Implementation of Boolean Functions through Multiplexers with the Help of Shannon Expansion Theorem

Transcription

1 Internatonal Journal o Computer pplcatons ( ) Volume 62 No.6, January 23 Implementaton o Boolean Functons through Multplexers wth the Help o Shannon Expanson Theorem Saurabh Rawat Graphc Era Unversty. Dehradun nushree Sah The Unversty o Greenwch, London, UK, Sumt Pundr Graphc Era Unversty. Dehradun BSTRCT Implementaton o Boolean uncton through multplexer can be done by varous multplexers dependng upon the select lnes. Implementaton o Boolean unctons can be done by varous methods, but n ths partcular paper stress s more on multplexers. Through Shannon expanson theorem,t s easy or us to mplement the Boolean unctons n a smpler way. Upto three varables, can be handled by multplexers, and above that we have taken ad o look out table, and how t uses multplexers n ther operatons. Keywords Multplexers, 2 x, 4 x, 8 x, multplexers, Shannon Theorem... Multplexer multplexer crcut has a number o data nputs, one or more select nputs and one output. It passes the sgnal value on one o the data nputs to the output. Data nput s selected by the values o the select nputs.select nput S, chooses as the output o the multplexer ether nput x or x. Multplexer s unctonalty can be descrbed n the orm o a truth table. S s to Multplexer ( = ) and 4 to multplexer have our data nputs x,x,x 2, & x 3 and two select nputs S and S. The two bt number represented by SS select one o the data nput as output o the multplexer. x 2 S S x x x 3.4 Graphc symbol S S 2 x. Graphc symbol.2 truth table x x 2 x 3 S.5 Truth table.3 Sum o products crcut 8

2 Internatonal Journal o Computer pplcatons ( ) Volume 62 No.6, January 23 S S x x x 2 x 3.6 Sum o product crcut 4 to multplexer S Sx SSx SSx2 SSx3 multplexer that has n data nputs, requres log 2 n select nputs. Larger multplexer can be constructed rom small multplexers. 2. I- Usng 8x multplexer or mplementaton (,, 3 ) = (3,5,6,7) S S x x 8 x x x.7 Usng 2 to multplexer to buld a 4 to multplexer 2. Boolean uncton mplementaton o multplexer n=2 m n number o nput varables m- number o select nputs 2.. 8x multplexer mplementaton 2.2 II- Usng 4x Multplexer or mplementaton Connectng two varables wth selecton lnes o multplexer and remanng sngle varable o the uncton s used or the nputs o the multplexer. I s a sngle varable the nputs o multplexer are chosen to be ether or or to (a) (,, 3 ) = (3,5,6,7) MSB.e. s used as sngle varable, and, 3 as select nputs. 9

3 Internatonal Journal o Computer pplcatons ( ) Volume 62 No.6, January 23 Mnterms 2 (b) (,, 3 ) = (3,5,6,7) I LSB.e. 3 s used as sngle varable and, as select nputs Mnterm 3 4 = = = 2.2. Truth table 6 7 = Truth table I Multplexer Implementaton 4 x 4 x (3,5,6,7 ) x Multplexer Implementaton x Multplexer mplementaton Besdes usng such nputs, t s possble to connect more complex crcut as nputs to a multplexer allowng uncton to be syntheszed usng a combnaton o multplexers & other logc gates. 2

4 Internatonal Journal o Computer pplcatons ( ) Volume 62 No.6, January Shannon s Expanson theorem Shannon s expanson or the Shannon decomposton s a method by whch a Boolean uncton can be represented by the sum o two sub uncton o the orgnal. Shannon expanson develops the dea that boolean uncton can be reduced by means o the dentty. x x x x where s any uncton and x and x are postve & Three nput majorty uncton mplemented usng a 2 to multplexer For three nput XOR uncton = = Ā ( + 3 ) + ( + 3 ) negatve shannon coactors o respectvely. postve shannon coactor o uncton wth respect to varable x s deend as that uncton wth all nstances o x replaced by negatve shannon co actor s the same, but replaces all nstances o x by. ny Boolean uncton (,, 3, ---n) can be wrtten n the orm (, ---n) =Ā. (,, 3 ---n) +. (,, n) Ths expanson can be done n terms o any o the n varables. as (,, 3 )= (3,5,6,7) Functon can be expressed as sum o products orm = Ā 3 + Ā Ā It can be manpulated nto = Ā ( 3 ) + (Ā Ā ) = Ā ( 3 ) + ( + 3 ) 3.3 Truth table + 3 2x 2x 3. Truth table 3.4 Three nput XOR mplemented wth 2 to Multplexer 3 2 x In Shannon s expanson the term (,, ----n) s called the co-actor o wth respect to Ā, denoted as. Smlarly the term (, n) s called the co-actor o wth respect to, wrtten as, hence 3.2 Truth table Crcut 2

