The WayBack machine tells me I wrote some of this about 2002. I modify this in 2017 to use this terminology in contrast to this inconsistent terminology. Field strength is measured in volts per meter. Signal strength is measured in watts per square meter.

When I was a kid I wondered how radio stations avoided interfering with each other. “Different frequencies” was an obscure concept, when even audio frequency was still obscure.

Phase array radar and holograms are explained and probably invented in terms of wave concepts. They collectively seem in conflict with the inverse square law that seems like one of those laws of nature that you shouldn’t try to cheat on.

When FM radio became common in cars the phenomenon of fading and null spots became widely known.

There are germs of several ideas mixed in with the above comments. I want to introduce a scheme that draws on these ideas. What would it take to build a radio transmission system that put the strong spots where the only intended receiver was? By strong spot I mean where the signal is near a maximum—the opposite of a null spot. This is a bit like phase array radar where by regulating the phases of many transmitting elements, signal energy is delivered to a small part of the space that would receive the signal of any one of the elements transmitting in isolation. Think of it as phase array radar with ad-hoc empirical phases.

Instead of transmitters and receivers, I will refer to stations and mobile units (MUs) for we want to design a two way system. Both would transmit and receive.

The idea is to install many permanent stations each operating on the same very narrow band. Each station would be a transmitting and receiving element. This is like phase array systems except that the stations are distributed hundreds of meters apart, where ever sites are conveniently available. Their phase would be adjusted to maximize the signal at the intended MU. This is far more energy efficient than current schemes where the signal energy is delivered over the entire cell. It saves both bandwidth and energy. The stations can simultaneously transmit different signals to several MUs, each MU located near its own strong spot. As in phase array radar, receiving is a variation on transmitting.

My intuition is that the stations would need to be spread in clumps of just a few, a few clumps per cell. The stations of a clump would need to be a wavelength or so apart. If all of the stations are within a few wavelengths of each other and those distances are small compared with the distance to the MU, then the strong spots will all have at least one long dimension. It will be a lobe that is long along the radius vector. If stations are spread apart at distances comparable with the distance to the MU then the strong spots may be small in all dimensions. (Well if the stations are distributed in mainly two dimensions, then the lobes will have a large vertical extent.)

There are several obvious, perhaps fatal problems with this scheme. Three I address here are (1) the ability to control phase to a few picoseconds, (2) the discovery of the correct phase for each combination of element and MU, (3) tracking these phases as the transmission paths change slowly.

Several times each second, all stations but one would shut up and listen to the remaining element that would set the standard phase for the neighborhood. Still, a highly stable oscillator is required.

The determination of the phase for each combination of station and MU would be empirical. If the frequency is used half-duplex then by reciprocity the transmit and receive phases are the same. I think that the phase information would need to be recomputed or adjusted a few times per second due to changes in the multipath configurations. The simplest scheme would be to reserve 1 microsecond a few times per second for any particular active MU. During that time just that unit would transmit a pure carrier. Each base station would then know its phase relationship with the MU. The oscillator on the MU is not so good but the relative phases between the base stations, regarding that MU, would be accurate. I do not anticipate that this would work well for moving cars, but that it could compensate for a few people milling around an MU.

### Problem with Receiving

It may not be feasible to require such stable oscillators in the MUs. The stations would not know what phase to receive from an MU. The signal addition might have to be done centrally by digitally transmitting over land lines time averaged signal strength for two orthogonal phases. (2017: bad idea) Such transmission would cover the needs of several MUs, however. Such time averaging would be over durations of a large fraction of a baud. I think that the phase relationship between stations and MUs might be a byproduct of this calculation.

### Math Annex

There is some surprising math here. Consider a phase array radar with 100 elements. You are a cloud that the radar has decided to ping. The radar runs each element with the same frequency but with the phases adjusted so that the 100 signals will be all in phase as they reach you. If your distance from the radar is r then the signal strength of the individual signals is each 1/r2 and you might think that the total strength would be 100/r2. That would about right except that it does not take into account that the signals are in phase. The field strength due to each element goes down as 1/r, and the energy (power) is the square of the field strength. The collective field strength of the elements is thus 100/r and the square of that is 10000/r2. Big difference! The signals at the neighboring cloud are systematically out of phase and the signal strength there is much less than 100/r2. In some sense the signal strength averaged over all directions is 100/r2

These ideas could be applied to DSSS except I wouldn’t know how to come up with effective inverses of the transfer function. This would likely be required several times per second for each MU and each base station. Tricks with the precomputed FFT of the sequence might reduce this problem to the previous solution.

This does not seem promising with frequency hopping.

See this for some different ideas based on the same physics.
It would be good to integrate this with current proposals for 3G service.
A significant precursor
Sampler

The generalization is that the towers should observe the field strength from each client phone and use proportionate field strength when responding to the phone, also coordinate the phases so that the signals add at the phone. Of course the transmit phases are the conjugate of the received phases. That just means that if a signal from the phone gets to some tower with phase advanced by 15 picoseconds compared to some other tower, adjust the phases in the opposite direction when transmitting to the phone.

Note that the system need not know or deduce distance, or multi-path effects, or even delay on the back-haul links; for each tower and each phone a field strength and phase, several times per second, is all that is necessary. I suspect that the major computing is the detection of this information.

For the towers to hear the phones, signals must be received by several towers and the phase adjusted before adding the signals. Mathematically the signals for one phone via several towers must be weighted by that complex number that relates the phone to that tower. I suppose that today (2016) only the tower with the strongest signal is used to accept information from the phone. Before the phases can be adjusted between towers for better reception, those adjustments must be known. Some of the signal from the phone must be known to the receiver so that that part may be devoted to deducing that relative phase. This is not trivial but refutes a class of bogus impossibility proofs.

The above plan ignores how many cycles of the carrier frequency the various paths have; ‘distances’ are modulo the wave length. As the data rate increases so that the bit time becomes comparable with the transmission time we have a problem. The mechanism that learns the phase between one phone and one tower can learn to count full cycles of the carrier. Multi-path is more complicated. While multi-path does not impact the long bit logic, it may limit the data rate if the signal element (baud) is not long compared with the multi-path delay. This problem is shared with conventional cell tower technology. The solutions are similar (deconvolving with the transfer function) but the transfer function is even more computationally expensive to track.

It is good to remember that the field strength of a signal (the electric field) is a vector and that the vector addition is a 3D thing as well as a complex number thing that includes phase. Later versions of LTE presume more than one antenna in the phone and this scheme is set up to exploit that as well. At that point we are into MIMO. If the phone and towers are all approximately at the same elevation and the tower antennas are all vertical, as is conventional, the signal at the phone will be vertical as well. Multi-path may well upset this and reflections are often the dominant signal. Just now I don’t know whether there is an opportunity for synergy between the pCell and the later LTE beyond what envisioned by the LTE authors.

phase array light