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 34 By Nichola Lacey Representing Images Objective: Understand what a pixel is and be able to describe how many pixels relate to an image and the way images are displayed. Pixels Pixel stands for “Picture Element” and refers to the way bitmap images are made up of lots of tiny squares of colour which, when they are small, blend together to make one large image. In the example above, you can see when a small part of the image is expanded you can see the individual pixels in greater detail. In this instance the image was expanded by 400% so you can see the pixels. You may have noticed this happening if you copy an image from the internet and then expand it. This is known as the image becoming “pixelated”. The pixels are displayed in neat rows and columns and a computers screen will have millions of these pixels to make up the display. Even on a toolbar which you may assume only has 1 or two colours you notice how it can become pixelated when you zoom into it and you will see the variety of shades that are used to make up the image. A pixelated image is one where the individual pixels are clearly visible Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 35 By Nichola Lacey Objective: Describe the following for bitmaps: size of pixels, colour depth. Know that the size of a bitmap image in pixels (width x height) is known as the image resolution. As we have seen, when we zoomed into the images, the larger the pixels are, the grainier the image becomes, also the overall size of the image will become larger. Imagine this square is a very simple image. If every pixel was ½ a centimetre wide and ½ a centimetre tall the overall image would be 1 ½ centimetres wide and 1 ½ centimetres tall and the image would have 9 pixels in it (3 x 3) Now look at this image: It is the same image but in this example each pixel has doubled in size and is therefore 1 centimetre wide by 1 centimetre tall making the image 3 centimetres wide and 3 centimetres tall but is still only has 9 pixels in it, it is only the pixels that have changed size. The pixels on a computer monitor are very small and the smaller the pixel size the better quality the image will be. The size of the pixels is measure in DPI (dots per inch) and this is how many pixels can fit in a square inch. Task 18: WITHOUT looking it up on the internet what do you think the average resolution is for the following: Item Resolution (written a DPI) The average computer monitor iPhone 8 screen An image in a glossy magazine When people talk about the image size they are not meaning the width and height of the image but rather the number of pixels that make the width and height of the image. This is because, as we have seen, the image can be re-sized but the number of pixels will stay the same. The image size is described in pixels (width x height) Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 36 By Nichola Lacey Let’s have another look at the flower image. This image is 252 x 181 pixels (it is always width shown first and then height). If we make the image larger or smaller, it still has the same number of pixels that make up the image. This image still has 252 x 181 pixels. This image still has 252 x 181 pixels. Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 39 By Nichola Lacey This will double the number of pixels needed for the entire image to 00000001010000000000011010010000000110101010010001101110101110010110 10101010100101100110100110010001100101100100000001101001000000000001 01000000 The number of colours used in an image can greatly affect the quality. Here is the same image saved with varying colour depth: Image Colour depth Bits used per pixel 16, 777, 216 possible colours per pixel 24 65,536 possible colours per pixel 16 256 possible colours per pixel 8 2 possible colours per pixel 1 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 40 By Nichola Lacey Objective: Describe, using examples, how the number of pixels and colour depth can affect the size of a bitmap image. Task 19: As you can see from the image below the colour depth greatly alters the file size. Write, in your own words, why you think this may be. Task 20: As you can see from the image below the image size greatly alters the file size. Write, in your own words, why you think this may be. Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 41 By Nichola Lacey Calculating file size Objective: Calculate bitmap image file sizes based on the number of pixels and colour depth. To work out the file size you will need to know the following: W = image width H= image height D = colour depth (in bits) As long as you know these things you can make an approximate calculation of the file size. W x H x D = File size Lets take our simple image from earlier: The width is 8 pixels wide, the height is 9 pixels tall and a single bit is used for the colour depth. 8 x 9 x 1 = 72 bits Let’s try another one… The width is 252, the height is 181 and the colour depth is 24 bits per pixel. 252 x 181 x 24 = 10,944,688 bits (which is 136,836 bytes or 136 Kb) Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 44 By Nichola Lacey To convert a pattern into a binary number you just need to read across the rows and write down the 0 or 1 depending on the colour of each pixel. When you get to the end of a row simply move to the start of the next row but write the binary digits directly after the last you wrote. Let’s have a look at this pattern Start with the first row and the binary sequence would be 010, move down to the next row and the binary sequence is 110, however instead of writing this on a separate row you continue with the same line of digits. Eventually you would get 010110011010 Task 23: Convert the pattern into binary digits where 0 is white and 1 is black. Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 45 By Nichola Lacey End of chapter recap Task 23: Answer these questions. Question Your answer 1. What is a pixel? 2. How are pixels displayed in an image? 3. What is the difference between image size and file size? 4. What is meant by the term “image resolution”? 5. How can the number of pixels and the colour depth affect the file size of a bitmap image? 6. If you have an image 300 x 700 with a colour depth of 24 bits, what is the approximate file size? 7. Draw the following 4 x 5 1 bit image from the binary sequence where 0 is white and 1 is black. 01101001100111111001 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 46 By Nichola Lacey Representing Sound Objectives: Understand that sound is analogue and that it must be converted to a digital form for storage and processing in a computer. Sound is produced by something vibrating which makes the air particles around it vibrate. These vibrations travel through the air and the delicate hairs inside our ears pick up on these subtle vibrations in the air and pass these messages to our brains that interpret them as sound. These vibrations are known as “analogue” however computers work using “digital” messages. Here is an analogue sound wave that has been plotted on a graph. Computers, as we know, store data digitally as binary. Everything in a computer needs to be broken down and stored in a binary format so to save an analogue sound wave on a computer it must be converted into a digital format. Creating a digital sound wave Objective: Understand that sound waves are sampled to create the digital version of the sound. Describe the digital representation of sound in terms of: sampling rate, sample resolution. To store this as a digital sequence a sample must be taken of the sound way at set intervals. Using our wave from above we may say this is a sound wave over the period of 1 second so we can split that second down into smaller parts. Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 49 By Nichola Lacey This is the same sound clip but this time it has been split into 20 samples a second (20 Hz) and each sample is taken from numbers ranging from 0 to 21 to allow greater accuracy of the soundwave to be recorded. This would increase the number of bits required to store the image. Task 25: Fill in this table to record the decimal value of each sample and convert it to the binary equivalent using a 5-bit number. Sample Decimal number Binary number Sample Decimal number Binary number 1 11 2 12 3 13 4 14 5 15 6 16 7 17 8 18 9 19 10 20 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 50 By Nichola Lacey Calculate sound file sizes Objective: Calculate the sound file size based on the sampling rate and the sample resolution. To work out the file size of a sound clip you will need to know the following: rate = sampling rate res = sample resolution secs = number of seconds As long as you know these things you can make an approximate calculation of the file size. rate x res x secs = File size Lets take the following example: A sound clip lasts 15 seconds, and a sample is taken every tenth of a second. Each of those samples can be between 0 and 255 so this will require 8 bits per sample. 