# Phased non linear array antenna - First Sidelobe

I have a problem I cannot seem to solve and I REALLY need some help.

It's about phased-array antennas whose dipoles are not equally spaced, not equally phased, not equally fed (amplitude).
Let's assume that all the dipoles are the same and that always form a matrix of one or more columns with the same number of elements.
Every element of this matrix is made of 2 dipoles: one polarized by +45° and the other by -45°. Below an example:

X    X

X    X

X    X

X    X

X    X


Where every X is a couple of +/-45° polarized dipoles.

Provided that I'm currently interested only in the co-polar (I'm analyzing one pol. at time) vertical far-field pattern, I'm trying to find out a formula to use in an algorithm which would give the angular location of the first sidelobe above the main lobe.

You should also know that while in theory there MAY be a solution, the data I have at disposal (vertical pattern measurement of such antennas) suffers from noise at low dBs and interpolation errors (the antenna is measured every 1°).

Main issues here are:

• I can search for relative minima and maxima, but I cannot always say that the first sidelobe is where the first relative maximum above main beam is, because there are many sidelobes which get "integrated" in the first one (see picture below), which contain anything but relative maxima (in this case the algorithm should return "85°"):

• I cannot trust relative minima to really be minima, since sometimes a sidelobe is so low that will not be shown, or it's presence will be "overwritten" by interpolation of near points (here the sidelobe should be in both images below more or less by 70°):

• Even if I found some maximum/minimum in the range from -40dB (plot limit) to -80dB (measurement limit), noise would come into play and I would be led to believe that what I found could basically not be a mini sidelobe

Moreover, I keep hearing that there is an experimental formula which approximates the position of the first sidelobe and that would be: Main beam peak angular position - 1.5 * Vertical Half Power Beamwidth
Which is mostly correct... But sometimes fails with a lot of error (up to 3° in the GSM frequency range - I checked it on the visible sidelobes).
My question is: Is there a better formula (best if it is totally transparent to the algorithm - basically if I can find what I need only with the data I have avaiable and not need other inputs like amplitude, phase and position of dipoles)?