eTEACH presentation accessible player instructionseTEACH presentation accessible player instructionsThis eTEACH player requires that you have Flash 8 or above installed. If you do not, download the player from Adobe.
The next major heading marks the start of the presentation. Each next lower level heading thereafter marks the beginning of a slide. An audio control button that plays the audio track for the slide follows the contents and notes for each slide in the presentation. Pressing this button after the audio playback toggles pausing and resuming the playback. Before moving off the play audio button, complete listening to the audio track for the slide or pause the audio. Control + Shift + L increases volume. Control + Shift + Q decreases volume.
There is a table of contents for the presentation following the presentation. The table of contents can be used to access sections of the presentation in random order. Type Control + Shift + C to go to the table of contents.
When links to external resources are present in the presentation, a list of these resource links follows the table of contents. When following a link to a resource, you should remember which slide you were on. When you return to the main page, your screen-reader may not return you to that slide. You can use the table of contents or just click through the headings to return to the correct slide.
This presentation contains resources. Click here to proceed immediately to the resource list. A link to resources follows each slide.
If the presentation contains one or more quizzes, a button to take you to each quiz follows the slide at which the quiz is to occur. Each question in the quiz begins with a level three heading. The answer choices will be presented in a list. If your instructor has supplied a hint for the questions, it will follow the answers and is formatted as an anchor. The correct answer to the question follows the list of answers and the hint (if included). The correct answer is also formatted as an anchor.
Chapter 1 - PPT - Mano & Kime - 3rd EdSlide 1 Lecture 2 Binary CodesLecture 2 Binary Codes
Slide 2 OutlineOutline Binary Codes Decimal Codes Gray Codes Error-detection Codes Alphanumeric Codes
Slide 3 Binary Numbers and Binary CodingBinary Numbers and Binary Coding Flexibility of representation Within constraints below, can assign any binary combination (called a code word) to any data as long as data is uniquely encoded. Information Types Numeric Must represent range of data needed Very desirable to represent data such that simple, straightforward computation for common arithmetic operations permitted Tight relation to binary numbers Non-numeric Greater flexibility since arithmetic operations not applied. Not tied to binary numbers
Slide 4 Non-numeric Binary CodesGiven n binary digits (called bits), a binary code is a mapping from a set of represented elements to a subset of the 2n binary numbers. Example: A binary code for the seven colors of the rainbow Code 100 is not used Non-numeric Binary Codes Binary Number 000 001 010 011 101 110 111 Color Red Orange Yellow Green Blue Indigo Violet
Slide 5 Number of Bits RequiredGiven M elements to be represented by a binary code, the minimum number of bits, n, needed, satisfies the following relationships: Example: How many bits are required to represent decimal digits with a binary code? M = 10, hence n = ceiling (log2 10) = ceiling (4) = 4 Checking: Number of Bits Required
M = 10 Therefore n = 4 since: 24 =16 is 10 > 23 = 8 and the ceiling function for log2 10 is 4.
Slide 6 Number of Elements RepresentedNumber of Elements Represented Given n digits in radix r, there are rn distinct elements that can be represented. Can also represent fewer elements, m < rn Examples: You can represent 4 elements in radix r = 2 with n = 2 digits: (00, 01, 10, 11). You can represent 4 elements in radix r = 2 with n = 4 digits: (0001, 0010, 0100, 1000). This second code is called a "one hot" code.
Slide 7 Binary Codes for Decimal DigitsBinary Codes for Decimal Digits Decimal 8,4,2,1 Excess3 8,4, - 2, - 1 Gray 0 0000 0011 0000 0000 1 0001 0100 0111 0100 2 0010 0101 0110 0101 3 0011 0110 0101 0111 4 0100 0111 0100 0110 5 0101 1000 1011 0010 6 0110 1001 1010 0011 7 0111 1010 1001 0001 8 1000 1011 1000 1001 9 1001 1 100 1111 1000 There are over 8,000 ways that you can chose 10 elements from the 16 binary numbers of 4 bits. A few are useful:
Slide 8 Binary Coded Decimal (BCD)Binary Coded Decimal (BCD) The BCD code is the 8,4,2,1 code. This code is the simplest, most intuitive binary code for decimal digits and uses the same powers of 2 as a binary number, but only encodes the first ten values from 0 to 9. Example: 1001 (9) = 1000 (8) + 0001 (1) How many invalid code words are there? Answer: 6 What are the invalid code words? Answer: 1010, 1011, 1100, 1101, 1110, 1111
Answer 1: 6 Answer 2: 1010, 1011, 1100, 1101, 1110, 1111
Slide 9 Excess 3 Code and 8, 4, 2, 1 CodeWhat interesting property is common to these two codes? 9 s complement codes Excess 3 Code and 8, 4, 2, 1 Code
Answer: Both of these codes have the property that the codes for 0 and 9, 1 and 8, etc. can be obtained from each other by replacing the 0 s with the 1 s and vice-versa in the code words. Such a code is sometimes called a complement code.
Slide 10 Gray CodeWhat special property does the Gray code have in relation to adjacent decimal digits? Only one bit position changes with each increment Gray Code Decimal 8,4,2,1 Gray 0 0000 0000 1 0001 0100 2 0010 0101 3 0011 0111 4 0100 0110 5 0101 0010 6 0110 0011 7 0111 0001 8 1000 1001 9 1001 1000
Answer: As we counts up or down in decimal, the code word for the Gray code changes in only one bit position as we go from decimal digit to digit including from 9 to 0.
