Synthesizing Functions using K-maps

[shamieh]

For a timing diagram - synthesize the function $f$(x1,x2,x3) in the simplest sum of products form.

So I have a picture of this timing diagram, which I can't really show on here unless i physically took a picture and uploaded it, but it's really irrelevant because I know I have the correct truth table, so hopefully we can work with that.

So my Truth Table reads:

1. x1 x2 x3 | f
2. 0 0 0 | 1
3. 0 0 1 | 0
4. 0 1 0 | 0
5. 0 1 1 | 1
6. 1 0 0 | 0
7. 1 0 1 | 1
8. 1 1 0 | 1
9. 1 1 1 | 0

So now I know I have $f$(x1,x2,x3) = \(\displaystyle \sum\)m(0,3,5,6)

Which means I have:

x!x2!x3! + x1!x2x3 + x1x2!x3 + x1x2x3!

So I need to put this function in the simplest sum of products form.. So I'm assuming i need to minimize the function that I just got above? If I am on the right track- then I now need to use a K-Map to find the minimization.

So here it goes.. (This is my K-Map)

... x2 x3
.. 00 01 11 10
x1 0[1) 0 1 0]
.. 1[0 1 0 (1]

So my question Is what now? How should I group all these 1s? Just group each of them by themselves? And if so, How do I read off what is going on here?
Would I read it like this ? x1!x2!x3! + x1x2!x3 + x1!x2x3 + x1x2x3! ?
Thanks for your time.

If this is something you can't explain or think I should just read more up on, please let me know, because I can take constructive criticism. I just want to make sure I know how to do these.

 

[Ackbach]

Yeah, diagonal clumping isn't allowed on K-maps. There's no simplification possible for that function, and you already have the simplest SoP. That's what I say.

 

[shamieh]

Your help is greatly appreciated. If only you guys knew how much you really help me. I seriously live on this forum - thanks to teachers who machine gun through chapters and T.A.s who can barely speak English. You all are very helpful. Appreciate everything you have supplied and helped me with.

Sham

 

[Ackbach]

Your help is greatly appreciated. If only you guys knew how much you really help me. I seriously live on this forum - thanks to teachers who machine gun through chapters and T.A.s who can barely speak English. You all are very helpful. Appreciate everything you have supplied and helped me with.

Sham

Thanks very much for those kind words! I can assure you, it works both ways. When we get courteous users who ask interesting questions, that makes it all worth-while!

 

[LudusRex]

Hi there,

First of all, great job on setting up the truth table and identifying the function in its simplest form. You are on the right track!

Next, let's talk about grouping the 1s in the K-Map. The goal is to group as many 1s as possible in the K-Map to minimize the function. In order to do this, we need to look for groups of 1s that are adjacent to each other, either horizontally or vertically. We cannot group diagonally.

In your K-Map, we can see that there are two groups of 1s - one group of two 1s and one group of four 1s. So, we can group these 1s as follows:

x1!x2!x3! + x1x2!x3 + x1!x2x3 + x1x2x3!

= x1!x2!x3! + x1x2!x3 + x1!x2x3 + x1x2x3!

= x1!x3!(x2! + x2) + x1x3(x2 + x2!)

= x1!x3 + x1x3

= x1(x3 + x3!) (since x! = 1)

= x1

Therefore, the simplest sum of products form for the function $f$(x1,x2,x3) is x1.

I hope this helps! Let me know if you have any further questions. Keep up the good work!