4 Switching Algebra 4.1 Axioms; Signals and Switching Algebra
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1 4 Switching Algebra 4.1 Axioms; Signals and Switching Algebra To design a digital circuit that will perform a required function, it is necessary to manipulate and combine the various input signals in certain ways that are in agreement with mathematical logic. Appropriate methods especially to simplify logical expressions are provided by Boolean algebra. In 1854, George Boole ( , English mathematician) introduced a logical discourse. The objects of his study were true and false and their expressions were propositions. We will use axioms (i. e. commutivity) which were presented by E. V. Huntington in The first technical application of two-valued Boolean algebra to the properties of electrical switching circuits was demonstrated by C. E. Shannon in Because logical behaviour of semiconductor based electronic switching elements can be described in the same way, the term switching algebra is still appropriate. We use 1 and 0 instead of true and false. The two elements can be operated on by the basic operations: AND ( ), OR ( ) and NOT ( ) The special mathematical logic of two-valued Boolean algebra can be represented by using axioms, theorems and corollaries. The laws of Boolean algebra are few and can be deduced from truth tables for NOT, AND and OR. Digital Circuits I 4-1
2 NOT and AND Laws NOT Laws: NOT Truth Table A A = Not Laws 0 = 1 1 = 0 = A The NOT operation is designated by the overscore (or bar). This inverter function implements the complement concept in switching algebra. A double bar of a function, called involution, is the function itself. AND Laws: AND Truth Table A B A B AND Laws A 0 = 0 A 1 = A 1 is identity element A A = A idempotency A = 0 complement of A is The AND operation is also written with the dot operator ( ) ( A B). Because it has algebraic similarities to multiplication, the ANDing of two or more binary signals is often called the Boolean product or the product term. Digital Circuits I 4-2
3 OR Laws and Priorities OR Laws: OR Truth Table B A A B OR Laws A 0 = A 0 is identity element A 1 = 1 A A = A idempotency A = 1 complement of A is The OR operation is also written with the plus operator (+) ( A + B). Because it has algebraic similarities to addition, the ORing of two or more binary signals is often called the Boolean sum or the sum term. Priority of operations - The NOT operator has the highest priority. - Following the German Industry Standard (DIN) AND and OR Operators have the same priority. Parentheses have to be used in order to give a desired precedence. - Standards which are used in the USA and therefore are implemented in most of the hardware development tools, state: When AND and OR appear in the same expression, then AND operations have priority and should be carried out at first. If parentheses are encountered, then the normal precedence of AND over OR is overruled. - Parentheses are recommended in order to avoid misinterpretations. - AND and OR have higher priority than XOR and XNOR (Chapt. 4.3) Digital Circuits I 4-3
4 4.2 Truth Tables A truth table defines an input-output relationship of a digital system by listing all binary input combinations and their corresponding outputs. Each input combination is a separate row in the table. Normally truth table rows will have the orderly appearance of a series of binary numbers in increments of 1 from 0 to 2 m 1 ( left hand side columns), where m is the number of the distinct input signals to the system, each represented by a bit. The resulting outputs of the system are listed to the right of the input columns. If a system with m = 2 inputs has a single output Y, then 16 output combinations can be obtained with a truth table. In general it follows: Number of inputs: m Input combinations: N = 2 m Number of possible results: K = (2) N All 16 Truth Tables with m = 2 Input Signals and one Output Y i Decimal B A Y 0 Y 1 Y 2 Y 3... Y 14 Y Truth tables are implemented as Lookup Tables in memory based programmable devices like FPGAs. Input signals represent addresses of memory bits which are connected to the output. Digital Circuits I 4-4
5 Example 1 of a Truth Table Problem: Detecting two adjacent buttons pressed. Suppose three switches are arranged left to right as shown. If a switch is closed, its associated output signal (A, B or C) is connected to voltage and becomes high. If a switch is open, no current flows through its associated resistor (R1, R2 or R3)and therefore the signal is connected to ground and is low. Write a truth table that describes an output Y = 1 if two, and only two, adjacent switches in the row of three switches are closed at the same time. Solution: The inputs A, B and C provide 2 3 = 8 input combinations. The input combinations for which the output Y becomes 1 are A = 1 and B = 1 or B = 1 and C = 1. Decimal C B A Y Digital Circuits I 4-5
6 Example 2 of a Truth Table Problem: If any two adjacent switches out of four on-off switches in a row are on (A+D is invalid), light 1 is on. If exactly two out of four switches are on, light 2 is on. We assume that a switch which is on produces a logical 1 馚 as input. Solution: With DCBA as inputs and Light 1 and Light 2 as outputs the number of columns is 6. Since the m = 4 input switch signals are independent there are 2 4 = 16 rows in the truth table. The order of switches in the row is fixed. There is no wrap around from D to A. Therefore in in row no. 9 light 1 is off. Result: The writing of all rows for a truth table clarifies the complete understanding of a problem. Decimal D C B A Light 1 Light Digital Circuits I 4-6
7 4.3 Boolean Theorems No. Name Theorems for two and three signals 1 Commutivity A B C = B A C = A C B =... 2 A B C = B A C = A C B =... 3 Associativity (A B) C = A (B C) = A B C 4 (A B) C = A (B C) = A B C 5 Distributivity (A B) (A C) = A (B C) 6 (A B) (A C) = A (B C) 7 Adsorption A ( A B) = A B 8 A ( A B) = A B 9 Absorption A (A B) = A 10 A A B = A 11 Adjacency A B A B = B 12 (A B) ( A B) = B 13 DeMorgan (A B C D...) = A B C D (A B C D...) = A B C D... Digital Circuits I 4-7
8 Proof of Associativity and Distributivity withtruth Tables Associativity No. 3: Truth table with m = 3 input signals Decimal C B A A B B C (A B) C A (B C) A B C Distributivity No. 5: Truth table with m = 3 input signals Decimal C B A A B A C B C (A B) (A C) A (B C) Digital Circuits I 4-8
9 Laws of XOR Algebra Exclusive OR XOR Truth Table B A A «B XOR Laws A «0 = A A «1 = A «A = 0 A «= 1 XNOR Laws A «1 = A A «0 = A «A = 1 A «= 0 Equality EQV/XNOR Truth Table B A A «B No. Name XOR theorems for two and three signals 1 Distributivity (A B) «(A C) = A (B «C) 2 (A B) «(A C) = A (B «C) 3 Adsorption A ( A «B) = A B 4 A ( A «B) = A B 5 DeMorgan (A«B) = A «B = A «B 6 (A «B) = A «B = A «B Digital Circuits I 4-9
10 4.4 Basic Logic Library German standard symbols are defined in DIN part 12. Logic Functions and Symbols Function Equation DIN-Symbols old US-Symbols INVERTER Y = A US standard symbols are defined in ANSI/IEEE Std The symbols are the same as in DIN with the difference that the inverter bubble is substituted by a triangle. US logic circuit schematics and development software tools are not always adapted to the ANSI standard. Old US symbols are very often in use. AND OR NAND NOR XOR Exclusive OR EQV(XNOR) Equality Y = A Y = A B B Y = (A B) Y = (A B) Y = ( A B) (A B) Y = ( A B) (A B) Digital Circuits I 4-10
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