cs2100-5-Karnaugh-Maps

# cs2100-5-Karnaugh-Maps - CS2100Computer Organisation...

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CS2100 Computer  Organisation http://www.comp.nus.edu.sg/~cs2100/ Karnaugh Maps (AY2009/2010) Semester 2

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CS2100 Karnaugh Maps 2 WHERE ARE WE NOW? Number systems and codes Boolean algebra Logic gates and circuits Simplification Combinational circuits Sequential circuits Performance Assembly language The processor: Datapath and control Pipelining Memory hierarchy: Cache Input/output Preparation: 2 weeks Logic Design: 3 weeks Computer organisation
CS2100 Karnaugh Maps 3 KARNAUGH MAPS Function Simplification Algebraic Simplification Half Adder Introduction to K-maps How to use K-maps Converting to Minterms Form Prime Implicants and Essential Prime Implicants Example on Finding Minimal SOP Expression Finding POS Expression Don’t-care Conditions

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CS2100 Karnaugh Maps 4 FUNCTION SIMPLIFICATION Why simplify? Simpler expression uses fewer logic gates. Thus cheaper, uses less power, (sometimes) faster. Techniques Algebraic Using theorems Open-ended; requires skills Karnaugh Maps Easy to use Limited to no more than 6 variables Quine-McCluskey Suitable for automation Can handle many variables (but computationally intensive)
CS2100 Karnaugh Maps 5 ALGEBRAIC SIMPLIFICATION (1/4) Aims to minimise Number of literals, and Number of terms But sometimes conflicting, so let’s aim at reducing the number of literals for the examples in the next few slides. Difficult – needs good algebraic manipulation skills.

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CS2100 Karnaugh Maps 6 ALGEBRAIC SIMPLIFICATION (2/4) Example 1: Simplify (x+y) (x+y') (x'+z) (x+y) (x+y') (x'+z) = ( x x + x y' + x y + y y' ) (x'+z) (associativity) = (x + x ( y'+y ) + 0 ) (x'+z) (idemp, assoc., complement) = (x + x 1 ) (x'+z) (complement, identity) = ( x + x ) (x'+z) (identity) = x (x'+z) (idempotency) = x x' + x z (associativity) = 0 + x z (complement) = x z (identity) Number of literals reduced from 6 to 2.
CS2100 Karnaugh Maps 7 ALGEBRAIC SIMPLIFICATION (3/4) Example 2: Find minimal SOP and POS expressions of F(x,y,z) = x' y (z+y' x) + y' z x' y (z+y' x) + y' z = Minimal SOP: Minimal POS: a

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CS2100 Karnaugh Maps 8 ALGEBRAIC SIMPLIFICATION (4/4) Example 3: Find minimal SOP expression of F(a,b,c,d) = a b c + a b d + a' b c' + c d + b d' a b c + a b d + a' b c' + c d + b d' = a
Karnaugh Maps 9 HALF ADDER (1/2) Half adder is a circuit that adds 2 single bits (X, Y) to produce a result of 2 bits (C, S). The

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cs2100-5-Karnaugh-Maps - CS2100Computer Organisation...

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