Apply filter before or after Downsampling?
I haven't looked at the reference yet , but just to answer the question, yes filtering is performed prior to down sampling to avoid aliasing back in the out of band signals. However if your signal is already appropriately bandlimited you could forego the filter entirely. There's various reasons you might want to do other, unrelated, filtering after the downsampling operation. In fact, doing such filtering after the downsampler is preferable due to the lower sample rate and thus lower computational complexity.