Questions on Need of filtering before decimation?

Question

This is a question related to the need for filtering before decimation. I understand that there are many questions likewise in this forum. I have gone through these and still little confused (or) may be overthinking about it.

• I have a test sequence(1x1000)sampled at 176.4 KHz(example) and I need to decimate it to let's say 44.1KHz. (the maximum frequency component in the sequence is 20KHz, I am taking examples from audio freqs).

• The way I understand decimation is, any sequence with sample rate Fs is decimated to Fsd(=Fs/D) ... then there is a higher chance that it violates the Nyquist sampling theorem and information is lost in the original sequence(if we don't think about filtering and just decimate it).

• In my example, if I decimate (by factor 4) 176.4Khz to 44.1Khz, it still satisfies the Nyquist sampling, so there will be no concept of aliasing since Fsd>2*20Khz ------ **In this case, do I need to apply LPF before Decimation. If Yes why ?*

• If I want to decimate by factor 16, i.e. 176.4KHz to 11.025Khz, it doesn't satisfy Nyquist sampling (as my new Fsd<2*20KHz).. aliasing occurs.So, we (undesirably) remove the higher frequency components in my test sequence by applying LPF (with cutoff freq ≤ 5.515 (F_coutoff ≤ Desired_sample_rate/2) )

LPF is applied such that there are no frequency components higher than 5.515KHz Is this how I need to choose my cut-off ??

Answer 1

If you think for a moment why you filter, it is to remove frequencies that would fold to the remaining bandwidth when decimating.

So if these frequencies that would fold do not exist in the audio then it will have same bandwidth whether you filter it or not. So in the first case, there is really no need to apply LPF first (unless there is noise that would fold and cause problems).

In the second case, it needs to be filtered, and you calculated the cut-off correctly. However, in practice it won't be a brick-wall filter, so depending on needed filter performance and allowable transition bandwith, you might want to select for example cut-off of 4.5 kHz and allow the filter to have about 1kHz transition band to have some reasonable attenuation level at 5.515 kHz.