[music-dsp] the art of oversampling? aka Newbie question...

[Nigel Redmon]

Ross gave you the proper order. I'll elaborate a bit so you can see 
why. The upside to understanding "why" is that you'll never be 
confused again about the order or wether you even need to filter in 
special cases.

First, the main reason we upsample/process/downsample is for cases 
where we need more bandwidth for the process step, and a typical 
reason we need more bandwidth is that the process step is non-linear, 
creating higher frequencies that would otherwise alias significantly 
into the audio band, so my example here assume that. Upsampling lets 
significant alias products stay high enough in the oversampled band 
that they can be removed before returning to the original sample rate.

We start with a spectrum that looks something like this--a sample rate 
of Fs, and our unaliased audio in the band from 0 to Fs/2, with alias 
mirror images extending beyond that:

____     _________     _________
     \ | /    |    \ | /    |    \
      \|/     |     \|/     |     \...
      Fs/2    Fs           2Fs

If we up-sample by a factor of two (by placing a 0 between each 
sample), our new Fs (Fs') is the old 2Fs, and some of the alias image 
is now below Fs'/2--within the new audio band:

____     _________     _________
     \ | /    |    \ | /    |    \
      \|/     |     \|/     |     \...
             Fs'/2          Fs'

So, we usually need to get rid of the undesired image with a lowpass 
filter before further processing (if that undesired image will have 
consequences in the processing):

____                   _________
     \ |      |      | /    |    \
      \|______|______|/     |     \...
      Fs'/4  Fs'/2  3Fs'/4  Fs'

Then we can do some processing that might create significant content 
up to 3/4Fs'. Anything between Fs'/2 and 3Fs'/4 will alias, mirroring 
about Fs'/2 and reflecting back between Fs'/4 and Fs'/4. This gets 
harder to draw in ascii, so I won't, but suffice to say that the nice 
empty-ish space we made with the lowpass gets filled in.

When we're done processing at the higher rate and it's time to go back 
to the original sample rate, we use a lowpass filter again to get back 
to something like that third figure, and decimate by retaining every 
other sample. If we don't clear out that area first, the content there 
will fold back into our audio band at the lower rate (as usual, 
because Nyquist said we need to sample at twice the frequency that we 
need to reproduce, and after decimating, that band will be 
under-represented and alias).

The same applies for 4x and any other oversampling ratio.