What the heck is on that guyz roof?

I often see the strangest antennas strapped to someone's chimney or poking out of someone's trunk of the car. Usually it belongs to someone who's a ham radio enthusiast, or buddy searching for extra terrestrials or simply someone grabbing HDTV off the air. I've always been fascinated by the symmetry or the mathematical basis behind the aesthetic design, and "what does that antenna do?"

Image Source

I came across a Log Periodic Antenna design, while looking up the Blonder-Tongue Doctrine in US patent law oddly enough; the case set a precedent. Sometimes LPA is referred to as Isabel (in honor of one of the creators) at the University of Illinois.  The repeating nature also gives it a fractal design. The design parameters are simplified into four design criteria

1) Upper and lower operating frequencies set the shortest and longest dipole length
2) Number of elements
3) Apex angle of antennae
4) Each successive element is a scaled-down length of its immediate predecessor down the array.
5) The scaling factor τ (tau) derived from a log function.

The longest dipole is 1/4 wavelength of the lowest frequency; the shortest dipole is 1/4 wavelength of the highest frequency. The geometry diagram shows only the top half of the antenna; the bottom half is a mirror image of it. I don't understand why they have to make it a pyramidal design for aesthetics. They must do this to achieve a symetrical RF footprint instead for two lobe pairs instead of just a single side.  This article explains in detail how the formulas are used and the magic math involved.

I found an online calculator for the LPA design parameters but I just find it creepy that the page displays your IP address and makes snide comments about the browser you're using. Other than that, it's very accurate, I matched the numbers I punched in with a real Kathrein-Scala dual band wireless antenna spec sheet.

Cross Polarization

The LPA can be quite small to cover a fairly broad radiation footprint with a reasonable power gain. What's neat is only a part of the array is active at a given frequency. Therefore the antenna can cover a wide frequency band without the need of a switching system. This is good for television reception or as advertised, dual band wireless applications.

To the untrained eye, the Log Periodic Antenna could be confused as a Yagi or have many similarities but I'd say that the LPA is more triangular while a yagi is more rectangular.

Image Source