1. ECE 331 – Digital System Design
Minterms and Maxterms,
Incompletely Specified Functions,
and
Design and Analysis of Combinational Logic Ckts
(Lecture #5)
The slides included herein were taken from the materials accompanying
Fundamentals of Logic Design, 6th Edition, by Roth and Kinney,
and were used with permission from Cengage Learning.

2. ECE 331 - Digital System Design
Minterms and Maxterms
Fall 2010
ECE Digital System Design

3. ECE 331 - Digital System Design
Minterm
In general, a minterm of n variables is a product (ANDing) of n literals in which each variable appears exactly once in either true or complemented form, but not both.
A literal is a variable or its complement.
For a given row in the truth table, the corresponding minterm is formed by
Including the true form a variable if its value is 1.
Including the complemented form of a variable if its value is 0.
Fall 2010
ECE Digital System Design

4. ECE 331 - Digital System Design
Minterms
Fall 2010
ECE Digital System Design

5. ECE 331 - Digital System Design
Minterm Expansion
When a function f is written as a sum (ORing) of minterms, it is referred to as a minterm expansion or a standard sum of products.
aka. “canonical sum of products”
aka. “disjunctive normal form”
If f = 1 for row i of the truth table, then mi must be present in the minterm expansion.
The minterm expansion for a function f is unique.
However, it is not necessarily the lowest cost.
Fall 2010
ECE Digital System Design

6. ECE 331 - Digital System Design
Minterm Expansion
Exercise:
Determine the minterm expansion for the function specified by the following truth table. A B C F 1
Fall 2010
ECE Digital System Design

7. Minterm Expansion Exercise:
Determine the minterm expansion for each of the following Boolean expressions.
F1 = ABC' + AB'C + A'BC + A'BC' + AB'C'
F2 = ABCD + ABC'D' + A'BC'D + AB'C'D
Fall 2010
ECE Digital System Design

8. ECE 331 - Digital System Design
Minterm Expansion
Exercise:
Determine the minterm expansion for each of the following Boolean expressions.
F3 = AB + BC + A'C'
F4 = A.(B + C) + (A' + B).(B + C')
Fall 2010
ECE Digital System Design

9. ECE 331 - Digital System Design
Minterm Expansion
The minterm expansion for a general function of three variables can be written as follows:
Denotes the logical
sum operation
This is the truth table for a general function of three variables.
Each ai is a constant with a value of 0 or 1.
Fall 2010
ECE Digital System Design

10. ECE 331 - Digital System Design
Maxterm
In general, a maxterm of n variables is a sum (ORing) of n literals in which each variable appears exactly once in either true or complemented form, but not both.
A literal is a variable or its complement.
For a given row in the truth table, the corresponding maxterm is formed by
Including the true form a variable if its value is 0.
Including the complemented form of a variable if its value is 1.
Fall 2010
ECE Digital System Design

11. ECE 331 - Digital System Design
Maxterms
Fall 2010
ECE Digital System Design

12. ECE 331 - Digital System Design
Maxterm Expansion
When a function f is written as a product (ANDing) of maxterms, it is referred to as a maxterm expansion or a standard product of sums.
aka. “canonical product of sums”
aka. “conjunctive normal form”
If f = 0 for row i of the truth table, then Mi must be present in the maxterm expansion.
The maxterm expansion for a function f is unique.
However, it is not necessarily the lowest cost.
Fall 2010
ECE Digital System Design

13. ECE 331 - Digital System Design
Maxterm Expansion
Exercise:
Determine the maxterm expansion for the function specified by the following truth table. A B C F 1
Fall 2010
ECE Digital System Design

14. Maxterm Expansion Exercise:
Determine the maxterm expansion for each of the following Boolean expressions.
F1 = (A+B+C')(A+B'+C)(A'+B+C)(A'+B+C')(A+B'+C')
F2 = (A+B+C+D)(A+B+C'+D')(A'+B+C'+D)(A+B'+C'+D)
Fall 2010
ECE Digital System Design

15. ECE 331 - Digital System Design
Maxterm Expansion
Exercise:
Determine the maxterm expansion for each of the following Boolean expressions.
F3 = (A + B')(A + C)(B + C')
F4 = A'.B' + A'.C' + B'.C
Fall 2010
ECE Digital System Design

16. ECE 331 - Digital System Design
Maxterm Expansion
The maxterm expansion for a general function of three variables can be written as follows:
Denotes the logical
product operation
This is the truth table for a general function of three variables.
Each ai is a constant with a value of 0 or 1.
Fall 2010
ECE Digital System Design

17. Conversion between Forms
Fall 2010
ECE Digital System Design

18. ECE 331 - Digital System Design
Circuit Design
A function f can be represented by either a minterm expansion or a maxterm expansion.
Both forms of the function can be realized using logic gates that implement the basic logic operations.
Minterm Expansion (Standard SOP)
Consists of the sum (OR) of product (AND) terms.
Realized using an AND-OR circuit.
Maxterm Expansion (Standard POS)
Consists of the product (AND) of sum (OR) terms.
Realized using an OR-AND circuit.
Fall 2010
ECE Digital System Design

