Spreading of a pulse in time domain
Simon is right except there is one more assumption required. If the transmitter and receiver are moving toward or away from each other, then Doppler effects will cause the pulses to be closer together or farther apart in time.

One other kind of weird thing to consider. The channel between TX and RX could be time varying. The propagation speed of EM waves depends on the dielectric properties of the space in between. If a large dielectric solid material was somehow inserted between TX and RX after the first pulse, then the second pulse would be substantially delayed due to the slower EM propagation in the dielectric material. I don't think it is possible to do this in the space of 5ns, but since I don't know what your final application is, I thought I would mention it. Other things like humidity or rain or snow can have an effect also, but normally these are not high-speed processes.

I suspect multi-path will be your biggest problem. You will need to look into some kind of correlation detection scheme.

Also, you will have a hard time getting approval from FCC or similar agencies. In other words, have fun experimenting, but don't think you are going to market a product in the developed world based on broadband pulse transmission unless you are willing to go through a lot of trouble.

One last thing. The shape of your received pulse will depend somewhat on the channel bandwidth, which will depend on antenna bandwidth. The more bandwidth your antennas have, the closer the RX pulse will look to the TX pulse.