Algebraic Methods for the Analysis and Synthesis

Transcription

1 lgebraic ethods for the nalysis and Synthesis Fundaentals of oolean lgebra asic Postulates. oolean algebra is closed algebraic syste containing a set K of two or ore eleents and two operators and, ND and OR respectively (a) (b) (a) (b) a a a a a b b a a b b a 4. a ( b c) ( a b) (b) a ( b c) ( a b) c outativity of the and operations. (a) c ssociativity of the 5. (a) a ( b c) ( a b) ( a c) (b) a ( b c) ( a b) ( a c) 6. (a) (b) a a a a and operations. vaihdantalaki Distributivity of over and over. liitäntälaki osittelulaki a and b belongs to K a b a b belongs to K belongs to K In binary syste K{,} []

2 lgebraic ethods for the nalysis and Synthesis Fundaentals of oolean lgebra Duality If an expression is valid in oolean algebra, the dual of the expression is also valid. In the Expression In the Dual expression Exaple : (a) ( bc) ( a b) ( a c) a ( b c) ab ac a [] 2

3 Postulates and theores and its duals D D D D Y Y Y Y Deorgan T8(a) and T8(b) lgebraic ethods for the nalysis and Synthesis Fundaentals of oolean lgebra [] 3

4 lgebraic ethods for the nalysis and Synthesis Now we focus on the oolean algebra in which K{,} Two-valued oolean lgebra Variables : X,X 2,X3,X4,, X n The variable can have value : or Switching function of X : f ( X,X,X,X,, ),X 2,X3,X4,, X n Switching function f can have value : or X n ssuption : n2 F F F 2 F 3 F 4 We have two variables and F 5 F 6 F 7 F 8 F 9 F F F 2 F 3 F 4 F 5 n variables in binary syste n 2 2 different functions [] 4

5 lgebraic ethods for the nalysis and Synthesis F F F 2 F 3 F 4 F 5 F 6 F 7 F 8 F 9 F F F 2 F 3 F 4 F 5 f f f f f f f f (, ) f8 (, ) (, ) f9 (, ) (, ) f (, ) (, ) f(, ) (, ) f2 (, ) (, ) f3 (, ) (, ) f4 (, ) (, ) f (, ) 5 [] We can write 6 different functions of the two variables. 5

6 lgebraic ethods for the nalysis and Synthesis F F F 2 F 3 F 4 F 5 F 6 F 7 F 8 F 9 F F F 2 F 3 F 4 F 5 f 8 f 6 ND & or Exclusive OR f 4 OR f NOR f 9 Exclusive NOR NND f 7 XOR is odd function [] 6

7 lgebraic ethods for the nalysis and Synthesis We have oolean Operators : OR ND NOT OR-operator ND-operator opleent operator Operator Precedence :. Parenthesis 2. NOT 3. ND 4. OR Exaples : ( x, y) ( x y) First ( x y) f then copleent f ( x, y) x y First x, y then ND operation [2] 7

8 lgebraic ethods for the nalysis and Synthesis Truth Tables NOT function F x Input Output Truth table : NOT operation OR function F x y Inputs Output Truth table : OR operation ND function Inputs Output F x y Truth table : ND operation [2] 8

9 lgebraic ethods for the nalysis and Synthesis Truth Tables The validity of oolean functions can be shown easily with truth tables Exaple : Deorgan's theore X Y X Y X Y X Y X Y [2] 9

10 lgebraic ethods for the nalysis and Synthesis The opleent of Function oolean function : F (,, ) ( ) Truth Table for F F F Dual of the function and copleent each literal The copleent of a function ay be derived algebraically through Deorgan's theores. F F F ( ) ( ) ( ) ( ) The copleent of function F [2]

11 lgebraic ethods for the nalysis and Synthesis lgebraic Fors of ssuption : n oolean variables X, X 2, X 3, X n Standard Product n X 2 X3 X n X n Xi j i X (ND ter) inter n variables can be cobined to for 2 n inters If oolean function is with all possible different input cobinations (F ), we can write : n F 2 n ( X, X, X ) X where 2 n i j i j [2]

12 lgebraic ethods for the nalysis and Synthesis lgebraic Fors of ssuption : n oolean variables X, X 2, X 3, X n Standard Su n X 2 X3 X n X n Xi j i X (OR ter) axter n variables can be cobined to for 2 n axters If oolean function is with all possible different input cobinations (F ), we can write : n F 2 n ( X, X, X ) X where 2 n i j i j [2] 2

13 inters and axters for Three inary Variables Ter Designation Ter Designation inters axters ( ) ( ),4,5,6 7, ) F(,,,2,3 ) F(,, The set of inters The set of axters lgebraic ethods for the nalysis and Synthesis lgebraic Fors of 3

14 lgebraic ethods for the nalysis and Synthesis lgebraic Fors of Standard for oolean function can be expressed in standard or nonstandard fors. F Y XY XY Z ( Y Z )( X Y Z ) F X 2 SOP Su of Products POS Product of Sus F(,, ) 2 3 (,2, ) 3 F(,, ) 7 SOP (,4,5,6 ) anonical for , POS anonical for Nonstandard for ( D E) F 3 [2] 4

