I want to locate a small ball in the air 1000 times a second. The ball is thrown from the ground and then flies in the air for 0.5 seconds until it lands. Distance from the location instrument is 1 meter
That probably means you're not satisfied with a range resolution of a couple meters – you need a much finer measurement.
That means your system must have a very high bandwidth to allow for sufficient localization accuracy.
And if yes what signal should it be (sine?),
lol, no. A sine's foremost property is that it repeats – so you can't do location with it.
You need a high-bandwidth signal that is as aperiodic as possible – white noise would be perfect, but there's technical and legal limits to how white noise can be.
what radio frequency should I use
You'd need to use a high frequency, because
- The higher the frequency, the smaller the antennas
- The higher the frequency, the easier it is to get a high bandwidth around that frequency
and what circuit should I use to generate the signal?
Since you'd need to produce a high-bandwidth signal that should be easy to detec by your receivers, probably a well-conditioned noise source, the legally required band filter, power supply … you're trying to build a very complex system! This question is HUGE!
That quickly becomes very complicated to build, even if you don't make your life unnecessarily hard by putting the transmitter (instead of a reflector) inside the ball.
You'd really rather want a monostatic or bistatic radar instead of a triangulating a transmitter, and a bit of target tracking.
Anyway, wrong technology at this point (there's excellent microwave radars in the 24 to 200 GHz range, but they are most certainly out of your price and complexity range, if you think image processing is hard).
For a handful dollars you can get ultrasonic transducers. These are usually used for industrial ranging applications. Getting 1 kS/s for multiple receivers might be a challenge with the cheap boards you can get from China, but really, a megasample multi-channel receiver isn't that hard to build (compared to the radio system you were sketching).
By the way, I'd heavily contest your "1000 measurements per second" number. That sounds very unnecessary; the system you're observing is physically not able to change its state unpredictably 500 times per second, making this number of measurement unnecessary.
As an engineer:
You're trying to solve something. Don't start that by finding a technology and then trying to roughly match the problem to it.
Instead, write down what you need to achieve. What is it that you want to achieve with that measurement? Is it predicting where the ball will land? Is it estimating side winds? Estimating irregularities in the ball's weight distribution?
From that follows which parameters you actually care about – for example, in a windstill room with balls of known mass and air drag, you really only care about exit velocity and angle of the ball – the rest of the ball's trajectory is fully defined by these two parameters. No need to 1000s of measurements, at all!
In a less well-isolated situation, you might need to figure out the parameters of a more complex physical model, and incorporate distortion and measurement noise. In any case, I really do not see your measurement requirement come out of that, anywhere.
Explicitly deriving your measurement requirements is the first, necessary step in system design.
Based on that, you'd then sit down again and consider how you can estimate these parameters. That will make your system way, way, way easier than what you're planning to do now!
If dealing with uncertainty and disturbances, you'll want to track/update your estimation. There's many state modelling tools that are employed for such tasks – a classic would be the Kalman Filter, which you'll see can work with very sparse updates and can predict complex trajectories from few noisy measurements.