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8/18/2019 CS for RL 1st Session
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Digital Electronics
CS for RL 1st session
rktiwary
4/4/16 1RKTiwaryBITS,ilani
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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 (
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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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$ 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
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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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$ 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: 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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$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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$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" @<ily 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#
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 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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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
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
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
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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4/4/16 RKTiwary BITS,ilani 1
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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4/4/16 RKTiwary BITS,ilani 0
Example Let us determine thecanonical product-of-sums
form of ! (x, y, z) " x’ 8 y’ 9 z). #sing the
abo$e procedure! (x, y, z) " x’ 8 y’ 9 z). 7 8 x’ 9 yy’ 9 zz’ :8 y’ 9 z + xx’ :
7 8 x’ 9 y + z)(x’ 9 y + z’)8 x’+y’ 9z)(x’ 9 y’ 9 z’ :F G8 x + y’ 9 z)(x’ 9 y’ 9 z)%
Example Let us determine the
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Example Let us determine thecanonical product-of-sums
form of ! (x, y, z) " x’ 8 y’ 9 z). #sing theabo$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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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