CS for RL 1st Session

8/18/2019 CS for RL 1st Session

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Digital Electronics

CS for RL 1st session

rktiwary

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T1 !"1#$or t%e Boolean f&nction

a" '(tain t%e tr&t% ta(le of $(" Draw t%e logic )iagra* &sing t%e original Boolean

E+ressionc" -se Boolean .lge(ra to si*lify t%e f&nction to a

*ini*&* n&*(er of literals)" '(tain t%e tr&t% ta(le of t%e f&nction fro* t%e

si*lie) e+ression an) s%ow t%at it is t%e sa*eas t%e one in art a

e" Draw t%e logic )iagra* for t%e si*lie)e+ression an) co*are t%e total no" of gateswit% t%e )iagra* of art (

4/4/16 !RKTiwaryBITS,ilani


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'(tain t%e tr&t% ta(le of $



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+y57 8w9w5:+y5 7 111 9 11 7 *10 9 *



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4/4/16 !#RKTiwaryBITS,ilani


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Si*ilarly+5y5 7 8w 9 w5: +5y57 *1 9 *3

.n) w5+y 7 w5+y8 9 5:7 111 9 11 7*2 9 *6



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T%enw+5y 7 ;w+y7 ;

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T1< !"13E+ress t%e following f&nction as a s&* of

*inter*s an) ro)&ct of *a+ter*s$ 8.,B,C,D:7 B5D 9 .5D 9 BD



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$ 8.,B,C,D:7 B5D 9 .5D 9 BD sol< B5D 9 .5D 9 BD 7 D8B5 9B:

9 .5D



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9 .5D 7 D 9.5D



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$ 8.,B,C,D:7 B5D 9 .5D 9 BD sol< B5D 9 .5D 9 BD 7 D8B5 9B: 9

.5D 7 D 9 .5D 7 D81 9 .5:

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9 .5D 7 D 9 .5D 7 D81 9

.5: 7 D



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4

$8.,B,C,D:7=8*1,*0,*,*2,*3,*11,*

10, *1:

$58.,B,C,D:7=8*

1

,*0

,*

,*2

,*3

,*11

,*

10,

*1:5

$8. B C D: =8


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41

$8.,B,C,D:7 =8*15*05 *5*25 *35*11 >*10 ,*1:

$58.,B,C,D:7 8*1 9 *0 9 *9 *29

*39 *119 *10 9*1:5

7 =8,!,4,6,#,1,1!,14:


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4/4/16 RKTiwary BITS,ilani 4!

$8.,B,C,D:7 =8*15*05 *5*25 *35*11 >*10 ,*1:

$58.,B,C,D:7 8*1 9 *0 9 *9 *29

*3

9 *11

9 *10

9*1

:5

7 =8,!,4,6,#,1,1!,14: 7 * 9 *! 9 *49 *69 *#9

*19 *1!9 *14

$8. B C D: =8


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4/4/16 RKTiwary BITS,ilani 40

$8.,B,C,D:7 =8*15*05 *5*25 *35*11 >*10 ,*1:

$58.,B,C,D:7 8*1 9 *0 9 *9 *29

*39 *119 *10 9*1:5

7 =8,!,4,6,#,1,1!,14: 7 * 9 *! 9 *49 *69 *#9

*19 *1!9 *14

$8.,B,C,D:7 8$58.,B,C,D::5

7 8*5 ? *5! ? *54 ? *56 ? *5# ?


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$8.,B,C,D:7 =8*15*05 *5*25 *35*11 >*10 ,*1:

$58.,B,C,D:7 8*1 9 *0 9 *9 *29

*3

9 *11

9 *10

9*1

:5

7 =8,!,4,6,#,1,1!,14: 7 * 9 *! 9 *49 *69 *#9

*19 *1!9 *14$8.,B,C,D:7 8$58.,B,C,D::57 8*5 ? *5! ? *54 ? *56 ? *5# ? *51

? *51! ? *514:

$8. B C D: =8


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$8.,B,C,D:7 =8*15*05 *5*25 *35*11 >*10 ,*1:

$58.,B,C,D:7 8*1 9 *0 9 *9 *29

*39 *119 *10 9*1:5

7 =8,!,4,6,#,1,1!,14: 7 * 9 *! 9 *49 *69 *#9

*19 *1!9 *14$8.,B,C,D:7 8$58.,B,C,D::57 8*5 ? *5! ? *54 ? *56 ? *5# ? *51

? *51! ? *514:

C i l $


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Canonical $or*. si*ler an) faster roce)&re for

o(taining t%e canonical s&*ofro)&ctsfor* of a switc%ing f&nction iss&**arie) as follows"1" E+a*ine eac% ter* if it is a *inter*,

retain it, an) contin&e to t%e ne+t ter*"!" In eac% ro)&ct t%at is not a *inter*,c%eck t%e aria(les t%at )o not occ&r

for eac% x i that does not occur, multiplythe product by (x i + x’ i ).

