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CT455: Computer Organization K-Map

Learn how to simplify functions using Karnaugh Maps and algebraic techniques. Understand half adders, SOP, POS expressions, and more.

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CT455: Computer Organization K-Map

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  1. CT455: Computer OrganizationK-Map

  2. Lecture 5: Karnaugh Maps • Function Simplification • Algebraic Simplification • Half Adder • Introduction to K-maps • Venn Diagrams • 2-variable K-maps • 3-variable K-maps • 4-variable K-maps • 5-variable and larger K-maps

  3. Lecture 5: Karnaugh Maps • Simplification using K-maps • Converting to Minterms Form • Simplest SOP Expressions • Getting POS Expressions • Don’t-care Conditions • Review • Examples

  4. Function Simplification • Why simplify? • Simpler expression uses less logic gates. • Thus: cheaper, less power, faster (sometimes). • Simplification techniques: • Algebraic Simplification. • simplify symbolically using theorems/postulates. • requires skill but extremely open-ended. • Karnaugh Maps. • diagrammatic technique using ‘Venn-like diagram’. • easy for humans (pattern-matching skills). • simplified standard forms. • limited to not more than 6 variables.

  5. Algebraic Simplification • Algebraic simplification aims to minimise (i) number of literals, and (ii) number of terms • But sometimes conflicting. • Let’s aim at reducing the number of literals.

  6. Algebraic Simplification • Example: (x+y).(x+y').(x'+z) (6 literals) = (x.x+x.y'+x.y+y.y').(x'+z) (assoc.) = (x+x.(y'+y)+0).(x'+z) (idemp.,assoc., compl.) = (x+x.(1)+0).(x'+z) (complement) = (x+x+0).(x'+z) (identity 1) = (x).(x'+z) (idemp, identity 0) = (x.x'+x.z) (assoc.) = (0+x.z) (complement) = x.z (identity 0) Number of literals reduced from 6 to 2.

  7. Minimum SOP and POS Representations • The minimum sum of products (MSOP) of a function, f, is a SOP representation of f that contains the fewest number of product terms and fewest number of literals of any SOP representation of f. • Example -- f(a,b,c,d) = m(3,7,11,12,13,14,15) = ab + acd + acd = ab + cd • The minimum product of sums (MPOS)of a function, f, is a POS representation of f that contains the fewest number of sum terms and the fewest number of literals of any POS representation of f. • Example -- f(a,b,c,d) = M(0,1,2,4,5,6,8,9,10) = (a + c)(a + d)(a + b + d)(b + c + d) = (a +c)(a + d)(b + c)(b + d)

  8. Algebraic Simplification • Find minimal SOP and POS expressions of f(x,y,z) = x'.y.(z + y'.x) + y'.z x'.y.(z+y'.x) + y'.z = x'.y.z + x'.y.y'.x + y'.z (distributivity) = x'.y.z + 0 + y'.z (complement, null element 0) = x'.y.z + y'.z (identity 0) = x'.z + y’.z (absorption) = (x' + y').z (distributivity) Minimal SOP of f = x'.z + y'.z (2 2-input AND gates and 1 2-input OR gate) Minimal POS of f = (x' + y').z (1 2-input OR gate and 1 2-input AND gate)

  9. Algebraic Simplification • Find minimal SOP expression of f(a,b,c,d) = a.b.c + a.b.d + a'.b.c' + c.d + b.d' a.b.c + a.b.d + a'.b.c' + c.d + b.d' = a.b.c + a.b + a'.b.c' + c.d + b.d' (absorption) = a.b.c + a.b + b.c' + c.d + b.d' (absorption) = a.b + b.c' + c.d + b.d' (absorption) = a.b + c.d + b.(c' + d') (distributivity) = a.b + c.d + b.(c.d)' (DeMorgan) = a.b + c.d + b (absorption) = b + c.d (absorption) Number of literals reduced form 13 to 3.

  10. Algebraic Simplification • Difficulty – needs good algebraic manipulation skills. • Advantage – very open-ended (to your desired form!)

