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slide-1
SLIDE 1 CSE 370 – Spring 2001 - Combinational Logic - 1

&RPELQDWLRQDOORJLFWRSLFV

❚ /RJLFIXQFWLRQVWUXWKWDEOHVDQGVZLWFKHV ❙ 127$1'251$1'125;25 ❙ PLQLPDOVHW ❚ $[LRPVDQGWKHRUHPVRI%RROHDQDOJHEUD ❙ SURRIVE\UHZULWLQJ ❙ SURRIVE\SHUIHFWLQGXFWLRQ ❚ *DWHORJLF ❙ QHWZRUNVRI%RROHDQIXQFWLRQV ❙ WLPHEHKDYLRU ❚ &DQRQLFDOIRUPV ❙ WZROHYHO ❙ LQFRPSOHWHO\VSHFLILHGIXQFWLRQV ❚ 6LPSOLILFDWLRQ ❙ %RROHDQFXEHVDQG .DUQDXJK PDSV ❙ WZROHYHOVLPSOLILFDWLRQ

CSE 370 – Spring 2001 - Combinational Logic - 2

; < SRVVLEOHIXQFWLRQV))

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❚ 7KHUHDUHSRVVLEOHIXQFWLRQVRILQSXWYDULDEOHV ❙ LQJHQHUDOWKHUHDUHQIXQFWLRQVRIQLQSXWV

CSE 370 – Spring 2001 - Combinational Logic - 3

&RVWRIGLIIHUHQWORJLFIXQFWLRQV

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CSE 370 – Spring 2001 - Combinational Logic - 4

; < ; QDQG <

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≡ not ( (not X) nor (not Y) ) X nor Y ≡ not ( (not X) nand (not Y) )

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CSE 370 – Spring 2001 - Combinational Logic - 5

$QDOJHEUDLFVWUXFWXUH

❚ $QDOJHEUDLFVWUXFWXUHFRQVLVWVRI ❙ DVHWRIHOHPHQWV% ❙ ELQDU\RSHUDWLRQV^` ❙ DQGDXQDU\RSHUDWLRQ^` ❙ VXFKWKDWWKHIROORZLQJD[LRPVKROG

WKHVHW%FRQWDLQVDWOHDVWWZRHOHPHQWVDEVXFKWKDWD E FORVXUH DELVLQ% DELVLQ% FRPPXWDWLYLW\ DE ED DE ED DVVRFLDWLYLW\ DEF DEF DEF DEF LGHQWLW\ D D D D GLVWULEXWLYLW\ DEF DEDF DEF DEDF FRPSOHPHQWDULW\ DD DD

CSE 370 – Spring 2001 - Combinational Logic - 6

%RROHDQDOJHEUD

❚ %RROHDQDOJHEUD ❙ % ^` ❙ LVORJLFDO25LVORJLFDO$1' ❙ LVORJLFDO127 ❚ $OODOJHEUDLFD[LRPVKROG

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SLIDE 2 CSE 370 – Spring 2001 - Combinational Logic - 7

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❚ $Q\ORJLFIXQFWLRQWKDWFDQEHH[SUHVVHGDVDWUXWKWDEOHFDQEH ZULWWHQDV DQH[SUHVVLRQLQ%RROHDQDOJHEUDXVLQJWKHRSHUDWRUVDQG

CSE 370 – Spring 2001 - Combinational Logic - 8

$[LRPVDQGWKHRUHPVRI%RROHDQDOJHEUD

❚ LGHQWLW\ ; ; '; ; ❚ QXOO ; '; ❚ LGHPSRWHQF\ ;; ; ';; ; ❚ LQYROXWLRQ ; ; ❚ FRPSOHPHQWDULW\ ;; ';; ❚ FRPPXWDWLYLW\ ;< <; ';< <; ❚ DVVRFLDWLYLW\ ;<= ;<= ';<= ;<=

CSE 370 – Spring 2001 - Combinational Logic - 9

$[LRPVDQGWKHRUHPVRI%RROHDQDOJHEUDFRQWG

❚ GLVWULEXWLYLW\ ;<= ;<;= ';<= ;<;= ❚ XQLWLQJ ;<;< ; ';<;< ; ❚ DEVRUSWLRQ ;;< ; ';;< ; ;<< ;< ';<< ;< ❚ IDFWRULQJ ;<;= ';<;= ;=;< ;=;< ❚ FRQFHQVXV ;<<=;= ';<<=;= ;<;= ;<;=

