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

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.

## 2 thoughts on “Physics of Wireless Broadband”

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