dig3KarnghMapsWithXOR_C

dig3KarnghMapsWithXOR_C - Digital Circuit Engineering...

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© John Knight dig3KarnghMapsWithXOR_C.fm p. 0 Revised; February 17, 2009 Digital Circuit Engineering Carleton University 2009 MAPS (X + A)(X + B) = X + AB YX + X = X 2nd Distributive Simplification General DeMorgan F(a, b,. .. z,+, . ,1,0) F(a , b , . .. z , . ,+,0,1) Absorption Y + XY = X + Y 00 01 11 10 01 a bc ab + bc + ca b c 1 1 1 1 00 01 10 a bc = ab + ca b c 1 1 1 1 Consensus, use Karnaugh ab + bc + ca KARN G A U H 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 111 1 1 Easily Forgotten Wrap-Arounds Slide i Carleton University © John Knight Digital Circuits 3 p. 1, dig3KarnghMapsWithXOR_C.fmRevised; February 17, 2009 Chapter 3 Comment on Slide i Karnaugh Maps Another form of the truth table Labelling Maps Circling Maps Minimizing Algebra by Maps Precautions when circling Don’t Cares Where don’t cares come from - Binary-Coded Decimal (BCD) digits Don’t cares on Karnaugh Maps Appendix XORs logic manipulation Using Karnaugh Maps with XORs
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© John Knight dig3KarnghMapsWithXOR_C.fm p. 2 Revised; February 17, 2009 c b 00 01 11 10 00 01 10 bc F bc F b · c b · c b · c aa 00 01 10 01 a bc a A truth table Redraw table Label abc on Another way Abbreviated of labelling the sides labelling 00 01 10 a bc b c Combined labelling b · c This order is important 0 00 0 01 0 0 10 1 00 1 01 1 1 10 abc F half for a= 0 half for a= 1 This order is important Compact 00 01 10 bc F a= 0 1 F 00 01 10 a bc Label inputs on the with F not yet filled in. with halves side by side the table 2 bits change 1 bit changes Layout for a Karnaugh map Axis Labelling Conventions Check that moving one square only changes one input bit sides map of F abc Slide 2 Karnaugh Maps; Equations From Truth Tables Carleton University © John Knight Digital Circuits 3 p. 3, dig3KarnghMapsWithXOR_C.fmRevised; February 17, 2009 Karnaugh Maps; Equations From Truth Tables Karnaugh Maps Karnaugh Maps The map is like a truth table Each square on the map represents a different input combination. All possible input combinations are represented on the map. The inputs are labelled around the edges of the map. Not inside the squares as shown on the right. Arrangement of the squares As one steps from one square to the next, either up, down, left or right, only one bit should change in a single step. If one goes to the nearest diagonal neighbour, two bits will change. 100 101 111 110 000 001 011 010 abc abc one bit change one bit change two bit change Comment on Slide 2
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© John Knight dig3KarnghMapsWithXOR_C.fm p. 4 Revised; February 17, 2009 00 01 11 10 01 a bc 00 01 10 a bc b c Different Functions Using AND F=abc 1 00 01 10 a bc F=a bc (010) 1 b c 00 01 10 a bc F=a b b c 00 01 10 a bc F=b 1 b c 1 1 1 1 All the squares where a=0, b=1. All the squares where b=1. 1 1 1 1 F=b 00 01 10 a bc b c 00 01 10 a bc b c 1 1 1 1 F=? Wrap around 1 1 F=? c b a 1 1 1 1 1 1 1 1 F=? All the squares where 1 Slide 3 Karnaugh Maps; Representing AND Terms Carleton University © John Knight Digital Circuits 3 p. 5, dig3KarnghMapsWithXOR_C.fmRevised; February 17, 2009 Karnaugh Maps; Representing AND Terms Representing AND Terms Representing AND Terms Any single square (Top row, first two maps)
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This note was uploaded on 02/07/2010 for the course ELEC 2607 taught by Professor Lee during the Winter '10 term at Carleton CA.

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dig3KarnghMapsWithXOR_C - Digital Circuit Engineering...

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