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CS 121 Digital Logic Design Gate-Level Minimization Chapter 3.

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Presentation on theme: "CS 121 Digital Logic Design Gate-Level Minimization Chapter 3."— Presentation transcript:

1 CS 121 Digital Logic Design Gate-Level Minimization Chapter 3

2 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 3.1 Introduction (1-1)  Gate-Level Minimization refers to the design task of finding an optimal gate-level implementation of the Boolean functions describing a digital circuit.  Notes about simplification of Boolean expression:  Minimum number of terms and literals in each term (minimum number of gates and inputs in the digram).  Reduce the complexity of the digital gates.  The simplest expression is not unique.  Simplification Methods:  Algebraic minimization  lack on specific rules. (section 2.4).  Karnaugh map or K-map.

4  A Karnaugh map is a graphical tool for assisting in the general simplification procedure.  Combination of 2, 4, … adjacent squares  The relation is: Logic circuit ↔ Boolean function ↔ Truth table ↔ K- map ↔ conical form ↔ satndrad form.  Conical form: ( sum of minterms, product of maxterms.  Standrad form: ( simplifier : sum of product, product of sum 3.2 The Map Method (1-12)

5 3.2 The Map Method (2-12) Two-variable maps: Y’Y XX’

6 3.2 The Map Method (3-12)  Rules for K-map: o We can reduce functions by circling 1’s in the K- map o Each circle represents a minterm reduction o Following circling, we can deduce minimized and- or form.  Rules to consider o Every cell containing a 1 must be included at least once. o The largest possible “power of 2 rectangle” must be enclosed. o The 1’s must be enclosed in the smallest possible number of rectangles.

7 Example 1: F(X,Y) = XY’ + XY Two-Variable maps (cont.) 3.2 The Map Method (4-12)  From the map, we see that F (X,Y) = X. Note: There are implied 0s in other boxes.  This can be justified using algebraic manipulations: F(X,Y) = XY’ + XY = X(Y’ +Y) = X.1 = X 1X

8 Example 2: G(x,y) = m1 + m2 + m3 Two-Variable maps (cont.) 3.2 The Map Method (5-12)  G(x,y) = m1 + m2 + m3 = X’Y + XY’ + XY  From the map, we can see that : G = X + Y 1 11 X Y

9 Example 3: F = Σ(0, 1) Two-Variable maps (cont.) 3.2 The Map Method (6-12)  Using algebraic manipulations:  F = Σ(0,1) = x’y + x’y’ = x’ (y+y’) = x’ 1 1 X’ xyF xyF

10  3 variables  8 squares ( minterms).  On a 3-variable K-Map: ◦ One square represents a minterm with three variables ◦ Two adjacent squares represent a product term with two variables ◦ Four “adjacent” terms represent a product term with one variables ◦ Eight “adjacent” terms is the function of all ones (logic 1). 3.2 The Map Method (7-12) Three-variable maps:

11  using algebraic manipulations: F = X’Y’Z’ + X’YZ’ + XY’Z’ + XYZ’ = Z’ (X’Y’ + X’Y + XY’ + XY) = Z’ (X’ (Y’+Y) + X (Y’+Y)) = Z’ (X’+ X) = Z’ 3.2 The Map Method (8-12) Three-variable maps (cont.): Example 1: F(X,Y) = X’Y’Z’ + X’YZ’ + XY’Z’ + XYZ’ Y Z x

12 Example 2: F=AB’C’ +A BC + ABC +A BC + A’B’C + A’BC’ three-Variable maps (cont.) 3.2 The Map Method (9-12)  From the map, we see that F=A + BC + BC B C A

13 Example 4 : F (x, y, z)= Σm (2, 3, 6, 7) three-Variable maps (cont.) 3.2 The Map Method (10-12)  using algebraic manipulations:  F(x, y, z) = x’yz + xyz + x’yz’ + xyz’  = yz (x’ + x) + yz’ (x’ + x)  = yz + yz’  = y (z + z’)  = y y z x Y

14 Example (3-1), (3-2) : three-Variable maps (cont.) 3.2 The Map Method (11-12)

15 Example (3-3), (3-4) : three-Variable maps (cont.) 3.2 The Map Method (12-12)

16 3.3 Four-Variables Map (1-9)  4 variables  16 squares ( minterms).  On a 4-variable K-Map:  Two adjacent squares represent a term of three literals.  Four adjacent squares represent a term of two literals.  Eight adjacent squares represent a term of one literal.  Note: The larger the number of squares combined, the smaller the number of literals in the term.

