# 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

With a title like the one I’ve chosen, I’m sure only the die hards will get even this far, so this post will be a somewhat technical follow-up to my informal post yesterday that was picked up by Felix Salmon and also noted by Paul Krugman.

In graduate school, I had fun building large-scale simulations where I modeled the propagation of radio waves transmitted from a cell phone through a 1 cubic millimeter resolution model of a human head

SAR distributions for the sagittal slice

SAR distributions for the frontal slice

SAR distributions for the coronal slice

These simulations are generally extremely accurate. The physics is well understood and can be modeled with a high degree of confidence. One of my favorite stories is when my office mate was modeling an aircraft and comparing results to measurements, he noticed the angle seemed off by .5 degrees. He called up the lab and, sure enough, the measurement was off by .5 degrees from what was specified. The simulation was more accurate than the measurement.

As mentioned yesterday, the behavior of a radio wave depends on its frequency and hence its wavelength. The two are related by

$\lambda = \frac{c}{f},$

where $\lambda$ is the wavelength, $f$ is the frequency, and $c$ is the speed of light. To simplify things, we can write that formula as

$\lambda (cm) = \frac{30}{f(GHz)}.$

For example, if the frequency is 1 GHz, i.e. $f(GHz) = 1$, then the wavelength is 30 cm, i.e. $\lambda(cm) = 30.$ When we double the frequency to 2 GHz, the wavelength reduces by half to 15 cm.

The wavelength is an important number to keep in mind because radio waves interact more strongly with objects whose size is roughly on the order of the wavelength of the radio wave. When I was in grad school, the frequency we were looking at was 900 MHz (.9 GHz) with a corresponding wavelength of roughly 33 cm. In a comment on Felix’s blog, Mark states:

It’s amazing that Phorgy can make so many technical errors and still make you worry that he’s right. For example:

“That is why the…7-800 MHz range is so valuable for cell applications.”

The 700 MHz band has never been used for cell phone communications in the US. It was auctioned off, but the spectrum is unused at present.

There is a big difference between what is valuable and what is available. Things are valuable sometimes precisely because they are not available. My statement was about the frequencies at which cell applications would be better off. I could have and maybe should have made the range a little broader, say 700-900 MHz, but that was not the point I was trying to make. The point is that radio waves propagate nicely, i.e. they ooze well, in the 700-800 MHz range. Higher than that and we begin to see directionality creep in, i.e. the radio waves begin to have preferred directions and the coverage becomes less uniform. For a very nice interactive demonstration of this, have a look at this:

Radiation Pattern of a Linear Antenna

The important number in that demonstration is the “Dipole Length (Wavelength)”. This is the length of the antenna relative to the wavelength. So in that demo, setting the slider t0 .5 means the antenna is half the length of the wave. For numbers up to 1.0, the pattern is fairly uniform, but once you get above 1.0, you start to see nulls where there is no radiation. This is one source of directionality in wireless signals.

From Felix’s article, we have:

For one thing, Phorgy’s limit of 1,000mps in total for a few city blocks is I think far higher than anything AT&T is currently able to provide. With what Baruch calls “compression, prioritisation, all that level 4-7 stuff you can do at the packet level” (don’t ask me), you can serve a lot of people with that kind of bandwidth.

The number I gave (1,000 mbps) was a “guestimate”, but it wasn’t a wild guestimate. When/if the formal studies are done, I am confident the number will not be too far off from this. This number includes “compression, prioritisation” and even polarization and modulation. I’m not talking about spectrum here, I am talking about the total availability of bits to everyone within a given vicinity. The number available to any one person will be simply this number divided by the number of people simultaneously downloading stuff within this vicinity. This is similar to the early days of cable modems. You could tell your neighbor was downloading a pirated movie because your connection drops to a crawl.

To be sure, my note was meant to convey an important message and sometimes a degree of license is warranted. No one should believe that any phone company has built out enough towers to reach the ultimate wireless speed limit, but how many people knew there was a wireless speed limit?

A couple years ago, I was telling people that in a few years, we would begin to see the limits of wireless broadband. I think we are beginning to see it, but we still have a way to go before we truly hit that ultimate speed limit. But we will.

There is no number of towers or “Wi-Fi” hot spots that will overcome this physical limitation. For one thing, you cannot put Wi-Fi hotspots too close together or they start to interfere with one another. You can be clever and switch to a neighboring channel, but that again only delays the inevitable.

