I know that this is about low cost scopes but below is a link to a good introduction to FFT on LeCroy scopes.
Deep memory on a software based FFT is a must. The Chinese scopes are no match for the FFT and math functions on even the older 9300 series LeCroy scopes with deep memory. LeCroy had deep memory scopes with FFT probably 10 years before anyone else. Even before HP got into the business with their scopes that were lucky to do 6 bits and did not have FFT, and before Tek had anything that would do FFT.
With the advancement in processing speed they are more responsive than ever.
Scopes that do the FFT in hardware (dedicated hardware memory with dedicated FFT processing in hardware as opposed to a mixed approach) have the advantage of speed (Rohde & Schwarz does this on their high line scopes) but limitations in effective resolution bandwidth due to the fixed number of bins.
The ability to adjust the depth of the sample memory to optimize the required frequency resolution and or the processing speed is quite handy. Once you get your hands on a deep memory scope with FFT and see the advantages of variable depth sample memory you will never want to go back to wimpy or fixed memory scopes again.
My old 9354AL can vary the sample memory from 500 samples to 2M samples and I can do an FFT to zoom memory and then do additional math on the FFT like FFT average. Math on Math on Math is one of the great features of these scopes and they were built in the 1990's. The follow on scope generations are built on the same proven architecture with faster processing, more acquisition memory and even more math power.
These scopes trace their lineage back to Walter LeCroys' involvement with measurements in high energy physics and that is why they were way ahead of the curve in all matters of mathematical processing, including FFT.
See:
http://teledynelecroy.com/doc/setting-up-an-fftIt is still true that there are many occasions where you do need a good RF spectrum analyzer but for a lot of uses FFT on a good scope is still quite handy.
Sam
W3OHM