Slide 11 Gray Code (Continued)B 0 111 110 000 001 010 011 100 101 B 1 B 2 (a) Binary Code for Positions 0 through 7 G 0 G 1 G 2 111 101 100 000 001 011 010 110 (b) Gray Code for Positions 0 through 7 Gray Code (Continued) Does this special Gray code property have any value? An Example: Optical Shaft Encoder
Yes, as illustrated by the example that followed.
Slide 12 Gray Code (Continued)Gray Code (Continued) How does the shaft encoder work? The encoder disk contains opaque and clear areas Opaque represents 0 Clear represents 1 Light shining through each ring strikes a sensor to produce a 0 or a 1. Encoding determines rotational position of shaft
Answer 1: The encoder disk contains opaque and clear areas. Opaque represents a 0 and clear a 1. Light shining through each ring of the encoder corresponding to a bit of the code strikes a sensor to produce a 0 or a 1. Answer 2: In addition to the correct code, either 011 or 100, the codes 000, 010, 001, 110, 101, or 111 can be produced. Answer 3: Yes, the shaft position can be completely UNKNOWN!
Slide 13 Gray Code (Continued)Gray Code (Continued) For the binary code, what codes may be produced if the shaft position lies between codes for 3 and 4 (011 and 100)? {011,100} are correct, but {000,010,001,110,101,111} also possible Is this a problem? Yes, shaft position can be UNKNOWN B 0 111 110 000 001 010 011 100 101 B 1 B 2
Answer 1: The encoder disk contains opaque and clear areas. Opaque represents a 0 and clear a 1. Light shining through each ring of the encoder corresponding to a bit of the code strikes a sensor to produce a 0 or a 1. Answer 2: In addition to the correct code, either 011 or 100, the codes 000, 010, 001, 110, 101, or 111 can be produced. Answer 3: Yes, the shaft position can be completely UNKNOWN!
Slide 14 Gray Code (Continued)Gray Code (Continued) For the Gray code, what codes may be produced if the shaft position lies between codes for 3 and 4 (010 and 110)? Only the correct codes: {010,110} Is this a problem? No, either is OK since shaft is between them Does the Gray code function correctly for these borderline shaft positions for all cases encountered in octal counting? Yes, no erroneous codes can arise G 0 G 1 G 2 111 101 100 000 001 011 010 110
Answer 1: Only the correct codes, either 010 or 110 Answer 2: No, the shaft position is known to be either 3 or 4 which is OK since it is halfway in between. Answer 3: Yes, since an erroneous code cannot arise. This includes between 0 and 7 (000 and 100).
Slide 15 Warning: Conversion or Coding?Warning: Conversion or Coding? Do NOT mix up conversion of a decimal number to a binary number with coding a decimal number with a BINARY CODE. 1310 = 11012 (This is conversion) 13 0001|0011 (This is coding)
Slide 16 Error-Detection CodesError-Detection Codes Redundancy (e.g. extra information), in the form of extra bits, can be incorporated into binary code words to detect and correct errors. A simple form of redundancy is parity, an extra bit appended onto the code word to make the number of 1 s odd or even. Parity can detect all single-bit errors and some multiple-bit errors. A code word has even parity if the number of 1 s in the code word is even. A code word has odd parity if the number of 1 s in the code word is odd.
Slide 17 4-Bit Parity Code Example4-Bit Parity Code Example The codeword "1111" has even parity and the codeword "1110" has odd parity. Both can be used to represent 3-bit data. Even Parity Odd Parity Message - Parity Message - Parity 000 - 0 000 - 001 - 1 001 - 010 - 1 010 - 011 - 0 011 - 100 - 1 100 - 101 - 0 101 - 110 - 0 110 - 111 - 1 111 - 1 0 0 1 0 1 1 0
Even Parity Bits: 0, 1, 1, 0, 1, 0, 0, 1 Odd Parity Bits: 1, 0, 0, 1, 0, 1, 1, 0
Slide 18 ASCII Character CodesASCII Character Codes American Standard Code for Information Interchange This code is a popular code used to represent information sent as character-based data. It uses 7-bits to represent: 94 Graphic printing characters. 34 Non-printing characters Some non-printing characters are used for text format (e.g. BS = Backspace, CR = carriage return) Other non-printing characters are used for record marking and flow control (e.g. STX and ETX start and end text areas). (Refer to Table 1 -4 in the text)
Slide 19 ASCII PropertiesASCII Properties ASCII has some interesting properties: Digits 0 to 9 span Hexadecimal values 3016 to 3916 . Upper case A - Z span 4116 to 5A16 . Lower case a - z span 6116 to 7A16 . Lower to upper case translation (and vice versa) occurs by flipping bit 6. Delete (DEL) is all bits set, a carryover from when punched paper tape was used to store messages. Punching all holes in a row erased a mistake!
Slide 20 UNICODEUNICODE UNICODE extends ASCII to 65,536 universal characters codes For encoding characters in world languages Available in many modern applications 2 byte (16-bit) code words See Reading Supplement Unicode on the Companion Website http://www.prenhall.com/mano
Slide 21 SummarySummary Binary Codes Decimal Codes Gray Codes Error-detection Codes Alphanumeric Codes
Table of ContentsThe object immediately following this sentence is the media player.