5 Internatonal Journal o Computer pplcatons ( ) Volume 62 No.6, January 23 In general the expanson s done wth respect to varable, then denotes (,,,,,, n ) and (,, ) n Complexty o the logc expresson may vary, dependng on The most commonly used logc block s a lookout table (LUT), whch contans storage cells that are used to mplement a small logc uncton. Each cell s capable o holdng a sngle logc value ether or. LUTs o varous szes may be created, where the sze s dened by number o nputs. nputs. whch varable, s used. Takng another example, to mplement the uncton = Ā Ā a) Usng 2 to multplexer, Shannon s expanson usng gves = Ā (Ā 3 ) + ( + 3 ) / / 2x 2x / 3 I 2 x / 2x I 3.7 Crcut or a two nput LUT 3.5 Usng 2 to multplexer b) Usng 4 to multplexer, urther...usng gves = Ā Ā 2 ĀĀ2 + Ā Ā2 + Ā 2 Ā2 + 2 = Ā Ā 2 (Ā 3 )+Ā (Ā 3 )+ Ā 2 (Ā 3 )+ () ) 4 x ( ) 3.6 Usng 4 to multplexer 22

6 Internatonal Journal o Computer pplcatons ( ) Volume 62 No.6, January 23 / / 2x 2x / / / / 2x 2x 2x 2x / / 2x 2x three nput LUT Usng shannon s expanson any our varable uncton can be realzed wth at most three 3- LUTs (look up tables). Consderng the uncton = Ā 2 3 +Ā Ā 3 + Ā Ā 2 4 Expanson n terms o produces = Ā Ā + = Ā (Ā Ā 3 + Ā 3 4 )+ (Ā Ā 3 4 +Ā 2 Ā 4 ) = Ā (Ā Ā 3 )+ (Ā Ā 3 4 +Ā 2 Ā 4 ) crcut wth there 3 LUTs that mplements ths expresson Usng there 3 LUTs 23

7 Internatonal Journal o Computer pplcatons ( ) Volume 62 No.6, January 23 Decomposton o the uncton usng, nstead o, gves = Ā = Ā 2 ( 3 + Ā 4 )+ (Ā Ā 3 +Ā 3 4 ) as Ā2 = 2, hence only two 3 LUTs are needed Usng two 3 LUTs. 4. CONCLUSION In ths paper we have seen that Boolean unctons can be mplemented usng derent multplexers, 2x, 4x or 8x. Wth the help o Shannon expanson theorem,complcated Boolean unctons can be made easy,n mplementng through multplexers and LUTs (look up table ),agan ormed wth derent combnaton o multplexers. Ths study wll be very helpul or researchers and ntellectuals to easy understandng and practcng o mplementaton o Boolean unctons through multplexers n the eld o computer scence and technology. 5. REFERENCES [] en.wkpeda.org/wk/multplexer [2] en.wkpeda.org/wk/shannon's_expanson [3] rturo Hern andez gurre, Bll P. Buckles, and Carlos Coello Coello. Evolutonary synthess o logc unctons usng multplexers. In C. Dagl,.L. Buczak, and et al., edtors, Proceedngs o the th Conerence Smart Engneerng System Desgn, pages 3 35, New York, 2. SME Press. [4] R. L. shenhurst, The decomposton o swtchng unctons, n Proc.Int. Symp. Theory o Swtchng Functons, pr. 957, pp [5] stola, J.T., Stankov c, R.S., Fundamentals o Swtchng Theory and Logc Desgn, Sprnger, 26. [6] M. MORRIS MNO Dgtal Logc and Computer Desgn 2nd edton [7] D. Nasb S. Gll, J.B. Dxt Dgtal Desgn and Computer Organsaton 24