# Phorgy Phynance

## The Ultimate Wireless Broadband Speed Limit

This is a follow-up to my previous posts

as well as some of the comments those posts generated.

First of all, I was pleased that loganb (Comment on 12/29, 9:27 PM EST) brought Shannon into the picture because information theory is important. As Rod pointed out, “information is physical” so the limits of wireless broadband communications will involve a knowledge of both Maxwell and Shannon. In fact, if any grad student out there was interested in both information theory and wireless communications, it would be a fun project to try to determine this ultimate wireless broadband speed limit.

### The Problem

If two computers were to communicate via laser beam and they had direct line of sight access, the speeds at which those two computers could communicate over the airwaves should approach those possible via fiber optics.

Of course that is not what one means by wireless broadband. We do not think of a PC communicating to a tower via laser beam with direct line of sight. So this is not the “ultimate speed limit” I’m talking about.

Instead, we want to think of a region of space containing multiple wireless broadband devices. What is the maximum “density of information” available in the air within this region. For concreteness, we could consider

$\text{1 unit} = 10 m\times 10 m\times 10 m$

and

$\text{1 test region} = \text{10 units}\times\text{10 units}\times\text{10 units}$

What is the ultimate physical maximum number of bits that can be communicated via wireless broadband devices within one test region?

### My Guestimate

My “guestimate” is that in a test region the ultimate wireless broadband speed limit, i.e. the maximum number of bits that can be communicated via wireless broadband devices is 1,000 Mbps. This is the total number of bits available to everyone within the test region. To get the speed available to any one wireless broadband user, we simply take this speed limit and divide by the number of users simultaneously downloading (or uploading) content.

For example, if my guestimate is correct and there were 100 people simultaneously using their wireless broadband devices, then with a perfectly design wireless network (*cough*) each person would have access to speeds of 10 Mbps. This is still pretty high and most in the US would drool for these speeds (although people in Korea, Japan, and parts of Europe would yawn).

I don’t ever expect routine wireless broadband access in excess of 10 Mbps in crowded tech savvy cities and even this speed assumes perfect network design. It will take a while for us to reach this ultimate speed limit as determined by fundamental physics, but we will eventually. With bandwidth demands doubling every couple of years, it is easy to imagine running into this speed limit within 5 years.

### A Note on Radio Wave Oozing

Here is a comment from Zathras:

“The behavior of a wave depends on its frequency. At low frequencies, radio waves are kind of like molasses. They can ooze around corners and through buildings.

As frequencies increase, the waves start acting more like laser beams.”

This is absolute nonsense. For starters, go into a parking garage and compare your AM radio reception to your FM. The AM won’t come in, despite being lower frequency than the FM. There is no “molasses effect” here. Yes, I realize that these are lower frequencies than the cell phone ones, but since the above quote is stated in such absolute terms, it is still an effective counterexample.

The reasons for the interference are more complex. It has to do with the interference with vibrational frequencies that the molecules in the barriers have. The multi-GHz range is full of resonant frequencies for molecules. It also has to do with the fact that higher frequencies attenuate more in conducting metals than lower frequencies do. It certainly has nothing to do with any laser beam/molasses nonsense.

The actual physics of radio waves is more complex than can be communicated in a few paragraphs.  However, the “oozing” analogy does take you some distance in understanding what is going on. Unlike molasses, radio waves can ooze through some materials such as glass, dry wall, wood, etc. One material that radio waves cannot penetrate is metal. In fact, the better the material is at conducting electricity, the worse radio waves are at penetrating it. For example, the earth is fairly good at conducting electricity so it is difficult to send radio waves through the earth.

Now consider two people trying to communicate via radio waves, but they are separated by a wall of metal. No dice. The radio waves cannot penetrate. Now, puncture a small hole in the wall. To be concrete, lets say we are communicating at a frequency of 3 GHz with a corresponding wavelength of 10 cm. If the hole is 1 cm in diameter, it is difficult for the radio signal to “ooze” through that tiny hole. Remember, the “size” of anything as far as a radio wave is concerned is only as a ratio of its wavelength. In this case, the hole is $.1 \lambda$.

The ability of a wave to ooze through the hole depends on the size of the hole relative to its wavelength. Roughly speaking, when the size of the hole is greater than $.5\lambda$ it has a much easier time oozing through it.

Keep in mind that most buildings have some kind of metal exterior and especially underground parking lots are surrounded by rebar, etc.

Now, an AM radio wave is roughly 1,000 kHz (or .001 GHz) with a corresponding wavelength of 300 meters. For an AM radio wave to ooze through a hole, that hole would have to be more than 150 meters wide. This is why AM does not penetrate very far into a tunnel.

On the other hand, an FM radio wave is roughly 100 MHz (or .1 GHz) with a corresponding wavelength of 3 meters. Although still fairly big, FM radio waves can still ooze through garage doors in underground parking lots, through building windows, into tunnels, etc.

Wireless broadband at 2.5 GHz or 12 cm can  “ooze” though windows etc, but as I said, they ooze too fast and have a harder time turning corners. Hence, at these higher frequencies, we begin to see shadowing etc.

