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XOR Operator

XOR. XOR Operator. A short digression… … to introduce another Boolean operation: exclusive-OR (XOR). XOR Operator. Also referred to as an “odd” function since it returns a 1 only when an odd number of 1’s are input. Simplification.

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XOR Operator

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  1. XOR XOR Operator • A short digression… • … to introduce another Boolean operation: exclusive-OR (XOR)

  2. XOR Operator • Also referred to as an “odd” function since it returns a 1 only when an odd number of 1’s are input

  3. Simplification • Using the axioms to “prove” that a simplified version of a circuit is equivalent to the complex version takes a special kind of person… • …of which I’m not one • Fortunately, there’s another way…

  4. Karnaugh Maps • Also known as K-Map • Recall that an expression can be written in the form F(A,B,C) = Σ(0,2,4,5,6) • Which means the functional value is 1 at binary input patterns 0, 2, 4, 5, 6 and 0 at all other input patterns • What does the truth table look like?

  5. K-Maps • F(A,B,C) = Σ(0,2,4,5,6) is called a “sum of minterms” representation • The expression for such a representation is F(A,B,C) = A’B’C’ + A’BC’ + AB’C’ + AB’C + ABC’ • We could simplify this via the axioms, right? (assuming we were that special kind of person) • It’s painful!!!

  6. BC B A A A 0 1 0 1 00 01 11 10 0 1 0 0 1 0 1 3 2 0 1 3 2 4 5 7 8 1 CD AB 00 01 11 10 0 1 3 2 00 01 4 5 7 8 11 12 13 15 14 10 8 9 11 10 K-Maps • A K-Map is a grid (map) where each square corresponds to a minterm Note the ordering here is Gray code, not binary

  7. K-Maps • Notice how neighboring squares (minterms) differ by a single bit…this is the key to the whole thing • Consider minterms 1 and 3 • 1: A’B’C • 3: A’BC • If we were to OR these together • (A’B’C + A’BC) would simplify to A’C via the axioms

  8. K-Maps • Great, now what do we do with them? • Place 1’s on the squares that correspond to minterms in the truth table • Place 0’s on all other squares • Group adjacent 1’s into the largest group whose size is a power of 2 • See example

  9. K-Maps • Notes: • Adjacencies wrap top-to-bottom and left-to-right • 1’s can be part of more than one group • When you are grouping adjacent squares you’re essentially applying axiom 4 (x + x’ = 1) so the variable that is being “spanned” can be removed from the minterm

  10. K-Maps • We’ll do more of this next time so continue reading in Chapter 1

  11. Assignment #4 • Implement the method BooleanOperator in the given class public class BooleanOperator { static final int AND = 0; static final int OR = 1; static final int NOT = 2; static final int NAND = 3; static final int NOR = 4; static final int XOR = 5; static String BooleanOperator (String _lhs, String _rhs, int _oper) { String result = new String(); // -- YOUR CODE TO ASSIGN SOMETHING TO result GOES HERE return result; } }

  12. Assignment #4 • The method should perform as follows • _lhs is a string of 0’s and 1’s corresponding to the left hand side of a Boolean operation • _rhs is a string of 0’s and 1’s corresponding to the right hand side of a Boolean operation • _oper specifies the Boolean operation to be performed • The method should perform the specified operation on each of the “bit-pairs” in the two strings

  13. Assignment #4 • For example… • …should return the string “0001” • The other operators should work in a similar manner returning the appropriate bit patterns in the string • The NOT operation should return the inverse of the _lhs operand (ignore the _rhs operand) • If _lhs and _rhs are not the same length, your method should pad the shorter one with ‘0’s in the MSBs prior to performing the operation • i.e. _lhs = “11”, _rhs = “1” becomes _lhs = “11”, _rhs = “01” String result = BooleanOperator.BooleanOperator(“0011”, “0101”, BooleanOperator.AND);

  14. Assignment #4 • Create a test class (e.g. TestBoolean.java) to test your code by • Print out truth tables for each of the operators • Answer questions 1-1 and 1-2 on page 37 of the text. • Note that you may have to “chain” operations • i.e. ABC must be done in two steps (AB)C which is allowed by the axioms

  15. Assignment #4 • Also, do problems 1-3, 1-4, 1-5, 1-6, and 1-7 on page 37 of the text • You won’t be using your program on these problems

  16. Assignment #4 • Turn in • A print out of your BooleanOperator.java • A print out of your TestBoolean.java • A print out of a screen shot of your results • Your answers to problems 1-3 through 1-7 • Due next lecture (Tuesday)

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