Lecture #9 - Karnaugh Maps

Lecture #9 - Karnaugh Maps - ECE 301 Digital Electronics...

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Unformatted text preview: ECE 301 Digital Electronics Karnaugh Maps (Lecture #9) Learning Objectives Karnaugh Maps 2-variable 3-variable 4-variable How to use K-Maps Simplifying a minterm expansion (minimum SOP) Simplifying a maxterm expansion (minimum POS) Spring 2012 2 ECE 301 - Digital Reading Roth & Kinney Sections 5.1 5.3 Spring 2012 ECE 301 - Digital 3 Simplification Boolean Algebra is useful for general proofs. And can be used to simplify logic functions. However, two problems arise 1. Algebraic procedures are difficult to apply in a systematic way. 2. Difficult to determine when a minimum solution has been achieved. Karnaugh maps are generally faster and Spring 2012 ECE 301 - Digital 4 Simplification: Example Find the minimum SOP expression for the following logic function: F(A,B,C) = m(0,1,2,5,6,7) Spring 2012 ECE 301 - Digital 5 Simplification: Example F(A,B,C) = m0 + m1 + m2 + m5 + m6 + m7 F(A,B,C) = ABC + ABC + ABC + ABC + ABC + ABC F = (ABC + ABC) + (ABC + ABC) + (ABC + ABC) + (ABC + ABC) F = AB + AC + BC + AB Spring 2012 ECE 301 - Digital 6 Is this the minimum SOP expression? Simplification: Example F(A,B,C) = m0 + m1 + m2 + m5 + m6 + m7 F(A,B,C) = ABC + ABC + ABC + ABC + ABC + ABC F = (ABC + ABC) + (ABC + ABC) + (ABC + ABC) F = AB + BC + AC Spring 2012 ECE 301 - Digital 7 Unfortunately, it is not! Instead, Karnaugh Maps Like a truth table, the Karnaugh map of a function specifies the logical value of the function for all combinations of the input variables. Spring 2012 ECE 301 - Digital 8 Two-variable K-map Spring 2012 ECE 301 - Digital 9 Row # A B minterm m0 1 1 m1 2 1 m2 3 1 1 m3 m0 m2 m1 m3 1 1 A B m0 = AB m1 = AB m2 = AB m3 = AB A = 0, B = 1 A = 1, B = 0 Two-variable K-map Spring 2012 ECE 301 - Digital 10 Row # A B maxterm M0 1 1 M1 2 1 M2 3 1 1 M3 M0 M2 M1 M3 1 1 A B M0 = A + B M1 = A + B M2 = A + B M3 = A + B Two-variable K-map: Example Spring 2012 ECE 301 - Digital 11 Row # A B F 1 1 1 2 1 1 3 1 1 1 1 1 1 A B Minterm Expansion: F(A,B) = AB + AB = m(0, 2) Maxterm Expansion: F(A,B) = (A + B).(A + B) = M(1, 3) 2 1 3 Can this logic function be simplified?...
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Lecture #9 - Karnaugh Maps - ECE 301 Digital Electronics...

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