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# lecture10 - Lecture 10 Digital Logic Design Basics Logic...

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Unformatted text preview: Lecture 10 Digital Logic Design -- Basics Logic Design  From now, we’ll study how a modern processor is built starting with basic logic elements as building blocks.  Why study logic design?   Understand what processors can do fast and what they can’t do fast (avoid slow things if you want your code to run fast!)   Background for more detailed hardware courses Logic Gates  Basic building blocks are logic gates.   In the beginning, did ad hoc designs, and then saw patterns repeated, gave names   Can build gates with transistors and resistors  Then found theoretical basis for design   Can represent and reason about gates with truth tables and Boolean algebra   Assume know truth tables and Boolean algebra from a math or circuits course.   Section B.2 in the textbook has a review Gates A transistor inverter or A NOT gate A NAND gate A NOR gate Gates and Boolean Algebra Five basic gates Truth Table Digit Display Building a Circuit   Use And/Or gates to implement this expression  How many gates do we use?  Can we use less Boolean Algebra 101 How to simplify Boolean expression? Boolean Algebra (1)  Using just ‘0’ and ‘1’s (after George Boole)  Truth Table   A Boolean function with n variables has only 2^n possible input combinations   A table with 2^n rows Boolean Algebra (2)  Algebra representations Boolean Algebra (3) Another example with 3 inputs Boolean Algebra (4)   All Boolean functions can be implemented by logic gates (AND, OR, NOT) and vice versa.  Boolean operations   AND: XY, X ^ Y, X &Y  OR: X+Y, X v Y, X | Y  NOT: X, X’, /X, \X  Order: NOT, AND, OR Basics  P1: X = 0 or X = 1  P2: 0 . 0 = 0  P3: 1 + 1 = 1  P4: 0 + 0 = 0  P5: 1 . 1 = 1  P6: 1 . 0 = 0 . 1 = 0  P7: 1 + 0 = 0 + 1 = 1 Boolean Laws  T1 : Commutative Law   A + B = B + A   A B = B A  T2 : Associate Law   (A + B) + C = A + (B + C)   (A B) C = A (B C) Boolean Laws  T3 : Distributive Law   A (B + C) = A B + A C   A + (B C) = (A + B) (A + C)  T4 : Identity Law   A + A = A   A A = A Laws  T6 : Redundancy Law   A + A B = A   A (A + B) = A  T7   0 + A = A   0 A = 0  T8   1 + A = 1   1 A = A Other Laws Some identities of Boolean algebra. Example Idempotent theorem + commutative law distributive law complementarity law identity law Question 1 Simplify the Boolean expression (A+B+C)(D+E)' + (A+B+C)(D+E) and choose the best answer.  A + B + C  D + E  A'B'C'  D'E'  None of the above Question 2 Given the function F(X,Y,Z) = XZ + Z(X'+ XY), the equivalent most simplijied Boolean representation for F is:  Z + YZ  Z + XYZ  XZ  X + YZ  None of the above Question 3 An equivalent representation for the Boolean expression A' + 1 is  A  A'  1  0 Question 4 Simplijication of the Boolean expression AB + ABC + ABCD + ABCDE + ABCDEF yields which of the following results?  ABCDEF  AB  AB + CD + EF  A + B + C + D + E + F  A + B(C+D(E+F)) Question 5 Given that F = A'B'+ C'+ D'+ E', which of the following represent the only correct expression for F'?  F'= A+B+C+D+E  F'= ABCDE  F'= AB(C+D+E)  F'= AB+C'+D'+E'  F'= (A+B)CDE Deriving Complement of a Boolean Expression   DeMorgan’s LAW: The complement of a Boolean expression is formed by replacing all literals by their compliments; ANDs become ORs, ORs become ANDs X + Y = X •Y X •Y = X + Y Using DeMorgan’s Law X + Y = X •Y X •Y = X + Y X +Y • Z = ? Question 5 Given that F = A'B'+ C'+ D'+ E', which of the following represent the only correct expression for F'?  F'= A+B+C+D+E  F'= ABCDE  F'= AB(C+D+E)  F'= AB+C'+D'+E'  F'= (A+B)CDE Question 6 Given the function F(A,B,X,Y) = AB + X'Y, the most simplijied Boolean representation for F' is  (AB)' + (X'Y)  A'B' + XY'  (A'+ B')(X + Y')  (AB + X'Y)'  (AB)'(X'Y)' Question 7 Given the function F(X,Y,Z) = XZ + Z(X'+ XY), the equivalent, most simplijied Boolean representation for F is  Z + YZ  Z + XYZ  XZ  Z  none of the above Question 8 Simplijication of the Boolean expression AB + A (BC)' yields which of following results?  A  BC  B  AB  (BC)' ...
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