Here are example.
Yes, beamforming which is also known as spatial filtering, along with pulse compression and Doppler filtering.
Note how those diagrams match with this physical example which happens to be a module for Phazotron's Zhuk AESA radar.
Note that T/R does not necessarily mean TRM. TRM means T/R per module. DTRM is Dual T/R per Module or two T/R in one module. QTRM is Quad T/R per Module. The above example is a QTRM, and is actually more common with high frequency band like X band. You may not need to put a FPGA behind every T/R, but one in four is good.
It doesn't tell what could be done with a DSP on each element.
There is a master clock on every phased array radar, phase shifters , and radiating elements.
Now, the difference between AESA/PESA is in the case of PESA the signal amplification happens before the phase shifter, in the chase of AESA it happens after the phase shifter.
It means the AESA has a good 3db advantage on the transmitter and 3db on the receiver side.
The current generation USA radars use GaAs , and that has good 4-10 w/element performance .
The Irbis-e can generate 20 kW, it is around the capability of the GaAs.
But the AESA has 6db advantage on the Tx/Rx.
But the IRbis use receiving elements on each antenna, means the disadvantage is only 3db
To make the situation even more complicated, the Irbis can be rotated, that negate all transmitting advantage of the AESA, considering the rotation can give max 3db signal increase.
Now, having computer on each element how can help ?
The element receiving signal from an omnidirectional dipole, means there is nothing really to process/analyse.
Only the central receiver behind the phase shifters "see" the spatial picture.
All that the DSP in the element can do is to do same AD transformation, maybe controlling the phase shifter behind the backplane, and there is nothing else really.
OF course it can be used to make sub arrays and different channels, but that dramatically decrease the radar performance, and the PESA can do that as well with multiple emitters or multiple pulse .
No one can forget that cutting to half the radiation surface to scan with two beam actually will decrease the search volume by 80%.
So, fifth of the volume can be searched twice fast. wow.
The multi beam and all other thing is as interesting from warfare standpoint like the discovery of a 9 leg spider.