Chapter_3_Gate-Level_Minimization

# Chapter_3_Gate-Level_Minimization - Digital Logic Design I...

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May 26, 2011 EASTERN MEDITERRANEAN UNIVERSITY 1 Digital Logic Design I Gate-Level Minimization Mustafa Kemal Uyguroğlu

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May 26, 2011 2 3-1 Introduction Gate-level minimization refers to the design task of finding an optimal gate-level implementation of Boolean functions describing a digital circuit.
May 26, 2011 3 3-2 The Map Method The complexity of the digital logic gates The complexity of the algebraic expression Logic minimization Algebraic approaches: lack specific rules The Karnaugh map A simple straight forward procedure A pictorial form of a truth table Applicable if the # of variables < 7 A diagram made up of squares Each square represents one minterm

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May 26, 2011 4 Review of Boolean Function Boolean function Sum of minterms Sum of products (or product of sum) in the simplest form A minimum number of terms A minimum number of literals The simplified expression may not be unique
May 26, 2011 5 Two-Variable Map A two-variable map Four minterms x' = row 0; x = row 1 y' = column 0; y = column 1 A truth table in square diagram Fig. 3.2(a): xy = m 3 Fig. 3.2(b): x+y = x'y+xy' +xy = m 1 +m 2 +m 3 Figure 3.2 Representation of functions in the map Figure 3.1 Two-variable Map

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May 26, 2011 6 A Three-variable Map A three-variable map Eight minterms The Gray code sequence Any two adjacent squares in the map differ by only on variable Primed in one square and unprimed in the other e.g., m 5 and m 7 can be simplified m 5 + m 7 = xy'z + xyz = xz ( y'+y ) = xz Figure 3.3 Three-variable Map
May 26, 2011 7 A Three-variable Map m 0 and m 2 ( m 4 and m 6 ) are adjacent m 0 + m 2 = x'y'z' + x'yz' = x'z' ( y'+y ) = x'z' m 4 + m 6 = xy'z' + xyz' = xz' ( y'+y ) = xz'

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May 26, 2011 8 Example 3.1 Example 3.1: simplify the Boolean function F( x , y , z ) = Σ (2, 3, 4, 5) F ( x , y , z ) = Σ (2, 3, 4, 5) = x'y + xy' Figure 3.4 Map for Example 3.1, F ( x , y , z ) = Σ(2, 3, 4, 5) = x'y + xy'
May 26, 2011 9 Example 3.2 Example 3.2: simplify F ( x , y , z ) = Σ (3, 4, 6, 7) F ( x , y , z ) = Σ (3, 4, 6, 7) = yz + xz ' Figure 3.5 Map for Example 3-2; F( x , y , z ) = Σ(3, 4, 6, 7) = yz + xz'

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May 26, 2011 10 Four adjacent Squares Consider four adjacent squares 2, 4, and 8 squares m 0 +m 2 +m 4 +m 6 = x'y'z'+x'yz'+xy'z'+xyz' = x'z'(y'+y) +xz'(y'+y) = x'z' + xz‘ = z' m 1 +m 3 +m 5 +m 7 = x'y'z+x'yz+xy'z+xyz =x'z(y'+y) + xz(y'+y) =x'z + xz = z Figure 3.3 Three-variable Map
May 26, 2011 11 Example 3.3 a Example 3.3: simplify F(x , y , z) = Σ (0, 2, 4, 5, 6) F ( x , y , z ) = Σ (0, 2, 4, 5, 6) = z'+ xy' Figure 3.6 Map for Example 3-3, F ( x , y , z ) = Σ(0, 2, 4, 5, 6) = z' + xy'

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May 26, 2011 12 Example 3.4 Example 3.4: let F = A'C + A'B + AB'C + BC a) Express it in sum of minterms. b)
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## This note was uploaded on 05/26/2011 for the course EE 211 taught by Professor Hasandemirel during the Spring '10 term at Eastern Mediterranean University.

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Chapter_3_Gate-Level_Minimization - Digital Logic Design I...

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