|signa 'is aphyaial quanHhy , whieh contains

whleh is a un cHon 9
SntormaNon 4
vaiablea,
more independent
signals
Analog ig nal Digitat 6iqna
-sig nal havinq conHnous -siqnals havlg ouy
valueQ Rntte numbev s
-have 1nnihe number of valueA,
different yaluea, - not cotthuous si
eq. m0st of the hingqs two dishnct alues
Obsesved in the. naural

onalag signad,
SOUrce computer, Ato D
s0urce - sinal 4enerabors,
rans ducer s ete. converhey
NooF valuep fite (2 8,16 etc)

4, Advantages of Diqikal siqnals

1.
Digltal sig nals can be
pro cess ed & brangmit bed mare yguenty 2 veliably than
analeg signaia.
2. 9t is possible to stove the digttal data,.
|3: Further proessinq % the digital date (ike playback
is possible.
|4. The ekeet 'noise' (unwan hed voltaqe ueuation)
is less. $o digta! data doeot not o compt.
 CH-O
NUM BeR SYSTEM

st Inro duetion
The psiniples 9. wosking compuers,
municaion
the
system6, tntesnet, emall et e
And
are baged
th ese
systems
pmnuiples
ave
9 dgital techniques. A
to Digltal bystem,
Elechronic cal w
atoy2
in this, the input is glven with the help
) Switche, Th is is conveted into elecm'col sig nalo
which have 2 d is crehe values, 0y levels -Low & HIGH
the 2 lewelo,
The
signad will always bees &Dhe
the achu al value6 the is immate tal as
signal.
tonq as it is within the speded angeLoo
HIGHIlevel, This ty pe o sign al is known as Digit a
& the ckt ingid e calc atoy sed to
signala the
Digital cisuits.
procer s th ese signals ave kn0 wn as
A cal catoy is an erample e dgtal systemo,

Dig ital systems vequires d'gital tnemaHon,

number
-oigital.3ngoma Hon is vepres ented by t*ed
hoh conhnuon Y discvete symbals caMed Digits.
2. valves
Pig'tal We binang syshems (bis hauiy
se

Or stahes as binay syshem is hauiy adu ntages as

tollows t
Moct inkormaion procensig syohems
1, roro switchea whi
which ane binay device,
are more eliabte.
ja. Bin any signals
deu'sion makin9 procergew Yequred
l3The bagic
digital syshe ms a re
binang
 By
bigital signals Has thwo oTs hehe
HrGH
-A digital sig nal
voluen lewels are
Digital
Analog sigma TH
als
A

tu
1

- There
dinerent gepresentañons digitat
signals
PosiTIVE Lo GIC NEGATIVE
LOGIe
SV

3:5V 3-sV

lower of two level s has .

-higher of tuwo levels han
been desighated as LOw lewels been desig nared as LoH
&the higher as HIGH lewels. lewel & the toweY as HIGH
lewel S.
Any vtq in the Tange 3SV to sV.will be consider
as HIGH I ewelin tve lo qic sy stery
Low level i5
-ve (ogic syshem,
similay, any valtage in the
6e considered as 4nge b o to iV will
Yange
H1GH lewel in the -ve
is tve loqic systemd
the -ve logic
Th e
atual voltage vang es system. to Lowd
comespondimg
 Nibble

word
Combinatnaihoonn 8 6its
Byte eombi 4 bls
9it- a binany diqo ort
HGH eel
ombimation o, 1GbiMbits.ag
are not
N! are
diherent or Sam e kor ald types ckts
The two
dinrnt logie tamlieg.
also 6e diserere bysiqnal lew els HIQHe Lo can
binavy epresenhed the binany ignap
A
- Sin ce dgit (o or )is &omp
two digital
possible lewel algnal reyed only
can have
one n the
can be w ed Gin avy Number 6yster
7oY the
syshem s. anagysis
|The two 1ewels:CoY s ta hes
can also be
as oN OFR
TRUE oY fALSE. designared
4
Numbesing ays hem i- A no.
OY
6yotem
num erals.
is
dgnedy oig'ts
(CDecim al Number yahemi
The numbe 64shem in which an
ten
osdered sel
Symbals ’ O,, 2, 3,4, S, G, 7, 8*& 9 known
digits a re used to speeky any number.
The adix oy bae this number 6ysrem is to
Any numbey is coltechon these digits.
e.g 1982 .365.
- Ha hou al pavt
3nhger Radix
Pavt Point/dewmal point
-Some the other commonly uoed number system
are Binay. Octal, herade im al
-The knowledqe s th ese number ysems is veny"
ewsenial koY undevstan ding. analy sing &desigaing
dig ital syshema,
-The in otmahon available the km s) numeras,
alphabets the sp eaal ehava chers ov in quy
 thee must be converhed n
tombin a hon'
komat te th a unigue eombinaH on Os 4
Co de.
a Co dinq sch eme, kno wn a The pvo cens
Co ding is knowh encodig
Numbe 5yshem ls an o rea' set symbols
knowh as) digits wi ules
anthm etic opeyahonS ike addiin
deyinedmulH°Y tipergormin,
canon et
A colletion ) thee digits makea numbey
which t
genesal has too parts stege &Hrava
Set apart by e radix point (-) te

dn dn-2 - di. . . d, do d d-zs.d.a.

iwhere numbeN
b adix 6y b e the
namber syshen
S digits i integer jportion
m = nsmber
raetional
dht = Most slgnijeant diqit Cmsd)
dom = least signicant digft ( lsd)

The digits th a numoer are placed side by side

each Posiion in the núm
oY index
númber is assigned a wegt
impotance by some predeaigned nwee.
 chara che sHo comnntu wed Number á ysems t

Numb er
ase oy symbols weght ansign
syshe Yadix(b) ws ed to posiH on
(di os d-)

Binany
3569.25
-

O,, 2,3,..?
becmal O,l,2,..g to 39 74,59

Hera ol,2,..9, 3 PA9S 6

demal A,B,C,D, E, F
-.

OBin any Number áyshem Sysheg with bae

The numb er
CoY vadix) two is known as the binay nuiaber syshemg
tth epe tw o ymbols ane sed to repres ent
numb ers shem,known bit6.
posiHo is
-9t is a P'osihonai Syshem, ie eveny
assigned speupe numbey
we'gbt
a

system legt- most bitt is

- S imilay to de eimal
the ngh t
knowh as most- si'gni'eant b1t CMsB) &
as he. least Sighicant bit ( LS3).
most blt is kno wn
Os Cane added ed to tt,e let the
-Any numben 6) 0s
value
the valu the number,
number with ou t changing
9p 6
- A 9p 4 bits is known qs nibble & a
8 6its is known as byre.
tombin a hon' these must be
converhed tnto
binany
unigue ombinak on 0sd ls uo
kolmat te th a
s Cheme. kno wn as
Co de. The
a co ding
Co ding is known as encodingi
Numbe Syshem is an et g(8ym bols
kn owh as) digits with ules degined oY perggrming
anthm etic opeyahons ike addihen mulhp ti canon etc,

A collecion these digíts makes a number

which tb genesal h as tw0 parts tnteqen & raciohal
set apavt by radix point (-) te

(ND5= dn dniz d.*.. d, do ddzd_ed

where a. number
Yadix the number syshe
n= number s
digits i nteger pohos
m = number digtts aeional potio
dh= MOsL signieant diait (msd)
d-m= least signicant digft ( lsd)
o<(dË or d ) sb-I

The di'gits ih a number ave placed side by eide 4

each posihon ' in the númber is assig ned a weght
ov inde impotan ce by so me predeoignied we.
 ch arache visieo commonlu wed Number 43
Numb er Gase oy symbols Jweightansign Examp
syshem to posih on
Yadix(b) ws ed
(di os d -)

2
Binany
o ctal 8
2 O,I
O,,2,3,..7
3569.25
39 74,5
becmal O,1,2,..9
3FAg -S6
Hexa- ol,2,...3,
deeim al A,B, D,E, F

OBinay Nummber óyshem The number sysheg with

base
number systeg
wo is known as the bin aY
CoY va dix) wsed tD represent
th epe two symbols ane

S4shem,k nown as bits.

numbers t this is
posihonal syshem, eveny posiHon
- 9 t is a

assigned speue we'ghtsyshem legt- most bit is

eimal numbey
- Similay to de CMs B) & the ng ht
most- signiant bit bit(LsB).
knowh as as the. least sighig'eant
mos t bit is kn own added to the
the
| Os Can6e
-Any numbeh 6 alue n the numb
the Value
changing
number with oh t nibble & a gp4
4 6its is knowh q3
- 4A 9p is known 4S a byte
8 6its
 thelr comspondhg delma number
numbers &
4 bit binay Numbev
beuim al Numbe
Do
8inay - o o
B
83
o
2
o o- -
o
o-o-
- -o o - -
S

o
o o
7
oo
o-
-- 9
o
-o
o
-o 1 2

Con version
2) Binary - bo Deima|
converhed into it
Any binay number can be
eqwvalent deimal number uslng the welghts assignd
to eaeh bít posiHon.

DeheX mine the deumal numbers vepresented by the

numbers.
jollowlng bimany

(3)1o
 +{1*2)
=128+64 +32 4 |G+8t4t 2+|

= (53)o

too o0o o o0), (0),e

Cottot. toto)a
= (1x25) + (0x2) +(Ix2) +(ix) + (ox 2')+(1x2)+
(Ix2)+ (ox) + (I*2 )+ (ox a)+ ( 1*25)

: (45 (5625)ie

Cuoo. to1)2 +
((xa)+(ox
i .(Ix2 +(Ux)+ (oxa')+ (oxa') +
8 t 0 t o¢|+ ot O 2S+0t 0.o6 25
= (9. 3|2S)10

(o.totor)2
(in +(o*2y+(ixaS)
E (ox2°) + (i* 2')+ (o*i") +

OCio01. olo1), +(oxi+(r*)+oxa

(I*2)+ (o+22+ (ox3)+(r*2)
ADDeimal to 8inay Conv ersion
on is. obtained by
This conversihack s% the
ou
%em alndess
eonn
divicion by 2 4 keeping comversin is eche
Iwhlle toy he racrional pats, the
con Hnuous muiplicahon by 2 4 keeping
| by
he inhegers g enevard
quivala nt (or to a
Conyert given decimal to 6inay
equivalent ban e-2number)

13
2 CLSO)
2 3 = (oD,

CM S)

(o.65t2 s)io
O.Gs2s x 2 = I32s0

0312sOX 2
I2So00
(.6562Sho=
062so0x 2 =
02s000 X2 0.Soo00 CLSB) cfrachional conversid
is generay camie

2 58
:2 29
2 I4
2
CI1,, =(loto),
2

’1Cmso)
*or4ero= C?)4 0-84X2 l e 8
Coiroi)a

fracioal Pavt
9nteyer Pavt
2 625
p
2 2

(2s0 = (lldo)2
-:(25. s); =(I|oot. 2!

