Basics of digital logic

[WarFox]

I am currently about to finish a class in my course work that deals with digital logic at a very basic level. So, I would like to share a little bit of knowledge of what I have learned.

 

 

Data Representation in a Computer

 

 

Communicating digital can be traced back to the days of Samuel Morse and the invention of the telegraph. Communicating over long distances via wire required some sort of standardized system of communication. Samuel Morse developed the famous system that we know as Morse code to facilitate communication. On paper, the language is represented by a series of dots and dashes, coming from a speaking, it is represented by long (DAH) and short (DIT) beeps. By standard convention, a "dah" has a width of 3 "dits."

 

 

(A) .- [DIT - DAH]
(B) -... [DAH - DIT - DIT - DIT]

 

While used in the telegraph, it was not implemented in computers but have become a real world pre-computer example of how information could be stored. Morse code is created to travel across a wire by turning current on and off along a wire, which is generally created by a telegraph operator tapping a metal paddle on to a metal surface.Essentially, just being a switch. Morse code didn't become the standard of data representation, instead binary logic was chosen instead. Representing information in 1s and 0s instead of long and short audio beeps. A "HIGH" voltage is generally considered to be represented by "1" or also known as "ON/TRUE." "LOW" voltage is represented by a "0" or "OFF/FALSE."

 

Now, binary is more than just a convention, it is an actual way of doing mathematics. We conventionally use a "base ten", also known as the "decimal system" ( system of counting (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10). Binary is a "base two" system where there are essentially only two unique characters that make up the whole number system which can be repeated to make more complex number. Look at the bubble below:

 

0 (zero)
1 (one)
10 (two)
11 (three)
100 (four)

 

 

Each individual symbol is dubbed a "bit" and can only represent 2 possible values. So "10" is 2 bits in width. A quick way to see how many possible values a certain number of bits can represent, we can do a quick calculation of 2 to the power of some bits.

2^2 = 4
00
01
10
11

2^3 = 8
000 (zero)
001 (one)
010 (two)
011 (three)
100 (four)
101 (five)
110 (six)
111 (seven)

The bottom line is that information in computers can be represented as a switch or series of switched. Imagine we have a battery connected to 8 light bulbs with a switch between the battery and each light switch. So, we have 8 light bulbs and 8 light switches. Using binary, we can represent 2^8 numbers starting from zero by switching on lights. Light bulbs that are lit represent a 1, light bulbs that are not lit represent a 0. Computers at the very basic level are a system of switches that perform operations on switches to change the system's state.

 

Boolean Algebra and Truth Tables

Boolean was a man who had a goal of being able to relate human decision making to mathematical logic. He wanted to develop a mathematical way of expressing logic. Thus, he developed what we call Boolean Algebra. This form of algebra uses typical math symbols that we are all used to seeing, however they have a different meaning. In this form of math, a state of a machine, or a decision being made is equal to an equation of variables that take on certain behavior based on the state of inputs and their relation to one another. A gppd way to explain this is to take a look at an example and break it down.

F = ab + c'b

F is the output of an equation. On the right side of the equals sign, we have three variable (a b c). "ab" is an expression that represents multiplication, which in boolean algrebra is representative of "and." The addition symbol is representative of an "or." An apostrophe is means an inversion. Any value that is not inverted is assumed to represent true. An inversion of a value means false. This system will also assume that F is true. This is how we can read this.

 

F is true if a and b are true or c is not true and b is true

A quick table for reference:

 

* AND
+ OR
' NOT

Now, another way we can represent data is a more easy way is a truth table. In the follow link is a file named "truthtables.pdf" with three sample truth tables. Each column represents an input or output. The top row of each table is a label, then underneath is the state of that input or output.

 

Schematics of a Basic Digital Circuit

Now that some base information has been established, now basic circuits that a system can use to execute logic can be discussed. For creating digital logic circuits in class, I used Logisim, which is what I will be using for create examples.

 

There are three main basic components in a digital circuit, these are made up of transistors. The construction of them from transistors is outside of the scope of this post. These three main components are the ones discussed in the previous section; AND, OR, and NOT. Here are two images, the first is how the gate are represented. The second image are the truth tables to explain the logical operation that is being performed by each type of gate.

Using our boolean equation and a truth table is a quick way to prototype a digital circuit. Here is a drawing of logism of the circuit that would produce the same results as the equation "F = ab + c'b." Recreating this circuit in logism or a similar program, we can see that the behavior of this circuit matches what our truth table says. To do this for any boolean expression, simply take the inputs and connect them with their appropriate gates. In the case of "ab," the inputs named a and b are connected to the two input pins of the AND gate. The output is fed to an OR gate that is represented by the addition symbol. Input c is inverted by a NOT gate, the output of the NOT gate is fed into an input of another AND gate that takes another input that comes from b. The output of this second AND gate is also fed into the OR gate. If either of these AND gates outputs true, then the output of the circuit (F) will be true.

[theifyppl]

Nice! What are you studying? CS, Electrical Engineering, or both/neither? I had to take a few FPGA courses that required me to learn all of this stuff -- really enjoyed all of it.

 

[WarFox]

BS in software engineering is what I'm working towards.