1 ENGR 254 Lecture

2 DeMorgan Symbol Equivalence

3 Likewise for OR (be sure to check errata!) FIG 4-4

4 DeMorgan Symbols

5 Definitions (Sec ) Literal – A variable or the complement of a variable. Example X, Y, Y’, etc. Product term – A single literal or product of literals. Example: Sum-of-products expression – A logical sum of product terms. Example: Sum term – A single literal of sum of literals. Example: Product-of-sums expression – A logical product of sum terms. Example:

6 Definitions Normal term – A product or sum term in which no variable appears more than once. –Example non-normal: –Example normal: Minterm (n variables) – A normal product term with n literals. Example: Maxterm (n variables) – A normal sum term with n literals. Example:

7 Truth table vs. minterms & maxterms

8 Combinational analysis XYZF

9 Signal expressions Multiply out: F = ((X + Y)  Z) + (X  Y  Z) = (X  Z) + (Y  Z) + (X  Y  Z)

10 New circuit, same function F = X  Z + Y  Z + X  Y  Z

11 “Add out” logic function Circuit:

12 Shortcut: Symbol substitution

13 Different circuit, same function

14 Another example

15 Sum-of-products form AND-OR NAND-NAND

16 Product-of-sums form OR-AND NOR-NOR P-of-S preferred in CMOS, TTL (NAND-NAND)

17 Brute-force design Truth table --> canonical sum (sum of minterms) Example: prime-number detector –4-bit input, N 3 N 2 N 1 N 0 row N 3 N 2 N 1 N 0 F F(N 3, N 2, N 1, N 0 ) =   (1,2,3,5,7,11,13)

18 Minterm list --> canonical sum

19 Algebraic simplification Theorem T8, Reduce number of gates and gate inputs

20 Resulting circuit