Unsure about designing an ultra wide band antenna
I know this is not a good answer. But once the internal antena of my cell phone broken, so it showed no signal. I did cut a same size stip from a copper plate and soldered. I got full signal strenth. It was about 2 cm. I think you can start from a cell phone to experiment at GHz range because it will show the signal strenth. I dont talk about wave equation , because you said that you know all that. Good luck!!