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Minterms and Maxterms, Incompletely Specified Functions, and Design and Analysis of Combinational Logic Ckts (Lecture #5). ECE 331 – Digital System Design. The slides included herein were taken from the materials accompanying

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slide1
Minterms and Maxterms,

Incompletely Specified Functions,

and

Design and Analysis of Combinational Logic Ckts

(Lecture #5)

ECE 331 – Digital System Design

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.

slide2
ECE 331 - Digital System Design

Minterms and Maxterms

minterm
ECE 331 - Digital System DesignMinterm
  • 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.
minterm expansion
ECE 331 - Digital System DesignMinterm 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.
exercise determine the minterm expansion for the function specified by the following truth table
ECE 331 - Digital System Design

Exercise:

Determine the minterm expansion for the function specified by the following truth table.

Minterm Expansion
minterm expansion1
ECE 331 - Digital System Design

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

Minterm Expansion
minterm expansion2
ECE 331 - Digital System Design

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')

Minterm Expansion
minterm expansion3
ECE 331 - Digital System DesignMinterm 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.

maxterm
ECE 331 - Digital System DesignMaxterm
  • 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.
maxterm expansion
ECE 331 - Digital System DesignMaxterm 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.
exercise determine the maxterm expansion for the function specified by the following truth table
ECE 331 - Digital System Design

Exercise:

Determine the maxterm expansion for the function specified by the following truth table.

Maxterm Expansion
maxterm expansion1
ECE 331 - Digital System Design

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)

Maxterm Expansion
maxterm expansion2
ECE 331 - Digital System Design

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

Maxterm Expansion
maxterm expansion3
ECE 331 - Digital System DesignMaxterm 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.

circuit design
ECE 331 - Digital System DesignCircuit 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.
circuit design1
ECE 331 - Digital System DesignCircuit 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.
slide20
ECE 331 - Digital System Design

Incompletely Specified Functions

incompletely specified functions
ECE 331 - Digital System DesignIncompletely 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.
incompletely specified functions1
ECE 331 - Digital System DesignIncompletely 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.

incompletely specified functions2
ECE 331 - Digital System DesignIncompletely 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

incompletely specified functions3
ECE 331 - Digital System DesignIncompletely 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.

incompletely specified functions4
ECE 331 - Digital System DesignIncompletely Specified Functions

The minterm expansion for this example is:

“don't care” minterms

The maxterm expansion for this example is:

“don't care” maxterms

slide26
ECE 331 - Digital System Design

Design and Analysis

of

Combinational Logic Circuits

design
ECE 331 - Digital System DesignDesign
  • Combinational Logic Circuits
    • Outputs are functions of (present) inputs
    • No memory
    • Can be described using Boolean expressions
    • Can be described using Truth tables
design1
ECE 331 - Digital System DesignDesign
  • 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
design2
ECE 331 - Digital System DesignDesign
  • 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.
design3
ECE 331 - Digital System DesignDesign
  • 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.
design process
ECE 331 - Digital System DesignDesign Process
  • Identify requirements
    • circuit specifications
  • Determine the inputs and outputs.
  • Derive the Truth Table
  • Determine simplified Boolean expression(s)
  • Implement solution
  • Verify solution
design exercise
ECE 331 - Digital System DesignDesign: 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:

design exercise1
ECE 331 - Digital System Design

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.

Design: Exercise
analysis
ECE 331 - Digital System Design

Determining the behavior of a system given its description.

The description of the system is often provided

in the form of a circuit diagram.

Analysis
analysis1
ECE 331 - Digital System DesignAnalysis
  • 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.
analysis process
ECE 331 - Digital System DesignAnalysis 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.
slide37
ECE 331 - Digital System Design

Analyze the following combinational logic circuit:

1. Derive the Boolean expression

2. Determine the Truth table

Analysis: Exercise