Minterms

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Minterms

  1. 1. LOGIC DESIGN PART 3 Minterm
  2. 2. • Larger logic problems require a systematic approach for solution. Modern integrated circuit chips can use millions of logic devices. The sheer magnitude of these designs is a clear sign that a formal approach to the design is needed. • Given a Boolean function described by a truth table, – determine the smallest sum of products function that has the same truth table. – determine the AND-OR-NOT circuit that implements that smallest sum of products function.
  3. 3. Minterm Expansions • Different ways of expressing a Boolean function can have widely varying levels of complexity. • More complex circuits will require more gates and inverters, so it's a reasonable goal to learn how to devise circuits that are as simple as possible. • In this section we are going to look at how you can represent circuits differently using
  4. 4. • Let's look at a simple Boolean function of three variables. We'll describe this function with a truth table. The input variables are X, Y and Z, and the function output is F. Let's examine this function in some detail. The only non-zero entries are at: X = 0, Y = 1, Z = 0 and X = 1, Y = 0, Z = 1 The function is 1 for those two input conditions and zero for all other input conditions.
  5. 5. How we can implement this function • We want the output to be 1 whenever we have either – X=0 AND Y=1 AND Z=0 • OR when we have – X=1 AND Y=0 AND Z=1. • This word statement is very close to the function we want. We've highlighted the important aspects of the function. Here's the function: • This function is read as (NOT-X AND Y AND
  6. 6. Defining Minterms • This form is composed of two groups of three. Each group of three is a minterm. Important points about minterms include the following. – In a minterm, each variable, X, Y or Z appears once, either as the variable itself or as the inverse. – Each minterm corresponds to exactly one entry (row!) in the truth table.
  7. 7. • To build any Boolean function from minterms do the following. – Get a truth table for the function – For each entry of the truth table for which the function takes on a value of 1, determine the corresponding minterm expression remembering that every variable of its inverse will appear in every minterm. – OR all the minterms from the second step together.
  8. 8. IN SHORT ! • A truth table gives a unique sum-of-products function that follows directly from expanding the ones in the truth table as minterms.
  9. 9. An example using Minterms • Three young graduates have formed a company. The three graduates, Alisha, Ben and Corey have a system to minimize friction. For all minor decisions they want to use a circuit that will determine when a majority of the three of them has voted for a proposal. Essentially, they want a box with three inputs that will produce a 1 at the output
  10. 10. STEP 1 Get the truth table T1 T2 T3 T4
  11. 11. STEP 2 • Identify the minterms
  12. 12. STEP 3 ORing all together
  13. 13. • Now we need to simplify our equation further more. • For this we require basic Boolean algebra first
  14. 14. Simplifying the Circuit FIRST POSSIBILITY
  15. 15. SECOND POSSIBILITY If you OR anything with itself, you get the original quantity back (X + X = X).
  16. 16. • Here is the three-variable truth table and the corresponding minterms:
  17. 17. Example • Example: suppose a function F is defined by the following truth table Since F= 1 on rows 1, 2, 4, and 7, we obtain A compact notation is to write only the numbers of the minterms included in , using the Greek letter capital sigma to indicate a sum:

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