Gate-Level Minimization
Chapter 3
1
• Finding an optimal gate-level implementation of the Boolean
functions describing a digital circuit
• Difficult to execute by manual methods
• Computer-based logic synthesis tools
• Important to understand the underlying mathematical description
and solution of the problem
• Execute a manual design of simple circuits
2
3.2 Map Method
• Complexity of the algebraic expression in many different forms but
equivalent
• Algebraic means – lack specific rules
• Karnaugh Map (K-map) method – simple, straightforward procedure
for minimizing Boolean functions
• Pictorial form of a truth table
• K-map
•
•
•
•
•
•
Diagram made up of squares
Each square representing one minterm of the function
Function = Sum of minterms
Present a visual diagram of all possible ways a function
Standard forms: sum of products or product of sums
The simplest expression (not unique)
3
Two Variable K-Map
• Four minterms for two variables
• Mark the squares whose minterms belong to a given function
o
M M Ms
y ng y
a o
FIGURE 3.1
Two-variable K-map
FIGURE 3.2 Representation of
functions in the map
01 10 11
4
Three Variable K-Map
• Eight minterms for three binary variables
• Similar to the Gray code (only one bit changes in value from one
adjacent column to the next)
• Mark the squares whose minterms belong to a given function
둘다 내무시
fiori
o
i
FIGURE 3.3 Three-variable K-map
5
it
Example 3.1
롒
FIGURE 3.4 Map for Example 3.1, F(x, y, z) = Σ(2, 3, 4, 5) = x’y + xy’
6
Example 3.2
as
FIGURE 3.5 Map for Example 3.2, F(x, y, z) = Σ(3, 4, 6, 7) = yz + xz’
7
떨어져 있어도 묶을수
있다
Example 3.3
FIGURE 3.6 Map for Example 3.3, F(x, y, z) = Σ(0, 2, 4, 5, 6) = z’ + xy’
8
Example 3.4
FIGURE 3.7 Map of Example 3.4, A’C + A’B + AB’C + BC = C + A’B
9
3.3 Four-Variable K-Map
FIGURE 3.8 Four-variable map
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Example 3.5
FIGURE 3.9 Map for Example 3.5, F(w, x, y, z) = Σ(0, 1, 2, 4, 5, 6, 8,
9, 12, 13, 14) = y’ + w’z’ + xz’
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Example 3.6
FIGURE 3.10 Map for Example 3.6, A’B’C’ + B’CD’ + A’BCD’ + AB’C’ = B’D’
+ B’C’ + A’CD’
12
Prime Implicants
크게그린사각형
• All the minterms of the function are covered
• The number of terms in the expression is minimized
• No redundant terms
• A prime implicant: A product term obtained by combining the
maximum possible number of adjacent squares in the map
FIGURE 3.11
Simplification
using prime
implicants
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3.4 Product of Sums Simplification
늅곱낙다
F 를 門怡
use 0
F든
드모르간
FIGURE 3.12 Map for Example 3.7, F(A, B, C, D) = Σ(0, 1, 2, 5, 8, 9, 10)
= B’D’ + B’C’ + A’C’D = (A’ + B’)(C’ + D’)(B’ + D)
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Gate-Level Implementation
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3.5 Don’t Care Conditions
end
0도 되고
1도 시는 애
• Function is not specified for certain combinations of the variables
• = Incompletely specified functions
• Don’t care condition = unspecified minterms of a function
• Further simplification
• Marked as X
FIGURE 3.15
Example with don’tcare conditions
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3.6 NAND/NOR Implementation
• Easier to fabricate with electronic components
• Basic gates used in all IC digital logic families
• NAND circuits = Universal gate
• To obtain the simplified Boolean function in terms of Boolean
operators and then convert the function to NAND logic
FIGURE 3.16 Logic operations with NAND
gates
FIGURE 3.17 Two graphic symbols for a
three-input NAND gate
17
Two-Level Implementation
• The implementation of Boolean functions with NAND gates requires
that the functions be in sum-of-products form.
FIGURE 3.18 Three ways to implement F = AB + CD
18
承凹 口
Example 3.9
FIGURE 3.19 Solution to Example 3.9
19
Multilevel NAND Circuits
• F=A(CD+B)+BC’
• Converting a multilevel AND-OR diagram into an all-NAND diagram
FIGURE 3.20 Implementing F = A(CD + B) + BC’
20
FIGURE 3.21 Implementing F = (AB’ + A’B) (C + D’)
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NOR Implementation
• Dual of the NAND operation = Universal gate
• Converting OR-AND diagram to a NOR diagram
• Product of sums
FIGURE 3.23 Two graphic symbols for the
NOR gate
FIGURE 3.22 Logic operations with NOR
gates
22
FIGURE 3.24 Implementing
F = (A + B)(C + D)E
FIGURE 3.25 Implementing F =
(AB’ + A’B)(C + D’) with NOR gates
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3.7 Other Two-Level Implementations
• NAND and NOR gates
• Allow the possibility of a wire connection between the outputs of two
gates to provide a specific logic function = wired logic
• Wired-AND logic: open-collector TTL NAND
• Wired-OR logic: NOR outputs of ECL gates
i
FIGURE 3.26
gates
節灝
in
蛇站
Wired logic: (a) Wired-AND logic with two NAND gates (b) Wired-OR in emitter-coupled logic (ECL)
24
Nondegenerate Forms (1)
• Two-level combinations of gates
• AND/OR/NAND/NOR
En
• Degenerate forms: single operation
• Nondegenerate forms: implementation in
안되는
합체
e
• Sum-of-products form
• Product-of-sums form
• AND-OR-INVERT Implementation: NAND-AND, AND-NOR: Wired AND with
NAND
FIGURE 3.27 AND–OR–INVERT circuits, F = (AB + CD + E )’
25
Nondegenerate Forms (2)
• OR-AND-INVERT Implementation: OR-NAND, NOR-OR: Wired OR with NOR
FIGURE 3.28 OR–AND–INVERT circuits, F = [(A + B )(C + D)E]’
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Tabular Summary and Example
i
sfnDTTEF
1
.ae
후
2중과정
27
가
鄙
FIGURE 3.29 Other
two-level
implementations
F’=x’y+xy’+z :
F=(x’y+xy’+z)’ :
AND-OR-INVERT
AND-NOR
NAND-AND
F=x’y’z’+xyz’
F’=(x+y+z)(x’+y’+z)
F=[(x+y+z)(x’+y’+z)]’
OR-AND-INVERT
OR-NAND
NOR-OR
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3.8 Exclusive-OR Function
none
사용됨
A 빈번히
교환
이
Useful in arithmetic operations
Error detection and correction circuits
FIGURE 3.30 Exclusive-OR implementations
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Odd Function
• Multiple-variable exclusive-OR operation
• Odd number of variables be equal to 1 = odd function
FIGURE 3.31 Map
for a three-variable
exclusive-OR
function
FIGURE 3.32 Logic diagram of odd and even functions
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Four-variable Exclusive OR Operation
FIGURE 3.33 Map for a four-variable exclusive-OR function
31
Parity Generation and Checking
炎
떕
이許
32