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    Based on slides by:Charles Kime & Thomas Kaminski

    2004 Pearson Education, Inc.

    ECE/CS 352: Digital System Fundamentals

    Lecture 7 Karnaugh Maps

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    Chapter 2 2

    Outline

    Circuit Optimization Literal cost Gate input cost

    Two-Variable Karnaugh Maps

    Three-Variable Karnaugh Maps

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    Chapter 2 3

    Circuit Optimization

    Goal: To obtain the simplestimplementation for a given functionOptimization is a more formal approachto simplification that is performed usinga specific procedure or algorithmOptimization requires a cost criterion tomeasure the simplicity of a circuit

    Two distinct cost criteria we will use: Literal cost (L) Gate input cost (G) Gate input cost with NOTs (GN)

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    Chapter 2 4

    D

    Literal a variable or its complementLiteral cost the number of literalappearances in a Boolean expression

    corresponding to the logic circuitdiagramExamples:

    F = BD + A C + A F = BD + A C + A + AB F = (A + B)(A + D)(B + C + )( + + D) Which solution is best?

    Literal Cost

    DB CB B D C

    B C

    L=8

    L=11L=10

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    Gate Input Cost

    Gate input costs - the number of inputs to the gates in theimplementation corresponding exactly to the given equationor equations. (G - inverters not counted, GN - inverters counted)For SOP and POS equations, it can be found from theequation(s) by finding the sum of: all literal appearances the number of terms excluding terms consisting only of a single

    literal,(G) and optionally, the number of distinct complemented single literals (GN).

    Example: F = BD + A C + A F = BD + A C + A + AB F = (A + )(A + D)(B + C + )( + + D) Which solution is best?

    DB CB B D C

    B D B C

    G=11,GN=14G=15,GN=18

    G=14,GN=17

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    Chapter 2 6

    Example 1:F = A + B C +

    Cost Criteria (continued)

    A

    BC

    F

    B C L = 5

    L (literal count) counts the AND inputs and the singleliteral OR input.

    G = L + 2 = 7

    G (gate input count) adds the remaining OR gate inputs

    GN = G + 2 = 9

    GN(gate input count with NOTs) adds the inverter inputs

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    Example 2:F = A B C +L = 6 G = 8 GN = 11F = (A + )( + C)( + B)L = 6 G = 9 GN = 12Same function and sameliteral cost

    But first circuit has bettergate input count and bettergate input count with NOTsSelect it!

    Cost Criteria (continued)

    B C

    A

    ABC

    F

    C B

    F

    AB

    C

    A

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    Why Use Gate Input Counts?

    CMOS logic gates:

    Each input adds:P-type transistor to pull-up networkN-type transistor to pull-down network

    F

    +V

    X

    Y

    +V

    X

    +V

    X

    Y

    (a) NOR

    G = X + Y

    (b) NAND (c) NOT

    X . Y

    X

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    Boolean Function Optimization

    Minimizing the gate inputs reduces circuit cost.Some important questions: When do we stop trying to reduce the cost? Do we know when we have a minimum cost?

    Two-level SOP & POS optimum or near-optimumfunctionsKarnaugh maps (K-maps) Graphical technique useful for up to 5 inputs

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    Two Variable K-Maps

    A 2-variable Karnaugh Map:

    Similar to Gray Code Adjacent minterms differ by one variable

    y = 0 y = 1

    x = 0m 0 = m 1 =

    x = 1 m 2 = m 3 =

    yx yx

    yx yx

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    K-Map and Truth Tables

    The K-Map is just a different form of the truth table.Example Two variable function: We choose a,b,c and d from the set {0,1} to

    implement a particular function, F(x,y).