17 3.3 Four-Variables Map (2-9) Flat Map Vs. Torus

18 3.3 Four-Variables Map (3-9) Example 1 (3-5) : F(w,x,y,z) = ∑ ( 0,1,2,4,5,6,8,9,12,13,14) y z w x Y’ W’YZ’ XYZ’ F = y‘ + w‘yz‘ + xyz‘

19 3.3 Four-Variables Map (4-9) Example 2 (3-6) : F = A’B’C’ + B’CD’ + A’BCD’ + AB’C’ C D A B F = B‘D‘ + B‘C‘ + A‘CD‘ B’D’ B’C’ A’CD’

20 3.3 Four-Variables Map (5-9) Simplification using Prime Implicants o A Prime Implicant is a product term obtained by combining the maximum possible number of adjacent squares in the map. o If a minterm in a square is covered by only one prime implicant, that implicant is called Essential. o Prime Implicants and Essential Prime Implicants can be determined by inspection of a K-Map. o Notes: Two adjacent 1’s form prime implicant, if they are not within a group of four adjacent squares. Four adjacent 1’s form prime implicant, if they are not within a group of eight adjacent squares and so on.

21 3.3 Four-Variables Map (6-9) Simplification using Prime Implicants Example 1: F(A,B,C,D) = ∑ (0,2,3,5,7,8,9,10,11,13,15) Minterms covered by single prime implicant DB CB B D A ESSENTIAL Prime Implicants C BD CD BD DB B C D A AD BA

22 3.3 Four-Variables Map (7-9) Simplification using Prime Implicants Example 1: F(A,B,C,D) = ∑ (0,2,3,5,7,8,9,10,11,13,15) o Essential prim implicants: BD, B’D’ o Prime implicant: CD, B’C, AD, AB’. o The minterms that not cover by essential implicants are: m3, m9, m11.  The simplified expression is optained from the sum of the essential implicants and other prime implicants that may be needed to cover any remaining minterms. o So this function can be written with these ways: F = BD + B’D’ + CD + AD F = BD + B’D’ + CD + AB’ F = BD + B’D’ + B’C + AD F = BD + B’D’ + B’C + AB’

23 3.3 Four-Variables Map (8-9) Simplification using Prime Implicants Example 2: F(W,X,Y,Z) = ∑ (0,2,3,8,9,10,11,12,13,14,15) X Y Z W W X’Y X’Z’ Note: that all of these prime implicants are essential.

24 3.3 Four-Variables Map (9-9) Simplification using Prime Implicants Example 3: F(W,X,Y,Z) = ∑ (0,2,3,4,7,12,13,14,15) X Y Z W WX W’Y’Z’ W’X’Y W’YZ XYZ XY’Z’ Essential: WX Prime: XYZ, XY’Z’, W’Y’Z’, W’YZ, W’X’Y, W’X’Z’ W’X’Z’

25 3.5 Producut-of-Sum simplification (1-9) 1. Mark with 1’s the minterms of F. 2. Mark with 0’s the minterms of F’. 3. Circle 0’s to express F’. 4. Complement the result in step 3 to obtain a simplified F in product-of-sums form.

26 3.5 Producut-of-Sum simplification (1-9) Example 1: Simplify : F= ∑(0,1,2,5,8,9,10) in Product-of-Sums Form B C D A CD AB BD’ F’ = AB + CD + BD’ F = (F’)’ = (A’+B’) + (C’+D’) + (B’+D)

27 3.5 Producut-of-Sum simplification (1-9) Example 2: Simplify : F(x, y, z) =  (0, 2, 5,7)in Product-of-Sums Form y z x XZ X’Z’ F’ = XZ + X’Z’ F = (F’)’ = (X’+Z’) + (X+Z)

28 3.6 Don't Cares Condition (1-4)  Sometimes a function table or map contains entries for which it is known:  The input values for the minterm will never occur, or  The output value for the minterm is not used.  Functions that have unspecified outputs for some input combinations are called incompletely specified functions.  In these cases, the output value is defined as a “don't care” ( an “x” entry) assumed to be either 0 or 1.  The choice between 0 and 1 is depending on the way the incompletely specified function is simplied.  By placing “don't cares” in the function table or map, the cost of the logic circuit may be lowered.

29 3.6 Don't Cares Condition (2-4)  Example :  A logic function having the binary codes for the BCD digits as its inputs. Only the codes for 0 through 9 are used.  The six codes, 1010 through 1111 never occur, so the output values for these codes are “x” to represent “don’t cares.”

30 3.6 Don't Cares Condition (3-4) Example (3.9) : F(W,X,Y,Z) = ∑ (1,3,7,11,15) d(W,X,Y,Z) = ∑ (0,2,5) X Y Z W 1 1 x x 1 0 x F = YZ + W’Z X Y Z W 1 1 x x 1 0 x F = YZ + W’X’

31 3.6 Don't Cares Condition (4-4) Example (3.9) : F(W,X,Y,Z) = ∑ (1,3,7,11,15) d(W,X,Y,Z) = ∑ (0,2,5) X Y Z W 1 1 x x 1 0 x F’ = Z’ + WY’ F = Z ( W’ + Y)


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