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Digital Design, Kyung Hee Univ.

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Chapter 3. Gate-Level Minimization

Tong In Oh


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Chapter 3. Gate-Level Minimization• 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


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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)


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Two Variable K-Map• Four minterms for two variables• Mark the squares whose minterms belong to a given function

FIGURE 3.1 Two-variable K-map FIGURE 3.2 Representation of functions in the map


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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

FIGURE 3.3 Three-variable K-map


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Example 3.1

FIGURE 3.4 Map for Example 3.1, F(x, y, z) = Σ(2, 3, 4, 5) = x’y + xy’


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Example 3.2

FIGURE 3.5 Map for Example 3.2, F(x, y, z) = Σ(3, 4, 6, 7) = yz + xz’


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Example 3.3

FIGURE 3.6 Map for Example 3.3, F(x, y, z) = Σ(0, 2, 4, 5, 6) = z’ + xy’


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Example 3.4

FIGURE 3.7 Map of Example 3.4, A’C + A’B + AB’C + BC = C + A’B


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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’


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Prime Implicants• 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 3.11Simplification using prime implicants


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3.4 Product of Sums Simplification

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• 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 3.15Example with don’t-care 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 gatesFIGURE 3.17 Two graphic symbols for a three-input NAND gate


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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


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Example 3.9

FIGURE 3.19 Solution to Example 3.9


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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’


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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 3.22 Logic operations with NOR gates FIGURE 3.23 Two graphic symbols for the NOR gate


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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

FIGURE 3.26 Wired logic: (a) Wired-AND logic with two NAND gates (b) Wired-OR in emitter-coupled logic (ECL) gates


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Nondegenerate Forms (1)• Two-level combinations of gates• AND/OR/NAND/NOR• Degenerate forms: single operation• Nondegenerate forms: implementation in

• 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 )’


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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


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FIGURE 3.29 Other two-level implementations

F’=x’y+xy’+z : F=(x’y+xy’+z)’ : AND-OR-INVERT

AND-NORNAND-AND

F=x’y’z’+xyz’F’=(x+y+z)(x’+y’+z)F=[(x+y+z)(x’+y’+z)]’OR-AND-INVERT

OR-NANDNOR-OR


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3.8 Exclusive-OR Function

FIGURE 3.30 Exclusive-OR implementationsUseful in arithmetic operationsError detection and correction circuits


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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


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Parity Generation and Checking

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