    Function Table K-MapInput

    Values(x,y)

    FunctionValueF(x,y)

    0 0 a0 1 b1 0 c1 1 d

    y = 0 y = 1

    x = 0 a bx = 1 c d

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    Karnaugh Maps (K-map)

    A K-map is a collection of squares Each square represents a minterm The collection of squares is a graphical representation

    of a Boolean function

    Adjacent squares differ in the value of one variable Alternative algebraic expressions for the same functionare derived by recognizing patterns of squares

    The K-map can be viewed as

    A reorganized version of the truth table

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    Some Uses of K-Maps

    Finding optimum or near optimum SOP and POS standard forms, and two-level AND/OR and OR/AND circuit

    implementations

    for functions with small numbers of variablesDemonstrate concepts used by computer-aided design programs to simplify largecircuits

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    K-Map Function Representation

    Example: F(x,y) = x

    For function F(x,y), the two adjacent cellscontaining 1s can be combined using theMinimization Theorem:

    F = x y = 0 y = 1x = 0 0 0

    x = 1 1 1

    xyxyx)y,x(F

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    K-Map Function Representation

    Example: G(x,y) = x + y

    For G(x,y), two pairs of adjacent cells containing1s can be combined using the Minimization

    Theorem:

    G = x+y y = 0 y = 1

    x = 0 0 1

    x = 1 1 1

    yxyxxyyxyx)y,x(G Duplicate x y

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    Three Variable Maps

    A three-variable K-map:

    Where each minterm corresponds to the productterms:

    Note that if the binary value for an index differs in onebit position, the minterms are adjacent on the K-Map

    yz=00 yz=01 yz=11 yz=10

    x=0 m 0 m 1 m 3 m 2

    x=1 m 4 m 5 m 7 m 6

    yz=00 yz=01 yz=11 yz=10

    x=0

    x=1

    zyx zyx zyx zyx

    zyx zyx zyx zyx

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    Alternative Map Labeling

    Map use largely involves: Entering values into the map, and Reading off product terms from the

    map.

    Alternate labelings are useful:y

    z

    x

    10 2

    4

    3

    5 67

    xy

    zz

    y y z

    z

    10 2

    4

    3

    5 67

    x

    0

    1

    00 01 11 10

    x

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    Example Functions

    By convention, we represent the minterms of F by a "1"in the map and leave the minterms of blank Example:

    Example:

    Learn the locations of the 8indices based on the variableorder shown (x, most significantand z, least significant) on themap boundaries

    y

    x

    10 2

    4

    3

    5 67

    1

    111

    z

    x

    y10 2

    4

    3

    5 671 111

    z

    (2,3,4,5)z)y,F(x, m

    (3,4,6,7)c)b,G(a, m

    F

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    Combining Squares

    By combining squares, we reduce number of literals in a product term, reducing the literal cost,thereby reducing the other two cost criteria

    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 termwith one variable

    Eight adjacent terms is the function of all ones (novariables) = 1.

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    Example: Combining Squares

    Example: Let

    Applying the Minimization Theorem threetimes:

    Thus the four terms that form a 2 2 squarecorrespond to the term "y".

    yzyyz

    zyxzyxzyxzyx)z,y,x(F

    x

    y10 24

    3

    5 671 1

    11

    z

    m(2,3,6,7)F

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    Three-Variable Maps

    Reduced literal product terms for SOP standardforms correspond to rectangles on K-mapscontaining cell counts that are powers of 2.

    Rectangles of 2 cells represent 2 adjacentminterms; of 4 cells represent 4 minterms thatform a pairwise adjacent ring. Rectangles can contain non-adjacent cells due to

    wrap-around at edges

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    Chapter 2 22

    Three-Variable Maps

    Topological warp of 3-variable K-mapsthat shows all adjacencies:

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    Chapter 2 23

    Three-Variable Maps

    Example Shapes of 2-cell Rectangles:y0 1 3 2

    5 64 7x

    z

    XY

    YZ

    XZ

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    Chapter 2 24

    Three-Variable Maps

    Example Shapes of 4-cell Rectangles:

    Read off the product terms for therectangles shown

    y0 1 3 2

    5 64 7x

    z

    Y

    Z

    Z

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    Chapter 2 25

    Three Variable Maps

    z)y,F(x,

    y

    1 1

    x

    z

    1 1

    1

    z

    z

    yx

    yx

    K-Maps can be used to simplify Boolean functions bysystematic methods. Terms are selected to cover the1sin the map.

    Example: Simplify )(1,2,3,5,7z)y,F(x, m

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    Summary

    Circuit Optimization Literal cost Gate input cost

    Two-Variable Karnaugh Maps

    Three-Variable Karnaugh Maps