Chapter 3

NAND

- Universal gate: any logic circuit can be implemented with it. - The figure shows the equivalent NAND to NOT, OR, and AND.

three variables K-map squares distribution.

- one square represents one minter, giving a term with three literals. - two adjacent squares represent a term with two literals - four adjacent squares represent a term with one literal - eight adjacent squares encompass the entire three-variable map and produce a function that is always equal to 1.

Choosing adjacent squares

1.) all the minterms of the function are covered when we combine the squares; 2.) the number of terms in the expression is minimized, and 3.) there are no redundant terms.

Convert to NAND

A convenient way to implement a Boolean function with NAND gates is to obtain the simplified Boolean function in terms of Boolean operators and then convert the function to NAND logic.

In three-variable K-map

Any two adjacent squares in the map differ by only one variable, which is primed in one square and unprimed in other.

Incompletely specified function

Functions that have unspecified outputs for some input combinations.

two-variable K-map

In the map, the minterms at which the function is asserted are marked with a 1.

three-variable K-map

Just one bit changes in value from one adjacent column to the next

K-maps

To solve K-maps is necessary to make sure that the algebraic expression is in sum-of-products form. Each product term can be plotted in the map in one, two, or more squares. The minterms of the function are then read directly from the map.

Essential Prime Implicant

a minterm in a square covered by only one prime implicant. It cannot be removed from a description of the function. a prime implicant is essential if it is the only prime implicant that covers the minterm.

implicant

a product term in regard to the function it belongs

Don't care minterm

is a combination of variables whose logical value is not specified. In the map, they are indicated by a X. In choosing adjacent squares to simplify the function in a map, the don't care minterms may assume to be either, 0's or 1's. We choose based on which one will generate the simplest expression.

K-map

is a diagram made up of squares, with each square representing one minterm of the function that is to be minimized.

Prime Implicant

is a product term obtained by combining the maximum possible number of adjacent squares on the map. The prime implicants of a function can be obtained from the map by combining all possible maximum number of squares.

Simplified expression

is obtained from the logical sum of all essential prime implicants, plus other prime implicants that may be needed to cover any remaining minterms not covered by the essential prime implicants.

simplest algebraic expression

is one that has a minimum number of terms with the smallest possible number of literals in each term.

gate-level minmization

is the design task of finding an optimal gate-level implementation of the Boolean functions describing a digital circuit.

Two-level implementation

it requires that the functions be in sum-of-products form.

Don't care Conditions

unspecified minterms of a function