Simplifying Expressions

Module 5

Reading Assignment: (4^th Ed) pp: 54-72 K-map Tool as a Study Guide	Schedule of Lectures	Sample Problems 2,3,7
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Goal

• Every logic function can be implemented by a two level circuits à impractical
• Find a circuit with the lowest implementation cost
• Number of terms corresponds to the number of gates
• Number of literals corresponds to the total number of connections

Algebraic Methods

• Adjacency of SOP/POS terms and minimization

E.x + E.x' = E
(E + x)(E+x') = E

• Difficulty with algebraic methods

K-Maps: Structure

• K-maps are structural representations of algebraic methods
• Representation: translating truth tables to K-Maps
• 2-variable
• 3-variable
• 4-variable
• Recognition: reading terms off K-Map
• Minterms and maxterms: example
• Redundant terms example
• must circle in blocks of 2^N
• can wrap circles around edges
• larger circles eliminate more terms : example
• More examples

K-Maps:Minimization

• Implicants, prime implicants, and essential prime implicants
• Optimization: 
• Finding implicants
• example
• Finding prime implicants
• example
• Finding essential prime implicants
• definition: a prime implicant necessary to implement the function
• contains at least 1 implicant that is not covered by any other prime implicant
• example
• Final function contains ALL essential prime implicants and some subset of nonessential prime implicants
• Therefore, to ensure a minimum POS or SOP solution, first circle the essential prime implicants, then determine how to circle non-essential prime implicants with the least number of circles
• example
• Product of Sums optimizations
• Find essential prime implicants using the 0s: these minterms identify the complement of the function F.
• Optimize for F’ as we have for F
• Take the complement of F’ to obtain F.
• Presence of Don't Cares
• Don't care may be interpreted as either a 1 or 0
• Selection to optimize the resulting circuit
• Example