O(o625)io
frachiohal Payt
8nheger pavt
O62s Y2 i25o
2
O"250 x2 = 0-Soo O
2 S
0-SoOX2 l00o
2 2

(o-s25io =(o12
|Cio62ro (Qlo. lo1,

x 2 = 37S0
O:683

D SOo *2

(os8s)
 (37.3),o
O31X2 0 6 2
2 37
2 I8 O62% 2 =l24
0·24*2 =048
2
2 o.8X2 =
2 2
(o"31) (0.010 o),

(37))o = (00l0) 2

(37.3Die = ( (ool0).oto02

An o th er MeHhodi
(30),-=6 + 8+4 t 2

(301

(23)]Signed Binany Numbers

gh the decimal number syshem,
plus (+) sig n isis uged
used to den obe a t ve number d
to denote a -ve numbe,
a
minus ( )sign
+ sign is us ually dvopped & the absen e
mean sg that the numbey has tve value,. This
signed numbex
sepses entaio n o nu mbey s khohwn as
An addihon al. bit is used as the siqn bit & iE is
placed an the MSB.
A'o is wsed to seprenent a tve numbeY ¢
a-ve nymbey
is used to eprvent
 e, qo ol0ootoo
value ( magnihd e) 7epres ents a tve
is ( o00t oo),
Nemb ey its

MSB (C8)ro
0ndicaten that it is a tve number,

o00 0.0 veprnents a Ve numbey wih

mag
NSB in dicaten that iE
is a -ve numbe
This kind o
known as reprsen ta tion K
s9e numbers is
619n-magniude vepresentahon."
Cx. Find the
: | numb ers
dedmal eguival ent S Bhe 7olto lng bin ary
a85umin g sig magnitde rep veoentation q
binany numbers.
a) olloo
5lgn bit MS8 = I ind itates Numb es is -ve
Magninòe

6) Þ(o00

sign 'bit = Ms = o indicaton number s +ve

magnihuoe (1x23) (8)
(oo10002
e) Dt
MSB 0 fVe No.

MSB=| -Ve No.

Magnituoe =
.
cm)z (-),,
 Gne's Com plement Represen tahon

0 by the tesuI Hing numbey i's k

one's complemen t the prst numbex.
the umbess ae omplement seaeha
Hhen
3 one [7 hese numbers s tve, the other
Wumbeya will be -ve with the same maghiyoe
NieVersa,
- Sn this repreo entaion also,

g. nd the i's complement o the 7ollouing bineny

number $
a) ol 00l10ÞI
b) |0|0l0

*Reproen t the 7olleuing numbers in is

a) t7 &-7
(+) = (0l|)2
Is (o1)
6) + 8 b-8
(-1S)io (to0002
(+8)t e (0000)
(-8) 1o =
(lou)2
-Foy nbit
can be
be inumber, the maximu epresentarion
is (2* epresen hed ' l's tve
co rapleme numbe which
-)&the manimum
number is(2h) ic
 Two's Complement Rep reoentaHan
' is added to t's complemen
kuown a
a binay numbeY, the outHng num ber is
the two's complemtnt 9 the binay numbos.
o n tiis epesen taian als0, k MSBis o, the number
s +ve, MSB is e numb er is &ve.
o h n-bit numbey the maximum tve numbe
's
which can be repreoenred In a's complement
the maximum - e numbey is - ( 2 )
The a's complement q the a's complement &
number is h e numbe t tset

Siqn Magniu de, l's 2a's compiemert sepreoento Hon wsig

four bits
Binar Numbey
D6cim aj No. 6i n magwitude one's esmplem ent Two's conplemert
0000
o00)

2 oO10

-8
-?
1ol0
-5

-4
-3
-2
 the numbers.
eomplemen t
Fin d the 2's ) oololo|

glooto
Num ber

's complem.
Add

numbes 's . its. 2's: complesie

t. 91. the LS 3 the
s obtained by changin4 eaeh o to & to o ex ceo
the
least- signip'east bìt
S the LSB
the number is 0, its 25 compleme
nu mber LSB to
is obtained by scanning te

MSB bit by bit & se taininq the bits as

e in duding the the irs t 1 &

eom plemen t all oher bits.

Ex Oin d 2's complement th e numbe.

Using
aboveRule )o(o0tO0

Number Number
2's comple 2's comple.

-i) o 0o00)

O0tooo0

) magnih de i r's eomplemln

)) 2's complement preen ta ioh
Minimnum no. 4 bits seguves to neprnent (1?),o"
 Binan Subra chon us ing 2's complement Merhod i

Minu en d is added to 2's complement subhaheud.

generaed,vesut is pasihve & in its
i).97 camy is genera.
hise

its twe's
clmplement krm.
) eamy is adwaya dds vavded.

exP7m (6)io - (4 io Msinq e's co mplement mehod.

complement (4)ho to gt(-4o

(t4 (oI 00), complement of(4)o =

a's complement of(41o

(-40o
Ans iS = 00|6 = (+221
dis card So Ang is tn bue koYm,

Ex) feryrtm (t)-(*O1o using a's complement menod.

(+)16 +(-4210
's complement cs)io to get c-6o
is complement of(6),
a's complement of ($)is C(olo)

Ans is = 00O0
(0)o
discaid camy
 Advantag Digital drcuits a
aith meHc opera Hon
wed perjorming
the ckt designed binany
91 posstble to wwe subhahon also We

addi ion to pergom binoysubhoehon to hat ah

sy
ehang e thepro b'em ahes the need add iio nal u
addiioh, This eimin adde cktCH
the s ame
arheY
hong.
Can ed oy boh the opeya convene
ma ker desiqn arithme c ckt veny
This comples entrep
this purpose, 2's
4 ch eaper or
is wsed,

(er)) pergsvm slnang

subhaion ueing 2's corn plena
numbens.
epreneniarioh
-5 11

t2 5= 0 0| +

2's compl of S di'scard

(-s) ie (ooio)e),

5-0l0/

t(-4)
fo d in
NO camy ADsis 16 -ie foym
 Ex 1? Knd s complemant the polloung numbes.

(-12`)6

22s 2 125
2 12 2
2

2 S
2

i's camplem en t ot (2s),,=ooo).

ie (-5),=(0o1lo) 2
(t12s),, ’íscomplemeyt
(ooo0 ¡to)
-C-135), = (0000o0to

224&
2 148 com plementing a nuMber twice
2 74 is the
-oviqinal number ibse
2 37
2

2
2

(49,6=eolo1000)
's comple (110 01 ol)
 numb ers
Rnd ds complement % tollouhg

34 LSD
o,o.,O

MS6

(37)=(o010), ’'ooiÍ ol'o

(s'complement of (7),6=10 19lO
+

I ’ (-3)n

2 25
2
2:

(s)6= loo1!
's compl ement
of(si),
i 3 0 )t6

130
LSB
2

232

2 2

MIB
IO000010 = o000 I000 0 9.! 0
Cwe have to wwe l2 bis)(a the 8 6ik are equived to
vepreoen t 128)
i's eomplement ot (13),

.|(-130)16 19 2's com plemerte Ctru ollt ||o),

 * Binary subrati on ueing i's complement Metnod,.

Ruleg t- he minuer d is added to ones complemnt

the sub hahend &
in the Yesutt, camy is geneva ted it is ad
to tye seslt e the sesut is posiHve & in i

in the
in esut, camy is notgeneva hed
neg aHve din ts one's complemen t

Er Per povg (c)o- (42, 's complem ent meho

4)6= (ol00)2 (-416

Ccamy generared shoud be

added tovesu

(0 o10), (t)io Ans. is ln tue

tber
is prenent tve

E) peroY m(e)o- (C)lo using 's complemert mehod,

(-c)ho=(1001)2
camy ’ an is in is mplement

(toll(o000)2
(D, is also comct ao in is complemen t (-o),o
exOpeigormg CHio-(6)ro wsing ts eomp lement meid,
's comp)ement of cc),, = .

(4)
camy
ans is io is complemet
s coupl. of pve reonlt e(0o1o) ie (-21a

9pergerm (-8)6-4), utiug 's complement Memod.

(-o-(-421.2 G-4)t(sia

cany mea

lnis complenant
forrn,
 t Bmany Mulip lieaHon:
ErO MiNy (41 by (8
(9 =
(8 ),, - (I000) (7

( iootooo) dec'mal polnt is placedte

ie Anse (iooo.lol
(9) x(8)-(32o

Ginay Division
Ex tI1HTo.J4010}

ito)no .toi)uLo!

Ahs plae Ghe

one
()

( o o)t Cl61),=
Remainder e (olo 0),
 si'gned number
kormt
. (-9io is (otooo
sepresen hed by
) tn epre.
scomplemen
) In e's , complem ent
t

TBinay At th met'
mehc
Binany Addihon -
uleo
binay addihoy
Avqend Adden d
cang Result

numb ers
Fx Ad d the
i) |o(
binag
(+) ( be

ca my
camy

)
1

111|IOolo
 division, deima Same Binany
Division
produt : Knal *
. 9
c]
o) bytooj MuIHply Ex
), by
(mutnpti
cato muliplicand erecHy o) by
as SQme
bingy,each OY
licat zero ether product
s the paial -9
n
muliplia
hon, deimal
to similg is E
MulHplication: Bin
( I|oII Ex
38 ",
,,(I10),
((oollo (a8),o
Subhahend Minuend
(23),,1 (38),
9ubrat Ex)
choy Subha ay Bin *
 Ex 1) Divide I0olol by

(0o.)

o ckal Number 5ystn -

Number sys he msith base ( or radi) erght is
known as the octal Number sysem.
9 wes /mbols - O, l,2,3, 4, S; , 7
- Zimilár to decimal & binay num ber syshen,
also a posiional syshem
- J t has 2 parts integeY e achonal set apart
Iadix (o ctal) poiut ()

actat to Demal conversio n

'otal number can be
converhed into its eqwvalent
Any
deim at number using
the weights assigned to octal digit posiion,

. ExI. eonvert (G329. 40s1) g ihto 'ts eguivalen t decngal

o1b, (c327. 405Dg (6 8*)+ (3 x8*4 (2 *g94 (zx8)

+ (4 X8) +(ox8)+(5xý) + (1x 8
(3289.Slo0o98)1
c323:40s'= (3287.si00098)io
* Binay Subhaioy i
binany subha ei oy : Ex gubhaet

Min uen d Svbhaheyd

(38),4t29),,1
Bindny

', 38

Bin qy Muliplicaion
T is similar to deim al mutipliahon.
-9 n binay ,each þaial produet is ethey zero
SQme as the (uiwplio
by ), guliplieand (mulhplica
Ex 19 MuIHply

Knal produt = Uoloj

Binay Division i
Same
deimal division.
Ex 1) Diyide

ackal Number systm

Number syshem with base ( oy Yadi) erqht is
known as the octal NumbeY sy s tem.
8 symbols - O, l,2 3, 4, s, 7
- Similay to delmal & 5inany num ber sysen, Eis
also a posiHonal sysem
- J t has 2 parts inegey & fa chona, set apart
by adix (o cta) poin t ()

(3-1) ctat to Deeimal conversion:.

