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Algebraic Procedures for Canonical Function Representation

Learn how to obtain canonical representations of functions using algebraic procedures with step-by-step examples. Understand transforming from sum-of-products to product-of-sums forms and vice versa.

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Algebraic Procedures for Canonical Function Representation

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  1. Lecture 14: Obtaining Canonical Representations of Functions PROF. INDRANIL SENGUPTA DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING

  2. Algebraic Procedure to Obtain Canonical s-o-p • Examine each term of a given sum-of-products expression; if it is not a minterm, go to the next step. • For all missing variable xi, multiply the term by (xi’ + xi). • Multiply out all products and eliminate redundant product terms. Example: f (a, b, c) = a.b’ + b + a.b.c = a.b’ (c + c’) + b (a + a’)(c + c’) + a.b.c = a.b’.c + a.b’.c’ + a.b.c + a.b.c’ + a’.b.c + a’.b.c’ + a.b.c = a.b’.c + a.b’.c’ + a.b.c + a.b.c’ + a’.b.c + a’.b.c’

  3. Algebraic Procedure to Obtain Canonical p-o-s • Examine each term of a given product-of-sums expression; if it is not a maxterm, go to the next step. • For all missing variable xi, add the term xi’ xi • Obtain the sum terms, and eliminate redundant terms. Example: f (a, b, c) = a’ (b’ + c) = (a’ + bb’ + cc’) (b’ + c + aa’) = (a’ + b + c)(a’ + b + c’)(a’ + b’ + c)(a’ + b’ + c’)(a + b’ + c)(a’ + b’ + c) = (a’ + b + c)(a’ + b + c’)(a’ + b’ + c)(a’ + b’ + c’)(a + b’ + c)

  4. Transforming One Form to Another • Double complement the given function and apply De Morgan’s theorem. • Rule to be followed: • Complement of a sum of true minterms is the same as the sum of the false minterms. • Example: f (a, b, c) = a’.b’.c’ + a’.b’.c + a’.b.c’ + a.b.c’ + a.b.c f = (f’)’ = [(a’.b’.c’ + a’.b’.c + a’.b.c’ + a.b.c’ + a.b.c)’]’ = [a’.b.c + a.b.c’ + a.b’.c]’ (Consider remaining minterms) = (a + b’ + c’)(a’ + b’ + c)(a’ + b + c)

  5. Canonical s-o-p from the Truth Table • Consider rows of the truth table for which the output is 1. • For each such row, form a minterm. • If the input variable is 0, the corresponding variable will appear in complemented form in the minterm. • If the input variable is 1, the corresponding variable will appear in uncomplementedform in the minterm. • Take the sum of all such minterms. • We get the canonical sum-of-products expression.

  6. Example

  7. Converting to Gate Level Realization

  8. Canonical p-o-s from the Truth Table • Consider rows of the truth table for which the output is 0. • For each such row, form a maxterm. • If the input variable is 0, the corresponding variable will appear in uncomplemented form in the maxterm. • If the input variable is 1, the corresponding variable will appear in complemented form in the maxterm. • Take the product of all such maxterms. • We get the canonical product-of-sums expression.

  9. Example

  10. Converting to Gate Level Realization

  11. END OF LECTURE 14

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