0" @&ltily o&t all ro)&cts an) eli*inate

re)&n)ant ter*s"


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Example Determine thecanonical sum-of-products

form for T (x, y, z) = x’y + z’ 9xyz. Applying rules 1–3, e obtain

! " x’y + z’ 9 xyz


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4/4/16 RKTiwary BITS,ilani 4#


form for T (x, y, z) = x’y + z’ 9xyz. Applying rules 1–3, e obtain

! " x’y + z’ 9 xyz7 x’y(z + z’ : 9 8 x + x’ :8 y + y’ : z’ 9 xyz

E l D t i th


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form for T (x, y, z) = x’y + z’ 9xyz. Applying rules 1–3, e obtain! " x’y + z’ 9 xyz

7 x’y(z + z’ : 9 8 x + x’ :8 y + y’ : z’ 9 xyz

7 x’yz + x’yz’ 9 xyz’ 9xy’z’ 9 x’yz’ 9 x’y’z’ 9 xyz

Example Determine the


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4/4/16 RKTiwary BITS,ilani


form for T (x, y, z) = x’y + z’ 9xyz. Applying rules 1–3, e obtain! " x’y + z’ 9 xyz

7 x’y(z + z’ : 9 8 x + x’ :8 y + y’ : z’ 9 xyz

7 x’yz + x’yz’ 9 xyz’ 9xy’z’ 9 x’yz’ 9 x’y’z’ 9 xyz 7 x’yz + x’yz’ 9 xyz’ 9 xy’z’ 9

x’y’z’ 9 xyz.

T% i l ) f


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T%e canonical ro)&ctofs&*sfor* is o(taine) in a )&al *anner

(y e+ressingt%e f&nction as a ro)&ct of factors

an) a))ing t%e ro)&ct x i x’ i toeachfactor in w%ic% t%e aria(le x i is

missing.

E l L t d t i th


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4/4/16 RKTiwary BITS,ilani !

Example Let us determine thecanonical product-of-sums

form of ! (x, y, z) " x’ 8 y’ 9 z). #sing the

abo$e procedure! (x, y, z) " x’ 8 y’ 9 z).

E l L t d t i th


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form of ! (x, y, z) " x’ 8 y’ 9 z). #sing the

abo$e procedure! (x, y, z) " x’ 8 y’ 9 z). 7 8 x’ 9 yy’ 9 zz’ :8 y’ 9 z + xx’ :

7 8 x’ 9 y + z)(x’ 9 y + z’)8 x’+y’ 9z)(x’ 9 y’ 9 z’ :F G8 x + y’ 9 z)(x’ 9 y’ 9 z)%

Example Let us determine the


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form of ! (x, y, z) " x’ 8 y’ 9 z). #sing theabo$e procedure

! (x, y, z) " x’ 8 y’ 9 z). 7 8 x’ 9 yy’ 9 zz’ :8 y’ 9 z + xx’ :

7 8 x’ 9 y + z)(x’ 9 y + z’)8 x’+y’ 9z)(x’ 9 y’ 9 z’ :F G8 x + y’ 9 z)(x’ 9 y’ 9 z)%

7 8x’9 y + z)(x’9 y + z’:8x’9 y’9z)

E . l % l) l k


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4/4/16 RKTiwary BITS,ilani

Eg" . class roo* %as an ol) clock ont%e wall w%ose *in&te %an) (roke oH

a" If yo& co&l) t%e %o&r %an) to t%enearest 1 *in&tes , %ow *any(its of infor*ation )oes t%e clock

coney;(" If & know w%et%er it is ."*" or "*"

, %ow *any a))itional (its of

infor*ation )o yo& know a(o&tt%e ti*e;


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Eg" .n analog oltage is in t%e range


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Eg" .n analog oltage is in t%e rangeof to" If it can (e *eas&re) wit%an acc&racy of J*, at *ost %ow*any (its of infor*ation )oes itconey

Sol< .n acc&racy of J *in)icates t%at t%e signal can (e

resole) to 1* interals" T%ere are s&c%interals in t%e range of olts, so

t%e signal

E l


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4/4/16 RKTiwary BITS,ilani #

E+a*le

$ #

-se M.MD gatesan) M'T gates toi*le*ent

N7E5$8.B9C59D5:9OP .B

.B9C59D5

E5$8.B9C59D5:

E5$8.B9C59D5:9OP

Q t . t% E l


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Qet .not%er E+a*le

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CS for RL 1st Session