Any 'o tal num ber cah be
converred lnto its equ'valent deumat number weing
the weights assigned to ea eb octal digit posiho9,
Ex1. Convert (G329. 40si) thto its equ'val en t de cimal
humer.
Bo15, (ca 27. 40s)g (6r8°)+ (3x8 (2xg¢ (ax8)
+ (4 X8)4(ox8)+(5xg) + (Ix8
(8287.Sl00o98)1
323: 40SI = (3287.Sto0098io
32) beimal to Octal comversion |
The onversi'on
'on Bm
pm
decimat to octal (base-lo to base ) is sim ilay to
the con ver6i on proceduve base (o ko 6 as e 2 onves
sion.
The only difteren a is th at humber & 's used iy
ploce 2 k division ih case
muIipliea hon the case Ka Hon af nummbers.
Ex cohvert
7oll owing d ecimat to tal
a) C243)1o

247
30
3

SSoo o S

c) (3 2 87. 5Io0098)1o

Sn heger Pavt Geetto hal Pe

328? O- Sl0 o0g8x8 4.98o0784
40 O.0800 784X8 064oG2 2
51
0
2 os4 oc228 =5. 2561G
8 3 ol25 6146X 8 ,o o0)4 Þ8

(328?2to = (6323)8 (0-sl 00o g8), = (o.4osi8

 JC328. 5100 o98) o = (632.40s))R

inarg Convevsion :
Octal numbers can be
eonerted nto equival en t binay numbers by replacng
each o ctat
digit y its 3 - blt euvalent

BnanY
binarg
& de im al équivatent of otat Numbers

eum al
Biuey

3
4
S
6
7

(2

I2
13

En convert (73 6)8 lh to equivalent bin numtber

36) = (
 to Conversion:
numb eYs an be
einay
eonveted into equivalent octal wumbers by making
& m
three bits shahn9_krom LSB
the numbey2
towav ds MSB k tnte 9ev part
| th.en vepleun9 ath gvoup three b'ts by rs

Tepres entaion.
FoY kYa eion al pavt, the qvoupi4 6 Hh ree bits are
mad e sfein q rom the binany paiht

7ollowin9 binag to its octal egwvad eut

a) Co0o)2

(o. 5
= C0:si4) 2

3 6

3 . 2

oo1|110. Uoolol oa1) (33G.G240s

 3 2

S0crat. Arth metic -

octa) Arith ehic Yules ane simlay

deinaf.or binany anthmeie

- Ånth mehc opesahong can bepekned by
1
.convebnq: the octal numbevs
then usih g the sules binay avtthsn eic.sU
Add ( 23) 8 & cc 7)8 (4)8
(I3)8
(o 2

Ex 2) SubhaL t
a)(37) Prom (s32g b) (95)g hm (2¬)8

(s3)8 Oololo|
(2¢)3
compl. comp
of (a
(4)8 Ttake a's Lomp
ais caid eamy of euit

S38- (37)g = 00 ool too

 MulHplieahon B divisioh can also be perormed uging
binay vepresentaHoy o octal numbevs then nkin
Use mul hplieanon division YWe
num b ers.

Applicahan Octai Number system -

u3ed to ehheo
Th is num b ey system is homally 1ong
string s binay da ta into digital syshm tike a
MiwocomputeÝ.
This makes the task enterin binay data n a
mirogohap uer eanier.
Thererone, the knowledge ok octal numb er Sgsem is
vey impstant the' uent use micopvo cens ors
other digital ckts.

Hexa de cima) Number. kyshem

The base hexade cimal
numbey syshem is l6 which requres 16 dis Hnet symbolJ
rerent the nuinbers.
to
- These are numeYalg
thvoug 9 & alphabets A
'.
6fnee numec digits & alphabets botb are used to
epresent the digits tn the hexadeumal number sy she
therege this is an alphanumen'c Number
- system.
nuimbevs
-There
ts
'sets
are

4
combinai ons n 4obit
binang
numbeYs can be enheed tn he
binany
COmphe x 1'5 the 6rim hewadecimaw (hex) digits.
 Bnay&deimal e wvalents o hexadecim al nubers

He*adeemal Deimal Binary

2
3
Ot0.o
5

7 7

g
A

B
C 12

D 13
E

Hexadeu'mal- to - Deeimal Convers ion -

-The hexadedmal
numb ex can be conerhed to its equivalent deual
mby
nuliplying e atgts by thelv hex wetsts.
- The bits to the le t de ci mal point ave mutipleed
power 16°, te te s0 kom LSD
by
MSD reAp.
- The bits to khe sight are mulNplied ' wtt negaHve poWer
6k 8 ie. t6 on. so
 Ex ) ohvert (2APc to egivalent decimal.
@AF2te e (a xie°)+ (A XIeb+(ExIe)

2) (0.F232,e
(oxie +(ExI+ (2x1+(?x1E

(o .443 0216.

3) (3A- 2F)c = (3x1c) +(AVI6+(2x 1e+(Fxt )

= (3*I'+ (10 x16)+ (2 xi)+ (1s X1)

(sB.834)o

(42) Decimal - to- Hexadecimal conversion

FoY con Ver sion
ro decmal to hexade cmal the pvO edue
O tal sy shem is appli ca ble, ugin (6 an bioy
the
dividing o 9nhger partD & multiplying
pavk) katho. Ckahon a
E ) convevt ollowng numbers to h oYmat.
a) (233 8)10
1C{ 2338

2
9

(2338)io = (g 22).. 922

 ):(464

28 13 ’ D

)95-s216
(S ’ F

(9s)1o ( SF)ic Co-sj e (0:8)ic

|(95-s)io = (s F 8)e

42 3
2
2 ’ 2

(61so (2A3) e

(9.3) Hex ade cimal to Biygny conversian t

Hexadecimal numbers
can be convexted inb equva ent binqny,,numbers by
Yeplacng ea ch her digit by its egvawest 4bib
nymbev.
bin ay
Ex ) convert gollowng Hex numbers to egoyualent btnany
numer.
a) (2F9A)
:2

(oo10 too toto2

A 2

oDl,oo10:
) (oA25) 1c= A 2 S

2
d) IA 3. BA)1G
A
A

3inary to Hexadecimal
convers ion
can be co
Gin avy numbers
nveshed into the eg wival en t
hexade cmu
hum ber6
by mak ing eroups ku bits sta vhnq Bom
LSO movinq to wards M8B
Tepladngea ch gnp 5% kou bis b its
vepresen ta ion . hezadeum al
Foy racHonat
pavt, th e above
stahn9 Rom the bit hext to theprocedne is repeate=
moving towards the vight. 6inqy point &
erOneonvert the. elte wing bmany numbevs to they

guivalent hex hymbers.

3 2 (32 B E)s

2 A (2gAF) I6
d) (0.0001L1O01 I6192

4
o1EB4ic

-
-

D A
2 6 - (2DE. CAS),

9) 00ol. lo0| (oo 0|

A
(IEL-99 A)
 Ex O Add (F2e & BADIC

bu

subhat (sc), from (3Fc Se! ,

3 1S
-S.C 2's com plement of (s C)
3
4looo|e2's complement oF(ren

CO

atsca vd cany
Muliplicaion & division can also be perkomed wing
the
6inany Yepren entaho heradecmal numb ers &
then m akin 4 & mulhplica h'on d divisioh gules .
numbev6.
biaany
cODES

Weig hted code welghted |Alphanumeic

code
Ewov de tecmg t
( in which'weight' code Comein code
has been a ssigned Er ens-3 O Hamming
t each symbol pass
BD'des
aa Parity
Holle ith
Binamy codes
 gbina
code
shhaight eusing
th
than de co
ma code
a tobits numbey ore egiYw
m
drvidually. incode
bit KuY this hed
byyepresen er deu'mal
is numb
each bits
d using
4 tents equiva inany 6
it, decimal throug
Ccoded) ted enepre are 9
uoal na
thedy by
igíts
O decmal code, this 9n
BcD Natuyal 2)
:-code
ihon, pos evevy
d weght
is weighbed a'
ince
a code
table
O. navg tnBinvend g
-
ne a 15 tu mberS
0 deemal
nu
peyom opesaHons etic
be can usiaith
ngm-Vanous
toYm. ed sha'ght)
a b
y in natiral
(ay numbers'
represen toysed isThis
t ht
Gian Osraig
code welgh
- decoted
8yshem; digital 'th' cioh coe
hechon deemoy used also are Codes
e e
wh. 's
kno avllable maion
is goy in ls
onyBpossible injovmahas
is 6inay
in de co the
this taHon
4 herpye inthe
maHon
i3 kor inthe used Althou95in
code evevy
aracerS. spedal roy alphab
ts e mene, nu be
veprent ave
tousedbinanYtodes Vaious
ich data
cal
de Co an ed
up
qro symbols,
the gvoup speual -hu:à
bolg
is sym
en ane words tetters
Ûv numbers, ben. h
 conven len t
- Snepire ,is ais advamtag e lt rs
wegd code ov input d olp opexa HonY uà
8-42- ode 6y
known a 6
Th ig to de a also

-8,4, 9 4| are the weg hts 4bite biay sde

dedm al dgb thevegore this s werg hhes vode

various 6many co der -

Deamal 6in any
N
C

4.0

8
9

|12
14
15:
P

odel,
weghhed
oh
Nh¡h
3 ' ( s -3) Co
de
Ex CESS Th is is anathe
type Bco ode,
bich each de dmal d'git s code d into -blt bin any

Th e code Yèach decmal 'gjt is dbtained by atlina

naural
delmal 3 t u the agit.
is ot wegh hed co de.
-gt
Th is Code is a sel -complegen hn code, whico
mean i's compleroent the cod ed number

g's complemeot the mumbes itsey.

- The sely complemenHng prope hy ths code helos
considesa bly tn perrm ing subtba ehon opeya ron in
digtal sysem s
Ex CeàS -3 eode 2' is O01, Sss i's cemplement
ol0, which is e*ceAs- 3 codeY dedma , wb ics
is 's's complemeot 2.
ST ay code - only
Untoishance ed SE is a Cyclie ode.
a
9t is veny useut code v hich a ded
Kepeod t bina ny to such a t
S0 thqt each co de number d ' s hm the
pr eeding 4 the succeeding number by e single bit.
his code is used xhensively sha7t en todevs
becane th is propey.
G code oY decimal numbey 5 is
G is , rhese two todeo dier by ony
posrhian (3re fm legt).
-9tis not weigahe d code.
he
ray code is YeHected co de and can be
con shuetd
ehed using this propety gen belov:
gen
Rule A -bit gr ay eo de h43 two ode wordg o . .

epresenHM9 dedm al num bens o d | esp.

may co des 6) (n) bits

ode' will bave
wdtten th oxder wihh

leading last
appended,
h ray todes w| be egual to th e
Gray tode woxds G an (nl) blt gyay code, ) tten
ord er Cassuminq miy plaud bet
Prst Gray codes) wth a

pperded,
Ex) Detesmlne a) 1 blt 6) 2 bit c) 3 bit
Gray cod es &
tabuw are along with theiy eguvalen t decdmal numbevs
a) bt ay code is comshu Ched
Decimal No.
Oe (above)

