The Sum of Products (SOP) logic equation implements a logic function using two levels, where the first level consists of AND gates. The second level takes all the outputs of the AND gates and ORs them together.
The MSOP, or Minimal Sum of Products, is a form of SOP where the number of gates (number of products) is minimal and the number of inputs to each gate (number of literals) is minimal. In essence, it implements a logic function using the 2-level AND-OR structure, where the circuit complexity is as small as possible.
Produce a ‘1’ for the following binary inputs:
The following binary inputs are defined as Don’t Care:
This gives the following Karnaugh map:
The aim is now to find a minimum set of the largest blocks (implicants) that will cover all the binary ‘1’s in the Karnaugh map. Each block will translate into an AND gate in the final implementation. The larger the block, the fewer inputs the AND gate will have.
We must use the ‘don’t cares’ to increase the size of a block if possible, as this will then reduce the number of inputs.
Now let’s start covering:
Here X is always ‘1’ and Y is always ‘0’, this gives the following product: XY’
Now, we try to cover the lower right ‘1’s. We can’t cover them using a single product so we end up doing something like this:
In blue: W=1, X=0, Y=1 which gives the product WX’Y.
In green: W=1, X=0, Y=0, Z=1 which gives the product WX’Y’Z.
Our SOP solution is now: XY’ + WX’Y + WX’Y’Z.
We must now ask ourselves the question: can we in any way make the blocks larger?
The answer is: YES. (surpise!)
First, we can extend the blue block to include the top don’t cares:
Now we can reduce the product WX’Y to X’Y. The AND gate now only needs 2 inputs instead of 3!
We can also extend the green block up one cell. This will cause one output ‘1’ to be covered by two blocks. This is not a problem as it doesn’t change the output of the function but it does reduce the number of inputs for the green AND gate:
The green block (W=1, Y=0, Z=1) now gives the following product: WY’Z
Our minimal sum of products implementation then becomes: XY’ + X’Y + WY’Z
Here is the logic circuit:
Hope this helps a little.