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Lecture 24: Logic Design (Part 1)

Lecture 24: Logic Design (Part 1). PROF. INDRANIL SENGUPTA DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING. Introduction. We shall discuss some of the common functional blocks used in logic design. We shall discuss the following: Multiplexer Decoder / Demultiplexer Encoder Comparator.

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Lecture 24: Logic Design (Part 1)

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  1. Lecture 24: Logic Design (Part 1) PROF. INDRANIL SENGUPTA DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING

  2. Introduction • We shall discuss some of the common functional blocks used in logic design. • We shall discuss the following: • Multiplexer • Decoder / Demultiplexer • Encoder • Comparator

  3. Data Selectors or Multiplexers • A multiplexer is an electronic switch that has: • A set of n data input lines D = {D0, D1, …, Dn-1} • A set of m select input lines S = {S0, S1, …, Sm-1} • One output line f • One of the input data lines is connected to the output depending upon the value of the select lines (f = D0 if S = 0, f = D1 if S = 1, f = D2 if S = 2, and so on). • Usually, n = 2m. • Also called a 2m-to-1 multiplexer.

  4. D0 2m-to-1 MUX D1 f Dn-1 S0 S1 Sm-1

  5. Implement a 2-to-1 MUX using Gates D0 2-to-1 MUX f D1 S0 f = S0’.D0 + S0.D1

  6. Implement a 4-to-1 MUX using Gates D0 4-to-1 MUX D1 f D2 D3 S0 S1 f = S1’.S0’.D0 + S1’.S0.D1 + S1.S0’.D2 + S1.S0.D3

  7. Implement a 4-to-1 MUX using 2-to-1 MUX-es D0 2-to-1 MUX D1 2-to-1 MUX f S0 D2 2-to-1 MUX S1 D3 S0

  8. Implement a 8-to-1 MUX using Smaller MUX-es D0 D4 4-to-1 MUX 4-to-1 MUX D1 D5 D2 D6 2-to-1 MUX D3 D7 f S0 S1 S2

  9. Implement Logic Functions using MUX • Implement an n-variable function using 2n-to-1 MUX. • Connect the n variables to the select lines. • Connect the truth table output column values to data inputs.

  10. Implement an n-variable function using 2n-1-to-1 MUX. • Connect (n-1) variables to the select lines. • Apply the remaining variable, its complement, 0 or 1 to the data inputs. • For this, we partition the truth table into groups of two rows each. D0 4-to-1 MUX D1 f D2 D3 S0 S1

  11. D0 8-to-1 MUX D1 D2 D3 f D4 D5 D6 D7 S2 S0 S1

  12. Implement Arbitrary Functions using 2-to-1 MUX • We repeatedly split a function into two smaller sub-functions by using Shannon’s decomposition theorem. • Shannon’s Decomposition Theorem: • An n-variable function f (x1, x2, …, xn) can be decomposed with respect to any of the n variables (say, x1) as: f (x1, x2, …, xn) = x1’.f (0, x2, …, xn) + x1. f (1, x2, …, xn) = x1’.f10 + x1.f11 • Each application of Shannon’s decomposition theorem is like using a 2-to-1 MUX: • Apply x1to the select inputs, and f10 and f11 to the data inputs.

  13. An Example Worked Out Given function: f = A’.B + B.C + A.C’ Expanding w.r.t. A, we get: f = A’.(B + B.C) + A.(B.C + C’) = A’.(B) + A.(B + C’) Expanding (B + C’) w.r.t. B, we get: B + C’ = B’.(C’) + B.(1)

  14. END OF LECTURE 24

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