Purpose of the Experiment
-
Write out the function represented by the kmap in SOP form.
-
Implement the function using no more than three gates
-
Prepare a truth table for the function
-
Using a simulation set up the circuit and demonstrate that the circuit
does indeed represent the function described by the map and the truth table
-
Write out the function represented by the kmap in POS form.
-
Implement the POS function using the least number of gates to explain why
the resulting circuit differs, if it does, from the circuit of (2) above.
-
Prepare a truth table from the function obtained in (5) and see if it differs
from (3) above.
-
Check out this circuit using simulation.
Theory
Any Karnaugh map can represent the sums of product
for a function and it's product of sums. Truth tables and simplified functions
can be derived from these K-maps.
Data
KMAP GIVEN
|
AB\CD
|
00
|
01
|
11
|
10
|
|
00
|
1
|
1
|
0
|
1
|
|
01
|
0
|
0
|
0
|
0
|
|
11
|
0
|
0
|
x
|
x
|
|
10
|
x
|
1
|
1
|
x
|
-
The function represented by the KMAP via SOP is:
AB' + B'D' + B'C' (below is the grouping in
ROW/COL)
AB' was obtained
from : 10/00, 10/01, 10/11, 10/10
B'D' was obtained from
: 00/00, 00/10, 10/00, 10/10
B'C' was obtained from
: 00/00, 00/01, 10/00, 10/01
B'(A + D' + C') is the above function in its
minimized form.
Truth Table For Above Formula (A)
|
A
|
B
|
C
|
D
|
AB' + B'D' + B'C'
|
|
0
|
0
|
0
|
0
|
1
|
|
0
|
0
|
0
|
1
|
1
|
|
0
|
0
|
1
|
0
|
1
|
|
0
|
0
|
1
|
1
|
0
|
|
0
|
1
|
0
|
0
|
0
|
|
0
|
1
|
0
|
1
|
0
|
|
0
|
1
|
1
|
0
|
0
|
|
0
|
1
|
1
|
1
|
0
|
|
1
|
0
|
0
|
0
|
X
|
|
1
|
0
|
0
|
1
|
1
|
|
1
|
0
|
1
|
0
|
X
|
|
1
|
0
|
1
|
1
|
1
|
|
1
|
1
|
0
|
0
|
0
|
|
1
|
1
|
0
|
1
|
0
|
|
1
|
1
|
1
|
0
|
X
|
|
1
|
1
|
1
|
1
|
X
|
-
The implementation for this function by using less than or equal to three
gates: (A)
![[ implimentation of above equation with less than three gates ]](design1.gif)
-
The function represented by the KMAP above in its POS (product of sum)
form:
B'(A+(C'+D')) (below is the grouping in ROWS/COLS)
B'
was obtained from : 01/00, 01/01, 01/11, 01/10
11/00, 11/01, 11/11, 11/10
A+C'+D'
was obtained from : 00/11, 01/11
(Since there is no order of operations for
sums/sums then the inner parenthesis can be dropped.)
B'(A+C'+D') Now this is already a minimized
form
Truth Table For Above Formula (B)
|
A
|
B
|
C
|
D
|
B'(A + C' + D')
|
|
0
|
0
|
0
|
0
|
1
|
|
0
|
0
|
0
|
1
|
1
|
|
0
|
0
|
1
|
0
|
1
|
|
0
|
0
|
1
|
1
|
0
|
|
0
|
1
|
0
|
0
|
0
|
|
0
|
1
|
0
|
1
|
0
|
|
0
|
1
|
1
|
0
|
0
|
|
0
|
1
|
1
|
1
|
0
|
|
1
|
0
|
0
|
0
|
X
|
|
1
|
0
|
0
|
1
|
1
|
|
1
|
0
|
1
|
0
|
X
|
|
1
|
0
|
1
|
1
|
1
|
|
1
|
1
|
0
|
0
|
0
|
|
1
|
1
|
0
|
1
|
0
|
|
1
|
1
|
1
|
0
|
X
|
|
1
|
1
|
1
|
1
|
X
|
-
Logic circuit simulation of the above function.(B)
Results
The two circuits above were designed to implement
the KMAP given. From the KMAP two functions were derived via SOP (sums
of products) and POS (product of sums.) The sums of products method yielded
the formula, F(a,b,c,d) = AB' + B'D' + B'C'. The product of sums method
yielded the formula, F(a,b,c,d) = B'(A + C' + D').
In figure A and B, the three input or gate
receives A, and the not of C, and not of D. The result of
this gate gets sent into the two input and gate. The other input
into this and gate is the not of B. The output from this
and gate is the result that complies with the truth table derived from
the formula. Conversely the formula was derived from the KMAP given as
part of the experiment.
When the simulation was applied and executed under
LogicWorks ( a software package used to simulate simple to complex circuitry
) all cases of input variants held true to the truth table. Further demonstrations
of the this circuit's results is available upon request.
Conclusions
This simulation showed that given a KMAP, it's functionality
can be derived and implemented via an algebraic function and conversely
it's truth table. The functionality was demonstrated via LogicWorks and
upheld the findings in the truth table and functions derived from the KMAP.
This simulation also demonstrated the fact that by taking the POS of any
particular K-MAP, with such a configuration, would yield results that would
not need simplification.
Although this is, indeed, a fully operational circuit
for this particular KMAP, and the results yielded true throughout the truth
table, this is not a unique construction of this KMAP's implementation.
This is especially true of this KMAP due to the number of its "dont care"
conditions.
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