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Consider a circular antenna array (UCA) with $N$ antenna elements as part of the CW bistatic radar system.

For each element we receive a signal of length $L$: $x_n[k]$, $k=1...L$, $n = 1...N$.

Оne of the antenna elements is directed towards the transmitter. Thus, we can find cross ambiguity functions (CAF) as cross-correlation functions between the reference channel and the remaining $N-1$ channels. Further, in each CAF, we detect targets and determine the ranges to them. My question is how to define directions on a target.

Standard explanations include beamforming. The way I understood this method: the samples from each of the channels, except for the reference for same time point, are used to form the directional diagram, i.e. from the $N \times L$ of the signal samples, $L$ diagrams are formed. Each directional diagram includes lobes directed to all targets as well as side lobes.

How to find out which direction corresponds to each target with its own range and how to exclude side lobes if their amplitude can be higher than the amplitude of the lobe directed at a target with a weak reflected signal?

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You use complex symbols and at the same time for me it seems very strange, the concept you describe. Its hard to guess if you are a professor and actually want to know exactly what you are asking, or a student and want to know general concept about finding targets with antenna array.

If you want general answer, then your whole aproach is far from what is used today. You cant effectively detect direction from signal's strength alone. And having a working antenna array you are much better off using phase shifters that are supposed to be in that array, that will perform a convolution for you in hardware almost for free. https://en.wikipedia.org/wiki/Convolution https://en.wikipedia.org/wiki/Phase_shift_module

In simple terms, you change signal delay until signal added up from two antenas becomes most powerful. This value of delay will tell you the direction of the target. Do it for every pair and you can get the direction very precisely. This is what beamforming usually is.

If you ment exactly what you are asking, then it is very complex. You can collect the signal strength for each antena, then try to make a simulation, simulate what strength would each antena show if signal would be in a particular location. You use your array data, such as each antenna's direction and position to make this simulation. And you add each particular antena's radiation pattern. In simulation you are likely better off using monte-carlo method, calculate the collection of signal strengths for a few points. Select the best fitting point, calculate a few points nearby that point, with more compactly placed points. Proper antenas wont work at all for this, regardless of solution, even with the one i offered because they have symmetrical radiation pattern and will together only show how far the target's direction is from the array's center direction. Best antenas for this tasks are likely half-covered, to provide very unsymetrical radiation pattern, that will help with detecting the direction. I dont think anything like this exist today, and I've never seen anything like this in history. It could be beneficial only before phase shifter were popularised. I agree that it can be beneficial for very low tech radar, if you are actually making this, i would like to read it.

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