  11. S X Half adder (X+Y) C Y Half Adder • Half-Adder is a circuit which adds two single bits (called X,Y) together, to produce a result of two bits (called C, S). • A black-box representation of this circuit is: Truth table representation is:

  12. X Y S C Half Adder • In sum-of-minterms forms: C = X.Y S = X'.Y + X.Y' • Algebraic simplification could simplify S to: S = X'.Y + X.Y' = XY • Giving:

  13. Introduction to K-maps • Systematic method to obtain simplified sum-of-products (SOPs) Boolean expressions. • Objective: Fewest possible terms/literals. • Diagrammatic technique based on a special form of Venn diagram. • Advantage: Easy with visual aid. • Disadvantage: Limited to 5 or 6 variables.

  14. Figure 3.1 Venn diagram and equivalent K-mapfor two variables

  15. Figure 3.2 Venn diagram and equivalent K-mapfor three variables

  16. Figure 3.3 (a) -- (d) K-maps for four and five variables

  17. a'b' ab' ab a'b a b Venn Diagrams • Venn diagram to represent the space of minterms. • Example of 2 variables (4 minterms):

  18. a'b' ab' ab a'b a b Venn Diagrams • Each set of minterms represents a Boolean function. Examples: { a.b, a.b' }  a.b + a.b' = a.(b+b') = a { a‘.b, a.b }  a‘.b + a.b = (a'+a).b = b { a.b }  a.b { a.b, a.b', a‘.b }  a.b + a.b' + a‘.b = a + b { }  0 { a‘.b',a.b,a.b',a‘.b }  1

  19. 2-variable K-maps • Karnaugh-map (K-map) is an abstract form of Venn diagram, organised as a matrix of squares, where • each square represents a minterm • adjacent squares always differ by just one literal(so that the unifying theorem may apply: a + a' = 1) • For 2-variable case (e.g.: variables a,b), the map can be drawn as:

  20. Alternative 1: Alternative 2: OR OR a a Alternative 3: a'b' ab' m0 m2 OR b a a b b ab m3 a'b m1 m0 m1 b ab a'b m3 m1 b a m3 m2 a'b' m0 ab' m2 b a'b' a'b a ab ab' 2-variable K-maps • Alternative layouts of a 2-variable (a, b) K-map and others…

  21. equivalent to: a a 1 0 b equivalent to: 0 1 b b 0 1 b a 0 1 a 2-variable K-maps • Equivalent labeling:

  22. C = ab S = ab' + a'b b b 0 0 0 1 a a 1 0 0 1 2-variable K-maps • The K-map for a function is specified by putting • a ‘1’ in the square corresponding to a minterm • a ‘0’ otherwise • For example: Carry and Sum of a half adder.

  23. b bc a 00 01 11 10 b 0 1 a'b'c' a'b'c a'bc a'bc' bc 00 01 11 10 a a ab'c' ab'c abc abc' 0 1 m0 m1 m3 m2 OR c a m4 m5 m7 m6 c Note Gray code sequence 3-variable K-maps • There are 8 minterms for 3 variables (a, b, c). Therefore, there are 8 cells in a 3-variable K-map. Above arrangement ensures that minterms of adjacent cells differ by only ONE literal. (Other arrangements which satisfy this criterion may also be used.)

  24. bc a 00 01 11 10 0 1 m0 m1 m3 m2 m4 m5 m7 m6 3-variable K-maps • There is wrap-around in the K-map: • a'.b'.c' (m0) is adjacent to a'.b.c' (m2) • a.b'.c' (m4) is adjacent to a.b.c' (m6) Each cell in a 3-variable K-map has 3 adjacent neighbours. In general, each cell in an n-variable K-map has n adjacent neighbours. For example, m0 has 3 adjacent neighbours: m1, m2 and m4.

  25. Figure 3.15 -- Example 3.13f(A,B,C,D) = m(1,2,3,6) = AC + BC

  26. b bc a 00 01 11 10 0 1 1 0 0 1 a 0 1 0 0 c Quick Review Questions (1) Textbook page 104. 5-1. The K-map of a 3-variable function F is shown below. What is the sum-of-minterms expression of F? 5-2. Draw the K-map for this function A: A(x, y, z) = x.y + y.z’ + x’.y’.z

  27. y yz 00 01 11 10 wx 00 01 11 10 m0 m1 m3 m2 m4 m5 m7 m6 x m12 m13 m15 m14 w m8 m9 m11 m10 z 4-variable K-maps • There are 16 cells in a 4-variable (w, x, y, z) K-map.