2 bit Gry code is con shu ched

9Yay co de s 6it
Deimal No. ay cod e

c) 3 6it Gay code is con s hu eed alny 2 bit ay eote

4
 Repreeen t the. deimal number 2nin
) E cess3 ta de
binany gor nm
c0de
V)
Hexade mal
1) GCD Code 2

(0.olo: o )
Cers-3).. Co de

fiv) Ga ode.
,i 8 bits a ne eq uined to epenent 24.
GYay code is conshy cted e27i's bheryore s bit
I010
represented
V)octai code
c221 = (33)8 = (011 o)

3 3

v) Hexadecina
((8), =(0o0. o

the dec'mal numbers 39 6 & 6 4o9

Ex ) Repreent i) BCD Code
iv)o ctal co de V) Hex coae
in Ex Cens -3 co de

441 : )396

U) Octal eode
 6) ) 409c - I000 oo0000 000
) 4o9c e ot o00000 1001olt0
) 4094 = o|lo00 |o0|00|
iv) 4o9C Io000)8 e o ol 00o 000 000 oo0
) 409G (Io00)., 000t. oD00 0000: 0000

Alphanumevc co des
-St is u6ed in
many computers to
Yepresent alphanumenc chesaehers & 6ymbols nresn
can be caued htexn al code.
-Friquehtty, thete is a need to represent more
than
64harachess ineludinq the lower case letters &
6peial conhol ch ara Chers- Koy the
dgita! ta erma hon. brangmis6ion ).
-For this Teason, the ollowihg 2 code are
used, 1ormaliy
n

1. Amerian stan dard ode k in te

8n ormanon hange
(Asa)
2. Ex ten ded BCD 3h
terch ange code (EBcDIC)
OTe ASCII ode
 Applicahon
code is popudary uson te sheyt potikon
encedevs.
ode wo1d w eh
A shat posih oh enoder pro du en
vepresents the ongdey posih on oy e shoyt
- The eneoder con si ss e ght sovra, w opa dit
dete t r os shown 15 9
Patte sn6 6k opoque 2 hansbarent segments is etesed
o t on the ophica disc. So ceresponding he blsck
pomon, the ph oto deetr podu en
is produud.
the bran spaent pomoh
- The patteYns´on the dise ave acoding to he ode
seguired to be produced at the dete r olp
Gg b) shows the pathesn o binag co de d Ko
showS the patesn boY producing he 5Tay cede.
igit
Soure
3 bit code t ep suent
the shagtt posey
photo-derect
OpHcal disc
H a shat pos íhon enoder

olo

co de
Hg b paHern or bih ang code
 conversionh
Gray- to- Ginany
are sa me. 80 wie
S hep The MSB B aray
net biC k 4Tay ode.
Add binony MS8 to the mes,
Record the result b ghove he ca
this pYocerS unH LSB is Yeached
3) corinue

Bi B6

B3 G2
araycode
 ay Convers
cor ion]:
8iMay tb G
is
Reecord the MS3:as it
shep O netposihon, re cording the
.Add this bit to the
sum &hglegy e caunHl my
completed.
Rcid s cesslve sum s

convert binay to 'g

Ex
t t
entt
Cdi'scard

JL

(otto)gra
Ctolt), ’
 Codea -
Comeehng
ErroY DetecHn9 angmihed frm
arr t be
When binay signalato another lo ca Hon Cneel vey)
hansmitev) ele cal
One lo cahon ( ocuY beeause
ra nsmiee
bransmission emrs
noise in
byansmission
th e
m.
channel. Due o
bransmited as a
beecelved
a
sig nau
vie versa. e ceived
i5 Ehe
to dere et the emY
t is desised ¢ comeet it.
data woyd lo cae its bit postHon

EroY Derechng Cod es i

to deimal g'
Con sider a eco code (comeoponding
tool is ransmtbed is eceved a lo|.
- - -

6e
sin ce an invalid Bco co de, there7ore it To
detected by the Yeceive. But 91 it is eeelued as
which is a valid %c0 code koY deçmal I, he
sereier may inter pret rt as deim a t &th e
is ndt dere ched.
to0t ( =(9),)’o0gr ( (6)
To th at Che

ery ald ay
avoid its
inorect in
herpre taion by he eceveY,
the code muet poss es the
o cu Yen ce Sk any propety that the
code woYd inb
single rans koYMs a valid
Foy n-6it
inval'd eode woYd.
ode = 24
desired to make possible comnbinaions
this code
code, owy halk 5 he
shoul d be Jndvd ed to possible 2 comb
inqhons,
he code.
 eans, Ranerha. bit ie t bbit
attq ch ed tb
SLtode to make the bits
number k
wa b o n t o make he, number
ones In the n
sesut n q Cn+12 bit code een ov odd , it will ceranly
benas e o d ehe chihg code.

beminimum sdis tance.

nlumtevs two c o de words
any is

minimum dishan ce is: tw' oy.m ore.

Foy d ete ehn9 % YO Y an exha bit known.as p a 4

it is code meke the
attaehed to erch
mumber sohes in the code even ( ren pahy oY
edd kiadd sparihy.
Tabie- sho ws G co code.wih pa oty bit oteehed

BCO co de G Co co de wi ben G Co Code wih o dd

Pashy Pahy
D P D C B A B A

O,

1
 oy?:
bit CP)'o
padhy
a S to make
a
so
Table showS code word the
at tach ed to eeY odd
even & o dd

numbev o
especHely. emor, the
th ere is
ony
One
end by Pahy -code
eheek.
the Yecei vih4
is dete cted a t check mehod onty
Can d'etet emy in
The pay
mitted wod at ther Ye eceivhng en d. 91

the ons
chauged 4 h e r e the
lo cate the bit which has
does Hot emse.
queshon come on

1
Erroy cone in g Co den
adding singie pasihy it
messag e be9 ransmih
along with the ingomahon,
posihoh c n be dehe ched
bit
Qn emoY n single
- The pathy check giveg
in koYmai oh that The
income et. 9t can not lo cate
the teceiv ed mess age
th e bit þosihon in whi ch eor bas occuved &therekO
canh ot Comet the erov.

(ACh
Let us con sid ey e 4obit bin any word ol0 is
tronsihed along uls an ewen paMy 6 t . DUe to

bonsmission eror in one bit posih on the eroneous

word eceived may be o0100, o01I.o0O01
oto! Depen ding on the posiH on bo, b,,b,
bz veop
FoY a ode to be emY its minimum
distanu must be more than comengs
single b/t emoy Can too.
be dete chod lo caee
thls eode
ueing
A the
Said to be e y
come coreing cooe 'b
an be dedu ced
Gom
R

4
Ham minq Co de 2

EHDIN0nq Co de code.
comecH
Pahy bits to
conshmched by addin g
'n7ormaHon
es$g So S tu be able to lo cate
bihsp sihoh n Whi c emoy o CcuY 6.
(

Ashy e on
aHa mming co deto
QSUme k- pat hy bi'ts. P , , ..
added to
ticode.
n-bt message to (n +l<)- 6/t

-The Valu k must be hosenin SU a SO

ble .to des csibe the lo

he b +k po ssible
cahon amg k
eY bít posiHons lo caohs
emy condiH on.:
kmugt sta s74 the 'n eq
Th e lo ca ion uay |2*> ntk+ a eo de wod
is assigned a decimal number, staHng to MsB
LSB.
kpay bheck re
on see red bits
pergrmed
each ode word, Each pay heek. tndudes one
e
paihy bits .
-Thegesu!t checks e corded
Qs eroY hs been- oeteahe d
beer deteehed;
- let the esults padhy he eks involvlng
the
pahy (2, eop,
G is ik detecd
O ik here is no
The deinal valve of bin ary ward kevm ed
gives the decimq1 value
enY.
similady C2, 3.el
, 2,. Ck
the
emoneous bit. 9k th ere
aslgned o caHo

euma'
w)v6e
alyeo.
 Thg de cim al umbey is he posiHsn lo ca hon
The paihy bits P, Pa an placed in lo oa ong"
to the
- valve (o oy ) ane
as to make the Hammih9
aasigned
Co de pamny bits s
Panhy 0V odd Dnd wh en an emoY Occuye
the
paihy
number w}) take on the va lue
posihon
lo caioh the bib.
0ssigned b the
9n case BeD ode with three pavihy bits tye
are seve n eY posiHons.
Eror posHoy

o (no em)

Ta ble gives tege eroY positHons & the

volu es osiHon numbey.
,3, 5, }
crmeponaing
2,3, C,9 they
4, 5, S,3 then
P is sel eered $o as
Theeg;
in posíhons 3, S,?
toestablish en (or oda) po
P,'is sel ecred --.. (oY odd )
is sel eehed
$, 5,6,?
rDeremin@ anmmhq codes
ati s kaoe betn the. Use evtn 4 .
Bco, code Hamming eo de

2, 3, (, ? Yegwres P, e | 0 1c
Cuoolt

Hamming ode ollo = (0oIl0)

P, h,
fo, 3. S,?

Minim um distan co

bhe
Hammlng coda seguene o olto is hansmHed
l due to e Y in ohe poslH on it is rceved s
oIo
tocateeg the po Sfhon he emY 6't uie paihy chece
Agiven he method koY Obtalnin 4 cont sey ence.
P n

,3,5 posho
2, 3,
 lo cath of eoY is 1n 3posha
receed
To comt the
eny b t
s
Ceepl@mentd. .! comeet messge
eiled.

3) receed as I 4 o l what is the comet da ta

1 1.
".
2 s,

!3,S?
P, =
2 3

po

4) veceved datais IOLO what is the hanml

data ?

n,
2 3

(olo) = oosihon (u
CometAato
S
9 Encode data fio0 in pauhy
3bits even poM Hammiing.

P,

2 3

I3

Pa

1
 Chapher 11(lart A) Boolean Algebra
Gasic Diqltal circuits]
9n the digjtal syshem there
ave
baaic apevations perkot med , yYeapeutve
complexities of the system. Th ese operaHons m
be yeqwred to be
peromed a number of hmes
ege digta sysem like dia)tal computer ov a
h a

jta conwol system etc.