15 lgebraic ethods for the nalysis and Synthesis lgebraic Fors of Standard for If ND-ters or OR-ters contains one or ore variables, the SOP or POS is standard for. Exaples Two-level Ipleentations Y X Y Z X Y F Y XY XY Z oolean function expressed in su of products standard for F su of products X Y Z X YZ ( Y Z )( X Y Z ) F X 2 oolean function expressed in product of sus standard for F 2 product of sus [2] 5

16 lgebraic ethods for the nalysis and Synthesis lgebraic Fors of Exaple ( D E) F Nonstandard for 3 oolean function expressed in nonstandard for D E F 3 Three-level Ipleentation F D E 3 oolean function expressed in su of products standard for D E F 3 Two-level Ipleentation [2] 6

17 lgebraic ethods for the nalysis and Synthesis lgebraic Fors of anonical standard for The oolean function can be described in Su of Products (SOP) or Product of Sus (POS) fors. If all ND-ters ( i ) contains all variables, the SOP for is canonical. If all OR-ters ( i ) contains all variables, the POS for is canonical. With all other input cobination, F : {, 4, 5, 6, 7 } F(,, ) 2 3 (,2,3 ) SOP anonical for F(,, ) (,4,5,6 7, ) ( )( )( )( )( ) POS anonical for [2] 7

18 lgebraic conversion to the canonical for ( ) ( ) F,, F ( ) ( ) F F F anonical for 2 3 lgebraic ethods for the nalysis and Synthesis lgebraic Fors of anonical standard for 8

19 lgebraic ethods for the nalysis and Synthesis lgebraic Fors of Incopletely Specified Functions ertain input cobinations ight never be applied to a particular switching network. ll input cobinations do occur for a given network, but the output is required to be or only for certain cobinations. Don't-care inters Don't-care axters d i D i [] 9

20 lgebraic ethods for the nalysis and Synthesis lgebraic Fors of Incopletely Specified Functions Exaple The function (,, ) f has inters :,3, 7 and don't care conditions : d, d 4 5 inter list : axter list : di Di because f on don't cares f f f Function f (,, ) (,3 7, ) d( 4, 5) or (,, ) (,2,6 ) D( 4,5 ) f The copleent of function f (,, ) (,2,6 ) d( 4, 5),3 7, D 4, 5 (,, ) ( ) ( ) i d d i D D f / / [] 2

21 lgebraic ethods for the nalysis and Synthesis Logic Gates Introduction nalog devices and systes process tie-varying signals. Signal can take an any value across a continuous range of voltage or current. Voltage tie Digital devices and systes process also tie-varying signals, UT Signal can take " only two " values. In binary syste K{,} Voltage Static states tie Two perissible states Undefined logic level during transition 2

22 lgebraic ethods for the nalysis and Synthesis Logic Gates Introduction Definition of binary signal in dc logic (level-logic) Positive-logic The ore positive voltage is level (thrue). V ( ) thrue Thrue False The lower voltage is level (false). ( ) V ( ) V > V ( ) false T t High Low State transition Negative-logic The ore positive voltage is level (false). V ( ) false Thrue False The lower voltage is level (thrue). ( ) V ( ) V < V ( ) T thrue t Low High 22

23 lgebraic ethods for the nalysis and Synthesis Logic Gates Introduction n active-high signal is asserted when it is high (positive logic). sserted ctive True Signal is set to logic Deasserted Negated False Signal is set to logic n active-low signal is asserted when it is low (negative logic). ctive-low signal naes : a,,run,run * ctive-high signal naes : a,,run [2] 23

24 lgebraic ethods for the nalysis and Synthesis The ost basic digital devices are called gates NOT gate (Inverter) NSI/IEEE Standard Input Output Input Output Truth table : NOT gate Logic Gates Introduction OR gate Inputs Output Truth table : OR gate ND gate Inputs Output & Truth table : ND gate 24

25 lgebraic ethods for the nalysis and Synthesis Logic Gates Introduction NOR gate Inputs Output NSI/IEEE Standard Truth table : NOR gate NND gate Inputs Output & Truth table : NND gate 25

26 lgebraic ethods for the nalysis and Synthesis Logic Gates Introduction XOR gate x y F Inputs Output F x y xy NSI/IEEE Standard x y Truth table : XOR gate 26

27 lgebraic ethods for the nalysis and Synthesis Logic Gates Extension to ultiple Inputs The binary operation of gate is coutative and assosiative. ND-, OR-operation gate can be extended to have ultiple inputs. ultiple Input OR x y z F OR-operation x y y x coutative ( x y) z x ( y z) x y z assosiative ultiple Input ND x y z F ND-operation x y y x coutative ( x y) z x ( y z) x y z assosiative [2] 27

28 lgebraic ethods for the nalysis and Synthesis NND-, NOR-operation NND and NOR operators are coutative NND and NOR operators are not associative NOR-operation x y y x (coutative) Logic Gates Extension to ultiple Inputs ( x y) z x ( y z) x y z ultiple Input NOR x y z F Ipleentation of the ultiple input NOR gate ultiple Input NND x y z F Ipleentation of the ultiple input NOR gate [2] 28

29 lgebraic ethods for the nalysis and Synthesis Logic Gates Extension to ultiple Inputs ultiple Input XOR XOR operator is associative and coutative x y z F Ipleentation of the ultiple input XOR gate "Odd function" Inputs Output Truth table of the XOR gate [2] 29

30 The End 3