Written by Eric

December 31, 2009 at 12:34 pm

### 7 Responses

1. Still on EM wave propagation:

One phenomenon that is very interesting is the propagation of EM waves in saltwater. Since saltwater contains ions, it is a conductor and, hence, it absorbs energy from the EM field. In other words: saltwater attenuates EM waves.

This is a serious problem if we want to communicate with submerged submarines. Low frequency radio waves are less attenuated and, thus, can be used to communicate with submerged submarines, but the price to pay is that the data rate will be very low. At frequencies of interest, the penetration depth of EM waves in saltwater is on the order of 1 meter!! However, as anyone who has ever snorkelled in the Caribbean can tell you, the visibility underwater can be tens of meters (with a diving mask). Well, the seawater in the Caribbean is also saltwater, and if the attenuation is proportional to the frequency, how can it be that radio waves are brutally attenuated in saltwater, while EM radiation in the visible range is a lot less attenuated?

A semi-classical model is required, as expected

Rod Carvalho

January 1, 2010 at 3:19 pm

• Hi Rod (Happy New Year!)

http://en.wikipedia.org/wiki/Water_absorption

The conductivity of a material is usually expressed as a function of frequency, i.e. in the Fourier domain. Modeling the time domain response to pulses (and bits) requires convolution, etc.

The conductivity spectrum of water is particularly complex

http://en.wikipedia.org/wiki/File:Water_absorption_spectrum.png

http://people.seas.harvard.edu/~jones/es151/gallery/images/absorp_water.html

with a large spike in the microwave region (which explains how microwave ovens work). Note that the absorption is at a minimum in the visible spectrum.

It could very well require semi-classical models to try to “explain” the conductivity spectrum, but once you have it, from the perspective of computer simulations, all we need is the spectrum given as an input.

phorgyphynance

January 1, 2010 at 4:22 pm

• My question was meant more as a challenge than as an actual question. I remember studying this phenomenon many years ago, when I was an undergrad. Unfortunately, I don’t remember the exact explanation right now.

One possible sketch of an explanation would be to think of the ions in saltwater interacting with the EM field: at radio frequencies, the ions interact with the field, and the higher the frequency, the more energy is transferred from the field to the medium (the greater the attenuation); by contrast, at optical frequencies, the photons are too energetic to interact with the (heavy) ions and, hence, there’s relatively little attenuation. Since I am merely an electrical engineer, not a physicist, I am fully aware that my conjecture is probably nonsense.

PS: I don’t think the H20 absortion spectrum is that interesting in this case, because saltwater is much more than just H20.

Rod Carvalho

January 2, 2010 at 2:59 pm

2. I think this is something that awaited by the Internet service users, because it will get a breakthrough in the field of internet.

Danny Poetra

February 27, 2010 at 1:15 am

3. Now I am curious, and have some practical interest in the subject. I am a ham radio operator. For a couple of years, I have been trying to get some accurate information on how frequency of radiation affects attenuation in various materials. Every study I have ever seen concentrates on teh high frequencies used by WiFi. As a practical matter, I woudl just liek to know how much attenuation might be expected in an unfinished wood structure at HF, VHF, and UHF frequencies. The consensus among other ham operators is that at HF frequencies wood is mroe or less transparant, but that it grows mroe opaque as frequency increases. I suspect it is more complicated than that, but no one seems to have anything but opinion or heresay.

Neal

June 11, 2010 at 4:50 pm

4. Why do you think you’re the only one who is talking about this? I’m just curious. Your logic seems to make sense but your guesstimate seems vague. It is certainly not something we can trust as being a true representation of the limit in my opinion. The only thing I can conclude that there must be an “ultimate wireless speed limit” but that this limit must be so high that it really isn’t a concern for anyone in the near future (the next 10 years). I know in my city 3G at lunchtime crawls to a hault and thus we have been sold 4G as a solution but other cities far larger seem to manage 3G. I have to assume that the limitation is rather due to the unwillingness of the telecommunications company to install new towers to meet this peak demand rather than actually reaching the ultimate speed limit. Otherwise in larger cities it would be impossible to get any useable speed with 3G. Is this the case?

I also think you’ll be able to overcome this limit through the use of directional antennas that can focus on specific areas without interferring with other antennas. For example, if you have instead of one omni-directional antenna servicing all the customers in a circle. You have 4 directional antennas servicing each quadrant. You should be able to service 4 times the clients in the original circle as each quadrant has its own wireless link and fibre backbone. Is that right?

I’m just interested because where I am the opposition party is pushing for a national wireless network instead of the incumbants fibre to the premise network. It would seem the oppositions party’s network would be a mistake but I can’t fathom if this ultimate wireless speed limit is actually a significant concern for the decision. Perhaps down the line, 20 years from now when we all want terrabyte access this limit would be sigificant but both systems will fail at that point.

Concerned Citizen

November 25, 2012 at 1:24 pm

5. [...] frequency creep is partially due to the issues related to the ultimate wireless bandwidth speed limit I outlined, but is also driven by a slight misconception that can be found on [...]