International Conference on Engineering Humanities & Science (ICEHS-2017)
Seventh Sense Research Group www.internationaljournalssrg.org Page 134
Implementation Of Full Adder Using Mux By
Applying Shannon Expansion Theorem Ms.Shankha Mitra Sunani
Assistant Professor In The Department Of Electronics And Telecommunication Engineering
Government Engineering College Pmec Berhampur
ABSTRACT:
Shannon expansion theorem can be applied in
terms of more than one variable. For designing
any function using MUX we can use Shannon
expansion theorem for any variable function. If
the Shannon expansion is done on one variable,
two variable, three variable then the resulting
expression could be implement using 2:1
mUX,4:1 MUX,8:1 MUX and so on. Depending
upon the select line we choose the variable for
expanding Shannon expansion theorem but in
this paper I will discuss how we design a full
adder using multiplexer (MUX) with the help of
Shannon expansion theorem. Here I have given
more stress on Shannon expansion theorem.how
we will apply shanon expansion theorem for any
function.In this paper I have discussed design of
Full Adder using 2:1 MUX,4:1 MUX and 8:1
MUX with the help of Shannon expansion
theorem.if Shannon expansion function is done
in terms of all n variable then that particular
result is the canonical sum of product(SOP).
Keywords:Shannon Expansion Theorem,Full
Adder,Multiplexer,2:1 MUX,4:1MUX,8:1
MUX.
1.Introduction :
Full adder is a combinational logic circuit that
performs addition of three binary bits and
perform sum and carry as its output.it has three
input (x,y,z)and two output(sum(s),carry(c)).
Truth Table of Full Adder:
Input output
x Y z s c
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
Sum(s)=x y z + x y z + x y z +x y z
Carry(c)=x y z + x y z + x y z +x y z
International Conference on Engineering Humanities & Science (ICEHS-2017)
Seventh Sense Research Group www.internationaljournalssrg.org Page 135
In this paper I want to discuss the design of full
adder using multiplexer with the help of
Shannon expansion theorem. That‟s why first I
will give brief description about Shannon
expansion theorem. Shannon expansion theorem
allows any Boolean function „f‟ can be
expressed in the form:
f(m1,m2,m3,…………mn)= m1
.f(0,m2,...................mn) + m1 .
f(1,m2,……………….mn)
(1)
this expression is for by considering one variable
Shannon expansion theorem by considering two
variable can be expressed in the form :
f(m1,m2,m3,…………mn)= m 1 m
2 f(0,0,m3) + m1
m2f(0,1,m3) +m1m2 f(1,0,m3)+m1m2f(1,1,m3)
(2)
The above expression described in the form of
one variable and two variable Shannon
expansion theorems. One variable function gives
two combinations that is 0 and 1 similarly two
variable gives four combination that is 00, 01,
10 and 11.According to that combination the
Shannon expansion theorem expand.
Here I want to design full adder using
multiplexer with the help of Shannon expansion
theorem. Multiplexer is a combinational logic
circuit that receives information from many
inputs and directs this information to one output.
Selection line is used to select one input at the
output line. In general we can say it has 2n input,
one output and n number of selection line. In 2:1
MUX 2 input,1 output, and 1 selection line ,in
4:1 MUX 4 input,1 output and 2 selection line
and so on .
2.Block diagram ,truth table and expression of 2:1 MUX
I0
Y
I1 22:
(2:1 MUX BLOCK DIAGRAM)
S0
TRUTH TABLE
Selection line(s0) Output(y)
0 I0
1 I1
Y=s I0 + s I1
International Conference on Engineering Humanities & Science (ICEHS-2017)
Seventh Sense Research Group www.internationaljournalssrg.org Page 136
Similarly in 4:1 MUX,let‟s consider 4 input are
I0,I1,I2 and I3,two selection line are S0 and S1 ,one
output y then the expression for 4:1 MUX is
written in the form:
y=s1 s0 I0 + s1 s0 I1 + s1 s0 I2 +s1s0I3 (4)
In 8:1 MUX ,let‟s consider 8 inputs are
I0,I1,I2,I3,I4,I5,I6,I7,three selection line s0,s1,s2 and
one output Y ,then the expression for 8:1 MUX
is written in the form
y= s0 s1 s2 I0+ s0 s1 s2 I1+ s0 s1 s2 I2+ s0 s1 s2 I3+
s0 s1 s2 I4+ s0 s1 s2 I5+ s0 s1 s2 I6+s0 s1S2I7
(5)
in the above I have written the expression
directly because in this paper I will give more
stress on Shannon expansion theorem.Atlast I
will compare the final expression of Shannon
expansion theorem with the above expression.
