There is one important problem: your shot signal must be periodically repeated to be indicated as a spectrum on your analyzer. Only then the above procedure can work.
If you have only one shot run, the best way would be to use an oscilloscope with memory to record it. From the real pulse shape you can calculate the spectrum by the Fourier Transform.
Modern digital oscilloscopes include this function in "MATH"; you can record one shot and see its spectrum, too.