The basic opesahons ave AND, oR, NoT 2
FLIP-fLOP.
OThe AND opesahon):
The AND' aperahon
th e o|p S% an AND
is
gate is t 'k & ohly ik a the
eghed
inputs ave t.
A B

VeA AND B AND

C... AND N
= A. RC..... N
SYMBOL
YsA-B.c...N
(Goolean Eq) or Y= AB (fo YNe2)
logical e)
Thth Table lag'cal Eqn
Tuth Table i
Any togical operaton can be dey'red in termg
ov fovms a table containlhg u possible input
Combin aHons (2 combinaion6
comespond ing outputs.
are
- Ginay vaiables also eemd to alo gical Vatable,
- The berm lrer m gate S used because the
ber the operaHon a digta
simlaihy
eg. For ah
Por AND opeohon the g ae opens (Ye t)
onty when al the lPs a ng preo ent 'e at lo q)c ( 1e
AlTh e oR peraHon - a s ' the
The OR opesa Hon is deyned o move
otput anoR g:ahe Is 1k
inpu ts qhe .
Y A OR B
Y= A + B
B
(Y equals A aR
logjc Eq
ymbol 1

Tuh Table

The NOT Opes ion Snvehey. 9t bas

9t is also kn o wn as

one /p (A) one olp Cy)

NOTA

= A
Yeruals NOT A

(Yegudo complemen
Truth table
lo 'calEq
Symbal
tnverey
bD ag an
The N 0T opesaHon is alsa reyered
ay comp leme ntaHon. bubble,

The pres erats.

denoeg
tnvession in digtal
always
 De

N R ooleen Coy logi

NAND Any AND R
can be
realized by
by using
eapression
move
NOT g ates three operah'o ns, two opesaH on
these
-Erom NANo
deved
bave bee
one tYpe Rghes eihe
Yeelzhon nyg
-.oniy NANDL NOR
are
Geeause n th is season
erpressian, Univers a Gare.
.9ahes <re kn oAb

NANb Operaton NOT A ND

operaha n is
- Th e

NAND operahoh.
g ahe 7o llo wed by
known aS
CN2) AND
fig shows an N ile
a NOT gate. destbed in the
is ekt can be
The operaion 9, th

ghe
B.. N
(A- B...
olp NOT g ae Y = Y'=

A
A

N AND symbol
NOTaperaion 4s NOT- AND 6peraho
fig
A B

C'Yeguals NoT(A AND 8) ")

Trut Table
 Desivahah
S
Ba3ic log2c
gaheo
opexatiang usiy owy NAND
A

Realizaton of N OT
qate uglu
A
NAND gare
Ye AB
RealizaHon o
ND 9are ualug NANDare
A

Ye AtB

Realizaion of OR
2ahe NAND gate
NOR opesaH on:
The N0T- OR
opesa hGh is kyown as
INOR operahon.
A

NOR Opevaion as
NOT OR Operaion
NORSymbol
Tuth Teble
olp of oR gate y'= At 8t. N
(Y euals NoT
O1P of N0t qahe Y =.Ý'= At6+.. N (A ORG))

A bubble gn the olp side ot NORgate repres ents NOT

operaH an.
 operaho
0Kealizaho ot basic log'e
Y A
A
erahon velHg wOR
op
Realizahon of NO7
Ye At8

of oR o peraHon uslkg
RealizahHon

YA8
A

slu AND A ate

operahon
Realizahan of AND

N
#|EXCLUSIVE- dR- dPERATIO GR
The EXCLUSI VE
used in dig ital ck ts.
widely
CEX-OR?opesaton is
-9t is not a basic aperahon e can be
untvers
perjomed
qaes AND, OR l NOT OV
using the be sic
NAND 0Y NOR.
:gates A Y= A X-0R 8

LogBe,egvahas
symbg
Truth fable
+ From the uth table it is obsesved th gt wh en boh
Hhe ipS ave same (o ort) t e olp is o, whereas
when th e i/ps are not same Cone ) them s 0 &
oth ev he is ) the op
is 1,
EX CLUSIV E NoR
oPeRATIO N
BY

log tc Eg2

Twth Fg ble
-9t is inversi oh
when both the Et-OR.
ps ave
d wh en bo th the
/ps are Mdt same th e o p is 'o

Postulates of a Mathemati eal sysrem

k m the bagic assum piong pom which it
iS possible to de du ee
the rules, theorems 4 propetes s
the sysem.
OAsSod aHve law

( co mmuta Hve ta -

3) Dishi buive lqw x-Cyt 2) y+2

Cover )
Cover ( )

Dualiy: Some theoemg can be derved m othey Th%

by 0 inhevch anging t <sigas & Dintrch angigo

ThesYems -Whieh e e lated his way
Duals.
 (oR law A+0 A (Snversian la
AND La w A,0 =0
A:|= A
A+A A
AAe A

8ooleRn Algebra
9n the middle 4 (9" cen ny, an
mameaiu'an George Goole deweloped
Eng lish veables, khowy
rules kY
Menipula hon & % binay
the bq 81s
GooleQn Algebva '. This 's
caleul atoys ete.
compues ,
d'gtal syems ke can be represen ted by 4 leltey
8 inavy verables The vayiables ean
an A, , X,,Y, .:
Symbo) such he twoposslble
values qt
have one k

Álgebsaic Theorems
6oolean
Theores
THm No.
A +o A
A-| A
.2
A+1=
|·3

A + A = A
A:A = A
A+A =|

1.8
A.(B+ ) = A8 +AC
A+ Bc: (A 8)(A t c)
At A8 = A

12 A (Atg) =A
I.13 A+ A8 = (A+B)
A (At3)= Ag
1.15 AB+ AB = A
CA+ 8). CA +3) =A
AB + e (A +) (A+ B).
I-18 (A+B) (A+ ) AC t t
AB + +B C= AB +Ac
 120
t. 2)
|22
(AtB)(
A.B.C. e)Bt c) (A+B) (Ã+c)

TheOYe ms
Atto Bto+,.
(.8
ach S he involve a single Vadable
theovems can be yroved by
eweny p05STble value
TM the vaiable. consida g
AtocA
hene pro ved,
Theonems 1.9 to l20 in
ean 6e
proved volve moYe
makihgBCe (A+
A+
a brut table.
B) (A+ ) can be
by makín the uth table pYoved
A C BC
AtBC A+B
(A+B) (A+c)

|lheorems 2 . 22 ane knOWn as DeMovgan's Theorems.

y'ng the
These Theovems can be proved by rst conside
Vaiables case then this. esut.
the
- bm the truth Table given in table l 8, We 9ee
selaions A+=
 EXamples oE
Trth Table to pove De-Mo vq ads Theovens
B NOR, N
circult
740 O
avalable ih
-9t has

arr ang d

Prove the logjcal Equahon ot aAND & NoR

law
) NAND OpeSahon :
3va qble,
vslng
ABC (A B).c
CA.B +c
usiy.Demorgets la
NOR operaion

A+BtC= (A#B) +C I

(A tB), c 74
74
A4 B+C

6et Boolean Algebya & ovdary Algebra 7

*Difteren e
7

eals with eal numbers (whicth cons 7

U ordin gry algebYa d elements) 7
an in kinle set deals wihy deghed Set o F2 elements 7
Booleen Algebra 4over is valid owy Y
of
i disbbuHve aw era.
but not foy age
osdinany muthplcaie
Boolean Algebra have adaltve or
not
0Boolean A1geba does subhaehon o divison operah
Ehere Re no algesre
Invere;
available m avdinay
plement operahvy is not
4Com
(ie x+a'
 ENamp) es of Ic Gatos
A) the ogeun eriong tke OR A ND NA ND
NOR NO T EXoR ah
com mesally avallable tn inheg rate d

740 O Chip is a quadruple 2 i/p NA ND gare

Qval able in pin D1p.
-9t has
4iden tica, inde penden t 2 ilp NA ND 8 re
arr ang ed as shown in g.
i-9t eqweD tsV dc
supply

-PIN cUr /B10 ck diagsam of Ic 74 o0 -

of the Available IC q areO
ICNO DlpHon
7400 Çuad 2 i/p NAND g ates
7402 Quad 2 i/e NoR ate

7404 Hax INvexter/NOT gate

7408 Quad 2 ilp AND 9 ate
7410 Tiple 3 ilp NAND
74| Triple 3 ilp AND gate
7420 Dul 4 ilp NAND 9ahe
7421 Dual 4 ilp AND g ahe
7427 Talple 3 i/p N OR gate
8 lp NAND
743O
7432 Guad 2 1/p OR 9ahe
7486. 7438C Quad ENOR 9ate
 74133 13 ip NAND 9ahe
74)3S quad tx OR (NORgare
742c0 Dual s ilp N6R 9are

Reduchon of Goolean Ex pressi dn :

A (2) A+A B = AtB
Oprove At AB
LHS LHS At B=
A¢A B
= A (I +8 =At AB+AB (: A At

A At B. 'RH,3.
(4) A+ B+ AB AtB
LHS (A +6) (A+ C) LHS A+A8+ AB
A At AC+ 3A+3 C A+ B (A+A)
A + AC + BAt8C
R,s

A +BC = ACD (G+8

=ACD

Prove (A+ B+ABD (A+ B)

LHS i- (At BtAB)(AtB)
< (At B+AB) (ÁAB+ Ãs)
- a+BtAB) (oto)
 BD
B8+
B.(B+e2-(3tDD Ye
BC=B B+ But
stmpliyfolla
wing
epress
ion
+A
D ABp Cot BCtA
AD= as(A
D. AD.AD BD+ AACo t BC+
A)tADJ [c,ct
Dt ADIAt6)t. t[B( =
Ctp.c AD+ (CAt AD) (8{AB+
GDt
CetDO D+ CAtA
Bb+A) ( AB+ BB+ =
LHS:
=B+Å+
AB
 yzt xgz+ xyityz
(+z =)

Simpliky wlny Gòoleen laws AG + AB

(becoi,
=A8.A. (A+B)
AB (A tG) (AA A)
A ABt A8B
(AÃ= 0, B½= o)

simpliy
(12) z=A+ B) (A+ G+0)D
-(+) (a5ts Dt oD)
= ABDt B tD
A(B+B)
BD

Dug

(a
A-B-c (A +B+c)
(c) A fAgc). AGc = A+B + (A-B-8)

A +8+c
 (2) ( A B+AG )ABC)
hed
ABTABC)
A BGC

) y= A
(Gtc) (AB+AC)
A

CcÃ+B).(6+))
AB+AC
AG.Ac
A+6) (â+¬)

AG

y= cC ABC + ABc)
AB+ A (8+c)
AB+ AB
A(G+ 3 E ) E )
ACG+) A(st6)(et?)
=a()(6tT))

C,AB
O show that
(A+8) 6t c) (¢ tA) = Bc+ AC+AB
(AB+B+ ACt Be](tA)
AG+6+AC+ BC]¢tA)
-fettat)3ttt)
- (8+ GctAC) (e +A)
=(6 (I++AC)CctA).
B(+ AcCtAi3t AAC.

-Bt ABt AC.

