I am trying to take a signal (as a list) and separate it into the low frequency and high frequency parts around some split frequency f.

Given that what little signal processing I ever studied is now decades in the past, I’m probably doing the wrong thing anyway but I am puzzled by the results and maybe someone can explain them and how I should do it.

I thought the easy way would be: Fourier my signal, clear all the components above (below) f and InverseFourier back…

The test signal is a 1s chord of C (CEG in equal parts C @261Hz) for f=300 the low part seems to give the C, but the high part is not just E & G as the following graphs show.

The code (including debug for completeness) is as follows

oneWave[f_, sampleRate_] :=   Table[Cos[i 2 Pi], {i, 0, (f - 1)/f, f/sampleRate}] nWaves[f_, sampleRate_, n_] :=   Flatten[Table[oneWave[f, sampleRate], {i, 1, n}]] ceg4 = ListPlay[a = nWaves[261.63`, 44100, 262];    1/3 (a + Take[nWaves[329.63`, 44100, 330], Length[a]] +       Take[nWaves[392.`, 44100, 392], Length[a]]), SampleRate -> 44100]  requencySplitDebug[signalList_List, splitF_?NumberQ, myDebugParams_List]:=      Module[         (* "Some common choices for {a,b} are {0,1} (default), {-1,1} (data analysis), {1,-1} (signal processing). " *)         {signalLen, lfPart, hfPart, fourierData, myDebugGraphics, myDebugData, lfIF, hfIF, fourierA=0, fourierB=1,rPad, lPad},         signalLen = Length[signalList];         fourierData = Fourier[signalList, FourierParameters->{fourierA, fourierB}];         rPad = PadRight[Take[fourierData,{1,splitF}],signalLen];         lPad = PadLeft[Take[fourierData,{splitF + 1,signalLen}],signalLen];         lfIF = InverseFourier[rPad,FourierParameters->{fourierA, fourierB}];         hfIF = InverseFourier[lPad,FourierParameters->{fourierA, fourierB}];         lfPart = Chop[Re[lfIF]];         hfPart = Chop[Re[hfIF]];         If[Length[myDebugParams]==3 && myDebugParams[[1]]==True && myDebugParams[[3]] > myDebugParams[[2]],             Print["Debugging On"];             myDebugGraphics =                  GraphicsGrid[{                     {ListLinePlot[Take[signalList,{myDebugParams[[2]],myDebugParams[[3]]}], PlotRange->All,PlotLabel->"Signal Input Detail"],                     ListLinePlot[Take[Abs[fourierData],{myDebugParams[[2]],myDebugParams[[3]]}], PlotRange->All, PlotLabel->"Signal Input Spectrum"]},                      {ListLinePlot[Take[lfPart,{myDebugParams[[2]],myDebugParams[[3]]}], PlotRange->All,PlotLabel->"Lower Part Signal Detail"],                     ListLinePlot[Take[Abs[Fourier[lfPart,FourierParameters->{fourierA, fourierB}]],{myDebugParams[[2]],myDebugParams[[3]]}], PlotRange->All, PlotLabel->"Lower Part Spectrum"]},                     {ListLinePlot[Take[hfPart,{myDebugParams[[2]],myDebugParams[[3]]}], PlotRange->All,PlotLabel->"Upper Part Signal Detail"],                     ListLinePlot[Take[Abs[Fourier[hfPart,FourierParameters->{fourierA, fourierB}]],{myDebugParams[[2]],myDebugParams[[3]]}], PlotRange->All, PlotLabel->"Upper Part Spectrum"]}                 }];             myDebugData = {fourierData, lfIF, hfIF};         ];          If[Length[myDebugData]==0,             Return[{lfPart,hfPart}],             Return[{lfPart,hfPart,myDebugGraphics, myDebugData}]         ];     ];  asplit = frequencySplitDebug[AudioData[ceg4], 300, {True, 1, 600}]; Show[asplit[[3]]] 

I really do want to get as close as possible to an ideal filter with execution time on a modest PC (i3) of ~1s for a list of length ~1000 because my real signal is data not audio and both parts need to be as clean as possible for the next processing steps.

I also tried a 7th order Buttwerworth filter on ceg4 (40k points) with high attenutation and a hour later it was still running

So, to summarise: what methods would you recommend for efficiently performing a signal split like this on uniformly sampled data? What if the signal were not uniformly sampled (i.e. a proper time-series) – fit and resample or…?

And why did my hf part still have the lf component?