CS 121 Digital Logic Design Gate-Level Minimization Chapter 3

Outline  3.1 Introduction  3.2 The Map Method  3.3 Four-Variable Map  3.4 Product of sums simplification  3.5 Don‘t Care Conditions  3.7 NAND and NOR Implementaion  3.8 Other Two-Level Implementaion  3.9 Exclusive-OR function

3.7 NAND and NOR Implementation (1-15)  Digital circuits are frequently constructed with NAND or NOR gates rather than with AND and OR gates.

3.7 NAND and NOR Implementation (2-15)  NAND gate: a universal gate.  Any digital system can be implemented with it. NAND Implementation

3.7 NAND and NOR Implementation (3-15)  To facilitate the conversion to NAND logic, there are alternative graphic symbol for it. NAND Implementation

3.7 NAND and NOR Implementation (4-15)  Procedures of Implementation with two levels of NAND gates: 1. Express simplified function in sum of products form. 2. Draw a NAND gate for each product term that has at least two literals to constitute a group of first-level gates 3. Draw a single gate using AND-invert or invert-OR in the second level 4. A term with a single literal requires an inverter in the first level. NAND Implementation Two-Level Implementation

3.7 NAND and NOR Implementation (5-15) NAND Implementation Two-Level Implementation F = AB + CD = [(AB + CD)’]’ = [(AB)’*(CD)’]’

3.7 NAND and NOR Implementation (6-15) NAND Implementation Two-Level Implementation Example (3.10): F(X,Y,Z) = ∑ (1,2,3,4,5,7) y z x Z X’Y 1 1 XY’ F = XY’ + X’Y + Z

3.7 NAND and NOR Implementation (7-15)  Procedures of Implementation with multilevel of NAND gates: 1. Convert all AND gates to NAND gates with AND- invert graphic symbols 2. Convert all OR gates to NANDgates with invert- OR graphic symbols 3. Check all the bubbles in the diagrams. For a single bubble, invert aninverter (one-input NAND gate) or complement the input literal NAND Implementation Multilevel Implementation

3.7 NAND and NOR Implementation (8-15) NAND Implementation Multilevel Implementation EXAMPLE 1: F = A(CD + B) + BC’

3.7 NAND and NOR Implementation (9-15) NAND Implementation Multilevel Implementation EXAMPLE 2: F = (AB’ + A’B).(C + D’)

3.7 NAND and NOR Implementation (10-15)  The NOR operation is the dual of the NAND operation.  The NOR gate is anothar universal gate to implement any Boolean function. NOR Implementation

3.7 NAND and NOR Implementation (11-15)  To facilitate the conversion to NOR logic, there are alternative graphic symbol for it. NOR Implementation

3.7 NAND and NOR Implementation (12-15)  Procedures of Implementation with two levels of NOR gates: 1. Express simplified function in product of sums form. 2. Draw a NOR gate for each product term that has at least two literals to constitute a group of first-level gates 3. Draw a single gate using OR-invert or invert-AND in the second level 4. A term with a single literal requires an inverter in the first level. NOR Implementation Two-Level Implementation

3.7 NAND and NOR Implementation (13-15) NOR Implementation Two-Level Implementation Example : F = (A+B).(C+D).E E

3.7 NAND and NOR Implementation (14-15)  Procedures of Implementation with multilevel of NOR gates: 1. Convert all OR gates to NOR gates with OR-invert graphic symbols 2. Convert all AND gates to NOR gates with invert- AND graphic symbols 3. Check all the bubbles in the diagrams. For a single bubble, invert aninverter (one-input NAND gate) or complement the input literal NOR Implementation Multilevel Implementation

3.7 NAND and NOR Implementation (15-15) NOR Implementation Multi-Level Implementation Example : F = (A B’ + A’B).(C+D’) A B’ A’ B

3.8 Other Two-Level Implementations (1-7)  16 possible combinations of two-level forms with 4 types of gates: AND, OR, NAND, and NOR  8 are degenerate forms: degenerate to a single operation.  (AND-AND, AND-NAND, OR-OR, OR-NOR, NAND-NAND, NAND-NOR, NOR-AND, NOR-NAND)  8 are generate forms:  NAND-AND = AND-NOR = AND-OR-INVERT  OR-NAND = NOR-OR = OR-AND-INVERT Nondegeneratd forms Implementation

3.8 Other Two-Level Implementations (3-7) Nondegeneraetd forms Implementation NORNANDORAND 2 nd level 1 st level #NAND(3.4)AND NOR$OR(3.4)OR AND(3.6)NAND# (3.6)OR$NOR Discussed before Generated forms Discuss now

