DeMorgan’s Law and Bubble Pushing A B A B A B A B + = • • = + SI232 Slide Set #7: Digital Logic (more Appendix B) 1 2 Bubble Pushing Example Representing Combinational Logic Truth Table Boolean Formula Circuit For combinational logic, these three: - are equivalently _____________ - straight-forward to ____________ - have no ______________ 3 4
2-Level Logic Example • Show the sum of products for the following truth table. • Represent ______ logic function(s) • Strategy: _________ all the products where the output is ________ – Utilizing just two types of gates A B C z 0 0 0 0 0 0 1 1 0 1 0 0 (assuming we get NOT for free) 0 1 1 0 – Two forms 1 0 0 1 1 0 1 1 • Sum of products 1 1 0 0 • Product of sums 1 1 1 1 – Relationship with truth table • Generate a gate level implementation of any set of • z = logic functions • Allows for simple reduction/minimization • Is this optimal? 5 6 Exercise #1 Exercise #2 • Show the sum of products for the following truth table. • Simplify the following equations (use Boolean laws discussed earlier) B ( A + ) 0 = A B C f 0 0 0 1 0 0 1 0 B ( A A ) = 0 1 0 1 0 1 1 1 ( A B )( A B ) + + = 1 0 0 0 1 0 1 1 1 1 0 0 ( A B ) ( A B C ) + • + + = 1 1 1 0 A B the same as AB ? Is 7 8
Exercise #3 Exercise #4 • A) Show the sum of products for the following truth table. • Use bubble pushing to simplify this circuit A B C f 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 • B) Simplify this equation 9 10 Reduction/Minimization Minimization by Hand A B C z 0 0 0 0 0 0 1 1 • Sum of Products: Truth Table: • Reduction is important to reduce the size of the circuit that performs 0 1 0 0 0 1 1 0 the function. This, in turn, reduces the cost of, and delay through, 1 0 0 1 z ( A B C ) ( A B C ) ( A B C ) ( A B C ) the circuit. = • • + • • + • • + • • 1 0 1 1 1 1 0 0 1 1 1 1 • What? – Less power consumption – Less heat – Less space – Less time to propagate a signal through the circuit – Less points of possible failure • It makes good engineering and economic sense! • Okay to duplicate terms while minimizing 11 12
Karnaugh Maps (k-Maps) Karnaugh Maps (k-Maps) Example #1 • Lets create a k-map table A B C z • A graphical (pictorial) method used to minimize Boolean – Borders represent all possible conditions 0 0 0 0 expressions. – NOT in counting order 0 0 1 1 0 1 0 0 • Don’t require the use of Boolean algebra theorems and equation – Be consistent 0 1 1 0 1 0 0 1 manipulations. • -What are the values for the map? 1 0 1 1 1 1 0 0 – The values of ___ • A special version of a truth table. 1 1 1 1 • To reduce, circle our powers of 2! • Works with two to four input variables (gets more and more difficult with more variables) B C B C BC B C • Groupings must be __________________ A • Final result is in _____________________ form A • Result: 13 14 K-Maps Example #2 Truth Table and Logical Circuit Example • Suppose we already have this k-Map. Minimize the function. • How does a truth table and subsequent sum of products equation create a logic circuit? C D C D CD C D • From the earlier example: A B 1 0 0 0 z = + + B • C A • B A • C A B 0 0 0 1 • Lets build the logical circuit: AB 0 1 1 0 – Which gates do we need? A B 1 1 1 1 – How many inputs do we have? – How do we connect the circuit? • Every “1” must be ____________ by at least one term • Larger blocks in k-Map produce smaller product terms 15 16
Example Circuit Exercise #1 A B C f • 1. Fill in the following K-Map based on the truth z = + + B • C A • B A • C 1 0 0 0 table at right 0 0 1 1 • 2. Minimize the function using the K-map 0 1 0 0 0 A 0 1 1 1 0 0 1 1 0 1 0 1 1 0 1 z B 1 1 1 0 C 17 18 Exercise #2 Exercise #3 • Suppose we already have this k-Map. Minimize the function. • Draw the two-level circuit for the function from Exercise #1 C D C D CD C D A B 1 0 0 1 A B 1 1 1 1 AB 1 1 0 0 A B 0 0 0 1 19 20
Exercise #4 General Skills • Does the function from Exercise #3 have a unique, two-level circuit of • Make sure you can populate a K-Map from a truth table minimal size? Will this always be the case? • Make sure you can populate a truth table from a K-Map • Given a circuit, know how to construct a truth table • Given a truth table, know how to produce a sum-of-products, and how to draw a circuit • Be able to understand minimization and use it • Know DeMorgan’s Law and other Boolean laws 21 22
Recommend
More recommend
