1. ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN Lecture 6 Dr. Shi Dept. of Electrical and Computer Engineering

2. KARNAUGH MAPS

3. Overview  K-maps: an approach to minimize Boolean 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 Boolean algebra.

4. Example: One-bit Adder Adder Cin Cout S B A Cout = A’*B*Cin + A*B’*Cin+A*B*Cin’ + A*B*Cin Requires: 8 ANDs, 3 Ors How to simplify???

5. xyF001011100110xyF001011100110 Karnaugh maps 01 y x 0 1 1 00 1 01 y x 0 1 x’y’ xy’xy x’y F = Σ(m 0,m 1 ) = x’y + x’y’

6. xyF001011100110xyF001011100110 Karnaugh maps  Alternate way of representing Boolean function  All 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 1’s in K-map equals number of minterms (products) in SOP  Optimized form: reduced number of minterms 01 y x 0 1 1 00 1 01 y x 0 1 x’y’ xy’xy x’y x y F = Σ(m 0,m 1 ) = x’y + x’y’

7. Karnaugh Maps  A Karnaugh map is a graphical tool for assisting in the general simplification procedure.  Two variable maps. 0 A 10 1 B 01 0 1 F=AB +A ’ B 0 A 11 1 B 01 0 1 °Three variable maps. 0 A 11 1 0001 0 1 BC 0 11 1 1110 F=AB’C’ +A B C + ABC +A BC + A’B’C + A’BC’ F=AB + AB + AB AB C F 0000 0011 0101 0110 1001 1011 1101 1111 +

8. Rules for K-Maps  We can reduce functions by circling 1’s in the K- map  Each circle represents minterm reduction  Following circling, we can deduce minimized and-or form. Rules to consider 1. Every cell containing a 1 must be included at least once. 2. The largest possible “power of 2 rectangle” must be enclosed. 3. The 1’s must be enclosed in the smallest possible number of rectangles.

9. Karnaugh Maps  A Karnaugh map is a graphical tool for assisting in the general simplification procedure.  Two variable maps. 0 A 10 1 B 01 0 1 F=AB +A ’ B 0 A 11 1 B 01 0 1 F=A + B °Three variable maps. F=A + B C + BC 0 A 11 1 0001 0 1 BC 0 11 1 1110 F=AB + AB + AB F=AB’C’ +A B C + ABC +A BC + A’B’C + A’BC’

10. Karnaugh maps  Numbering scheme based on Gray–code  e.g., 00, 01, 11, 10  Only a single bit changes in code for adjacent map cells  This is necessary to observe the variable transitions 0001 AB C 0 1 1110 C B A F(A,B,C) =  m(0,4,5,7) G(A,B,C) = 0011 C B A 100010000 011101111 C B A A = AC + B’C’

11. More Karnaugh Map Examples  Examples g = b' 01 0 1 a b c ab 00011110 0 1 01 0 1 a b c ab 00011110 0 1 01 01 f = a 0010 0111 cout = ab + bc + ac 11 00 0011 0011 f = a 1. Circle the largest groups possible. 2. Group dimensions must be a power of 2. 3. Remember what circling means!

12. Application of Karnaugh Maps: The One-bit Adder Adder Cin Cout S B A ABCinSCout 00000 00110 01010 01101 10010 10101 11001 11111 + S = A’B’Cin + A’BCin’ + AB’Cin’ + ABCin Cout = A’BCin + A B’Cin + ABCin’ + ABCin = A’BCin + ABCin + AB’Cin + ABCin + ABCin’ + ABCin = BCin + ACin + AB = (A’ + A)BCin + (B’ + B)ACin + (Cin’ + Cin)AB = 1·BCin + 1· ACin + 1· AB How to use a Karnaugh Map instead of the Algebraic simplification?

13. Application of Karnaugh Maps: The One-bit Adder Adder Cin Cout S B A Karnaugh Map for Cout Now we have to cover all the 1s in the Karnaugh Map using the largest rectangles and as few rectangles as we can. A B Cin 0 0 0 111 0 1

14. A B Application of Karnaugh Maps: The One-bit Adder Adder Cin Cout S B A 0 0 0 0111 1 Karnaugh Map for Cout Now we have to cover all the 1s in the Karnaugh Map using the largest rectangles and as few rectangles as we can. Cout = ACin

15. A B Cin Application of Karnaugh Maps: The One-bit Adder Adder Cin Cout S B A 0 0 0 0111 1 Karnaugh Map for Cout Now we have to cover all the 1s in the Karnaugh Map using the largest rectangles and as few rectangles as we can. Cout = Acin + AB

16. A B Cin Application of Karnaugh Maps: The One-bit Adder Adder Cin Cout S B A 0 0 0 0111 1 Karnaugh Map for Cout Now we have to cover all the 1s in the Karnaugh Map using the largest rectangles and as few rectangles as we can. Cout = ACin + AB + BCin