19. ECE 331 - Digital System Design
Circuit Design
However, the two combinational logic circuits realized for function f do not necessarily have the same cost.
Objective: minimize the cost of the circuit to be built.
Synthesis Process
Determine the minterm expansion.
Determine the maxterm expansion.
Use Boolean Algebra (and K-maps) to find an optimal, functionally equivalent, expression for each.
Compare the cost of the two expressions.
Build the “cheaper” circuit.
Fall 2010
ECE Digital System Design

20. Incompletely Specified Functions
Fall 2010
ECE Digital System Design

21. Incompletely Specified Functions
A function f is completely specified when its output is defined (i.e. either 0 or 1) for all combinations of its inputs.
However, if the output of a function f is not defined for all combinations of its inputs, then it is said to be incompletely specified.
Those combinations of the inputs for which the output of function f is not defined are referred to as “don't care” outputs.
Fall 2010
ECE Digital System Design

22. Incompletely Specified Functions
Consider the following example in which the output of circuit N1 drives the input of circuit N2:
Assume the output of N1 does not generate all possible combinations of values for A, B, and C.
In particular, assume there are no combinations of values for w, x, y, and z which cause A, B, and C to assume values of 001 or 110.
Fall 2010
ECE Digital System Design

23. Incompletely Specified Functions
The truth table representing an incompletely specified function includes an “x” (or a “d”) in each row corresponding to an input combination for which the output is not defined.
For the example under consideration:
“don't care” for ABC = 001
“don't care” for ABC = 110
Fall 2010
ECE Digital System Design

24. Incompletely Specified Functions
When we realize the function, we must specify values for the don’t-cares. It is desirable to choose values which will help simplify the function. If we assign the value 0 to both X’s, then
If we assign 1 to the first X and 0 to the second, then
If we assign 1 to both X’s, then
The second choice of values leads to the simplest solution.
Fall 2010
ECE Digital System Design

25. Incompletely Specified Functions
The minterm expansion for this example is:
“don't care” minterms
The maxterm expansion for this example is:
“don't care” maxterms
Fall 2010
ECE Digital System Design

26. Combinational Logic Circuits
Design and Analysis of
Combinational Logic Circuits
Fall 2010
ECE Digital System Design

27. ECE 331 - Digital System Design
Combinational Logic Circuits
Outputs are functions of (present) inputs
No memory
Can be described using Boolean expressions
Can be described using Truth tables
Fall 2010
ECE Digital System Design

28. ECE 331 - Digital System Design
Hierarchical design
Used to solve large design problems
Break problem into smaller (sub-)problems
Solve each sub-problem (i.e. realize design)
Combine individual solutions
Fall 2010
ECE Digital System Design

29. ECE 331 - Digital System Design
Specification
Describes the problem to be solved.
Describes what needs to be done,
not how to do it.
Implementation
Describes how the problem is solved.
Fall 2010
ECE Digital System Design

30. ECE 331 - Digital System Design
Issues
Most solutions are not unique.
More than one solution may meet the specifications.
Cannot always satisfy all of the requirements.
Must identify (and study) design tradeoffs.
Cost
Speed
Power consumption
etc.
Fall 2010
ECE Digital System Design

31. ECE 331 - Digital System Design
Design Process
Identify requirements
circuit specifications
Determine the inputs and outputs.
Derive the Truth Table
Determine simplified Boolean expression(s)
Implement solution
Verify solution
Fall 2010
ECE Digital System Design

32. ECE 331 - Digital System Design
Design: Exercise
We will design a simple binary adder that adds two 1-bit binary numbers, a and b, to give a 2-bit sum. The numeric values for the adder inputs and outputs are as follows:
Fall 2010
ECE Digital System Design

33. ECE 331 - Digital System Design
Design: Exercise
Design a combinational logic circuit to meet the following specifications:
1. 3-bit input (A = a2a1a0)
2. 1-bit output (z)
3. Output is high (logic 1) when 2 < A <= 5.
ECE Digital System Design

34. Analysis Determining the behavior of a system given its description.
The description of the system is often provided
in the form of a circuit diagram.
Fall 2010
ECE Digital System Design

35. ECE 331 - Digital System Design
Analysis
For two-level circuits, the analysis process is simple.
The Boolean expression representing the circuit can often be written by inspection.
For multilevel circuits, the analysis process is much more complicated.
Cannot write a Boolean expression by inspection.
Must follow a procedure to implement the analysis.
Fall 2010
ECE Digital System Design

36. ECE 331 - Digital System Design
Analysis Process
Identify inputs and outputs
Track circuit behavior from input to output
Determine Boolean expression for output(s)
Determine Truth Table
Examine circuit timing, power dissipation, etc.
Fall 2010
ECE Digital System Design

37. Analysis: Exercise Analyze the following combinational logic circuit:
1. Derive the Boolean expression
2. Determine the Truth table
ECE Digital System Design

38. ECE 331 - Digital System Design
Questions?
Fall 2010
ECE Digital System Design