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1 University of Technology Lecturer: Dr. Sinan Majid Course Title: microprocessors 4 th year
2 Lecture 5 & 6 Minimization with Karnaugh Maps
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, Kmap, and SOP Unoptimized form: number of s in Kmap equals number of minterms (products) in SOP Optimized form: reduced number of minterms x x y y x y x y y x xy xy F = Σ(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 KMaps We can reduce functions by circling s in the Kmap Each circle represents minterm reduction Following circling, we can deduce minimized andor 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.
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 for Four Input Functions Represent functions of 4 inputs with 6 minterms Use same rules developed for 3input functions Note bracketed sections shown in example.
8 Karnaugh map: 4variable example F(,,C,D) = Σm(,2,3,5,6,7,8,,,4,5) F = D C
9 Design examples D D D C C C Kmap for LT Kmap for EQ Kmap for GT LT = EQ = GT = ' ' D + ' C + ' C D ''C'D' + 'C'D + CD + 'CD C' D' + C' + D' Can you draw the truth table for these examples?
10 Physical Implementation C D Step : Truth table Step 2: Kmap Step 3: Minimized sumofproducts EQ Step 4: Physical implementation with gates D C Kmap for EQ
11 Karnaugh Maps Four variable maps. CD F= C + CD +C + C D +C + C F=C +CD + C+ D Need to make sure all s are covered Try to minimize total product terms. Design could be implemented using NNDs and NORs
12 Karnaugh maps: Don t cares In some cases, outputs are undefined We don t care if the logic produces a or a This knowledge can be used to simplify functions. CD C X X X D  Treat X s like either s or s  Very useful  OK to leave some X s uncovered
13 + + Karnaugh maps: Don t cares f(,,c,d) = Σ m(,3,5,7,9) + d(6,2,3) without don't cares  f = D + C D CD C X X X D C D f X X X
14 Don t Care Conditions In some situations, we don t care about the value of a function for certain combinations of the variables. these combinations may be impossible in certain contexts or the value of the function may not matter in when the combinations occur In such situations we say the function is incompletely specified and there are multiple (completely specified) logic functions that can be used in the design. so we can select a function that gives the simplest circuit When constructing the terms in the simplification procedure, we can choose to either cover or not cover the don t care conditions.
15 Map Simplification with Don t Cares CD x x x x x F= C D++C lternative covering. CD x x x x x F= C D+C +C+C
16 Karnaugh maps: don t cares (cont d) f(,,c,d) = Σ m(,3,5,7,9) + d(6,2,3) f = 'D + 'C'D without don't cares f = with don't cares 'D + C'D X X D by using don't care as a "" a 2cube can be formed rather than a cube to cover this node C X don't cares can be treated as s or s depending on which is more advantageous
17 Definition of terms for twolevel simplification Implicant Single product term of the ONset (terms that create a logic ) Prime implicant Implicant that can't be combined with another to form an implicant with fewer literals. Essential prime implicant Prime implicant is essential if it alone covers a minterm in the Kmap Remember that all squares marked with must be covered Objective: Grow implicant into prime implicants (minimize literals per term) Cover the Kmap with as few prime implicants as possible (minimize number of product terms)
18 Examples to illustrate terms C X D 6 prime implicants: ''D, C', C, 'C'D,, 'CD essential minimum cover: C + C' + ''D 5 prime implicants: D, C', CD, 'C, 'C'D D essential minimum cover: 4 essential implicants C
19 Prime Implicants ny single or group of s in the Karnaugh map of a function F is an implicant of F. product term is called a prime implicant of F if it cannot be combined with another term to eliminate a variable. Example: C D If a function F is represented by this Karnaugh Map. Which of the following terms are implicants of F, and which ones are prime implicants of F? (a) C D (b) D (c) C D (d) C (e) C D Implicants: (a),(c),(d),(e) Prime Implicants: (d),(e)
20 Essential Prime Implicants product term is an essential prime implicant if there is a minterm that is only covered by that prime implicant.  The minimal sumofproducts form of F must include all the essential prime implicants of F.
21 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!
22 + pplication of Karnaugh Maps: The Onebit dder dder Cout S S Cout How to use a Karnaugh Map instead of the lgebraic simplification? S = Cout = = = ( + ) + ( + ) + ( + ) = + + = + +
23 + pplication of Karnaugh Maps: The Onebit 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.
24 + pplication of Karnaugh Maps: The Onebit 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 =
25 + pplication of Karnaugh Maps: The Onebit 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 +
26 + pplication of Karnaugh Maps: The Onebit 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 = + +
27 + pplication of Karnaugh Maps: The Onebit dder dder Cout S S Cout Karnaugh Map for S S =
28 + pplication of Karnaugh Maps: The Onebit dder dder Cout S S Cout Karnaugh Map for S S = +
29 + pplication of Karnaugh Maps: The Onebit dder dder Cout S S Cout Karnaugh Map for S S = + +
30 + pplication of Karnaugh Maps: The Onebit dder Can you draw the circuit diagrams? dder Cout S S Cout Karnaugh Map for S S = No Possible Reduction!
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