In fact it is provable: mathematicians like to say, "you cannot comb a sphere".
Why is this applicable?
Light is polarized. It travels in some direction, and is polarized with respect to the directions perpendicular to the direction of travel. The polarizations can be phase shifted, so you get linear polarization of whatever angle, or circular of whatever ellipticity and axis angle. (Basically, polarization can be any Lissajous figure you can plot by varying the amplitude and phase of two sine waves of the same frequency.)
However we arrange an antenna array, we must emit some sort of polarization in every direction. The polarization and amplitude must change smoothly from point to point.
We can map these parameters onto the surface of a sphere, because we're looking for an isotropic -- spherically symmetric -- radiation pattern.
If we do this, then we must have the amplitude equal at all points. Fine. And we can have most of the polarization however we like, like say it's vertical all the way around the equator (like the case for a dipole's main lobe, a torus-shaped radiation pattern, the polarization is always axial).
We can keep polarization constant all the way to the ends, but then something happens
at the poles: looking down at the pole, to the left, polarization is aligned right; and to the right, polarization is aligned left. At the pole, the polarizations cancel, and there is an amplitude null at the pole (which violates our assumption of equal amplitudes at all points).
(In a real antenna, the width of the null corresponds inversely with the overall size of the antenna, in terms of wavelengths. We could make a very small null, with a carefully crafted, very large antenna; but that would be economically prohibitive.)
Or if we swirl the polarization, so it is vertical at the equator, and bends over (east-west) as we get closer to the poles, we get linear polarization around the equator and circular polarization from the top and bottom. But again we get a null at the poles, because polarizations are opposite on opposite sides of the poles.
A more accurate correction to our assumptions: we must have amplitude and polarization continuous at almost all points, except for finitely many discontinuities. A sphere with two poles (poles in the mathematical sense of an extreme point, as well as in the geographical sense) is such an example.
These are just two of many possible configurations, but it turns out
all possible configurations can be ruled out, and thus the
hairy ball theorem can be proven.
Incidentally, the example on there (a smoothly combed torus) is a dipole's pattern, down to differences in what angle the vectors are drawn at (the polarization is linear and axial), and the dimensions of the torus (a dipole's radiation pattern is a "horn torus", i.e., the inner diameter is zero). That's a perfectly consistent result.
In practical terms:
Not only will you always have nulls in the antenna response (and not be able to afford the space, and money, to shrink the nulls closer to the mathematically ideal points only), but you will have nulls due to unpredictable reflections in the environment as well.
Sidestep the problem entirely, by using several complementary antennas. Complementary in terms of polarization and radiation pattern, as well as spacing within the ambient multipath pattern. Do not combine them directly -- because that would create the same problem again -- but detect them separately. (The IF signals can be coherent -- that is, converted with the same LO, but passed through independent RF-mixer-IF-detector signal chains.) This is called diversity radio. The final output signals can be used directly, added together however you like (with whatever considerations might be necessary, of course).
The demodulated signals should be strong and coherent at this point, even if the antennas are in positions that would've otherwise interfered with each other (if simply connected together at RF). The case where diversity doesn't help much, is when multipath is strong (specular reflection off a metal building, say?) and the detected signal from one antenna, thus delayed, interferes with the other. (This kind of multipath was quite obvious on analog television, appearing as a shadow image, shifted right by the delay, overlaid on the main image.) A lot of digital comms include active delay compensation filters for this reason.
Tim
Home
Help
Search
Login
Register
