Simlification of Switching Functions

Transcription

1 Simlification of Switching unctions ( ) = ( 789 5) Quine-Mc luskey Original nonminimized oolean function m i m i m n m i [] m m m m m 4 m 5 m 6 m 7 m 8 m 9 m m m m m 4 m 5 m m m 7 m 8 m 9 m m 5 The number of 's 4 m m m m 8 m 9 m m 7 m 5

2 Simlification of Switching unctions Quine-Mc luskey n 4 m i m m m m 8 m 9 m m 7 m 5 Prime implicants (8) (89) (75) [] m m m m 7 m 8 m 9 m m 5 Essential prime implicants m i m j m m m m m 8 m 9 m m m m m 8 m m m 8 m m 8 m m 9 m m 9 m m 8 m m m m m 8 m 9 m 8 m m m 7 m 5 (75) overing Table ll minterms of the function must be covered ( ) = + + m i m j m k m l (89) (8) (75) (89) (8) m 4 m m m m m 5 7 m6 m m m 5 m4 m 8 m9 m m (89) (8)

3 Simlification of Switching unctions ( ) m( 56 7 ) d( ) = + Quine-Mc luskey ( simplification with don't care terms) d d d d mi () () () (5) (6) () () (7) () (mimj) () () (6) () (7) () (57) (67) () (mimj) (67) () (67) () prime implicants (67) () () (57) 5 6 overing Table 7 ( ) = + + +

4 Simlification of Switching unctions Quine-Mc luskey ( ) m( 56 7 ) d( ) = + We skip these ( simplification without don't care terms) mi () (5) (6) () (7) (mimj) (57) (67) prime implicants () () (57) (67) 5 6 overing Table 7 ( ) =

5 Simlification of Switching unctions ecause of propagation delay of gate the minimum SOP or POS epression is often modified to avoid undesirable events hazards. Propagation delay of gate depends on a local loading effects. t L N inputs gate t L t + t gate t + t gate wire wire N inputs Total signal traveling time from gate input to the net gate input includes wiring delay and propagation delay of the first gate. 5

6 Simlification of Switching unctions Two types of hazards Static ynamic Static -hazard Static -hazard ynamic hazards Third type (Essential Hazards) 6

7 Simlification of Switching unctions t = t t = = + t = t t = t = + Static hazard Static hazard Simple eamples of static hazard 7

8 Simlification of Switching unctions ssumption: Only one variable change value in each time. { } { } llowed transition { } { } { } { } Not allowed transition { } { } { } { } 8

9 Simlification of Switching unctions undamental Mode Only one input signal change at a time and only when the circuit is in the stable state. verage switching time of oolean variables > propagation delay of oolean function and individual gates. (>>) ombinational logic can be used as an eitation functions in (asynchronous) sequential circuits. (If it satisfied five requirements for proper operation : John. Wakerly page 68) 9

10 Simlification of Switching unctions ( ) = + + = m t f f f f ( ) = = m SOP f f ( f ) t ( f ) t > Simulated timing for transition : ()= to. f f Truth table

11 Simlification of Switching unctions SOP Static hazard can be prevented by adding a third product term which covers minterms: = and f f ( f ) t ( f ) t > Product term f covers transition to f f f f f In general with arbitrary functions : ( f ) > t ( f ) or t ( f ) t ( f ) t < The direction of transition during glitch generation depends on timing. f f Truth table f

12 Simlification of Switching unctions POS Eample (four variables: ) M i f f f f ( ) = M ( ) m m m m m 4 m m m 8 m5 m 7 m6 m 5 m9 m m ( ) = ( + )( + )( + )( + ) + f f f f4 m 4

13 Simlification of Switching unctions POS ( ) = ( + )( + )( + )( + ) + f f f f4 f f f f4

14 ( ) ( ) ( ) ( ) ( ) ( ) = = = = + + = + = Simplified functions: find common product terms Multiple Output Implementation Simlification of Switching unctions 4

15 Simlification of Switching unctions Multiple Output Implementation m m m m m 4 m m 5 7 m6 m m m 5 m4 m 8 m9 m m m m m m m 4 m5 m 7 m6 m m m 5 4 m 8 m m m m m m9 m m m 4 m m 5 7 m6 m m m 5 m4 m 8 m9 m m = + + = + + =

16 The End 6