Like another comment by loganb on Felix’s blog points out:

The real limit isn’t Heisenberg’s, it’s (Claude) Shannon’s, and those limits only apply to limits on the capacity of a given base station.

This is a great point, but I would say the two go hand in hand. Shannon and Heisenberg together determine the ultimate limit of how much information can by communicated via wireless broadband within a given vicinity.

One day, in the not too distant future, instead of going into a cafe and hopping online via a Wi-Fi hotspot, that same cafe will have a fiber optic plug next to the salt shaker.

Written by Eric

December 30, 2009 at 10:28 am

## Hello Heisenberg: “New York City not ready for the iPhone”

Interesting story from The Consumerist:

AT&T Customer Service: “New York City Is Not Ready For The iPhone”

Recall an earlier article of mine (from July 2007):

I chose a technically incorrect term “Wi-Fi” because that is what most people were talking about back then, but the subject was more generally about “wireless broadband”.

It is not that NYC isn’t ready for the iPhone. It is that NYC was the first to bump up against the inherent physical limitation of wireless broadband. There is no number of towers that will be able to accommodate hundreds or thousands of people within a small vicinity all expecting reasonable wireless bandwidth. There is a little thing called the Heisenberg uncertainty principle that no amount of marketing or engineering will be able to get around.

Edit: Here is a copy of a comment below in response to the questions:

I’ll try to write a separate article, but this is about the physics of waves.

Unlike finance, the physics of electromagnetic waves is well understood. Computer programs can be written to model radio waves to many digits of accuracy.

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. That is why the (relatively low frequency) 7-800 MHz range is so valuable for cell applications.

In recent years, the frequencies of cell phones and even more recently, smart phones, has increased from a little over 1 GHz to over 2.5 GHz (and beyond).

(Note: Your microwave oven operates at the same frequency as most smart phones now.)

As frequencies increase, the waves start acting more like laser beams. They no longer ooze around corners. You start to get “shadows” or dead spots with no signal. It becomes more difficult for the signals to penetrate walls etc. These problems get worse the higher you go in frequency.

An extreme case is an actual laser. Here, it becomes more difficult to distinguish the wave dynamics from particle dynamics. Like in Star Wars, the laser beams can bounce around like particles.

So we have two extremes: low frequency molasses waves and high frequency laser beams. As bandwidth demands increase, we begin moving the dial away from molasses (where we have good wireless signals) to laser beams (where we have dark spots, shadows, with no signal, etc).

There are many clever modulation tricks that delay the inevitable, but the basic rule is that you cannot defeat Heisenberg. This is an imprecise (but I hope effective) analogy that relates to the fact that at lower frequency (and longer wavelength, i.e. larger “effective size” of the wave), you have more certainty as to “where the photon is” (because it is coming from a relatively smaller antenna) you have more uncertainty about where it goes, i.e. it goes everywhere like a good wireless signal should. At higher frequency (and shorter wavelength), the antenna is relatively larger (compared to the wave) so we know less precisely where the photons are, hence we have more certainty as to where they are going, i.e. in a straight line instead of around a corner, which is undesirable for a wireless signal.

This physical fact does not deter marketing people. You can easily set up a demonstration on a van driving down the highway at 65 mph with an antenna mounted on top downloading web content at 100-1000 mbps. Don’t fall for this trick! They are essentially shining a laser beam at the van and tracking it down the road. Ask them to do the same demo with 1000 vans stuck in LA traffic. Forget about it.

I’d guestimate that a practical limit for the available wireless bandwidth in the air in a vicinity of say a couple square NYC blocks would be 1000 mbps. This is the TOTAL BANDWIDTH AVAILABLE FOR EVERYONE WITHIN A FEW NYC BLOCKS. So now divide 1000 mpbs by the number of people downloading stuff wirelessly taking into consideration highrise buildings, etc. That is probably a decent estimate of what the long term limits of wireless broadband would be.

So you can see, for the early adopters, wireless bandwidth is great! “Geez! This wireless is faster than my ethernet!” But once you start adding some real traffic, say 100 or 1000s of people all downloading stuff wireless within a small vicinity and you can imagine that we will easily bump up against the basic physical limitations as communicated by my good friend James Clerk Maxwell.

Written by Eric

December 27, 2009 at 4:42 pm