17. A B Cin Application of Karnaugh Maps: The One-bit Adder Adder Cin Cout S B A 0 1 110 01 0 Karnaugh Map for S S = A’BCin’

18. A B Cin Application of Karnaugh Maps: The One-bit Adder Adder Cin Cout S B A 0 1 110 01 0 Karnaugh Map for S S = A’BCin’ + A’B’Cin

19. A B Cin Application of Karnaugh Maps: The One-bit Adder Adder Cin Cout S B A 0 1 110 01 0 Karnaugh Map for S S = A’BCin’ + A’B’Cin + ABCin

20. A B Cin Application of Karnaugh Maps: The One-bit Adder Adder Cin Cout S B A 0 1 110 01 0 Karnaugh Map for S S = A’BCin’ + A’B’Cin + ABCin + AB’Cin’ No Possible Reduction! Can you draw the circuit diagrams?

21. 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’ = 1

22. MORE KARNAUGH MAPS

23. Overview  Karnaugh maps with four inputs  Same basic rules as three input K-maps  Understanding prime implicants  Related to minterms  Covering all implicants  Using Don’t Cares to simplify functions  Don’t care outputs are undefined  Summarizing Karnaugh maps

24. Karnaugh Maps for Four Input Functions  Represent functions of 4 inputs with 16 minterms  Use same rules developed for 3-input functions  Note bracketed sections shown in example.

25.  F(A,B,C,D) =  m(0,2,3,5,6,7,8,10,11,14,15) F = C+ B’D’ Karnaugh map: 4-variable example D A B 10011001 010001000 11 11 C + A’BD 04150415 12 8 139 3 7 2 6 1511 1410

26. A' B' D + A' C + B' C D B C' D' + A C' + A B D' LT= EQ= GT= K-map for LTK-map for GT Design examples 0 00100010 00 D A11 010001000 B C K-map for EQ 10011001 00 D A00 10011001 B C 010001000 11 D A00 0 00100010 B C Can you draw the truth table for these examples? A'B'C'D' + A'BC'D + ABCD + AB'CD’

27. ABCDABCD EQ Physical Implementation °Step 1: Truth table °Step 2: K-map °Step 3: Minimized sum-of- products °Step 4: Physical implementation with gates K-map for EQ 10011001 00 D A00 10011001 B C

28. Karnaugh Maps  Four variable maps. F=BC +CD + AC+ AD 0 AB 11 0 0001 00 01 CD 0 01 1 1110 F=ABC + ACD + ABC + AB CD + ABC + AB C 1 10 1 11 10 1 11 1 °Need to make sure all 1’s are covered °Try to minimize total product terms. °Design could be implemented using NANDs and NORs

29. Product of Sums

35. SoP from PoS

36. Sum of Products

37. SoP and PoS

38. Quine-McClasky Algorithm  Willard van Orman Quine 1908-2000  Professor at Harvard University  Edward McCluskey, Jr. 1929-  Preofessor at Stanford University  Hundreds of students in academia and industry  Jacob Abraham (UT Austin) W. Kent Fuchs (Cornell) Weiping Shi

39. DON’T CARES (ISFs)

40. Karnaugh maps: Don’t cares  In some cases, outputs are undefined  We “don’t care” if the logic produces a 0 or a 1  This knowledge can be used to simplify functions. 01 X0X1X0X1 D A 1 110X110X 00 B C CD AB 00 01 11 10 00 01 11 10 - Treat X’s like either 1’s or 0’s - Very useful - OK to leave some X’s uncovered

41. + C’D Karnaugh maps: Don’t cares  f(A,B,C,D) =  m(1,3,5,7,9) + d(6,12,13)  without don't cares f = 01 X0X1X0X1 D A 1 110X110X 00 B C A’D CD AB 00 01 11 10 00 01 11 10 Cf 00 01 10 11 00 01 1X 100110011100110011 D 0 1 0 1 0 1 0 101010101101010101 10100XX0010100XX00 A 0 0 0 0 0 0 0 011111111011111111 + B 0 0 0 0 1 1 1 100001111100001111 +

42. 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.

43. Map Simplification with Don’t Cares F=ACD+B+AC 0 AB xx 1 0001 00 01 CD 0 x1 0 1110 1 x0 1 11 10 1 11 x 0 AB xx 1 0001 00 01 CD 0 x1 0 1110 1 x0 1 11 10 1 11 x F=ABCD+ABC+BC+AC °Alternative covering.

44. Karnaugh maps: don’t cares (cont’d)  f(A,B,C,D) =  m(1,3,5,7,9) + d(6,12,13)  f = A'D + B'C'Dwithout don't cares  f = with don't cares don't cares can be treated as 1s or 0s depending on which is more advantageous 01 X0X1X0X1 D A1 110X110X 00 B C A'D by using don't care as a "1" a 2-cube can be formed rather than a 1-cube to cover this node + C'D