Finding the side lobe level of an antenna array

Question

I have a circular antenna array with N elements, for which the array factor can be defined as follows: $$AF(\phi)=\sum_{n=1}^N I_n \cdot e^{j \cdot(kr \cdot \cos(\phi-phi_n)+\beta_n}$$ I search a lot, but i couldn't found any exact and unambiguous method for finding the side lobe level or maximum SLL. As I understand, I can do one of the following to obtain the max side lobe level with some program:
A.

1. sample AF with small step sizes and \$-\pi<\phi<\pi\$
2. find max of \$|AF|\$, and the second peak, their difference is Max SLL
   

B.

1. sample AF with small step sizes and \$-\pi<\phi<\pi\$
2. normalize \$|AF|\$ by \$NormAF = |AF|/\max(|AF|)\$.
3. find max of NormAF , and its second peak, their difference is Max SLL
   

So the question is whether to use normalization or not. In some notes like (https://en.wikipedia.org/wiki/Side_lobe), looking at figures it appears they have used normalization, while in some others it is not.
finally, should we use 20*log10 (base 10 log) to obtain the result in decibels or it is already in decibels?

Answer 1

Regarding your two expressions for computing the sideline level, they are equivalent. Let \$AF(\phi_1)\$ be the largest peak, at \$\phi_1\$ and \$AF(\phi_2)\$ be the second largest peak in the pattern, at \$\phi_2\$. I will be assuming at this point that the array factors are in linear voltage units as described by your expression for \$AF(\phi)\$.

Case A:
"find max of \$|AF|\$, and the second peak, their difference is Max SLL"
$$SLL_A = \frac{|AF(\phi_1)|}{|AF(\phi_2)|}$$

Case B:
"normalize \$|AF|\$ by \$NormAF = |AF|/\max(|AF|)\$" $$AF_{norm}(\phi) = \frac{AF(\phi)}{\max(|AF(\phi)|)}$$ "find max of NormAF , and its second peak, their difference is Max SLL"
$$SLL_B = \frac{\max(|AF_{norm}(\phi)|)}{|AF_{norm}(\phi_2)|}$$ By the definition above \$\max(|AF_{norm}(\phi)|) = 1\$ $$SLL_B = \frac{1}{|AF_{norm}(\phi_2)|}$$ Similarly, by definition, \$AF_{norm}(\phi_2) = AF(\phi_2)/\max(|AF(\phi)|)\$ $$SLL_B = \frac{\max(|AF(\phi)|)}{|AF(\phi_2)|}$$ But, by your definition for the max value of the pattern \$\max(|AF(\phi)|) = |AF(\phi_1)|\$ $$SLL_B = \frac{|AF(\phi_1)|}{|AF(\phi_2)|}$$ Therefore we can see that \$SLL_A \equiv SLL_B\$

Regarding to conversion to decibels, the expressions for sideline level above are in linear voltage ratios. To convert to decibels: $$SLL_{dB} = 20\log_{10}(SLL)$$

The convention used here is that sidelobe levels are positive numbers in decibels, sometimes the convention used is opposite for the ratios and the sidelobe level is a negative number.

Also be aware that many tools for computing sidelobe level and antenna patterns will directly output the values in decibels rather than in linear voltage or power. Just something else to be on the lookout for.