CSE 370 – Spring 2001 - Combinational Logic - 10

$[LRPVDQGWKHRUHPVRI%RROHDQDOJHEUDFRQW

❚ GH0RUJDQV ;< ;< ';< ;< ❚ JHQHUDOL]HGGH0RUJDQV I;;;Q I;;;Q ❚ HVWDEOLVKHVUHODWLRQVKLSEHWZHHQDQG

CSE 370 – Spring 2001 - Combinational Logic - 11

$[LRPVDQGWKHRUHPVRI%RROHDQDOJHEUDFRQW

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CSE 370 – Spring 2001 - Combinational Logic - 12

3URYLQJWKHRUHPVUHZULWLQJ

❚ 8VLQJWKHD[LRPVRI%RROHDQDOJHEUD ❙ HJSURYHWKHWKHRUHP ;<;< ; ❙ HJSURYHWKHWKHRUHP ;;< ; GLVWULEXWLYLW\ ; <; < ;<< FRPSOHPHQWDULW\ ;<< ; LGHQWLW\' ; ;➼ LGHQWLW\' ;;< ;;< GLVWULEXWLYLW\ ;;< ;< LGHQWLW\ ;< ; LGHQWLW\' ; ;➼

slide-3
SLIDE 3 CSE 370 – Spring 2001 - Combinational Logic - 13

;< ;< 125LVHTXLYDOHQWWR$1' ZLWKLQSXWVFRPSOHPHQWHG ;< ;< 1$1'LVHTXLYDOHQWWR25 ZLWKLQSXWVFRPSOHPHQWHG ; < ; < ;< ;<

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❚ 8VLQJSHUIHFWLQGXFWLRQFRPSOHWHWUXWKWDEOH ❙ HJGH0RUJDQV

  • CSE 370 – Spring 2001 - Combinational Logic - 14

$VLPSOHH[DPSOH

❚ ELWELQDU\DGGHU ❙ LQSXWV$%&DUU\LQ ❙ RXWSXWV6XP&DUU\RXW

$ % &LQ &RXW 6 $ % &LQ 6 &RXW

  • &RXW $% &LQ $% &LQ $% &LQ $% &LQ

6 $%&LQ $%&LQ $%&LQ $% &LQ

CSE 370 – Spring 2001 - Combinational Logic - 15

$SSO\WKHWKHRUHPVWRVLPSOLI\H[SUHVVLRQV

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CSE 370 – Spring 2001 - Combinational Logic - 16

; < =

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❚ 127 ; ; a; ❚ $1' ;< ;< ;∧ < ❚ 25 ;< ;∨ <

CSE 370 – Spring 2001 - Combinational Logic - 17

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X xnor Y = X Y + X’ Y’ X and Y are the same ("equality", "coincidence")

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❚ 1$1' ❚ 125 ❚ ;25 ;⊕ < ❚ ;125 ; <

CSE 370 – Spring 2001 - Combinational Logic - 18

T1 T2 use of 3-input gate

$ % & ' 7 7 = $ % & ' =

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❚ 0RUHWKDQRQHZD\WRPDSH[SUHVVLRQVWRJDWHV ❙ HJ= $%&' $%&'

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SLIDE 4 CSE 370 – Spring 2001 - Combinational Logic - 19

WLPH FKDQJHLQ<WDNHVWLPHWRSURSDJDWHWKURXJKJDWHV

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❙ EXWQRWHKRZHGJHVGRQWOLQHXSH[DFWO\ ❙ LWWDNHVWLPHIRUDJDWHWRVZLWFKLWVRXWSXW

CSE 370 – Spring 2001 - Combinational Logic - 20

$ % & =

  • &KRRVLQJGLIIHUHQWUHDOL]DWLRQVRIDIXQFWLRQ

two-level realization (we don’t count NOT gates) XOR gate (easier to draw but costlier to build) multi-level realization (gates with fewer inputs)