3.8 Other Two-Level Implementations (4-7) Nondegeneraetd forms Implementation AND-OR-INVERT o AND-NOR= NAND-AND = AND-OR-INVERT o Similar to AND-OR, AND-OR-INVERT requires an expression in sum of products Example: F = (AB + CD + E) ‘

3.8 Other Two-Level Implementations (5-7) Nondegeneraetd forms Implementation OR-AND-INVERT o AND-NOR= NAND-AND = AND-OR-INVERT o Similar to OR-AND, OR-AND-INVERT requires an expression in product of sums Example: F = [(A+B). (C+D). E ] ‘

3.8 Other Two-Level Implementations (6-7) Nondegeneraetd forms Implementation To Get an Output of Simplify F’ into Implements the Function Equivalent Nondegenerate form ba F sum-of-products form by combining 0’s in the map AND-OR- INVERT NAND-ANDAND-NOR F product-of-sums form by combining 1’s in the map and then complementing OR-AND- INVERT NOR-OROR-NAND

3.8 Other Two-Level Implementations (7-7) Nondegeneraetd forms Implementation Example (3.11) : F(x,y,z) = ∑ (0,7) AND-OR-INVERT: F’ = x’y + xy’ + z F = ( x’y + xy’ + z ) ‘ OR-AND-INVERT: F = x’y’z’ + xyz’ F = [ (x’y’z’ + xyz’)’ ] ‘ F = [ (x+y+z). (x’+y’+z) ] ‘

3.9 Exclusive-OR Function (1-7)  Exclusive-OR (XOR) denoted by the symbol   x  y = xy‘ + x‘y  Exclusive-OR is equal to 1, when the values of x and y are diffrent.  Exclusive-NOR (XNOR):  (x  y )‘ = xy + x‘y‘  Exclusive-NOR is equal to 1, when the values of x and y are same.  Only a limited number of Boolean functions can be expressed in terms of XOR operations, but it is particularly useful in arithmetic operations and error- detection and correction circuits.

3.9 Exclusive-OR Function (2-7)  Exclusive-OR principles:  x  0 = x  x  1 = x‘  x  x = 0  x  x‘ = 1  x  y‘ = x‘  y = (x  y)‘  x  y = y  x  (x  y)  z = x  y  z)

3.9 Exclusive-OR Function (3-7)  Implementaion Exclusive-OR with AND-OR-NOT:  x  y = xy‘ + x‘y  Implementaion Exclusive- OR with NAND:  x  y = xy‘ + x‘y = x (x‘+y‘) + y (x‘+y‘) = x (xy)‘ + y (xy)‘ = [ (x(xy)‘ + y(xy)‘)‘]‘ = [ (x(xy)‘)‘ + (y(xy)‘)‘ ]‘

3.9 Exclusive-OR Function (4-7)  The 3-variable XOR function is equal to 1 if only one variable is equal to 1 or if all three variables are equal to 1.  Multiple-variable exclusive OR operation = odd function : odd number of variables be equal to 1.  (A  B  C) = (AB‘ + A‘B) C‘ + (A‘B‘ + AB) C = AB‘C‘ + A‘BC‘ + A‘B‘C + ABC = ∑ (1,2,4,7) Odd Function:

3.9 Exclusive-OR Function (5-7) Odd Function:

3.9 Exclusive-OR Function (6-7) Odd Function: A  B  C  D= ∑ (1,2,4,7,8,11,13,14)

3.9 Exclusive-OR Function (7-7)  Exclusive-OR function is useful in systems requiring error-detection and correction circuits.  A parity bit is used for purpose of detection errors during transmission.  Parity bit : an extra bit included with a binary message to make the number of 1’s either odd or even.  The circuit generates the parity bit in transmitter is called parity generator.  The circuit checks the parity bit in receiver is called parity checker. Parity Generation and Checking:

3.9 Exclusive-OR Function (8-7) Parity Generation and Checking: Example : Three-bit message with even parity Three-bit Massage Parity bit XYZ P o From the truth table, P constitutes an odd function. o It is equal 1 when numerical value of 1’s in a minterm is odd o P = x  y  z

3.9 Exclusive-OR Function (8-7) Parity Generation and Checking: Example : Three-bit message with even parity o From the truth table, C constitutes an odd function. o It is equal 1 when numerical value of 1’s in a minterm is odd o C = x  y  z  P