I had to put the .ini files somewhere, and as the user can create several .ini files of their own, the documents folder was the logical place to store them, along with any other output from the analyser.
I had a somebody with a similar problem when running Vista. It turned out to be permission settings for the documents folder although that may not be the case here.
A quick test: Uninstall the analyser and ensure the folder RSP Spectrum Analyser and anything it may contain has been deleted from your documents folder. Now install and run the analyser. At this point you say nothing appears on-screen. Once you have tried to run the analyser, check to see what folders, sub-folders and files (if any) have been created in your documents folder.
All the best