ENEE244-02xx Digital Logic Design Lecture 10. Announcements HW4 due 10/9 – Please omit last problem 4.6(a),(c) Quiz during recitation on Monday (10/13)

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Presentation transcript:

ENEE244-02xx Digital Logic Design Lecture 10

Announcements HW4 due 10/9 – Please omit last problem 4.6(a),(c) Quiz during recitation on Monday (10/13) – Will cover material from lectures 10,11,12.

Agenda Last time: – Prime Implicants (4.2) – Prime Implicates (4.3) This time: – Karnaugh Maps (4.4)

Karnaugh Maps

Three-Variable Maps Each cell is adjacent to 3 other cells. Imagine the map lying on the surface of a cylinder

Four-Variable Maps Each cell is adjacent to 4 other cells. Imagine the map lying on the surface of a torus

Karnaugh Maps and Canonical Formulas Minterm Canonical Formula

Karnaugh Maps and Canonical Formulas Maxterm Canonical Formula

Karnaugh Maps and Canonical Formulas Decimal Representation

Karnaugh Maps and Canonical Formulas Decimal Representation

Product Term Representations on Karnaugh Maps

Examples of Subcubes

Subcubes for elimination of one variable Variables in the product term are variables whose value is constant inside the subcube.

Subcubes for elimination of one variable

Subcubes for elimination of one variable

Subcubes for elimination of one variable

Subcubes for elimination of two variables

Subcubes for elimination of two variables

Subcubes for elimination of two variables

Subcubes for elimination of two variables

Subcubes for elimination of two variables

Subcubes for elimination of three variables

Subcubes for elimination of three variables

Subcubes for elimination of three variables

Subcubes for elimination of three variables

Subcubes for sum terms

Using K-Maps to Obtain Minimal Boolean Expressions

Example Both are prime implicants How do we know this is minimal? Need an algorithmic procedure.

Finding the set of all prime implicants in an n-variable map:

Finding the set of all prime implicants Subcubes consisting of 16 cells—None Subcubes consisting of 8 cells—None Subcubes consisting of 4 cells in blue. Subcubes consisting of 2 cells in red (not contained in another subcube). Subcubes consisting of 1 cell (not contained in another subcube)— None.

Essential Prime Implicants Some 1-cells appear in only one prime implicant subcube, others appear in more than one. A 1-cell that can be in only one prime implicant is called an essential prime implicant

Essential Prime Implicants Every essential prime implicant must appear in all the irredundant disjunctive normal formulas of the function. Hence must also appear in a minimal sum. – Why?

General Approach for Finding Minimal Sums Find all prime implicants using K-map Find all essential prime implicants using K- map **If all 1-cells are not yet covered, determine optimal choice of remaining prime implicants using K-map.

Next time: Lots of examples of finding minimal sums using K-maps