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Introduction to digital logic, fundamental logic gates, Boolean algebra & simplifications, 3-6 input Kanaugh Maps

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- 1. Digital Logic CS2052 Computer Architecture Computer Science & Engineering University of Moratuwa Dilum Bandara Dilum.Bandara@uom.lk
- 2. Outline Logic gates Boolean Algebra Kanaugh Maps 2
- 3. Logic Gates Every digital device is based on a set of chips designed to store & process information Basic building blocks of these chips are logic gates Implementation of gates can be different Different materials & fabrication technologies Different operating voltages e.g., 5v vs. 0v, 3+v vs. 0.5v Today, they go as low as 1.9 - 2.1v But their logical behavior is consistent across all computers 3
- 4. Common Logic Gates NOT/inverter AND 4 Truth table Algebraic function
- 5. Common Logic Gates (Cont.) OR NAND 5
- 6. Common Logic Gates (Cont.) NOR XOR/EXOR (Exclusive OR) 6
- 7. Common Logic Gates (Cont.) NXOR/EXNOR (Exclusive NOR) 7
- 8. Exercise Implement 3-input AND using 2-inputs ANDs Implement a NOT gate using a NAND 8
- 9. Fundamental Logic Gate 2-input NAND gate can be used to build any other gate 9
- 10. Exercise Implement AND using NANDs Homework Implement following gates using NANDs OR XOR 10
- 11. Boolean Algebra Boolean variable Takes only 2 values – either TRUE (1) or FALSE (0) Boolean function Mapping from Boolean variables to a Boolean value Application Can represent complex relationships in a digital circuit Boolean algebra Deals with binary variables & logic operations operating on those variables Application Can simplify a relationship among multiple inputs in a digital circuit 11
- 12. Basic Identities of Boolean Algebra Basic operations AND (.), OR (+), NOT (– or /) 0 – FALSE 1 – TRUE x + 0 = x x + x = x x · 0 = 0 x . x = x x + 1 = 1 x + x/ = 1 x · 1 = x x · x / = 0 12
- 13. Basic Identities (Cont.) Commutativity x + y = y + x xy = yx Associativity x + ( y + z ) = ( x + y ) + z x (yz) = (xy) z Distributivity x ( y + z ) = xy + xz x + yz = ( x + y )( x + z) 13 Textbook has a typo
- 14. Basic Identities (Cont.) DeMorgan’s Theorem ( x + y )/ = x/ y/ ( xy )/ = x/ + y/ Generalized DeMorgan's Theorem (a + b + … z) / = a / b / … z / (a.b … z) / = a / + b / + … z / Involution (x /) / = x 14
- 15. Exercise Simplify following functions using Boolean Algebra a + ab a(a + b) a(a / + b) a + a/ b (a + b.c) / a + ab = a(1+b)=a a(a + b) = a.a +ab=a+ab=a(1+b)=a a + a'b = (a + a')(a + b)=1(a + b) =a+b a(a' + b) = a.a' +ab=0+ab=ab (a + b.c)' = a'.b' + a'.c' 15
- 16. Homework Show steps for following simplifications (a + b)(a + b') = a ab + ab'c = ab + ac (a + b)(a + b' + c) = a + bc (a(b + z(x + a')))' = a' + b' (z' + x') (a(b + c) + a'b)'=b'(a' + c') (a + b)(a' + c)(b + c) = (a + b)(a' + c) No need to submit answers 16
- 17. Minterms Minterm Each combination of variables in a truth table As no of digital inputs (Boolean variables) increase no of minterms increase 2n minterms for n inputs As n increases Large truth tables Long Boolean functions Easier to represent using decimal equivalent of sum of minterms F(a, b, c) = Σ (2, 4, 5, 7) 17
- 18. Exercise Represent all odd numbers between 0 – 7 as a sum of minterms F(a, b, c) = Σ (1, 3, 5, 7) Use Boolean Algebra to simplify above function Σ (1, 3, 5, 7) = a‘b‘c + a‘bc + ab‘c + abc = a‘c(b‘ + b) + ac (b‘ + b) = a‘c + ac = c (a‘+ a) = c 18 Sum of Products
- 19. Karnaugh Map (K-Map) Pictorial representation of a truth table Map of squares – each square for each minterm Adjacent squares change in minterm only by one variable 19 bc a 00 01 11 10 0 0 1 3 2 1 4 5 7 6 Decimal equivalent of minterm
- 20. Example – 3 Variable K-Map Simplify represent of all odd numbers between 0 – 7 F(a, b, c) = Σ (1, 3, 5, 7) 20 ab c 00 01 11 10 0 1 1 1 1 1 0 0 0 0 = c
- 21. Exercise – 3 Variable K-Map Simplify following expression using a K-map 21 ab c 00 01 11 10 0 1 1 0 0 0 1 1 0 0 = a’c’ + a’b’
- 22. Kanaugh Map – Simplification Squares containing 1’s are grouped to get sum- of-product expression Group size must be 2n Group must be in rectangular shape Each group represents an algebraic term Its OK for groups to overlap OR of all those terms is the final expression Maximum group size better simplification A don’t care (X) can be interpreted as either 0 or 1 when needed, only if it contributes to simplification Don’t group Xs together! 22
- 24. Example – 4 Variable K-Map 24
- 25. Exercise – 4 Variable K-Map What would be a Boolean expression of a circuit that detects all even numbers between 0 – 15? Present as a sum of minterms 25
- 26. 5 Variable K-Map – Option 1 26 BC DE A 1 /0 00 01 11 10 00 01 11 10 0 5 3 1 8124 2 20 11 106 7 14 16 13 9 15 19 18 17 28 24 31 2921 23 3022 26 25 27
- 27. Example – 5 Variable K-Map What would be a Boolean expression of a circuit to detect all prime numbers between 0 & 31? 27 BC DE A 1 /0 00 01 11 10 00 01 11 10 0 5 3 1 8124 2 20 11 106 7 14 16 13 9 15 19 18 17 28 24 31 2921 23 3022 26 25 27
- 28. 5 Variable K-Map – Option 2 28
- 29. Example – 5 Variable K-Map – Option 2 29
- 31. Simplification as Product of Sum Some times it is useful to obtain the algebraic expression as a product of sums Can be obtained by finding sum of products for F/ & converting it to F To find F/ Group 0s in a K-map F(a, b, c) = Σ (1, 3, 5, 7) = Π (1, 3, 5, 7) 31
- 32. Exercise – Product of Sum Find sum-of-product & product-of-sum representation of following K-map 32 ab c 00 01 11 10 0 1 1 x 1 1 0 0 0 1 F(a, b, c) = ab’ + c F/(a, b, c) = a’c’ + bc’ (F/(a, b, c)) / = (a + c)(b’ + c) = F(a, b, c)

- ( x + y )( x + z) = x + xz + xy + yz = x(1 + z + y) + yz = x + yz