3.Design full adder using 2:1 mux with
the help of shanon expansion theorem
The output sum expression for full adder
Sum(s)=x y z + x y z + x y z +x y z
We can write the above expression as following
way:
f(x,y,z)= x y z + x y z + x y z +x y z
(6)
For designing the above expression using 2:1
MUX one selection line is required,so we will
write the Shannon expression by considering
one variable .let that variable is „x‟.so we can
write the Shannon expansion synthesizes in the
form of:
f(x,y,z)= x . f(0,y,z) + x.f(1,y,z) (7)
by using equation (6) we can write:
f(0,y,z)=1. y z + 1. y z + 0. y z +0.y.z
= y z + y z (8)
f(1,y,z)=0. y z + 0. y z + 1. y z +1.y.z
= y z + y z (9)
Put equation (8) and (9) in equation (7)
f(x,y,z)= x . ( y z + y z ) + x.( y z + y z)
=x (y X-OR z) + x(y X-NOR z) (10)
equation (10) is look like as a 2:1 MUX
expression, comparing equation (10) with
equation (3) then we will get selection line
s0=x,I0= y z + y z ,I1=y z + y z.
so sum can be design using 2:1 MUX as follows
y
z sum
y (2:1 MUX)
x
International Conference on Engineering Humanities & Science (ICEHS-2017)
Seventh Sense Research Group www.internationaljournalssrg.org Page 137
similarly we can also design carry by using 2:1
MUX,output carry can be expressed as
C(x,y,z)= x y z + x y z + x y z + x y
z,we can write this expression is in the form of
shannon expansion theorem in terms of one
variable function:
f(x,y,z)= x y z + x y z + x y z + x y z
(11)
f(x,y,z)= x . f(0,y,z) + x. f(1,y,z)
(12)
f(0,y,z) and f(1,y,z) will find with the help of
equation (11), so
f(0,y,z)=1.yz + 0. y z + 0. y z + 0. y z
f(0,y,z) =yz (13)
f(1,y,z)=0.yz + 1. y z +1. y z + 1. y z
f(1,y,z)= y z +y z + y z
=y(z + z )+ y z
=y + y z
=y(1 + z )
=y (14)
put equation (13) and (14) in equation (12)
f(x,y,z)= x . (yz) + x. y (15)
equation (15) is in the form of 2:1 MUX ,where
x is the selection line and I0=yz = y AND Z ,and
I1=y
y
z
carry(c) (2:1 MUX)
Y
x
4.DESIGN FULL ADDER USING 4:1 MUX
WITH HELP OF SHANNON EXPANSION
THEOREM
For designing full adder using 4:1 MUX with
the help of Shannon expansion theorem,we can
write the Shannon expansion theorem in terms
of two variable:
Sum(s)=x y z + x y z + x y z +x y z
We can write the above expression as following
way:
f(x,y,z)= x y z + x y z + x y z +x y z
(16)
f(x,y,z)= x y f(0,0,z) + x y f(0,1,z) + x y
f(1,0,z) + x y f(1,1,z) (17)
we will find f(0,0,z), f(0,1,z), f(1,0,z) and
f(1,1,z) by using equation (16)
f(0,0,z)=1.1.z + 1.0. z + 0.1. z +0.0 z
f(0,0,z)=z (18)
f(0,1,z) )= 1.0. z +1.1. z + 0. 0. z +0.1. z
f(0,1,z) )= z (19)
f(1,0,z) = 0.1. z +0.0. z +1.1. z +1.0. z
f(1,0,z) = z (20)
f(1,1,z) = 0.0.z + 0.1.z + 1.0.z +1.1. z
International Conference on Engineering Humanities & Science (ICEHS-2017)
Seventh Sense Research Group www.internationaljournalssrg.org Page 138
f(1,1,z) =z (21) put equation (18),(19),(20) and (21) in equation
(17)
f(x,y,z)= x y z + x y z+ x y z + x y z
(22)
equation (22) is look like as a 4:1 MUX
expression where two selection line are „x‟ and
„y‟,four inputs are I0=z,I1=I2= z ,I3=z
z
z sum
z
z (FULL ADDER SUM USING
X Y 4:1MUX )
Similarly carry can be expressed as C(x,y,z)= x
y z + x y z + x y z + x y z
f(x,y,z)= x y z + x y z + x y z + x y z
(23)
Shannon expansion theorem in terms of two
variable can be expressed as
f(x,y,z)= x y f (0,0,z) + x y f(0,1,z) + x y
f(1,0,z) + x y f(1,1,z) (24)
we will find f(0,0,z), f(0,1,z), f(1,0,z) and
f(1,1,z) by using equation (23)
f(0,0,z) = 1.0.z + 0.1.z + 0.0. z + 0.0.z
f(0,0,z) =0 (25)
f(0,1,z)= 1.1.z + 0.0. z + 0.1.z + 0.1. z
f(0,1,z)=z (26)
f(1,0,z) =0.0. z + 1.1. z + 1.0. z + 1.0. z
f(1,0,z) =z (27)
f(1,1,z)=0.1 z + 1.0. z + 1.1 z + 1.1. z
f(1,1,z)=1 (28)
put equation (25),(26),(27) and (28) in equation
(24)
f(x,y,z)= x y .0 + x y z + x y .z + x y .1
(29)
equation (29) is in the form of 4:1 MUX where x
and y are selection line,the four inputs are
I0=0,I1=I2=z,I3=1
International Conference on Engineering Humanities & Science (ICEHS-2017)
Seventh Sense Research Group www.internationaljournalssrg.org Page 139
0
z carry
z
1 Y ( CARRY USING 2:1 MUX)
x y
CONCLUSION
In this paper I have discussed that full adder can
be design using different multiplexer i.e 2:1
MUX,4:1 MUX with the help of Shannon
expansion theorem. This one will be very
helpful for researcher to easy understanding and
practicing of implementation of any function
through multiplexer in the field of engineering
and technology
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