>> Your signal returns are from the beam or side lobes
You're skipping a step -- where does that beam come from? For simplicity lets think about a scene illuminated uniformly (i.e. from a single element) so that we don't get hung up on the transmit beam. I think we agree you could still sweep a receiving phased array beam across that scene. Lets further assume it's digital beamforming, so you're storing a copy of the signal incident _at every element of the array_. Not a 'beam' yet, just a bunch of individual signals.
>> you get a return from everywhere all at once
Yes! Think about each of those elements of the phased array -- they're also receiving signals from everywhere all at once.
It only becomes localized into a beam when you combine all the elements with specific phase weights. That process of combining element returns to form the beam is mathematically identical to what you do in SAR as well -- combine all your individual 'element' (individual snapshot in space) responses with some phase weights to get the return in one direction. Repeat the math in all directions to form one dimension of the image (second dimension is the radar time-of-flight bit, which is unrelated to the beamforming).
Maybe not you specifically, but I think people don't understand the 'synthetic aperture' part. Specifically, that you can ignore the time between snapshots (because the transmitter and receiver are synchronized) and act like all the snapshots the platform took across the line of flight happened simultaneously. What you're left with is the element responses to a big phased array, and you can 'beamform' using those responses.
You're skipping a step -- where does that beam come from? For simplicity lets think about a scene illuminated uniformly (i.e. from a single element) so that we don't get hung up on the transmit beam. I think we agree you could still sweep a receiving phased array beam across that scene. Lets further assume it's digital beamforming, so you're storing a copy of the signal incident _at every element of the array_. Not a 'beam' yet, just a bunch of individual signals.
>> you get a return from everywhere all at once
Yes! Think about each of those elements of the phased array -- they're also receiving signals from everywhere all at once.
It only becomes localized into a beam when you combine all the elements with specific phase weights. That process of combining element returns to form the beam is mathematically identical to what you do in SAR as well -- combine all your individual 'element' (individual snapshot in space) responses with some phase weights to get the return in one direction. Repeat the math in all directions to form one dimension of the image (second dimension is the radar time-of-flight bit, which is unrelated to the beamforming).
Maybe not you specifically, but I think people don't understand the 'synthetic aperture' part. Specifically, that you can ignore the time between snapshots (because the transmitter and receiver are synchronized) and act like all the snapshots the platform took across the line of flight happened simultaneously. What you're left with is the element responses to a big phased array, and you can 'beamform' using those responses.