# What is the equation relating maximum data transfer rates for a given EM bandwidth

I was thinking, since visible light has a much higher EM frequency than radio wave light, its bandwidth also has a far higher max data transmission rate, so people have been researching how to use this "Li-Fi" for close-range data transfer. What is the equation for a maximum data transfer rate for a given bandwidth?

I have been referred to two Wikipedia pages here and here. Both I've tried extremely hard to understand, but they did not answer my initial question. I would SINCERELY appreciate if you could answer my question. Thanks in advance.

The formula you are seeking comes from the Shannon-Hartley theorem:

$$C=B\log_2\left(1+\frac{S}{N}\right)\tag 1$$

where C is the channel capacity in bits per second, B is the bandwidth in hertz, S is the signal power and N is the noise power.

The formula specifies the theoretical upper bound on the information rate for an arbitrarily low data error rate.

You can see from the formula that it is not simply a matter of the channel bandwidth but also the signal to noise ratio of the channel. Ham radio operators are enjoying digital exchanges where the noise power is greater than the signal power but this results in a very low channel capacity. It is still fascinating to make contact with a ham half way around the world with a signal than cannot even be heard above the static.

You also need to consider that a LiFi channel could have substantial overhead in the packets to accommodate routing, error checking, handshaking, etc. So the data portion of the transmission will be at a lower rate than the formula might first suggest.

Consider an LED light bulb where you can vary the color of the bulb from the color red (~450 THz) through to the color green (~550 THz) in order to transmit data. This provides an available bandwidth of 100 THz. That is 1 x 1014 hertz - a lot of bandwidth. The limitation in this case will not be the bandwidth of the channel but rather the capability of the transmit and receive electronics which will probably be limited to around 1 GHz of bandwidth or so with today's technologies. Whether or not this is a cost effective bandwidth for the consumer market is another matter.

In a LiFi application you must also consider that there is substantial interference from other light sources. Since these are not Gaussian noise sources, they cannot be plugged in the "N" part of equation 1. Instead, the system will likely need to include a robust CRC (Cyclic Redundancy Check) or equivalent in order to detect and correct for this interference. This feature will further reduce the bandwidth available for the actual data.

• I still don't understand one thing. What are the transmitter and receiver limited by? Can we not just use a photosensitive cell? Forgive me for my ignorance on the matter. – FOG.js Jul 28 '18 at 18:05
• @FOG.js It is not optimal that a photo cell would be used as it is too slow. Most likely a PIN photo diode or transistor would be used. The primary bandwidth limitation is on the receive side. – Glenn W9IQ Jul 28 '18 at 23:23
• The receiving photo cell or diode will still be limited by thermal noise and quantum effects. – hotpaw2 Jul 29 '18 at 7:34