CS151 Introduction to Digital Design

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CS151 Introduction to Digital Design. Chapter 2-4-1 Map Simplification. Circuit Optimization. Goal: To obtain the simplest implementation for a given function Optimization is a more formal approach to simplification that is performed using a specific procedure or algorithm

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### CS151Introduction to Digital Design

Chapter 2-4-1

Map Simplification

CircuitOptimization
• Goal: To obtain the simplest implementation for a given function
• Optimization is a more formal approach to simplification that is performed using a specific procedure or algorithm
• Optimization requires a cost criterion to measure the simplicity of a circuit
• To distinct cost criteria we will use:
• Literal cost (L)
• Gate input cost (G)
• Gate input cost with NOTs (GN)

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LiteralCost
• Literal – a variable or its complement
• Literal cost – the number of literal appearances in a Boolean expression corresponding to the logic circuit diagram
• Examples:
• F = BD + A C + A L = 8
• F = BD + A C + A + AB L =
• F = (A + B)(A + D)(B + C + )( + + D) L =
• Which solution is best?

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GateInputCost
• Gate input costs - the number of inputs to the gates in the implementation corresponding exactly to the given equation or equations. (G - inverters not counted, GN - inverters counted)
• For SOP and POS equations, it can be found from the equation(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 G = 11, GN = 14
• F = BD + A C + A + AB G = , GN =
• F = (A + )(A + D)(B + C + )( + + D) G = , GN =
• Which solution is best?

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GN = G + 2 = 9

L = 5

G =L+ 2 = 7

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• L (literal count) counts the AND inputs and the single literal OR input.
• G (gate input count) adds the remaining OR gate inputs

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• GN(gate input count with NOTs) adds the inverter inputs
CostCriteria(continued)
• Example 1:
• F = A + B C +

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CostCriteria(continued)
• Example 2:
• F = A B C +
• L = 6 G = 8 GN = 11
• F = (A + )( + C)( + B)
• L = 6 G = 9 GN = 12
• Same function and sameliteral cost
• But first circuit has bettergate input count and bettergate input count with NOTs
• Select it!

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Map Simplification
• Simplifying Boolean Expressions
• Algebraic manipulation
• Lacks rules for predicting steps in the process
• Difficult to determine if the simplest expression has been reached
• Map simplification (Karnough map or K-map)
• Straightforward procedure for function of up to 4 variables
• Provides a pictorial form of the truth table
• Maps for five and six variables can be drawn, but are more cumbersome to use

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Karnaugh Maps (K-map)
• A K-map is a collection of squares
• Each square represents a minterm
• Adjacent squares differ in the value of one variable
• Visual diagram of all possible ways a function may be expressed in standard forms
• Sum-of-Products
• Product-of-Sums
• Alternative algebraic expressions for the same function are derived by recognizing patterns of squares select simplest.

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Two-Variable Map
• 2 variables  4 minterms  4 squares.

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K-Map Function Representation
• Example: F(X,Y) = XY’ + XY
• From the map, we see that F (X,Y) = X.
• This can be justified using algebraic manipulations:

F(X,Y) = XY’ + XY = X(Y’ +Y) = X.1 = X

1 1

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K-Map Function Representation
• Example:G(x,y) = m1 + m2 + m3

G(x,y) = m1 + m2 + m3

= X’Y + XY’ + XY

From the map, we can see that G = X + Y

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Three-Variable Maps

3 variables 8 minterms (m0 – m7).

How can we locate a minterm square on the map?

 Use figure (a) OR  use column # and row # from figure (b)E.g. m5 is in row 1 column 01 (5 10 = 101 2)

Q. Show the area representing X’? Y’? Z’?

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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:

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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 F’ blank
• Example:
• Example:
• Learn the locations of the 8 indices based on the variable order shown (x, most significantand z, least significant) on themap boundaries

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