ENGIN 112 Intro to Electrical and Computer Engineering

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1 ENGIN 2 Intro to Electrical and Computer Engineering Lecture 8 Minimization with Karnaugh Maps

2 Overview K-maps: an alternate approach to representing oolean functions K-map representation can be used to minimize oolean functions Easy conversion from truth table to K-map to minimized SOP representation. Simple rules (steps) used to perform minimization Leads to minimized SOP representation. Much faster and more more efficient than previous minimization techniques with oolean algebra.

3 Karnaugh maps lternate way of representing oolean function ll rows of truth table represented with a square Each square represents a minterm Easy to convert between truth table, K-map, and SOP Unoptimized form: number of s in K-map equals number of minterms (products) in SOP Optimized form: reduced number of minterms y y x x y x y y x x xy xy F = S(m,m ) = x y + x y x y F

4 Karnaugh Maps Karnaugh map is a graphical tool for assisting in the general simplification procedure. Two variable maps. Three variable maps. C F= + F= + + C F F= C + C +C +C + C + C

5 Rules for K-Maps We can reduce functions by circling s in the K-map Each circle represents minterm reduction Following circling, we can deduce minimized and-or form. Rules to consider Every cell containing a must be included at least once. The largest possible power of 2 rectangle must be enclosed. The s must be enclosed in the smallest possible number of rectangles. Example

6 Karnaugh Maps Karnaugh map is a graphical tool for assisting in the general simplification procedure. Two variable maps. Three variable maps. C F= + F= + + F=+ F=+ C +C F= C + C +C +C + C + C

7 Karnaugh maps Numbering scheme based on Gray code e.g.,,,, Only a single bit changes in code for adjacent map cells This is necessary to observe the variable transitions C G(,,C) = C C C F(,,C) = Σm(,4,5,7) = C + C

8 More Karnaugh Map Examples Examples c a b f = a ab cout = ab + bc + ac b c a g = b' ab f = a. Circle the largest groups possible. 2. Group dimensions must be a power of Remember what circling means!

9 pplication of Karnaugh Maps: The One-bit dder dder Cout S S Cout How to use a Karnaugh Map instead of the lgebraic simplification? S = Cout = = = ( + ) + ( + ) + ( + ) = + + = + +

10 pplication of Karnaugh Maps: The One-bit dder dder Cout S S Cout Karnaugh Map for Cout Now we have to cover all the s in the Karnaugh Map using the largest rectangles and as few rectangles as we can.

11 pplication of Karnaugh Maps: The One-bit dder dder Cout S S Cout Now we have to cover all the s in the Karnaugh Map using the largest rectangles and as few rectangles as we can. Karnaugh Map for Cout Cout =

12 pplication of Karnaugh Maps: The One-bit dder dder Cout S S Cout Now we have to cover all the s in the Karnaugh Map using the largest rectangles and as few rectangles as we can. Karnaugh Map for Cout Cout = cin +

13 pplication of Karnaugh Maps: The One-bit dder dder Cout S S Cout Now we have to cover all the s in the Karnaugh Map using the largest rectangles and as few rectangles as we can. Karnaugh Map for Cout Cout = + +

14 pplication of Karnaugh Maps: The One-bit dder dder Cout S S Cout Karnaugh Map for S S =

15 pplication of Karnaugh Maps: The One-bit dder dder Cout S S Cout Karnaugh Map for S S = +

16 pplication of Karnaugh Maps: The One-bit dder dder Cout S S Cout Karnaugh Map for S S = + +

17 pplication of Karnaugh Maps: The One-bit dder Can you draw the circuit diagrams? dder Cout S S Cout Karnaugh Map for S S = No Possible Reduction!

18 Summary Karnaugh map allows us to represent functions with new notation Representation allows for logic reduction. Implement same function with less logic Each square represents one minterm Each circle leads to one product term Not all functions can be reduced Each circle represents an application of: Distributive rule -- x(y + z) = xy + xz Complement rule x + x =

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