CSE 370 – Spring 2001 - Combinational Logic - 21

:KLFKUHDOL]DWLRQLVEHVW"

❚ 5HGXFHQXPEHURILQSXWV ❙ OLWHUDOLQSXWYDULDEOHFRPSOHPHQWHGRUQRW ❘ FDQDSSUR[LPDWHFRVWRIORJLFJDWHDV WUDQVLWRUV SHUOLWHUDO ❘ ZK\QRWFRXQWLQYHUWHUV" ❙ IHZHUOLWHUDOVPHDQVOHVVWUDQVLVWRUV ❘ VPDOOHUFLUFXLWV ❙ IHZHULQSXWVLPSOLHVIDVWHUJDWHV ❘ JDWHVDUHVPDOOHUDQGWKXVDOVRIDVWHU ❙ IDQLQVRIJDWHLQSXWVDUHOLPLWHGLQVRPHWHFKQRORJLHV ❚ 5HGXFHQXPEHURIJDWHV ❙ IHZHUJDWHVDQGWKHSDFNDJHVWKH\FRPHLQPHDQVVPDOOHUFLUFXLWV ❘ GLUHFWO\LQIOXHQFHVPDQXIDFWXULQJFRVWV

CSE 370 – Spring 2001 - Combinational Logic - 22

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CSE 370 – Spring 2001 - Combinational Logic - 23

$UHDOOUHDOL]DWLRQVHTXLYDOHQW"

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CSE 370 – Spring 2001 - Combinational Logic - 24

,PSOHPHQWLQJ%RROHDQIXQFWLRQV

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slide-5
SLIDE 5 CSE 370 – Spring 2001 - Combinational Logic - 25

&DQRQLFDOIRUPV

❚ 7UXWKWDEOHLVWKHXQLTXHVLJQDWXUHRID%RROHDQIXQFWLRQ ❚ 0DQ\DOWHUQDWLYHJDWHUHDOL]DWLRQVPD\KDYHWKHVDPHWUXWKWDEOH ❚ &DQRQLFDOIRUPV ❙ VWDQGDUGIRUPVIRUD%RROHDQH[SUHVVLRQ ❙ SURYLGHVDXQLTXHDOJHEUDLFVLJQDWXUH

CSE 370 – Spring 2001 - Combinational Logic - 26

$ % & ) )

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6XPRISURGXFWVFDQRQLFDOIRUPV

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CSE 370 – Spring 2001 - Combinational Logic - 27

VKRUWKDQGQRWDWLRQIRU PLQWHUPV RIYDULDEOHV $ % & PLQWHUPV

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CSE 370 – Spring 2001 - Combinational Logic - 28

$ % & ) )

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$%& $%& $%&

CSE 370 – Spring 2001 - Combinational Logic - 29

$ % & PD[WHUPV

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CSE 370 – Spring 2001 - Combinational Logic - 30

6R33R6DQGGH0RUJDQVWKHRUHP

❚ 6XPRISURGXFWV ❙ ) $%&$%&$%& ❚ $SSO\GH0RUJDQV ❙ ) $%&$%&$%& ❙ ) $%&$%&$%& ❚ 3URGXFWRIVXPV ❙ ) $%&$%&$%&$%&$%& ❚ $SSO\GH0RUJDQV ❙ ) $%&$%&$%&$%&$ %& ❙ ) $%&$%&$%&$%&$%&

slide-6
SLIDE 6 CSE 370 – Spring 2001 - Combinational Logic - 31

FDQRQLFDOVXPRISURGXFWV PLQLPL]HGVXPRISURGXFWV FDQRQLFDOSURGXFWRIVXPV PLQLPL]HGSURGXFWRIVXPV

F1 F2 F3 B A C F4

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CSE 370 – Spring 2001 - Combinational Logic - 32

:DYHIRUPVIRUWKHIRXUDOWHUQDWLYHV

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CSE 370 – Spring 2001 - Combinational Logic - 33

0DSSLQJEHWZHHQFDQRQLFDOIRUPV

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CSE 370 – Spring 2001 - Combinational Logic - 34

$ % & ' : ; < =

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GRQWFDUH'&VHWRI: RQVHWRI:

CSE 370 – Spring 2001 - Combinational Logic - 35

1RWDWLRQIRULQFRPSOHWHO\VSHFLILHGIXQFWLRQV

❚ 'RQWFDUHVDQGFDQRQLFDOIRUPV ❙ VRIDURQO\UHSUHVHQWHGRQVHW ❙ DOVRUHSUHVHQWGRQWFDUHVHW ❙ QHHGWZRRIWKHWKUHHVHWVRQVHWRIIVHWGFVHW ❚ &DQRQLFDOUHSUHVHQWDWLRQVRIWKH%&'LQFUHPHQWE\IXQFWLRQ ❙ = PPPPPGGGGGG ❙ = Σ >PG@ ❙ = 00000'''''' ❙ = Π >0'@

CSE 370 – Spring 2001 - Combinational Logic - 36

6LPSOLILFDWLRQRIWZROHYHOFRPELQDWLRQDOORJLF

❚ )LQGLQJDPLQLPDOVXPRISURGXFWVRUSURGXFWRIVXPVUHDOL]DWLRQ ❙ H[SORLWGRQWFDUHLQIRUPDWLRQLQWKHSURFHVV ❚ $OJHEUDLFVLPSOLILFDWLRQ ❙ QRWDQDOJRULWKPLFV\VWHPDWLFSURFHGXUH ❙ KRZGR\RXNQRZZKHQWKHPLQLPXPUHDOL]DWLRQKDVEHHQIRXQG" ❚ &RPSXWHUDLGHGGHVLJQWRROV ❙ SUHFLVHVROXWLRQVUHTXLUHYHU\ORQJFRPSXWDWLRQWLPHVHVSHFLDOO\IRU IXQFWLRQVZLWKPDQ\LQSXWV! ❙ KHXULVWLFPHWKRGVHPSOR\HG HGXFDWHGJXHVVHVWRUHGXFHDPRXQWRI FRPSXWDWLRQDQG\LHOGJRRGLIQRWEHVWVROXWLRQV ❚ +DQGPHWKRGVVWLOOUHOHYDQW ❙ WRXQGHUVWDQGDXWRPDWLFWRROVDQGWKHLUVWUHQJWKVDQGZHDNQHVVHV ❙ DELOLW\WRFKHFNUHVXOWVRQVPDOOH[DPSOHV

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SLIDE 7 CSE 370 – Spring 2001 - Combinational Logic - 37

$ % )

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%UHPDLQV $KDVDGLIIHUHQWYDOXHLQWKHWZRURZV $LVHOLPLQDWHG ) $%$% $$% %

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CSE 370 – Spring 2001 - Combinational Logic - 38

FXEH ;

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  • CSE 370 – Spring 2001 - Combinational Logic - 39

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CSE 370 – Spring 2001 - Combinational Logic - 40

$ % &LQ &RXW

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  • CSE 370 – Spring 2001 - Combinational Logic - 41

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  • CSE 370 – Spring 2001 - Combinational Logic - 42

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SLIDE 8 CSE 370 – Spring 2001 - Combinational Logic - 43

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CSE 370 – Spring 2001 - Combinational Logic - 45

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CSE 370 – Spring 2001 - Combinational Logic - 46

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  • CSE 370 – Spring 2001 - Combinational Logic - 47

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CSE 370 – Spring 2001 - Combinational Logic - 48

F(A,B,C) = Σm(0,4,5,7) F’(A,B,C) = Σ m(1,2,3,6) F’ simply replace 1’s with 0’s and vice versa G(A,B,C) =

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SLIDE 9 CSE 370 – Spring 2001 - Combinational Logic - 49

& %' find the smallest number of the largest possible subcubes to cover the ON-set (fewer terms with fewer inputs per term)

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CSE 370 – Spring 2001 - Combinational Logic - 50

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CSE 370 – Spring 2001 - Combinational Logic - 52

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SLIDE 10 CSE 370 – Spring 2001 - Combinational Logic - 55

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CSE 370 – Spring 2001 - Combinational Logic - 56

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CSE 370 – Spring 2001 - Combinational Logic - 57

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SLIDE 11 CSE 370 – Spring 2001 - Combinational Logic - 61

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