How to make sense of LowpassFilter, SampleRate and the cutoff frequency ωc in different Mathematica versions?

Question

How do you make sense of the scale of "time" and "sampling" when using LowpassFilter and SampleRate ?

Given the test data with components at 1Hz and 5Hz

data1 = Array[N[Sin[2 Pi #] + Sin[2 Pi 5 #]] &, 100, {-1, 1}];

I expect to apply a low-pass filter at 2Hz and 10Hz

data2 = N @ LowpassFilter[data1, 2, SampleRate -> 50];

data3 = N @ LowpassFilter[data1, 10, SampleRate -> 50];

And see that data3 contains a significant part of the 5Hz component.

What am I doing wrong here?

In Win7 64 and Mathematica 10.2 and 10.1 I get

ListPlot[
  {
   data1
   , data2
   , data3
   }
  , Joined -> True
  , PlotRange -> {-2, 2}
  , PlotStyle -> {Gray, Red, Blue}
  ]

while versions 9.0 and 10.0 behave differently.

Should I understand that these new versions have a bug?

Answer 1

It looks like your problem comes from the fact that $\omega_c$ will be referring to an angular frequency, so $$\omega_c=2\pi f_c$$ Therefore, you can get the desired result by changing your data2 and data3 definitions:

data2 = N@LowpassFilter[data1, 2*Pi*2, SampleRate -> 50];
data3 = N@LowpassFilter[data1, 2*Pi*10, SampleRate -> 50];

Alternatively, you can define it in terms of Hz:

data2 = N@LowpassFilter[data1, Quantity[2, "Hertz"], SampleRate -> 50];
data3 = N@LowpassFilter[data1, Quantity[10, "Hertz"], SampleRate -> 50];

And you get

Finally, the difference people seem to be getting in pre-10.1 and 10.1/10.2 seems to be different filter types. Specifying LowpassFilter will use whatever lowpass filter Wolfram has programmed in. It wouldn't surprise me to see them change the filter type, if it made sense to them, especially since there are other functions like ButterworthFilterModel that can be used for specifying the filter type explicitly.