10 x 8 x 15 = 1200 bits Lets convert that into bytes: 1200 ÷ 8 = 150 bytes In reality, sample rates and sample resolutions are much higher. Task 26: WITHOUT looking it up, have a guess at the average sample rate and sample resolution would be to store these sounds digitally. Description Sample rate Sample resolution HDTV Cinema Film MP3 music track Task 27: Work out the approximate file sizes with the following data: Length of clip Sample rate Sample resolution Bits Shortest notation 60 seconds 48,000 Hz 24-bit 2 minutes 44,100 Hz 16-bit 1 hour 20 mins 96,000 Hz 24-bit Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 51 By Nichola Lacey End of chapter recap Task 28: Draw on the graph to show the digital samples that would be taken at the end of every tenth of a second. Fill in the table to show the samples that are taken and finally work out the smallest file size possible for this 1 second sound clip. Sample Decimal number Binary number Sample Decimal number Binary number 1 6 2 7 3 8 4 9 5 10 What would be the smallest file size for this sound file? Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 54 By Nichola Lacey Huffman coding Objective: Explain how data can be compressed using Huffman coding. Be able to interpret/create Huffman trees. Huffman coding provides a simple, unambiguous code by studying the frequencies that certain characters appear in a message. It is a lossless data compression method used with PKZIP files, JPEG image files and PNG image files. When you first start to learn Huffman coding it can seem like a bit of a magic trick. You may feel confused until you get to the end when you realise what you have just witnessed and are pleasantly astounded by the cleverness of it. I am going to help you by showing the final result first. It is ruining the magic trick but it will help you know what you are aiming for when performing the Huffman code algorithm. We are going to try to crack this code using the following table: Binary number Character 101  100  01  00  11  Working from left to right through the binary sequence below and using the table above, try to work out the correct sequence of characters. You may be surprised to find out there is only one way of cracking this code. 101010011100011100 The only way of splitting up the above binary number, when working from left to right, using the options from the table above are as follows: 101 01 00 11 100 01 11 00 This gives us the sequence…         Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 55 By Nichola Lacey You may notice that the three characters which crop up twice (,  and ) only use 2 binary bits and the others that crop up only once ( and ), use 3 binary bits to represent them. We have used a code that allows the most frequent characters to be represented using the least number of bits, but it still gives us a code that can only be cracked one way. Clever, isn’t it? This is what Huffman coding allows us to do. However, instead of using the strange characters we have just been using, Huffman coding is more commonly used with ASCII or UNICODE symbols or pixel colours on an image etc. The most frequent binary sequences in any data file can be given a new shortened binary number which makes the whole file take up less bits and therefore is faster to transmit over a network. You may be thinking, “But how can I possibly create a code that works that way?”. That is what Huffman coding does. It is an algorithm that allows you to create this binary code by following the steps as outlined on the next few pages. Creating your Huffman code We are going to have a look at how to compress a simple text file. Let’s think about a short sentence. “hello world” Using the example above there are 8 different characters (including the space) but there are 11 characters in the sentence. Using ASCII we would normally use 7-bits to represent each character which would mean we would need to use 77 bits (7 bits per character x 11 characters) for the message. We are going to use Huffman coding which will manage to compress this message down into only 32 bits. Not all the letters in the above sentence occur with the same frequency, some letters crop up a lot and others only occur once or twice. The idea is to lower the number of bits used to encode the data that occurs most frequently. If we count the number of times each letter occurs and sort them so the least frequent is at the top of the table and the most frequent is at the bottom of the table, we end up with a table like this: Character Frequency h 1 e 1 space 1 w 1 r 1 d 1 o 2 l 3 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 56 By Nichola Lacey Step 1: Take the top two items in the table able and draw nodes (round cornered rectangle as