Logic Function Optimization

Combinational Logic CircuitRegular SOP and POS designsDo not care expressionsDigital logic circuit applicationsKarnaugh MapsMinimization of logic functions

Combinational Logic CircuitThe function D=ABC+ABC+ABC+ABCcan be implemented in the sum of products (SOP) form as follows:Minterm:Product that contains all input variables or their complements for which function value is 1

Combinational Logic CircuitThe function D=(A+B+C)(A+B+C)(A+B+C)(A+B+C)can be implemented in the product of sums (POS) form as follows:Notice that SOP and POS forms can be implemented in such regular designs for any function.

Combinational Logic CircuitFind canonical sum of products (SOP) form for the following function F=AB+AC+ABCcan be implemented in the :We have:F=ABC+ABC+ABC+ABC+ABC

Combinational Logic CircuitTheorem:Each function can be represented in the unique form of SOP or POS.

Let us obtain POS from for the following function:D=ABC+ABC+ABC+ABC

Using DeMorgans lawD=(ABC+ABC+ABC+ABC)==(A+B+C)(A+B+C)(A+B+C)(A+B+C)

Two Implementations of XOR Logic Circuit

Do not Care - Logic CircuitDo not care condition is when the output function value is not important for a specific input combination

Do not Care - Logic CircuitDo not care can be used to simplify the circuit implementation

To design a digital logic circuit to control LED displaywe must first come up with a truth table.How to translate a desired circuit functionality into a truth table?How to Design a Digital Logic Circuit?

Digital Logic Circuit OutputThen each output ABCDEFG must be implemented as a separate logic function of 4 input bits xywz that represent BCD code of integer value

Decimal IntegerBCDxywzABCDEFGsegments000001111110100010110000200101101101300111111001401000110011501011011011601100011111701111110000810001111111910011110011

Digital Logic Circuit OutputFor instance to control the segment A the logic function is

FA=abcd+abc+abd+abc

or for segment BFB=ab+abcd+abcd+abc

Decimal IntegerBCDabcdABCDEFG000001111110100010110000200101101101300111111001401000110011501011011011601100011111701111110000810001111111910011110011

Binary-OctalDecoder CircuitDesign example

Karnaugh Maps in Logic CircuitKarnaugh maps can be used to simplify the logic function and its design

Karnaugh Maps in Logic CircuitKarnaugh map uses hypercube with the same function value in the whole cube to reduce the number of terms in the logic function

Karnaugh Maps in Logic CircuitThe whole cube in the Karnaugh map is represented by a single product term. For instance cubes in the example figure are.

(a) ab(b) ab(c) ab

Karnaugh Maps in Logic Circuit

Karnaugh Maps in Logic CircuitUse Karnaugh maps to obtain minimum SOP for these functions:

Z=WXY+WXY+WXY+WXYD=ABC+AB+ABCE=ABD+ABCD+ABCD

Do not Care - Logic CircuitF=L+DH

Digital Logic Circuit OutputUsing Karnaugh map we can minimize

FA=abcd+abc+abd+abc=bcd+abc+abd+abc

ACDB

Decimal IntegerBCDabcdABCDEFG000001111110100010110000200101101101300111111001401000110011501011011011601100011111701111110000810001111111910011110011

1111111

Segment A logic function isFA=abcd+abc+abd+abc

With do not caresFA=a+abc+bd+cb

ACDBOutput A of 7 Segments Decoder

Decimal IntegerBCDabcdABCDEFG000001111110100010110000200101101101300111111001401000110011501011011011601100011111701111110000810001111111910011110011

10011xx1xxx0111

Digital Logic Circuit OutputUsing Karnaugh map we can minimizeFB=ab+abcd+abcd+abc==bc+acd+acd+ ab

BACD

Decimal IntegerBCDabcdABCDEFG000001111110100010110000200101101101300111111001401000110011501011011011601100011111701111110000810001111111910011110011

11111111

Digital Logic Circuit OutputUsing Karnaugh map we can minimizeFB=ab+abcd+abcd+abc==ab+ bc+acd+acdWith do not caresFB=cd+cd+b

BACD

Decimal IntegerBCDabcdABCDEFG000001111110100010110000200101101101300111111001401000110011501011011011601100011111701111110000810001111111910011110011

11011xx1xx1011

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