Chapter 3. Table of content Chapter 1: Switching Algebra Chapter 2: Logical Levels, Timing & Delays
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1 hapter 3 Dr.-ng. Stefan Werner Tale of ontent hapter 1: Swithing lgera hapter 2: Logial Levels, Timing & Delays hapter 3: Karnaugh-Veith-Maps hapter 4: ominational iruit Design hapter 5: Lathes and Flip Flops hapter 6: Finite State Mahines hapter 7: asi Sequential iruits hapter 8: Numer Systems hapter 9: inary rithmeti hapter 10: inary odes Simplifiation of logial funtions Struture of KV-Maps Logial funtions in a KV-Map Minimization with KV Maps Minimizing aross the orders KV-Maps and Don t ares Produt-of-Sums and KV-Maps Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 2of 17 1
2 lternatives Original iruit Disjuntive Form onjuntive Form Whih iruit is the est implementation? What is the EST? Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 3of 17 iruit selet Original iruit Disjuntive Form onjuntive Form Result: hose iruit 1 or 3 for a ompat solution hose iruit 2 or 3 for a fast solution Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 4of 17 2
3 Try... Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 5of 17 KV-Map for multiple variales The KV-Map for multiple variales an e otained y mirroring the asi struture. The distriution of the variale s region does not need to e exatly like this. There are also other possiilities Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 6of 17 3
4 Two different valid setup for a 4 variale KV Map a d Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 7of 17 Neighours aross orders n 2-dimensional representation, it is not learly shown that the fields at the map s orders are neighours to eah other. This irumstane eomes learer when skething a KV-Map in 3 dimensions. Here we an easily see, that and are neighours. Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 8of 17 4
5 KV-Maps for more than 4 variales The assignment of areas in a KV-Map with 5 variales is ompliated. Using a KV-Map on more than 4 variales will lead to at least one area that is split up. This is the reason why the utilisation of KV-Map is somehow limited. Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 9of 17 Logial funtions in a KV-Map a KV-Map an e onsidered as a 2 dimensional truth tale. every possile input omination is found as one field of the KV-Map, every field orresponds to one row in the truth tale. a ell map helps to find eah row of a truth tale in a KV-Map (keep in mind that the position of the rows depends on how the KV-Map was drawn). Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 10 of 17 5
6 Truth tale and KV Maps To represent a funtion with the KV-Map we only have to fill the map s fields with the aording funtion values Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 11 of 17 Prime impliants and essential prime impliants any Prime mpliant that ontains at least one Minterm that is not overed y any other Prime mpliant must e part of the minimised funtion. these terms are therefore alled Essential Prime mpliants. all minterms overed y a a d the prime term a are also overed y at least one of the other prime terms a d a d ll other prime terms are essential prime terms f = d + a + a d Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 12 of 17 6
7 Minimising aross the orders When searhing for loks of 1s keep in mind that KV-Maps are irular => there might e impliants in the outer regions : Examples d a d a f = a d + d f = d Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 13 of 17 Example for don t are states W = D + D + D + D + D = D + D + D + D + D Y = D + D + D + D + D Z = D + D + D + D + D Don' t are = + D Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 14 of 17 7
8 Example: plaing the don t are states we have to draw four KV-Maps; all of them with the same ordering of don t are states Don' t are = D Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 15 of 17 Example: usage of the don t are states D D Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger W = + + D 1 Y = + + D 1 1 D D 1 = + D + 1 Z = D Leturer: Dr. ng. Stefan Werner 16 of 17 8
9 Example for produt of sums and KV-Maps f =( a + + )( a + + )( a + + ) a f = a + = ( a + )( + ) Fahgeiet Tehnishe nformatik Prof. Dr. ng. xel Hunger Leturer: Dr. ng. Stefan Werner 17 of 17 9
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9..7 hapter 6 Introduction to state machines Dr.-Ing. Stefan Werner Table of content hapter : Switching Algebra hapter : Logical Levels, Timing & Delays hapter 3: Karnaugh-Veitch-Maps hapter 4: ombinational
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