shown below) containing the frequency of the character and the character in round brackets. Join these together with a new node containing the total of the other two nodes. Add this value into your table in the correct sorted order and remove the top two values. Your table should now look like this: Character Frequency space 1 w 1 r 1 d 1 o 2 2 l 3 Step 2: Repeat step 1 until you only have 1 node left. Don’t forget to update your table by deleting the rows you have used and adding in the new node values in their correct sorted position in the table. Your table should now look like this: Character Frequency o 2 2 2 2 l 3 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 59 By Nichola Lacey Starting with the left-hand side of the Huffman tree, follow the branches to work out the unique binary number for each of the original characters and store them in a table. Character Binary code l 11 o 101 h 1001 e 1000 space 011 w 010 r 001 d 000 Step 5: Using this new binary table you can create a binary sequence. 10011000111110101101010100111000 This only has 32 bits rather than the 77 the message would have taken if each character was saved with 7-bits using ASCII. It may look like this could be misinterpreted but if you move through the binary sequence and check against the table with their new binary codes there is only one way this can be split into the correct characters Take the beginning of the sequence. There are no characters in the table that match 1, 10, 100 or 10011 etc. There is only one letter that could possibly be at the beginning and that is 1001. Once the correct letter has been found and decoded the computer will look at the next collection of bits in the sequence and decode them. Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 60 By Nichola Lacey Task 29: Complete the table for the following Huffman tree Character Frequency Bitmap Path Number of bits n 1 101 3 c 3 100 j 3 t r Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 61 By Nichola Lacey Calculating the bits used with data compression Objective: Be able to calculate the number of bits required to store a piece of data compressed using Huffman coding. Be able to calculate the number of bits required to store a piece of uncompressed data in ASCII. To calculate the number of bits used in a piece of data compressed using Huffman coding we need to add a couple of columns to our table we have produced so it now looks as follows. Character Binary code Frequency Number of bits l 11 3 2 o 101 2 3 h 1001 1 4 e 1000 1 4 space 011 1 3 w 010 1 3 r 001 1 3 d 000 1 3 To work out the total number of bits multiply the frequency by the number of bits for each character and then add them all together. (3 x 2) + (2 x 3) + (1 x 4) + (1 x 4) + (1 x 3) + (1 x 3) + (1 x 3) + (1 x 3) = 32 bits or 4 bytes. To work out the number of bits required for ASCII, remember ASCII is stored using 7-bits so you simply need to multiple the number of characters (including spaces) by 7. 11 x 7 = 77 bits or approximately 10 bytes. By performing the Huffman coding on this small piece of data we have managed to save 6 bytes of data in just 11 characters. This is quite impressive. Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 64 By Nichola Lacey Task 31: Use RLE to show the frequency and value for the following bitmap sequences in each row. Image sequence Run Length encoding 1 1 1 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 0 1 1 Task 32: Uncompress the RLE to shade in the correct boxes to recreate the bitmap image for each row Run Length encoding Image sequence 3 1 5 0 1 1 3 0 3 1 1 0 2 0 5 1 1 0 3 0 2 1 2 0 1 1 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 65 By Nichola Lacey End of chapter recap Task 33: Answer the following questions in your own words Question Your Answer 1. Why are files compressed? 2. Explain the difference between lossless and lossy data compression. 3. Explain how a text file can be compressed using Huffman coding 4. Explain how data can be compressed using run length encoding (RLE). Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 66 By Nichola Lacey Answers Task 1 Explanation Thousands Hundreds Tens Ones Decimal number One thousand, four hundred and fifty-six 1 4 5 6 1456 Nineteen ninety-nine 1 9 9 9 1999 Six thousand and twenty 6 0 2 0 6020 Task 2 Explanation Eight Four Two One Binary number Decimal value 1 x 8, 0 x 4, 1 x 2 and 1 x 1 1 0 1 1 1011 11 1 x 8, 1 x 4, 0 x 2 and 0 x 1 1 1 0 0 1100 12 1 x 2 and 1 x 1 0 0 1 1 0011 3 Task 3 Binary number Hidden Word 1011 1110 1110 1111 BEEF 1011 1110 1101 BED 1101 1110 1010 1111 DEAF 1010 1101 1101 ADD 1010 1100 1110 ACE 1011 1110 1010 1101 BEAD 1111 1110 1110 1101 FEED 1011 1010 1101 BAD 1100 1010 1011 CAB Task 4 Binary number 156 64 32 16 8 4 2 1 Decimal value 110 1 1 0 6 11001 1 1 0 0 1 25 10101 1 0 1 0 1 21 100111 1 0 0 1 1 1 39 1011000 1 0 1 1 0 0 0 88 1100001 1 1 0 0 0 0 1 97 10101100 1 0 1 0 1 1 0 0 172 11111111 1 1 1 1 1 1 1 1 255 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 69 By Nichola Lacey Task 13 1 1 0 0 + 0 1 1 0 1 0 0 1 0 1 1 0 0 1 + 1 1 1 1 1 1 0 0 0 1 1 1 1 0 1 1 1 0 + 1 1 0 1 1 1 1 0 1 1 1 1 1 1 0 0 0 + 0 1 1 0 1 1 0 0 1 0 1 1 1 1 0 0 0 1 1 0 1 0 + 1 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1 0 0 1 1 0 + 0 1 1 1 1 1 0 0 1 1 0 1 11110000 + 10011 = 1 1 1 1 0 0 0 0 + 1 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1101011 + 10111 + 1101001 = 1 1 0 1 0 1 1 1 0 1 1 1 + 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 1 1 1 1 1 1 1111 + 111111 + 11111111 = 1 1 1 1 1 1 1 1 1 1 + 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 1 1 0 1 0 1 0 1 1 1 Task 14 O ri gi n al Original binary number In st ru ct io n New binary number N e w 24 0 0 0 1 1 0 0 0 *2 0 0 1 1 0 0 0 0 48 34 0 0 1 0 0 0 1 0 /2 0 0 0 1 0 0 0 1 17 50 0 0 1 1 0 0 1 0 *4 1 1 0 0 1 0 0 0 200 60 0 0 1 1 1 1 0 0 /4 0 0 0 0 1 1 1 1 15 38 0 0 1 0 0 1 1 0 *8 0 0 1 1 0 0 0 0 48 76 0 1 0 0 1 1 0 0 /8 0 0 0 0 1 0 0 1 9 9 0 0 0 0 1 0 0 1 *16 1 0 0 1 0 0 0 0 144 168 1 0 1 0 1 0 0 0 /16 0 0 0 0 1 0 1 0 10 Task 15 Original message Converted into binary [STX]LOL[ETX] 000 0010 100 1100 100 1111 100 1100 000 0011 1 2 3 000 0010 011 0001 000 1101 011 0010 000 1101 011 0011 000 0011 Computers! 000 0010 100 0011 110 1111 110 1101 111 0000 111 0101 111 0100 110 0101 111 0010 111 0011 010 0001 000 0011 Task 16 Character Decimal value Character Decimal value J 74 e 101 T 84 m 109 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 70 By Nichola Lacey Task 17 1. ASCII allows for 127 different characters and can be represented with a single byte of data, Unicode allows for far more characters. 2. It allows more characters to be used so can incorporate alphabets from around the world and symbols which are used in maths and other specialist areas to be used. 3. 004116 4. ④ Task 18 Item Resolution (written a DPI) The average computer monitor 72 – 96 iPhone 8 screen 326 An image in a glossy magazine 300 Task 19 In order to use a greater variety of colours for each pixel, more bits would be needed to store these larger numbers. Therefore, as more bits are required for each pixel the file size would increase. Task 20 The image size is determined by the number of pixels for the width and height of the image. If more pixels are used to give a greater clarity to the image, this would increase the file size of the image. Task 21 Width Height Colour Depth Number of bits Shorthand 1,280 860 24 bits 26,419,200 3 Mb 2,500 2,000 8 bits 40,000,000 5 Mb 150 475 16 bits 1,140,000 142.5 Kb 15,000 20,000 64 bits 19,200,000,000 2.4 Gb Task 22 Task 23 0110100110010110 Teach yourself the fundamentals of data representation for AQA GCSE Computer Science Page 71 By Nichola Lacey Task 24 1. A single picture element, shown as a small square which can contain a single colour. These are combined to make one large image. 2. In rows and columns 3. The image size refers to how many pixels make up the width and height of an image but the file size is a calculation to find out how many bytes are needed to save the image (width x height x colour depth) 4. This is the number of dots per inch (DPI). In other words, how many pixels fit into a square inch of an image. The higher the DPI the higher quality the image will be. 5. To save more colours a large number of bits are needed to save the large number. If there are more bits needed for each pixel and lots of pixels required, then the file size will be larger than if there is a small number of pixels or not many different colours needto be saved. 6. 630 Kb 7. Task 25 Sample Decimal number Binary number Sample Decimal number Binary number 1 11 01011 11 9 01001 2 9 01001 12 17 10001 3 11 01011 13 8 01000 4 18 10010 14 10 01010 5 9 01001 15 3 00011 6 8 01000 16 18 10010 7 13 01101 17 17 10001 8 11 01011 18 14 01110 9 3 00011 19 12 01100 10 4 00100 20 7 00111 Track 26 Description Sample rate Sample resolution HDTV 48,000 Hz 24-bit Cinema Film 96,000 Hz 24-bit MP3 music track 44,100 Hz 16-bit Track 27 Length of clip Sample rate Sample resolution Bits Shortest notation 60 seconds 48,000 Hz 24-bit 69,120,000 8.6 Mb 2 minutes 44,100 Hz 16-bit 84,672,000 9.6 Mb 1 hour 20 mins 96,000 Hz 24-bit 11,059,200,000 1.3 Gb