Chapter 11 Binary Codes
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1 Chapter 11 Binary Codes Dr.-Ing. Stefan Werner Table of content Chapter 1: Switching Algebra Chapter 2: Logical Levels, Timing & Delays Chapter 3: Karnaugh-Veitch-Maps Chapter 4: Combinational Circuit Design Chapter 5: Latches and Flip Flops Chapter 6: Finite State Machines Characterization of Chapter 7: State Machine number codes Minimization Binary addition using Chapter 8: Basic Sequential Circuits Chapter 9: Number Systems Chapter 10: Binary Arithmetic Chapter 11: Binary Codes number codes BCD-Code Aiken Code 3-Excess Code Gray-Code 2of 17 1
2 Definition Definition: Code A code in general is a rule that maps a set of representations A ( the data words) to another set of representations B (the code words). Codes are used as means of data transfer and data storage for several purposes: Appropriate and simple data transfer and data processing Representation of information using the least number of symbols Security of data transfer 3of 17 Redundandant Codes Codes can be distinguished by certain fundamental properties. First of all, a code can be a redundant or a non-redundant code. In a non-redundant code the number of code words equals exactly the number of data words. Codes In a redundant code there are more possible code words than data words. Error correcting Redundant Error detecting Number Codes/ Binary Codes Non redundant the redundant code words can be used to build error detecting and even error correcting codes. Word Codes 4of 17 2
3 Characterization of Codes non redundant codes redundant codes letter codes number codes error detecting codes error correcting codes 5of 17 Characterization of Number Codes BCD Pseudo tetrads of 17 3
4 Overview Number Codes Hex. BCD Aiken 3-Exzess Gray keine keine Pseudo tetraden Pseudo tetraden Pseudo- A tetraden B C Pseudo D tetraden E Pseudo- Pseudo F tetraden tetraden 7of 17 BCD-Code Binary Coded Decimal Check and correction procedure +0II0 yes yes Sum =PT? carry no no No correction needed BCD Pseudo tetrads of 17 4
5 BCD-Code Binary Coded Decimal Example: Addition in BCD-Code 2nd Correction because Intermediate Result=PT of carry to next position1st correction needed Carry because of correction Addend Addend Carry = Intermediate t Result Correction 11 = Final Result BCD Pseudo tetrads of 17 Aiken Code Aiken Pseudo tetraden of 17 5
6 Aiken Code Check and correction procedure -0II0 Aiken ja Summe nein =PT? ja nein Übertrag +0II0 keine Korrektur Pseudo tetraden If result is not a PT => no corrections needed 11 of 17 Aiken Code Addition Addend Addend = Intermediate Correction = Final Result Aiken Pseudo tetraden of 17 6
7 3-Excess Code 3-Exzess keine Pseudo tetraden Pseudo tetraden 13 of 17 3-Excess Code Check and Correction Procedure +00II ja Übertrag nein -00II 3-Exzess keine Pseudo tetraden Pseudo tetraden 14 of 17 7
8 3-Excess Code Addition 3-Exzess keine Pseudo tetraden With carry Without carry (PT!!) = = Pseudo tetraden <- Correction (+ bzw. -3) <- Final Result 15 of 17 Gray Code Decimal (original) Gray-Code of 17 8
9 Gray Code by Glixon Decimal original Gray-Code Cyclic Gray Code byglixon of 17 9
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