  28. y yz wx m0 m1 m3 m2 m4 m5 m7 m6 x m12 m13 m15 m14 w m8 m9 m11 m10 z 4-variable K-maps • There are 2 wrap-arounds: a horizontal wrap-around and a vertical wrap-around. • Every cell thus has 4 neighbours. For example, the cell corresponding to minterm m0 has neighbours m1, m2, m4 and m8.

  29. Figure 3.16 -- Example 3.14f(A,B,C,D) = BD + BC + BCD

  30. 5-variable K-maps • Maps of more than 4 variables are more difficult to use because the geometry (hyper-cube configurations) for combining adjacent squares becomes more involved. • For 5 variables, e.g. vwxyz, need 25 = 32 squares.

  31. v ' v y y yz yz 00 01 11 10 00 01 11 10 wx wx 00 01 11 10 m0 m1 m3 m2 00 01 11 10 m16 m17 m19 m18 m4 m5 m7 m6 m20 m21 m23 m22 x x m12 m13 m15 m14 m28 m29 m31 m30 w w m8 m9 m11 m10 m24 m25 m27 m26 z z 5-variable K-maps • Organised as two 4-variable K-maps: Corresponding squares of each map are adjacent. Can visualise this as being one 4-variable map on TOP of the other 4-variable map.

  32. Larger K-maps • 6-variable K-map is pushing the limit of human “pattern-recognition” capability. • K-maps larger than 6 variables are practically unheard of! • Normally, a 6-variable K-map is organised as four 4-variable K-maps, which are mirrored along two axes.

  33. w b a‘.b' a‘.b ef 10 11 01 00 cd 00 01 11 10 m18 m19 m17 m16 ef 00 01 11 10 cd m22 m23 m21 m20 00 01 11 10 m0 m1 m3 m2 m30 m31 m29 m28 m4 m5 m7 m6 m26 m27 m25 m24 m12 m13 m15 m14 10 11 01 00 10 11 01 00 m58 m59 m57 m56 m8 m9 m11 m10 m40 m41 m43 m42 m62 m63 m61 m60 m44 m45 m47 m46 a m54 m55 m53 m52 m36 m37 m39 m38 m50 m51 m49 m48 m32 m33 m35 m34 cd 10 11 01 00 cd 00 01 11 10 ef a.b' a.b ef Larger K-maps Try stretch your recognition capability by finding simplest sum-of-products expression for S m(6,8,14,18,23,25,27,29,41,45,57,61).

  34. Simplification Using K-maps • Based on the Unifying Theorem: A + A' = 1 • In a K-map, each cell containing a ‘1’ corresponds to a minterm of a given function F. • Each group of adjacent cells containing ‘1’ (group must have size in powersof twos: 1, 2, 4, 8, …) then corresponds to a simpler product term of F. • Grouping 2 adjacent squares eliminates 1 variable, grouping 4 squares eliminates 2 variables, grouping 8 squares eliminates 3 variables, and so on. In general, grouping 2n squares eliminates n variables.

  35. Simplification Using K-maps • Group as many squares as possible. • The larger the group is, the fewer the number of literals in the resulting product term. • Select as few groups as possible to cover all the squares (minterms) of the function. • The fewer the groups, the fewer the number of product terms in the minimized function.

  36. y yz 00 01 11 10 wx 00 01 11 10 1 1 x 1 1 w 1 1 Simplification Using K-maps • Example: F (w,x,y,z) = w’.x.y'.z' + w'.x.y'.z + w.x'.y.z' + w.x'.y.z + w.x.y.z' + w.x.y.z = m(4, 5, 10, 11, 14, 15) (cells with ‘0’ are not shown for clarity) z

  37. y yz 00 01 11 10 wx 00 01 11 10 A 1 1 x 1 1 w B 1 1 z Simplification Using K-maps • Each group of adjacent minterms (group size in powers of twos) corresponds to a possible product term of the given function.

