CS 151: Introduction to Digital Design Chapter 2-9 Exclusive-OR Gates

CS 151 Complex Digital Logic Gates

CS 151 Exclusive OR/ Exclusive NOR The eXclusive OR (XOR) function is an important Boolean function used extensively in logic circuits. The XOR function may be;  implemented directly as an electronic circuit (truly a gate) or  implemented by interconnecting other gate types (used as a convenient representation) The eXclusive NOR function is the complement of the XOR function By our definition, XOR and XNOR gates are complex gates.

CS 151 Exclusive OR/ Exclusive NOR Uses for the XOR and XNORs gate include:  Adders/subtractors/multipliers  Counters/incrementers/decrementers  Parity generators/checkers Definitions  The XOR function is:  The eXclusive NOR (XNOR) function, otherwise known as equivalence is: Strictly speaking, XOR and XNOR gates do no exist for more that two inputs. Instead, they are replaced by odd and even functions. YXYXYX  YXYXYX 

CS 151 Truth Tables for XOR/XNOR Operator Rules: XOR XNOR The XOR function means: X OR Y, but NOT BOTH Why is the XNOR function also known as the equivalence function, denoted by the operator  ? X Y X  Y X Y or X  Y (X  Y)

CS 151 XOR/XNOR (Continued) The XOR function can be extended to 3 or more variables. For more than 2 variables, it is called an odd function or modulo 2 sum (Mod 2 sum), not an XOR: The complement of the odd function is the even function. The XOR identities:  X1XX0X   1XX0XX     XYYX    ZYX)ZY(XZ ) YX(        ZYXZYXZYXZYXZYX

CS 151 Symbols For XOR and XNOR XOR symbol: XNOR symbol: Symbols exist only for two inputs

CS 151 XOR Implementations The simple SOP implementation uses the following structure: A NAND only implementation is: X Y X Y X Y X Y

CS 151 Odd and Even Functions What about the case of more than 2 variables? A  B  C = (AB’ + A’B) C’ + (AB + A’B’) C = AB’C’ + A’BC’ + ABC + A’B’C Then, F is the logical sum of the four minterms having an index with an odd number of 1’s  F is called an odd function The other four minterms not included are 000, 011, 101 and 110, they have an index with even number of 1’s  F’ is called an even function. In general, an n-variable exclusive-OR function is an odd function defined as the logical sum of the 2 n /2 minterms whose binary index have an odd number of 1’s

CS 151 Odd and Even Functions

CS 151 Example: Odd Function Implementation Design a 3-input odd function F = X  Y  Z with 2-input XOR gates Factoring, F = (X  Y)  Z The circuit: X Y Z F

CS 151 Example: Even Function Implementation Design a 4-input even function F = (W  X  Y  Z)’ with 2-input XOR and XNOR gates Factoring, F = ( (W  X)  (Y  Z) )’ The circuit: W X Y F Z

CS 151 Parity Generators and Checkers Example: n = 3. Generate even parity code words of length 4: X Y Z P P=X  Y  Z

CS 151 Parity Generators and Checkers-Cont. X Y Z E P Check even parity code words of length 4 with odd parity checker: E should indicate an error (be true) when (X,Y,Z,P) have an odd number of 1’s.  E = X  Y  Z  P Example: operation (X,Y,Z) = (0,0,1) gives (X,Y,Z,P) = (0,0,1,1) and E = 0. If Y changes from 0 to 1 between generator and checker, then E = 1 indicates an error.