I know what you mean, but you are forgetting range in the doppler range map. MBC spot size in F increases but it’s center is on the (large) range gate(s) created where the center of the beam is. As you get farther from it, in the noise is reduced and with enough SNR, a traget can be found.
Look at this map. Range is 0 to N range bins(created by the FM) and doppler from 0 to M Doppler bina
You say spot size increases, I agree. But on the range axis the MLC ia centered in the beam center at range 5th bin. If the target is getting tracked and is beaming, near the ground. The target will be in the 4th range bin, its signal return will be lower than the clutter. Now lets go on my first example. The target is on the 3rd range bin. If the side aspect RCS is big enough and SNR is sufficient track can be kept.
Why is the beamwidth 14 degrees? I’m not getting this or misunderstanding what you want to say here.
Of course if the BW is as large as 14° the area illuminated will offer a larger return bandwidth.
Sin(7-7) is infinity, I don’t get your math or notation. But if I’m understanding what you say, a 14° beam centered at -7° in elevation will look from angle (-14° to 0°[horizon]). If its for example ok.
Of course you get a an integrated look area of infinity. But the signal return is not infinity. There’s a point the noise is bliss. And simply ia ignored due to the SNR.
On the doppler range map MBC at f0 is from the altitude return to infinity. But the intensity is not uniform but centered on the beam. Even if you might argue that the MBC is not on the horizon( which is what I think you want to point out), the sidelobes do and they don’t get infinity return because the intersected area goes to infinity.
Think of which ground return is stronger, that at 40km or the sum from near horizon to some x distance the signal isn’t even picked by the receiver!
When you range gate HPRF, you take into account the range bins created by the FM. Look at the doppler map again, if we use these (FM) range bins, the target(if enough SNR) will pop out between the rangebin(s) the MBC is and 0.
Let me answer the other post later, is 4h30am here