AB0t ABD +85 : AB

fB6+AD
=iABD+ D(Ag+)
 Switeh inq
The use
Vaiables d
mhe epptica Hon n blme are dem onshha hed
the simple switching ekt t'9. belon,

source

S6ura
a Swi tch eg t se'es 6witch es io parawel
log ic ANO
logic OR,

E1ecmnic diqital te ts are somemes caw ed swlFchi n

becanS e hey behove ke a switch, wlh the
aetive lement tyans is y eshhey condu
Cswith clos ed) oY not con dueinq lS witeh
"pen)
eing
Ex
ExprerS the oltow'ng Swithing ckt th big oy log ic
notaho

30urce

|L= CA+6).c|
 NAND
NOR eNOR UsinY gate
th Realizaian of ExOR

OwoR using NANb

A

A Y=

Y= À+ - eghot NORgahe
=A+B (De-Mo vq ass Th
take doub le inversian RHS

e*-oR 4sing NAND

Ex oR .Y= AG= ÀB+AB
Take double tnverin ot RHS
Ye AB + A B

Y=(e). (oB)

-Y- (AB.(a)
Yz A

-
ing NA
8Ex NOR using ND
NAND|

tak'ng double tnversi on ofRH 3

( (AB)
 t

AG

Ex NoR
D
usenq NAND gahe
NoR EOR
vAND

60olean er pressi on or
NANO
ta ke double invessi
RHS
this is the reuired
eapnasioh
A

_Y =

ExOR uSinq NoR gare

&oolean erpression oY EXOR gahe is Y= A8= ABt AB

take double invevsion 6 RHS,

(DeMovgads T)
-(
Y= (A +3).( At6)
=(A +)+ (Ãt8) take doublo lnverslo ot RHs we set
(A t2)+ A+) This is the xoa uivod odneian
 ( 6 + 6(
+ Ã+0)

(A+8)

A (A+O)+
(ExNOR ugina Nok gates

804lean ex pression EX NOR gae

ta ke double inve xsloy ot RHS to jet,
ABt A8
( .(nB) -be-Moxqans
(A +). (Ã+6)

take double inverslon ol RMs to g

This is the egured Cmprersi:
A
A

(A+D)
Ex- N0R
0R Melng NoR
N gares
 SA hn o Exprusion Gater:
mplementation using
Reduce
lowing eapnasion, inog lemest it uging baaic bgjc
the eliaing o
b Eh en implement E using only NAND gare.
Y= (AG+ A+6) A8

FoR NAND 94re

Rea zqion i
AAB + BABt
AABB epla ce inverter & AND
gate by its NAND egui vq
1eut.
Of A B+0

A AB

-D
Realizaton uginq
uslnq basic gakey

) Ye ABC+ BD+ simplihy the eg c realize usig banic

qare d ueing only woR gate. NAND Regli zari on
Ye ABC+ 8CD+_ Y= B(c+0)
= BC (A+ A) + BCD double inversion
8Ct BCD (as AtA=)
YcB ++0 -DeMor
A+AB eAtB

Y= 6le+D) Y=

Realiza Han usiuBasic gahe Real'zanen usny nNok

 Ex3 &implement usln 9 NAND A
Reduee the expression A+ A B +AB

al",.
(
- A+G
= A+B ()

foy NAND Reali zahgo

peMergoist

(R9) Reelize the erpren sion Ye AB+ t0 by NAND 9 e ony

let AB = M, Ye Mt

Takinq double inVe ssion ot both sid ea Y= MEN A

Ye (AB).(o)

(Er S Real'ze the erprnsion Y=(ABC +8)c using ny NoR

(': cc=C, Tc= 0).

usthy bemovg ads Th, A

8
-
Real'ze the
Kollo aing
enprnsion INVETER
then
sealize u'
it by ueing ng AOT ie AND oR,
Ye (AB + C). D owy NaR 9ahe,
yealizahon usinq A01

A8+C
(A6+c) D

sealiza hon usi4a NOR qate i-

k AB =AB = A+B -bemovger's T"

1et u put

agn sub shhuHhg valueof

Y= (A +¼+c) tb

Msiny NOR gate

 CH-)
COMBINATIONAL LO GIC

tiagrammo way
* Logic Diagram- this is a i/p-
showing th e olp zelahonghip.

Switch ing tquahons

The se laion beth ip & olps Can be
in the
toYm 4 eguaions cawed
suitching eg's. shouh
-The ilp vaables ane caled as swltching
a re algo cal ed as
vaviable,
8ooleen equahon 6.
s witching eg's
9t is algo caW ed as Syshm eg uaions.
The syshem eg's av sultching eg?s can be two difereg
types.
0 sUM oF PRODUCT (SO P) komat
PRODU CT oF sUM (POS) koy mat

Typeo a bigital Sysrem s)

O combin ehonal logic ck t
[eguen hal logic ckt

Combina Honal loqic druit|:

The olp combinahonal ckt
at awy instant time, depends anly ah the lewels
present at llp tesmìnats.
Combinhonal cde t do nat use any memsy. so h e
preious stare i/p does nat have
prsent stahe 6 the ckt.
De?:- t is logic ckt in which ol p
n the combinahon
depends
the ilps.
- The olp does not depend on past value ps oy op
hen ee donot Yeqwre any mem ny.
 A
Combi naHona
An ckt

A
combinaHonae1 ckt have
olps .

2erPs pS 1o9c qares are conn oste d go combinahgn

ckt basically kongists 9logic guea,
Adders, sub bracors,
en co des, deeo ders ompavatu,
MulHplex
Code covertevp,
es, demuliplex esg.
sOP t POS Kepresentahon

SOP ( Sum ot Pyodut) FORM t-

ASsu me logic epressioh Ye Ac+Bct Ae
-A 3,C a re caled as ibesalg, 0V LT Tproduet te ms
ilp S of combinaional ckt.
The erprsl on is ln the 7ovm s sum three teyms AB,
AC& AC wih eaeh individ ual evm is produet two
vaable say ABoY AC etci
pYoducts in the sop torm are not Be ahes
ad dih ong oY mulipliaH on .3h 7au they
POS (Product ot Sum) poRM :
produ ts
ASsu me
logic expnsien Ye (A+e'otc)at)6M herns
-The t#hesels are oRed toqethey to 7oym the sum byms &te
Suy terms are AND ed to get th e eoprersion t'n Pos
 Canoni cal SOp & POS
The word ean onical is
desbe cond)hon subching equaHon, uged b
lhe meqning o
-This sWe canonil is con bo a mng
stahes that eg eh tem us ed in a
gen eNal W
s
must contain al! the evailable /ps
Vavablea. witchlnq eg's
Sum tor mat
2. produt
sumi (PO6) 7o mab
Sim pliky-8ooleen egs, sometmes an ip
is
ae not
eminated to

sìmpled.
slmpliy the eg, But canonical
9t is
vaiable'r
it Con tains
redun dqn s
apposite sìmpligeahon. so
Many tima, swi tehing e's w then in s0P or Dos rm
ar n ot
cangnca. That means eb term may not contajn
l the ilp vaiab e,

A loqic ecpession is sadt be io the stan dard ov

1 ca nonical sop oY POS- orm} eas pvo
dvit teym tor So)
(Y P0s) consisks a Whexalg i eiy
Com pl em ented ov uncsmplem ented lrm.
Conversion om sOP to $0p (standavd kom -
she p O': for eech tberm nd the
shep : The AND eIm with the hem
missing iteval
the missing ihesal & s
kollowed by Okg
compl ernenb
Ex) convert Y ABtAC+ BC igh
4o]P:- shep nd the
standard POs ksm,
missing tihevat gov
Y AB+ AC + 8C
A
misys
 : ANO eth rm wlh (Missing
omplemen t) (enesliexols t
AB.(C +E) + AT(B+a) +8c(AtÁ)
-AB C+ ABtt+ ABT LA674 Agct ÁB C

But A+A =A

<- stan dard soP KOm.

version pom Pos to stand avd pog smit

(step 0 Foy ea ch berm knd the mis sihg itexal
: Then oR eech te Ym with the te rm tovm eo by ANDing
the missin ti heral the hevm with Sts complement.
jorsssion to a et standard POS.
Simpligy the

) convert the nprsiGh Y= CA+8)(A+c) lB+) int standerd

pos
shep Rnd the, missinq.iheval Y e eh herg

shep : OR ea ch term ih missing ie sal. 9t5 complem enb)

Ye (a tG+c c) (Atc+GG)+.(B +8+A
But A +6C= (AtS) ( A+c
Y= (A+ B+c) (At 6+ T(At CtB (A+C+G+( AtB+¿)(Áts+c)
A- A=
(AtB+ OCA#6+)e (at6+)
PoS 6om.
 Converb the koltowing erprersions hto hls standake
O Y= AB + AC+BC

= AB(c+ ) + Ac (B+5) + Bc(At.

A3C+ AB7+ ACB + A B+ ABCt ABC

ABC+ ACB + A BC ABC (': At AcA)

Yz (A+6) (B+ c)
z(A¢B+ ce) (8tc A

POs

Y A+BCt ABC
ACB+ BCC+) +Bc(At ) f ABC
= ABC+ ABt ABC+ ABCt B A+ GCA+
ABc

AB t AB + ABCt ABt BCA -3oP

Minbesm - Ea ch indivldual term in the

stand ard sop yYy
is caled as tMinterm.

standavd sup

Maxteing - Ea th in divldual term in the sta

ydavd PO6 komy
is cawed Ma
teym,
 s tand erd Po 5

n' number
FoY
vamablea
wiM be 2".
the number 9 mintevms

vaiable s
Minhe vms Mahms
AJ B.C
mi MI
A+B+C = Mo
ABC -M, Ä+64. =M,
m2
ABC m
1
my
ABC
1 A BC z
mL B C= M
A++= M.

Re presentaion k logical enprrssi ons usin Minteyms &

Y ABC, +t ÀBc+AB Cc

.:Ve m+ m3t my Y= Em (3,4,

Whee E= Sum of products

Ye (A +B +c) (A+8+ c) (A+B+ c

M2 Mo

Ye M2 Mo Mc

|Ye. TT. M(o13 )

 Simplicamon Techniaues 8ooleen

O Algevaic simpliaHen
pliyahen ving kevnough maps
9wne- M¢-dvskey Memod
OAlqekxoic sim pliarion -.
the given expressien in to soP omy vein
SOP

he Booleag 1ews s DesMerqans Th by

impliky mis sop by checkin9 the pro dvct bermt
6mm on

t* Ye AB+ (A+ 8) (Ã +6) simpliry

3ol2:- Y AB+ (AÂ+ AB+ AB+ BB>
AB + A AG+ + BB
= AB+ AG+ BB + AÃ + 9B
= AB + B t AB (:ABtAB AG

) Ye Eml4,) simpliy
Ye mgt my t m
= AB7t ABCt A BC

ABC + AC ( 8+8 =)
t3c)e (a+B) (A+)
7(At (At B) : ( A+ B) (A tÃ)(A+ B)

y A+ B)
 TMO3, s) simpliy

,M3. My

(A+

But AA=0,
(AtG+C(A

(AB+ ¿(A+ +
(A + +)
(AtG+

(A+B+O(A B+. 8+E)

CA+Bt c) ( AB+ + )
A AB+ AAB t ACt
BB+ ÀB. G+ 6t+ A88+ ag ¬+ 27
ABB+
but AAB AR, A,AG 0,
ABGz AB

ACtAB+8t

(A+8+

)envert th e qiven enpronslon tn to minherms &simpKY toget

Same ans
 be sop .
vwaiomship
n'anesteny
g

(A+Ael)

tt
(c+z)

e KarnaK¡h Mep[impli7cahan
K-Map
approech to simplica hon 60oleen eaprosion.
- Sn this technique, the in ormahon cmtoin ed in abut
table oY aValoble Ih PoS Ov SoP is epennd
K- Map.
This rechnique is 9eneray wsed up o six vaiable
g@shows the K-MaR two, three
-9n an n-vamable k-map there are 2
-Eac cel eomeop ons to one the combinahons s
vamable, sin ce there ane 2"
combinahons
For ea ch tow 9 the mth table, or each minterm ¢
o ea ch maxteym th ee is one speHe cel tb he
K-Map
Gray cod e hgs been wwed kY
the
the ideagano cltn.
 two vaiable k.Map 3- vayiable kmap

