A question about antenna gain and signal property
properties of the incoming signal to the antenna


You can't.

All you observe is the electric signal; there's infinitely many possible electromagnetic waves that can lead to an identical signal at the antenna output. 

Think about it: Say your wave comes in from the main lobe direction and has an amplitude of 1 V. 

Now imagine a wave with twice the power arrives from an angle with a gain of -3dB – the resulting signal will be identical.

So, you can't.