Channeling your inner capacity

Channel Basics

When Shannon developed his theory of information, it was in the context of improving communication. His goal was to determine if there was a way to maximize the transmission rates over some noisy channel. In this module, we will learn more about the channel and some formal terminologies that we will use in the succeeding modules. The GIF below recalls the story of Bob and Alice. Bob wants to send a message to Alice across some channel. Let's simplify the scenario. Let's say Bob just wants to send a love letter to Alice through some wireless (satellite based) channel.

Figure 1: Simplified communication model

Figure 2 shows a simplified model of their communication. Bob is the source. Sometimes we use the term sender or transmitter for the source. The medium where the message goes through is the channel (in this case the wireless channel). Alice is the receiver who probably gets the correct message at the other end of the channel. Let's take a look at each component formally.

The Source

The source is the component that produces messages or objects sent over a channel. We represent the source as a random variable $S$ which contains the outcomes $\{s_{1},s_{2},s_{3},...,s_{n}\}$ and each outcome has its own probability distribution $\{P(s_{1}),P(s_{2}),P(s_{3}),...,P(s_{n})\}$. The random variable $S$ is also called the source alphabet and the outcomes are the symbols of the source alphabet. A combinational sequence of outcomes forms the message that we send over the channel. Consider the following examples:

• We can let the source alphabet $S$ be the English alphabet where the symbols are all the letters from a to z including the space. Suppose we want to send the message "the quick brown fox jumps over the lazy dog". All letters of the English alphabet has a probability distribution associated with it. The second programming exercises did the exact same method for English, German, French, and Tagalog languages.

• We can let $S$ be the binary alphabet where the symbols are $\{1,0\}$. The message could be streams of 1s and 0s that could mean something. For example, a message could be a sequence of events when a light is on or off. It could also be an indicator for sending decimal values in the form of binary digits.

• In biology, we can let $S$ be the DNA bases whose symbols are $\{A,C,G,T\}$. $A$ is adanine, $C$ is cytosine, $G$ is guanine, and $T$ is thymine. These symbols combine to form DNA sequences that are messages to instruct what kind of protein synthesis should a cell do.

Make sure to understand carefully what source alphabets, symbols, and messages mean.

The Channel

The channel is the medium where the source message will be sent to get to the receiver side. In our Bob and Alice example, they used a wireless channel. The channel has a maximum capacity of the number of symbols that we can transmit per second. We call this the channel capacity. We will discuss this later. A channel is, most of the time, associated with noise. A noiseless channel is where the information of the message gets to the receiver "perfectly". We'll revisit this later too. A noisy channel is when any of the symbols of the source message gets flipped or when additional non-source symbols gets added to the message. The noise disrupts the message and thereby affecting the information that the receiver gets. Let's take a few examples:

• In the Bob and Alice story, the wireless channel can disrupt Bob's message. Suppose one of the messages Bob sends is "love can be as sweet as smelling the roses". Unfortunately, noise can either corrupt symbols or add unwanted symbols such the the message becomes "love cant bed as sweat as smelting the noses". This is a disaster if it gets to Alice. 😱

• Another example is memory. Suppose you're in Mars and you need to record audio logs and videos unto a solid-state drive (SSD). Memories can be corrupted due to cosmic-ray bit flips. These bit flips occur due to cosmic energies that zap some bits of memory. Say some data $1010$ gets flipped into $0010$. Both representations mean totally different things.

• Last example is social media. Good people (source) would like to spread factual news (message) that are noteworthy of broadcasting to ordinary citizens (receivers). Unfortunately, this does not stop evil citizens in creating fake news (noise). So since both factual news and fake news mix together in social media, fake news disrupts the intended messages for ordinary citizens. Don't you hate it when this happens? 😩 Innocent people fall into this trap.

Of course, if the channels were noiseless, the examples above won't have any issues with the received messages. We won't be dealing with the physics of channels in this course. Although, it is worth mentioning since channels can be modelled for different entities.