CS151 Introduction to Digital Design Chapter 2: Combinational Logic Circuits 2-9 Exclusive-OR Operator and Gates 1Created by: Ms.Amany AlSaleh

2 Exclusive-OR and Exclusive-NOR Circuits  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. Created by: Ms.Amany AlSaleh

3 XOR and XNOR (Cont.)  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  Created by: Ms.Amany AlSaleh

4 Exclusive-OR Circuit  Exclusive-OR (XOR) produces a HIGH output whenever the two inputs are at opposite levels.  The XOR function means: X OR Y, but NOT BOTH Created by: Ms.Amany AlSaleh

5 Exclusive-OR Circuit  XOR function can also be implemented with AND/OR gates (also NANDs). Created by: Ms.Amany AlSaleh

6 Exclusive-NOR Circuits  Exclusive-NOR (XNOR) produces a HIGH output whenever the two inputs are at the same level.  Why is the XNOR function also known as the equivalence function? Created by: Ms.Amany AlSaleh

7 XOR/XNOR (Cont.)  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 Created by: Ms.Amany AlSaleh

8 XOR/XNOR (Cont.) 3-input exclusive-OR (XOR) logic gate: F= X Y Z F x y z x F z y XYZF  Created by: Ms.Amany AlSaleh

9 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  This function is equal to 1 only if one variable is equal to 1 or if all three variables are equal to 1. This implies that an odd number of variables must be one. This is defined as an odd function. 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.  The complement of an odd function is 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 Created by: Ms.Amany AlSaleh

10 Odd and Even Functions (Cont.)  Map for 3-variable XOR function. Minterms are said to be distance 2 from each other. Created by: Ms.Amany AlSaleh

11 Odd and Even Functions (Cont.)  Map for 3- and 4- variable XOR functions. Created by: Ms.Amany AlSaleh

12 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 Created by: Ms.Amany AlSaleh

13 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 Created by: Ms.Amany AlSaleh

14 Odd and Even Functions  Why? Implementation feasibility and low cost Power in implementing Boolean functions Convenient conceptual representation  Gate classifications Primitive gate - a gate that can be described using a single primitive operation type (AND or OR) plus an optional inversion(s). Complex gate - a gate that requires more than one primitive operation type for its description  Primitive gates will be covered first Created by: Ms.Amany AlSaleh