4aable kMap
AB
A+B
AB
A+B
Minrerm
to 2 eonmesþon din g
Vamable kM ap
Maxexm

ABC
ABC ABC
Minteym comeopondig9
to,3 vanable kmap

A +B+C A+8+c +B+C

Maxhorm
A+ B+@ A++ 5+8+

Co
ABTDABB ooAt8tC+D|A+ãt C+ol A+8+ c+ Ato++0
ED ABCD ABTD al A+bFeD A+6+c +o A+8+c
At9+c+O
ABCD ABCD

im comopon ding to 4yaiable MAteYm

 on kMap
Trth able
Repeoen taion 6
vamable to You5
Suppose Yis tcomopending
olp y is |

SOp
Ye c+ A

Comesponding to.rows
suppose olp y is O

JFoy 4vaiables)i
o, $,?,9, l2, 14, 1S
AB
co

|S
13

+ Abco
= Em(0, 5, ,9, 12, 14, 15)
 sim plicahon logical unuions ugi k-Map

r t is based on he prindple ), terms in

adia cent cels, Comnng
-Twe cels ave saia to be aiaent ik th ey oE5
{ veiable.

neprotns simplieaion involves qpng <

mlntems .
idenhhyin 9 prime-implicants (P1)
e9s en ial pme -
impticonts (EPI)
-A pme lmpli cant is
90 up 9 mlnteyms that (annot
a
be conbined wit dw aher minrm or
gps.
- An es ential prime-implicent ig
prime implicAnt is
which dne oY more minhesms are uhique,
containhs at lest one minhem whi ch is not cntained
In emy oho pYime (mpti cant.

t Grouping two ady acen t onea -

91 the re are two adjacert
ones on the map, these can be g p e d togeey &
the esu tHng beym will heve one lens theral h an he

|odginat hwo herms,

Groupin g touY Adyacent one - YeuIts in teym with

2 Iie ralg l eos han nu
Groupinq 8 Ady acent ones ot va iableg
y xesu lY ln a hesm wih 3 Uhesalg less th n
number k vanabl,
 Minimlzahon 9k
the
enpression is simpic
cah mof be kher simbd
a stage beyond whieh itnumb er S 9aher wth

number
egure minimum
i/ps to be gat, sueh an nprnsion mingimu
npwsion,
b as the minimized
uy minimizlng a given onprnsion in soP
he kmen
9iven trut table, we have to prepae
kSF then look ksv combin ahion 6 me kmap
combine .the on en in SUch a
-We have t that
The veA Uin g enprnsion is minimum
- To ollowin9 agor) Can b wed
aehlee this, the
L
the ones which can not be eombinRd

pime implicants.
9dentiY the oneg that can be eomined in
grsups
Ss two in ohly One Encivde suei gYo ups k nes.
gden hty the ones th qt cah be combìned with three
o her Ohe
to make a group tour adj a cent oheg
Jn ohe way. enurle su ch gvoups onea,
4) 9den hky the ones that Can be com bin ed with seen
other gnes, to make a
gYoup 8 adyacent onea ih
Enirde such groups ohea,
Ager idenhhying the essenial gp % 2,4 a 8 on ea 'b
there sHI vemains SOme oh es which
have not been
en urd ed then these are tD be
comb in ed with eaeh oher
already encireded ane.
-The logic ncion consisHng the ensenhal pvime impliog
abtained in sheps I to 4 & the pime implicgns obBeinedio
shep S will be the minimi zed
kunetion,
 k-Map
two vanable
Vaiables in
Nying
Quad
Vaviab le
Paiv

in th ree vaviable k- Mapi

*YOpin q vamables

10 A

A8Co0

GrOupinq vata bles in uY vaiable k- Map

lo
 stinimìza on oF 2vaes)e logic fun ti on uslny k Map:-
- (A, S O:E (o,i, 2, 3,
5) Ex2E (A.8.c)= Eml, 2, 3,?)

r E(A8,c)= m(0,,3,5,(,) Ex 4) F (A,6,c) =Em (, 3,5,C)

AG

10
2

(6C + 6c) +A (6 +6 c
F(A, 8,c)= E m(o, l,4 3, 4 )
A

3
Éx) fA,6 , C)= Sm(o,i, 3, S,6, 7)
AS

(Ga
 logic uno9 4 k. Map
*;MinimtzaHon o 6r vanable
5, 7, 8,9,1", 4)
Sm (o,, 9,3,

ot

ABCD
GB+CD+ A D+ eD+
|+(A,G, G, 0)

5,13, 1s)
Ex2) Y= Em (0., 2,

A 3 2

E) Ye E mll, S1, 9,1, 13, 15) obtain the simpl'ed lgical

sealize th at b aaic
edprsion using aten,

(ate+c)
4

|o8
(4f8+T)D

|Y D(A+ 6+)
 Em (l, 3, 4, 5,7,
3,1, 13,s)

oo
C

ErS) E
m(l,2.9, o,, 4,1S)
A
O0

12 13

Bbo B(OD)
= Ac t D+ cD

EY 6) = E m(0,2, S,G, ?, 8,o,13,1S)

AB
 Miaimize the legic unu fea, or (D2m(o,, 2,3, s
Pospom
|,14)
F (A,, c. o) =TM[4, qio, l2, 13,15)

0) -

Y= (A +8 +D).(A+ B4c). (A+8+ D)(

+C+0)
Ex3) Minìmize the logic yunerion Msih Meateims.
Y:n M(0,4, S, 7, 10,1, 14,15)
olol

oolo

Y= (A+ e+ D)- (A+6 +5).( À+)

 B).(E+6) ( Y=
Pos
enpresi
on minimied
+D A+B
,p)= +(A,B, +
(A +D) +B
+ (A
+70. +9 (ã 6), +c+
con logi'c vaiable
un fouY the im)ze Min x2)
AT+ B+ Y=
12,13)
8.9,10,,
2,3,7, (0,1, DDEEm B
sA
ABO0
0
(D BÀ ABCD
unthon log' variaslo ouy thMinimire
e
tns/ rminto yod sped
ty not hions loqicol
un almizahon
of
Dant ca Con dihon s-i which do hat
The ceus
are aSSumed vevsa,
sinco here are can es whles

not. alwya ue vamables do

ilp
er tain combinaions bb
-A1s0, or soma uncion s vadables do n ot
Certa'n combinaions 6
i/p
has a
aeolgner
moher
-9n such siuaions the to ass ume a 0
to him WhetheY
it is lest. these comb
inaHons, This ondn ie
oY each 2can be eprnted,
l bou't care condihon
kn as cel,
coespohding The X ma
xMak ln the
K- Map as
be assumed dep ending ugon
may simplev enprension.
Which one lads to a dont cave cond,
mintetms
()The teYms k
15) + d(0,2,5)
f(AB,c, D) = Em(i, 3, 7, 11,
9ts K-Map Athe mintmized eaprsio gn gien in
fig
2
X

( 9n herms maxterm $ dont cane codidohg. E4

fCA,B, Co) = TT m(4,5,6, 7, 8,12), d (L2,3,9, , 14) -Pos
9ts k-Map &the minimized erpens0on ave

O6
D
13

X
 rerms 0 bruth table. eq, com eldey th e tnuth table:
9|ps olp
B C
A

10
4

12
13

12,13,142+ d (o2,s)
7T M (4, C,
8,9, lo,

(
Y- (õ+D). (
 5-VaYioble K-Map
23 22
2)
1
3) B0
o) |24
28 27 24
C
25
24
|2
13

20, 24, 2s,26,

n 11,16,
Ex Sim pliy the logie erpre s 8,9,(o,
Em (0, S,
c,
F(AB,c,o, E): 29,29,3 I)

18
BC DE
Bc
22
23
2!
30

40)
)3
28
27

ABE

2?)
F(A,B,c,D,E) - Sm (oi, 7,9,, 13,15,le,?, 23, 2 5,
Ex29 simpl
|2
oo1
22

28 29 3) 3

24
252?| 2(

F
 F(A,B, C,D,E) Em (o, 2,S, 7, 13, 1S, 18, 20, 2, 23, 258, 29, 3D
JA=0|

00

25 2.

G-Vanable K-Map

1.0
33
ool 39
39

13
|44
1)
-
lo/40

ico\oo ol 51
Oo/ic
S2 S3 SS
of B 6)
61,
|27
J24
 Desiq Eramples

A yithahc Ycuts t

ddIhon two dne-tip

OHal-Adder the
A addey.
eh hay r the olo.
numbers is veyerned t as 4 s(suM) camy CC)
nputs
A&8 are hwo for
S)
o
A
An
A S

s ÄB +AB
2

Truh sa ble

addev
ealizaHon af half

Futl Adder there is no

perormed,
When mu 1Hbit addhon is o dev bis in case
lower
prevision to add e

adder.
ip termìnol is add ed & this ckt
for this purpose, a third
is used add An Bnd chol
humbers A &8 op.
where An Bn = n o1der blts h-1)ord4
dihon
Cn Camy gene s a d trm the ad
bits.

vegmea tb as wl addev
This ckt is Yepmed
 foy Sug

9/pS
2
An Bn Sn Oo
o
0

ol ps.
trth tabie
An B

An
Bn

Bn
Cnl

NAND- NAND Reaizahan

NAND-NAND Reazahon of Sn

Hal Subbractos
A logic ckt the subhaion 4 8 (sub ba hend)
Ydi frr A(minuend where A eB ahe l-bit numbers is 8efered
hal-svbhrachry.
c(bono w) oe the two i/Ps.
 AO
A
Fu
D(bi) c chogo)
A

buth-tgble
A

Roatizaion of holF subhay

(4)Fuw Subhar:
Jugt like a
guwse 'a
ult adde, we
ming mulHbib subhachon wb
çubhe ctoy ck t oy pevo
bìt posiion alss be there.
a beON Bom the preulouo An (minuen d), Bnf&t,
have 3)ps
-A RWll subta ttoy Il) two lp
the presiou s shage)
hend) b Ch-i ( bonow om

dlp - The kemap k op Di is

same as the k -map
An ßn Co Dn
S the adder ckt

2 The kemap Ko Co is
 Adder
S
A HAI A S
3 HA 2

cin

SUM

D
futl addev two hale adders
Proof
same 93
obrained for FA
C= (A6) in t AB

= Bin t AB+ A in t ( + )
Co= & Gnt A + AGn, -proVed
 chrl
subha
hals
u8/mg
Subhachry A 0i,
Futl
H.S_.)
A 8
( G)8), pìteren
subhoe
futl subha doy

BI Same as
Pvoof: eg)Bin = A 3 expesSioh fo
DE (A olpof F-s.

