4 Variable Karnaugh Map I Video Lecture | Analog and Digital Electronics - Electrical Engineering (EE)

hi there now we're going to look at a
four variable karnaugh map of course
with four variables we have sixteen
possible combinations so there will be
sixteen lines in our truth table and
therefore also sixteen lines in our
karnaugh map so I'm just going to go
ahead and draw that
now hopefully you saw the previous video
or we did the three variable karnaugh
map still a lot of the information is
going to be the same as far as how
things map out I'm going to review it
slightly in this video as well but the
first thing we're going to need to do is
make sixteen cells one for each output
in our truth table now we're going to
have four variables again and they will
be of course D C B and a now the next
thing we're going to do is put our
variable information here 0 1 now don't
forget our gray code here or one bit
changes at any given time causes us to
have this pattern along the top and
along the bottom so don't forget this
here and this here and again remember
that these are in a slightly different
order than we would normally expect them
to be now the next thing we're going to
do again is just map out or transfer the
information from the truth table into
the karnaugh map and it's going to work
the same way it did last time so for
example the first line of the truth
table right here 0 0 0 0 will correspond
to box 0 0 0 0 and the karnaugh map
right here and the second line 0 0 0 1
of course would map right here
you just have to be careful as you go
down the truth table to make sure you
take the appropriate jumps in the
karnaugh map and we're just going to
take a look at those right now as I fill
it so we're going to start right here
with zero zero zero so I have 0 1 and
then 0 1 if I do it in the order of the
truth table then I have 1 1 1 1 if
they're all ones I guess it doesn't
matter
next we jump down so 1 0 0 0 is the next
one that's going to be down here and
we've got 4 zeros and finally 1 1 0 0 is
right here so we're going to pick up
where we left off with 0 0 and 1 1 all
right so now we're done the information
from the truth table is mapped to our
karnaugh map now again we're briefly
going to go over the rules we want our
loops or our groups to cover all of the
ones in the kernel map sorry and none of
the zeros our loops have to be powers of
two so groups of 1 2 4 8 16 and so on in
the biggest groups that you can to give
us the more simplified terms in the
expression and like the last time we're
going to overlap if that allows us to
make bigger groups we just have to make
sure that each one of the loops has a
one that is unique to that loop and that
we'll make sure that we don't have any
redundant terms now there's a few
options here I'm just going to start
with this one right here so we're just
going to make one loop of for now again
we're going to do what we did with the
other one and that is take a look at the
domain of that loop so if we look up we
take up this much real estate and if we
look to the left we take this much now
because we're in this row the D and C
are going to stay so the term is going
to be
do not see because the D is a zero and
the C is the one so that's going to be
basically the term for this loop is do
not see now my other option is to take a
loop up here and then we're going to do
the same thing as examine the domain or
the area that this loop covers and again
I can see the a here is one and one it
doesn't change so the a will be one of
the terms since the be changes it's zero
here and it's one here then that term is
redundant and the B disappears now if I
look over to the left here I see that
the C term changes so it's gone and the
D term stays the same so it stays and
it's a zero so that's going to be D not
so that means the term for this loop is
going to be D not a now my third option
is going to be again I'm going to make a
loop of four right here and again I'm
going to look up and to the left now if
I look up here I can see that the be
variable stays so we're going to have a
B in the term the a you can see here
changes so it will disappear and if I
look over to the left I can see that the
C stays the same so the C will be in
that term and since the D changes zero
and one it will disappear so the term
for this blue loop is going to be B C
okay so now let's look at it all
together and I'm going to put all three
loops back
so there's my green one and the
expression for that was
do you not see and after that we looked
at this one up here
and the expression for that was D not a
and the final loop we looked at was this
one here hopefully that's not too
confusing now that they're all in and
the term for this one was BC now you'll
notice there's overlap in the loops
you'll notice all the loops are in
powers of two which they're all up to
four in this case which makes for
smaller terms there is overlap but each
one of those loops the red blue and the
green all three of them have ones that
only belong to that loop they don't all
have to belong but there has to be at
least one of the ones that's unique and
if you've done that then you're
overlapping is okay so what that's going
to give us is a final boolean expression
for this output that is B C or C D
naught or a D naught I just rearranged
them a little but that's our simplified
boolean expression based on this four
variable karnaugh map