CS221: Digital Design. Indian Institute of Technology Guwahati
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1 CS221: Digital Design KMap LogicMinimizationContd.. Minimization Dr. A. Sahu DeptofComp.Sc.&Engg. Indian Institute of Technology Guwahati 1
2 Outline Karnoughmap simplification 4 variable karnaughmap Don t care condition Algorithm for better grouping Karnaughmapwith>=5variable 2
3 4 Variable KarnaughMap CD AB m0 m1 m3 m2 01 m4 m5 m7 m6 11 m12 m13 m15 m14 10 m8 m9 m11 m10 CD AB A CD B AC D F = AC D + A CD + B Note the row and column orderings. Required for adjacency
4 FindaPOSSolution Solution AB A D CD A B AB CD F = A D + A B + AB CD F = (A+D)(A+B )(A +B+C +D) Find solutions to groups of 0 s to find F Invert to get F then use DeMorgan s
5 Don tcare Adon't care termis an input to a function that the designer does not care about Because that input would never happen Example: BCD number(0 9 9, A F)are4 bits, dont don tcare about input A F Supposeasystemhave5typeofinput a system 5 type of Unfortunately we can t have 2 input line Make 3 input line and last 3 sequence as don t care S0, S1, S2,S3,S4, X,X,X == > 000, 001.,111 5
6 DealingWithDon tcares Don t F = Σm(1, 3, 7) + Σd(0, 5) A A B C+AB C+ A BC+ABC BC x x = C F=C Circle the x s sthat help get bigger groups of 1 s Don t circle the x s that don t
7 7SegmentDisplay a f b g e d c 7
8 f a g b ActivationofLEDs of e d c 0 : a,b,c,d,e,f 1: bc b,c 2:a,b,g,e,d 3:a,b,g,c,d 4:f,g,b,c 5:a,f,g,c,d 5f 6:a,f,g,c,d,e 7:a,b,c 8:a,b,c,d,e,f,g 9abcdfg 9:a,b,c,d,f,g 8
9 BCDto7SegmentDisplay Display BCDare4bit Design a decoder to drive 7 segment LED BCD input W X Y Z Decoder A B C D E F G 9
10 PrimeImplicants Agroupofoneormore1 swhichareadjacent of one or 1s adjacent and can be combined on a KarnaughMap is calledanimplicant an implicant. The biggestgroup of 1 s which can be circled tocoveragiven1iscalledaprimeimplicant is called a implicant. They are the only implicantswe care about.
11 PrimeImplicants AB CD Prime Implicants Non-prime Implicants Are there any additional prime implicants in the map that are not shown above?
12 AB All The Prime Implicants CD Prime Implicants When looking for a minimal solution only circle prime implicants A minimal solution will never contain non-prime implicants
13 Essential Prime Implicants AB Not all prime implicants CD are required A prime implicant which is the only cover of some 1 is essential a minimal solution requires it. Essential Prime Implicants Non-essential Prime Implicants
14 A Minimal Solution Example AB CD F = AB + BC + AD Minimum q Not required
15 Another Example AB CD
16 AnotherExample AB CD F = A B is not required Every one one of its locations is covered by multiple implicants ocat o s s cove ed by A D + BCD + B D Minimum After choosing essentials, everything is covered
17 Finding the Minimum Sum of Products 1. Find each essentialprime implicant and include it in the solution. 2. Determine if any mintermsare not yet covered. 3.Findtheminimal#ofremaining prime implicantswhich finish the cover.
18 Yet Another Example (Use of non essential primes) AB CD
19 Yet Another Example (Use of non essential primes) AB A C CD AD CD Essentials: A D and AD Non essentials: A C and CD Solution: A D + AD + A C or A D + AD + CD A D
20 5 Variable KarnaughMap BC DE m0 m4 m12 m8 01 m1 m5 m13 m9 11 m3 m7 m15 m11 10 m2 m6 m14 m10 BC DE m16 m20 m28 m24 01 m17 m21 m29 m25 11 m19 m23 m31 m27 10 m18 m22 m30 m26 This is the A=0 plane This is the A=1 plane The planes are adjacent to one another (one is above the other in 3D)
21 SomeImplicantsina5 Variable a KMap BC DE BC DE D E A=0 A BCD A=1 AB C D ABC DE B C DE Some of these are not prime
22 5 Variable KMapExample Find the minimum sum-of-products for: F = Σ m (0,1,4,5,11,14,15,16,17,20,21,30,31),,,,,,,,,,, BC DE BC DE A=0 A=1
23 5 Variable KMapExample Find the minimum sum-of-products for: F=Σ Σ m( ) (0,1,4,5,11,14,15,16,17,20,21,30,31),,,,,,,,,,, BC BC DE DE A=0 A=1 F = B D + BCD + A BDE
24 CD 6 Variable KarnaughMap CD EF m0 m4 m12 m8 AB=00 01 m1 m5 m13 m9 11 m3 m7 m15 m11 10 m2 m6 m14 m10 CD EF AB=01 00 m16 m20 m28 m24 01 m17 m21 m29 m25 11 m19 m23 m31 m27 10 m18 m22 m30 m26 EF m32 m36 m44 m40 01 m33 m37 m45 m41 11 m35 m39 m47 m43 10 m34 m38 m46 m42 CD EF m48 m52 m60 m56 01 m49 m53 m61 m57 11 m51 m55 m63 m59 10 m50 m54 m62 m58 AB=10 AB=11
25 CD EF CD EF AB= AB= CD CD EF EF AB= AB= Solution = AC D + CDEF = AC D = CDEF
26 KMapSummary AKmapissimplyafoldedtruthtable simply a table where physical adjacency implies logical adjacency KMaps are most commonly used hand method for logic minimization KMapshaveotherusesforvisualizingBoolean visualizing equations youmayseesomelater. some
27 Thanks 27
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Working with ombinational Logic Simplification two-level simplification exploiting don t cares algorithm for simplification Logic realization two-level logic and canonical forms realized with NNs and NORs
Combinational Logic Design/Circuits
3 ` Combinational Logic Design/Circuits Chapter-3(Hours : 12 Marks:24 ) Combinational Logic design / Circuits 3.1 Simplification of Boolean expression using Boolean algebra. 3.2 Construction of logical