Engineer By Design » The Toolbox

Minimizing an Expression to Sum-of-Products Form

Charlene — Wed, 30 Sep 2009 23:59:30 +0000

I was given a review homework assignment recently for my Embedded Systems class. Most of it was just basic things such as converting decimal numbers to binary and hex and the sort, however I came across a problem on minimizing an expression to sum-of-products (SoP) form where the function yielded a summation of a bunch of values. It certainly didn’t help that I’ve been out of school for a bit, and most of what I was finding online to help me figure this out was on reducing a boolean function to sum-of-products form instead. Now that I’ve solved the problem, I figured I’d be a model citizen of the internet and post on how to come up with a solution for the problem to save some wasted time and effort for others.

Let’s say you are given the expression,

To solve this you would go through 4 simple steps:

Change each of the numbers in the summation into binary
Map the binary numbers into a Karnaugh Map
Group the Karnaugh Map as necessary
Interpret the expression.

Change each of the numbers in the summation into binary. In order, this would give you:

0000, 0001, 0010, 0011, 0100, 0101, 0110, 1010, 1011

Map the binary numbers into a Karnaugh Map.

Group the Karnaugh Map as necessary.

Interpret the expression. Thus, the reduced boolean expression in sum-of-products form is

Representing Negative 16-Bit Hex Values in Hex

Charlene — Wed, 30 Sep 2009 13:34:47 +0000

If you’re given a positive 16-bit hex value and asked to provide the hex representation of its negative value, it is otherwise asking for the 16-bit 2’s complement of the given value.

So, for example if you’re given 1D7A in hex and asked for representing -1D7A simply take its 2’s complement.

Convert to binary: 1D7A = 0001 1101 0111 1010
Flip the 1’s and 0’s: 1110 0010 1000 0101
Add 1: 1110 0010 1000 0101 + 1 = 1110 0010 1000 0110
Convert back to hex: E286

Therefore, the representation of -1D7A in hex is E286.

Graphs of Exponential Functions

Charlene — Wed, 30 Sep 2009 05:00:54 +0000

I’ve put together a bunch of screenshots I’ve taken from my TI-89 on the graphs of exponential functions, from y = x, y = x^2, …, y = x^10. The important thing to take from it is that when remembering how these functions look like:

y = x is a line
even exponents yield parabolas
odd exponents yield cubic curves (“squigglies”)

This is very trivial knowledge, but I’m putting this up because graphs of functions tend to be one of those things that once learned, people toss aside by relying on their graphing calculators can do. And knowing this off-hand is good general knowledge, really. (I.e., determining if a function is injective or surjective without breaking a sweat.)

The Toolbox

Charlene — Wed, 30 Sep 2009 00:04:39 +0000

As I’m going through this semester, I’m thinking about posting on some topics for my classes to help myself on preparing for exams and doing projects, as well as for anyone who’s looking for more help too. I often find myself surprised on how scattered information on seemingly simplistic search terms on Google are, and I’d like to have something that could act as my own resource.

I don’t intend on making posts under the category “The Toolbox” in-depth, as I want them to be short and to the point. Of course, if there’s anyone out there in teh internets who’d like me to elaborate more on certain topics I’d certainly be willing to help.

Of course, they’ll been plenty of posts on my other endeavors in music and art, but hey — an engineer needs to have their own toolbox of sorts.