3. Minimization and K-Maps
So, why do we need to minimize our circuits? Well, they are costly, VERY COSTLY. So, MINIMIZE. It also improves speed, smaller chance of defects, etc.
Minimization
Use boolean algebra rules to help minimization.
And here's some examples.
Let's consider the one-bit full adder. How do we minimize that?
$C_{out}=\bar{A}BC_{in}+A\bar{B}C_{in}+AB\bar{C_{in}}+ABC_{in}$
$=AB(\bar{C}+C)+A\bar{B}C+\bar{A}BC$
$=AB+A\bar{B}C+\bar{A}BC$
Wait, what now? Well, lets add a term!
$=AB+A\bar{B}C+ABC+\bar{A}BC+ABC$
$=AB+AC(\bar{B}+B)+BC(\bar{A}+A)$
$=AB+AC+BC$
So that's a little weird, let's use another tool that is easier.
Karnaugh Maps
Gray code is a way of encoding binary by only changing one bit on each step.
Using this gray code, we can convert it to a K-map to minimize binary.
Then, find rectangles of 1s (or Xs but you do not need to include Xs, they are optional). of heights and widths of 1, 2, or 4. Overlapping is fine. Use the biggest retangles possible and the fewest rectangels as possible.
Then for each rectangle, look at the values of the variables along the axes. some variables change, and others don't.Each rectangle is an AND minterm. Write the variables that stay the same for that rect (keeping the NOT bars). Ignore the variables that change. OR all the terms together.
A K-map with more than 2 variables.
The K-map wraps around, left-right and top-bottom. Think of it like a donut.
