Data Coding Theory/Shannon capacity

Channels
A channel is a communications medium, through which data can flow through. In a wireless network, the channel is the open space between the sender and the receiver through with the electromagnetic waves travel. In a wired network, the channel is the wire through with the electrical signals flow.

Signal to Noise Ratio
Data flowing through a channel is subjected to outside noise interference. In a wireless system, this interference can be a competing transmittor, or even natural phenomena such as lightning or solar radiation. These same kinds of interference can affect wired transmissions as well.

The ratio, in dB (decibels) of the transmitted information power to the noise power in the received signal is known as the signal to noise ratio. The signal to noise ratio, abbreviated "SNR", is an important metric to learn when studying a transmission network, because it affects how much data can be sent and at what speed. If we have a signal with energy E, and a noise density of N, we calculate the SNR as:


 * $$SNR = 10 \log_{10}\left(\frac{E}{N}\right)$$

Likewise, if we have a signal with an amplitude of A, and a noise with variance &sigma;, we calculate the SNR as:


 * $$SNR = 10\log_{10}\left(\frac{A^2}{\sigma^2}\right)$$

The higher the SNR, the more faithful the received signal will be to the sent signal. The lower the SNR, the more likely the received signal is going to contain errors. We won't cover SNR in too much detail in this book. However, the SNR is a very important quantity in a communications system, and there are classes of codes that are designed to help reduce errors in a channel with a low SNR.

Bandwidth
Bandwidth is a measurement of a communication's channel, and determines what frequency ranges are transmitted faithfully through the channel, and which ranges are not. We don't need to discuss bandwidth in any detail in this book, but suffice it to say that it is an important quality of a communications channel.

Shannon's Channel Capacity
The Shannon Channel Capacity is a formula that relates the bandwidth of a channel, the channel's SNR, and the bit rate.


 * $$C = W \log_2 (1 + SNR)$$

The channel capacity is the bit rate at which data can be sent along a channel with a negligible error rate. Attempting to send bits faster than this rate will increase the error rate beyond a negligible value. This will have large negative effects on our communications system. Notice that for a fixed bandwidth, as SNR decreases, the bitrate must also decrease.

Ramifications of Channel Capacity
Because of Shannon's Channel Capacity, we can see a few important facts. The first of which is, the more noise a channel has, the slower our bitrate needs to be. This means that if we want to send data at or above the channel capacity, we will need to have a robust error checking and correcting mechanism.

What we can also see is that for a fixed SNR, more bandwidth means a faster data rate.

In short, to send more information over a channel more quickly, we need to:
 * Increase the SNR
 * Increase the bandwidth
 * Use better coding and error correcting.