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Kmap

Karnaugh Maps (K-Maps)


A visual way to simplify logic expressions
It gives the most simplified form of the expression

Rules to obtain the most simplified expression


Simplification of logic expression using Boolean algebra is awkward because:
it lacks specific rules to predict the most suitable next step in the simplification process
it is difficult to determine whether the simplest form has been achieved.

A Karnaugh map is a graphical method used to obtained the most simplified form of an expression in a standard form (Sum-of-Products or Product-of-Sums).

The simplest form of an expression is the one that has the minimum number of terms with the least number of literals (variables) in each term.

By simplifying an expression to the one that uses the minimum number of terms, we ensure that the function will be implemented with the minimum number of gates.

By simplifying an expression to the one that uses the least number of literals for each terms, we ensure that the function will be implemented with gates that have the minimum number of inputs. 

Design of combinational digital circuits


Steps to design a combinational digital circuit

From the problem statement derive the truth table
From the truth table derive the unsimplified logic expression
Simplify the logic expression 
From the simplified expression draw the logic circuit

Example: Design a 3-input (A,B,C) digital circuit that will give at its output (X) a logic 1 only if the binary number formed at the input has more ones than zeros.


K-mapping & Minimization Steps


Step 1: generate K-map
Put a 1 in all specified minterms
Put a 0 in all other boxes (optional)
Step 2: group all adjacent 1s without including any 0s
All groups (aka prime implicants) must be rectangular and
contain a “power-of-2” number of 1s
 1, 2, 4, 8, 16, 32, …
 An essential group (aka essential prime implicant) contains
at least 1 minterm not included in any other groups
 A given minterm may be included in multiple groups
Step 3: define product terms using variables common to
all minterms in group
Step 4: sum all essential groups plus a minimal set of
remaining groups to obtain a minimum SOP


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