EE101-U3-BooleanAlgebra-Nazarian-Spring12

# T4 x x t4 0 shahin nazarianee101spring 2012 1 0 23

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Unformatted text preview: 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 Outputs O1 O0 0 1 0 1 1 1 0 1 0 1 1 0 0 1 1 1 Output values when inputs are 010 28 Truth Tables to Equations/Circuits •  Given a circuit with n-inputs, 2n possible combinations exist => 2n rows in a T.T. •  A general approach to converting a T.T. to equation or circuit… •  Build a checker/decoder circuit for each combination and include the combinations where the function should be ‘1’ Shahin Nazarian/EE101/Spring 2012 29 Finding Equations/Circuits •  Given a function and checkers (called decoders) for each combination, we just need to OR together the checkers where F = 1 Checker for 000 Checker for 001 Assume we use ANDgate decoders that output a ‘1’ when the combination is found Checker for 011 3-bit number {x,y,z} Checker for 100 Checker for 101 Z F 0 0 0 0 0 1 0 0 F Y 0 Checker for 010 X 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1 Checker for 110 Checker for 111 Shahin Nazarian/EE101/Spring 2012 30 Boolean Algebra Terminology Shahin Nazarian/EE101/Spring 2012 31 Minterms decodes Minterm 3 011 = x’•y•z = m3 Minterm 5 101 = x•y’•z = m5 To make the minterm, complement the variables that equal 0 and leave the variables in their true form that equal 1 Shahin Nazarian/EE101/Spring 2012 X 0 0 0 0 1 1 1 1 Y 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 F 0 0 1 1 0 1 0 1 32 Minterms (Cont.) •  Only one minterm can evaluate to 1 at any time X 0 0 1 1 Y 0 1 0 1 F 0 1 1 0 Shahin Nazarian/EE101/Spring 2012 m0 m1 m2 m3 X’•Y’ X’•Y X•Y’ X•Y 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 33 Canonical Sums •  We OR together all the minterms where F = 1 •  (Σ = SUM or OR of all the minterms) F = m2+m3+m5+m7 F = Σxyz(2,3,5,7) List the minterms where F is 1 Shahin Nazarian/EE101/Spring 2012 m0 m1 m2 m3 m4 m5 m6 m7 X 0 0 0 0 1 1 1 1 Y 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 F 0 0 1 1 0 1 0 1 34 Example Shahin Nazarian/EE101/Spring 2012 35 Maxterms •  One maxterm for every combination of inputs •  Each maxterm is the OR’ing of variables that will evaluate to 0 for only one combination •  decodes A maxterm “checks” or “_______” for a specific input combination and outputs 0 when found Maxterm 1 001 = x+y+z’ = M1 Maxterm 3 011 = x+y’+z’ = M3 To make the maxterm, complement the variables that equal 1 and leave the variables in their true form that equal 0 Shahin Nazarian/EE101/Spring 2012 X 0 0 0 0 1 1 1 1 Y 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 F 0 0 1 1 0 1 0 1 36 Maxterms (Cont.) •  Only one maxterm can evaluate to 0 at any time X 0 0 1 1 Y 0 1 0 1 F 0 1 1 0 Shahin Nazarian/EE101/Spring 2012 M0 X+Y 0 1 1 1 M1 M2 M3 X+Y’ X’+Y X’+Y’ 1 1 1 0 1 1 1 0 1 1 1 0 37 Canonical Products •  We AND together all the maxterms where F=0 F = M0•M1•M4•M6 F = Πxyz(0,1,4,6) List the maxterms where F is 0 Shahin Nazarian/...
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## This note was uploaded on 03/13/2013 for the course EE 101 taught by Professor Redekopp during the Spring '06 term at USC.

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