  38. y yz 00 01 11 10 wx 00 01 11 10 A 1 1 x 1 1 w B 1 1 z Simplification Using K-maps • There are 2 groups of minterms: A and B, where: A = w'.x.y'.z' + w‘.x.y'.z = w'.x.y'.(z' + z) = w'.x.y' B = w.x'.y.z' + w.x'.y.z + w.x.y.z' + w.x.y.z = w.x'.y.(z' + z) + w.x.y.(z' + z) = w.x'.y + w.x.y = w.(x'+x).y = w.y

  39. Simplification Using K-maps • Each product term of a group, w'.x.y' and w.y, represents the sum of minterms in that group. • Boolean function is therefore the sum of product terms (SOP) which represent all groups of the minterms of the function. F(w,x,y,z) = A + B = w'.x.y' + w.y

  40. Simplification Using K-maps • Larger groups correspond to product terms of fewer literals. In the case of a 4-variable K-map: 1 cell = 4 literals, e.g.: w.x.y.z, w'.x.y'.z 2 cells = 3 literals, e.g.: w.x.y, w.y'.z' 4 cells = 2 literals, e.g.: w.x, x'.y 8 cells = 1 literal, e.g.: w, y', z 16 cells = no literal, e.g.: 1

  41. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 P P P Simplification Using K-maps • Other possible valid groupings of a 4-variable K-map include:

  42. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 O O Simplification Using K-maps • Groups of minterms must be (1) rectangular, and (2) have size in powers of 2’s. Otherwise they are invalid groups. Some examples of invalid groups:

  43. Converting to Minterms Form • The K-map of a function is easily drawn when the function is given in canonical sum-of-products, or sum-of-minterms form. • What if the function is not in sum-of-minterms? • Convert it to sum-of-products (SOP) form. • Expand the SOP expression into sum-of-minterms expression, or fill in the K-map directly based on the SOP expression.

  44. A AB 00 01 11 10 CD 00 01 11 10 1 1 D 1 1 C B Converting to Minterms Form • Example: f(A,B,C,D) = A(C+D)'(B'+D') + C(B+C'+A'D) = A(C'D')(B'+D') + BC + CC' + A'CD = AB'C'D' + AC'D' + BC + A'CD AB'C'D' + AC'D' + BC + A'CD = AB'C'D' + AC'D'(B+B') + BC + A'CD = AB'C'D' + ABC'D' + AB'C'D' + BC(A+A') + A'CD = AB'C'D' + ABC'D' + ABC + A'BC + A'CD = AB'C'D' + ABC'D' + ABC(D+D') + A'BC(D+D') + A'CD(B+B') = AB'C'D' + ABC'D' + ABCD + ABCD' + A'BCD + A'BCD' + A'B'CD 1 1 1

  45. Simplest SOP Expressions • To find the simplest possible sum of products (SOP) expression from a K-map, you need to obtain: • minimum number of literals per product term; and • minimum number of product terms • This is achieved in K-map using • bigger groupings of minterms (prime implicants) where possible; and • no redundant groupings (look foressential prime implicants) Implicant: a product term that could be used to cover minterms of the function.

  46. 1 1 1 1 1 1 P 1 O 1 1 1 1 1 Simplest SOP Expressions • A prime implicant is a product term obtained by combining the maximum possible number of minterms from adjacent squares in the map. • Use bigger groupings (prime implicants) where possible.

  47. 1 1 1 1 P O 1 1 1 1 1 1 1 1 1 1 1 1 Essential prime implicants Simplest SOP Expressions • No redundant groups: • An essential prime implicant is a prime implicant that includes at least one minterm that is not covered by any other prime implicant.

  48. b A bc AB a 00 01 11 10 00 01 11 10 CD 0 1 1 1 0 1 1 1 1 00 01 11 10 1 1 a 0 1 0 0 D 1 1 1 c C 1 1 1 B Quick Review Questions (2) Textbook page 104. 5-3. Identify the prime implicants and the essential prime implicants of the two K-maps below.

  49. Simplest SOP Expressions • Algorithm 1 (non optimal): 1. Count the number of adjacencies for each minterm on the K-map. 2. Select an uncovered minterm with the fewest number of adjacencies. Make an arbitrary choice if more than one choice is possible. 3. Generate a prime implicant for this minterm and put it in the cover. If this minterm is covered by more than one prime implicant, select the one that covers the most uncovered minterms. 4. Repeat steps 2 and 3 until all the minterms have been covered.

  50. Simplest SOP Expressions • Algorithm 2 (non optimal): 1. Circle all prime implicants on the K-map. 2. Identify and select all essential prime implicants for the cover. 3. Select a minimum subset of the remaining prime implicants to complete the cover, that is, to cover those minterms not covered by the essential prime implicants.

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