+
= (As tAB) ·Bjn

(A+A3 + A 8
= AB8in + B 8in

pnved
 6egment beoder
- A drgital display that consists t
is mormall y used to delmal
LED
segmen ts icplay
rals ib digital 8yshm. Most pam1l)ler emamples are
econl
c cal aulahoys d watcses where one- 7-
*gment displag
One numwal o th vouyh 3.
to be
8i
ie fo usiny s aisplouy (data) device, the data has
cenverhed some Gde to the co de yeq wired k
binavy
dlsploy
cede uged ig nayval B cD.
Sout, Usually . the oinay
Aa a) 6hows the disploy deyice
6) shows the "segments which must be illy minahed or
eA ch Syifhe numeYals 4 fi'g ) gives the display sys em

eas
sion f
f-s, el lc
ot umexals
a)
1se, 'splay
6) display
A(MSB)
B CD to
BCD Sewen
mentb eeo
dev

) dlsplay syahey
ABCD are 4ip S & it is the náhsal Bo code
numerals 0 Hrough 9.
-k-Maps or ea ch ot the olps through 9 are geo
below.
- The enbies io the K-Map to six binary
combin ahons hot used tn
compohdiMg
he truth ta ble are x-dant care.
 stgment Decode
Tnts To ble of ptD to 7
olp s
b C d
BC
1

K-Maps o F Above 19 ble

foya foYb

A8 L0 A8
1 3/ 1

X 14
91 To

a - A + BD4 BD t CO
forc
fod
ol

6) 1
X X
 forf

AB 11
A

11 X
X

Ac

X
1 X

Reaizaioh: usivg NAND :

D

ce)

(a)

A Cf)

<g)
 cOnverey):-
Code
Binay to
aray
Decina NO Ginany 43

7 12

8
J

14

|2
1

2 2
2

13 2
1

10

o)
G,Bo

6
0)
317

.
B. Bo
 ay code
P olp
B)

D Go

Binany tO- Gay eo de convener

Gvcy- to- Binay code converter

(6, ,5,4 to,!, 9, 8)
Ba

G,Go
2 3 2

06
s

E,°, , )51

B1(23, 9, S, 1s,1s,3,9 GoCl; 2, 4 13, 19,1, 8)

11 11
(
04
13
12

Bo 4 Q44,çot',64,40
 B3

9,

4Tay, t BinanyCo de convenrer

* u n e MeCluskeu MinimizaHs n Teehnique |

Tabwav menod
Modern digital syshems designed usihg pLDs, FPGA.
druts which can Ge
& otheY veny lovge scaJe integsated
Con'gured by the end useY
Th es e dewices highy complex 2 therege the
techniques reg uired tor denigning. digital systems usi vg
hee desice have to be computer dnven Yaher than
manua,
A log tc minimizahon he chnique which
chayaue isHea is herejove requ'red,
O 9 showd in ave he capabiliky handling lavg e
numbeN 6k vaviabl eo,
2) 9t show d not depend o the abiliy
7rr ve cognlsing ptme -implicauts.
9t houd ensuve minimlzed exprensio,
9t shawd be swtable or compute ysoluHoh.

The une -Me Cluskey minimizahan technique saHspe

above vegu'rements hence can be wwed
 design loglc ckts.
the

Jla0dyautog es of k.Map tochique vgìgbla

Mare Han 6
minimiza hon %logic unchian s ihvolving
is unwidely.
Recogwihon pime- im plicants that may govm pevt lt
the humon
the simpll ed Yeles an the abiliy k
whethey the best selecHah
Making E d l y t tu sure
has beeh made.

two pats -
The wne MeCluekey mehod con sishs k

all the prime

O To nd by an xchave exhaushve seerch
implican ts that May jom pavt gthe simplied
kuehon.
prme innplicants obtained pavt
O To idenHy wsenial
amona the yem alnin q prime i plicants th 05e tha t
Choose
gve an expressioh wlhh the teast
numbY S Whexals.
 tne Qune. MeClutey
ysing
Ex1
Pky the togie uni on
mìnimiza ion teethnique. , Is)
7, 8, 9,
Sm (0, I,3,

unhan in
SRpl aneuge au me
minrevm
in
s
table
he
accordlng to t%e numb sinany
Yepoente ion oY m groupS contani Mg
d gorm bh e
Ohes con tain ed
hozon tal lin es
by
The gro ups ore separahd
to he n4mber
Table Gvoupin a minheym aordn g

Yaniables matched pair

Group Minterm
A 3

fable combi nahon of miuterm 96 S two

0,1
matth ed par
of minhems
O8 in the adjaunt
9P (mek
on e
maehedpar

J)|
3,|
rable combinahon n mìnterm gjp , u

Gvoup Min heY ms Vaiables

A

O,, 8, 9
O,8,|,9
, 3,9,|
PIs
,9,3,l|
2 3,7, , IS
3,!, 7, lS
rom tnis tableb'
YEA,,C,D) BTt BD +CD

PI Table
Deimal Nm ers MiHhevm s
3 9

Or1,8,
X
BD
X
3,?,||, 1S
ep1) od 8 mìnhery
B co are egsenial prme iapl'cons.
are con tained One PI Bd minherm 7 &s are contgined

- minhetm |d 9 are contained i'b &T 3 4I| con tained th co.

all he mintevms of 6nginal uneion are included in wo EPI.
s Minimiaed exponsion
YCAB, C,p) B +Cp
we cen veusy this wny k_Map te chnique
 8,9,",1S)
Y(A.B,c,o) e Sm (0,1, 3, ,
rehnig
ue
AB 01

10

E2) Simpliy he logic prwon uslhg Qune Me Cukoy mehod,

+(AG, C, D) = 8,3,1, 14)
s m(0,2,3, S, 7,

number of I's
Tasl a®of 9roups of miutems accor ding to

Vaiables
Group Minhetm
A

14

Nert tabl e 1s ceahd by comparn g the

minterms i'n odyoent goups metbing
Jn 9able , al the minteYmg r est m,, are ch ecked.
Table (6
Gvoup MineYms Vayiableo
A

O,)
O, 2
O, 8
!,3

, S

2, 3
8,9
3,7
3,0
S,7
9,0

Tabl e is cveaed by compering & matehihg

V)
minhesm s 1 ta ble b) a ve che cked (

Minhe Vanables
Group A

0, ;2, 3
O, , 8,9
0,2, !,3

I,3, S, 7 AD
I,3, 9,1
t,5, 3,7

Table
 . omatrhinq posstble in table 9.
The Mnhion f consists q, the prime- implicayt (m,)
Cunehe tk od minem th table )) the te vm
tb he pne mplican ts table O. 9t is given

The pI tasle comnodiny to above Cj? T6 prep9vod a

hert table :
PI kms Decimal Numbey Miuteyms
789 )

o,l,2, 3 X X

O, 8, l,9
AD I,3, S, 3
I,9,3,)| X x

observe the PI table c hd the ealumns con tainng oly

slhgle eoss ( x),
There ae. six minhesm whose cotumne have
CNOSS ! 2,6,7, 8,11 d I4, singa
Pime- implicants that cover min hems wlhh a
in thelr column ane ersen hal pme-ip
single cnss
liants. The
CNOSSe are en ëd ed, Check mavks qve pleed
below these
siagie
column. A eheek mqrk ls als o pla ed in
the ta ble on
the nsen thia primeimplicon ts.
Since al the prime. im plicants are nselohal p1 A EaO Tabl
is the minimi? ed
Veifieto) neion, AB
 Mehod,
simpliy the logic funoion using Quine- Mecluskey
t(A,B,C,D = Em(l,3,7, ",1s) + dta, 2, S)

MIu eym/ venable

Gvoup
j e

Dont ca ne teYmt A

7
14.

, 3

23

3, U (

3 7, 1S

Table 2,3

1, 3, S
, 5, 3,
2 3,7, , 1S
3,1l, ?1s
 Next Table ive
Pfable
t
belmal Numbevs
3
(Minteyms)

X
, 2,5"71

1,
Ex4 f(A, G,C,0) 0,, s, l0,12, 13,14,'S

| 100
Group Minheim
vanable.
32 3

12

|3

Dable
,3 À8D

3,|
S, 3

l2, 14
12,13
 I3, 1S

A B C b
10,1, 14, 1S AC

t2, 4, I31S AB
12, 13 , 14,S
I0,14, ,1s

PI Tevm MIeym S
Decima NO. -0,
3

10, )4, ,1s

I2, 14,13,1S
X
I, S,
3,|
5,13

Z m(2 3, <,?, 8, 9,3 ,)s) t d (4, lo ,12)

fCA,B,C, b)=En Cl,2, 4, 8.10,1, 12,13, 1s)

(Em FCA,D,c,D) = E m(2,9,0,, 13,15)

(62 8) (A g, C,o)= Em(i,3,

9,l0,, 13,19
|HAB, a0= Act 3Df cotAD
 COde Converter contne
demad)to Erers-3 (*s-3) -
3CD(inay coded

Deima
Ea En E, Eo from(i0e to
1
will bex'
dot are
2 eonds.
3
1

7
8

10
(6

tX

Eg 6,,60 6,B6t 6,8,

= G5, 8t 6,(Got8)

0)
11 X

Eo = Bo
 Eo

be 'x')
t cene

BCD--t- xs-3 co de converteY

),14!s
2).d ( o,
Io

9/ps
Eg E2 E, E 83 2 B 86
3
|0

6,6,

00 X

62 (8,9,10)
81*
6 ,2
X 00 K

X
 Bo
B

X$-3 to B cD Code Con verhey

BOto
GTay
Valld Bco ls x'd oH t
Ca re

7
8
 8o, ol

Ga Da

D Pe

Go =0 Do t D, Do

D, D,

P2 Ga
)

Code onwehey
8 CD to Gray
Deuign com binahonal ck t Whose i/p is a 4 bit numbev d
olp ls he two's omplem en t P number.
 bit Numbel to 2'6 complement of mat ilp numbt,

o|p
9| p
Dlma A3 An A Ao

|2
13
0

00 B2

o)471)7
2

t Ag,fiAo
3

S
2
|2

8
 ¤1 ( Ao t A) + A, Âo A2
81 = A, Ao
Bo Ao
A Ao

AotA

D
(AotA)A
B

A, Ao

Bo
Number
Number to 2's complement of that ilp
4 bit

ckt which will eceept 4 bit

12 convert bìn avy to oco. Design
& will pyo vide s bit BCD code.
binag
63828, Bo Dy DgD, Do
Decimal| B B 8Bo

I2

4 13

1S
7
+
8
 |0

00. o|

00

10

Da 82 + Þ,6)

Do

Do =Bo Bo

D4

D|
 simpli y the loge kuneion using Quine Me cusky Meho d
sIA,8,c,D) = Em(2 4, 811, 1s)t dl, lo, 12, 3)
Table!
MinheYms Vauableo
ABC B

1
2

|2

13

Vanable
Gyoup mìnteYm s B

(2, to

(810) ABD

(1 2,13
(I,1s)

2,
4, 12
8,10
812 coveihy ( . 9

A |2,13
 163 t02,
S2,0
|02 8) 3
06 8, 2 3
S2 20, Table
tedueion
8 20;2 2
ms
Minte Group
-arieble
|29
lo S
|02
60
39
S2
38
28
20 Table
ehim <edu
A
31
Minrerm3 roup
Vaiable 46
TAB,C,D,E)=
(2Im
2) Gor
l02, 89,52, 28,3 m SOP
cluskey0,
dereomine
the Method
Quine-
Me
mm
per
 MinheYMs
20,28, G2, 6o
PI Table
able

PÍs Minhe Yms M IneNmg

20 28|3839 s2 co o2o
39
AGCDEFG I29

38,102
t02, 103
2012 8-5s2 60