Why doesn't HighpassFilter fully remove the DC component? [duplicate]

Question

The high pass filter is not completely removing the DC component. See the picture. Blue is the signal before filtering, green is after.

Here is some background on the subject matter. I'm studying the beats of the heart in order to get something called HRV. The important aspects of HRV are between about 0.07 and 0.15 Hz. The standard practice in the literature is to pass the data through a band pass before analysis. I'm passing it through a low and high pass filter because I wanted to inspect the affects of both operations on the data. It is a bit of an art, because I want to smooth the data, but not distort the important information.

As requested here is the code. The data points are not evenly spaced in time, so the first step is to use interpolation to make evenly spaced points. Also, below that is 500 points from the data set.

RRData = RRDataCorrected;
 (*Below is plot of raw data before interpolation or filtering.*)
 (*For this plot I subtract out the DC signal.*)
 dumplot = 
 ListLinePlot[
  MapAt[# - Mean[RRDataCorrected[[All, 2]]] &, #, 2] & /@ 
   RRDataCorrected, PlotStyle -> {Blue}, 
  PlotMarkers -> Graphics@{Disk[{0, 0}, [email protected]]}, 
  AspectRatio -> 1/10, ImageSize -> Full];

myInterpolation = Interpolation[RRData, InterpolationOrder -> 3];
filteredUniformRRData = 
  Table[{i*beatinterval, myInterpolation[i*beatinterval]}, {i, 0, 
    Length[RRData] - 1}];
filteredUniformRRData = 
  Transpose[{Table[i*beatinterval, {i, 0, Length[RRData] - 1}], 
    LowpassFilter[filteredUniformRRData[[All, 2]], 1.5000000]}];
filteredUniformRRData = 
  Transpose[{Table[i*beatinterval, {i, 0, Length[RRData] - 1}], 
    HighpassFilter[RRData[[All, 2]], 00.1]}];

Show[ListLinePlot[filteredUniformRRData, PlotStyle -> Green, 
  PlotRange -> {All, All}], dumplot, AspectRatio -> 1/10, 
 ImageSize -> Full, PlotRange -> {All, All}]




RRDataCorrected = {{1058.403`, 1.025`}, {1059.4905`, 
   1.1500000000000001`}, {1060.6830000000002`, 
   1.235`}, {1061.9095000000002`, 1.218`}, {1063.1030000000003`, 
   1.169`}, {1064.2330000000004`, 1.091`}, {1065.2795000000003`, 
   1.002`}, {1066.2455000000002`, 0.93`}, {1067.1585000000002`, 
   0.896`}, {1068.0600000000002`, 0.907`}, {1069.0095000000001`, 
   0.992`}, {1070.0900000000001`, 1.169`}, {1071.303`, 
   1.2570000000000001`}, {1072.545`, 1.227`}, {1073.731`, 
   1.145`}, {1074.8295`, 1.052`}, {1075.844`, 0.977`}, {1076.796`, 
   0.927`}, {1077.7195000000002`, 0.92`}, {1078.6645`, 
   0.97`}, {1079.6915000000001`, 1.084`}, {1080.832`, 
   1.197`}, {1082.0485`, 1.236`}, {1083.2810000000002`, 
   1.229`}, {1084.4685000000002`, 1.1460000000000001`}, {1085.5585`, 
   1.034`}, {1086.5575000000001`, 0.964`}, {1087.5020000000002`, 
   0.925`}, {1088.428`, 0.927`}, {1089.3880000000001`, 
   0.993`}, {1090.4410000000003`, 1.113`}, {1091.6165000000003`, 
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   0.91`}, {1579.9910000000013`, 1.081`}, {1581.1335000000013`, 
   1.204`}, {1582.3710000000012`, 
   1.2710000000000001`}, {1583.6280000000013`, 
   1.243`}, {1584.8395000000012`, 1.18`}, {1585.988500000001`, 
   1.118`}, {1587.0805000000012`, 1.066`}, {1588.118000000001`, 
   1.0090000000000001`}, {1589.100000000001`, 
   0.9550000000000001`}, {1590.019500000001`, 
   0.884`}, {1590.888000000001`, 0.853`}}

Answer 1

I don't like LowpassFilter and HighpassFilter. Somewhere there is a discussion on how to make them work, but I can't find it now. (Relevant posts are here and here.) I prefer Butterworth filters. These are standard filters.

The only problem is that they take a bit of work to use. First you make the filter, then you have to convert it to an infinite impulse response filter and then apply it.

To get started I have called your data data and as you had spotted it is not equally spaced sampled data so I interpolate and resample. I work out the average increment from your original data.

 {t1, t2} = {data[[1, 1]], data[[-1, 1]]};
    mean = Mean[data[[All, 2]]];
    inc = Mean[Differences[data[[All, 1]]]];
    int = Interpolation[data];
    d1 = Table[{t, int[t]}, {t, t1, t2 - inc, inc}];
    plotd1 = ListLinePlot[d1, PlotStyle -> {Blue},
      PlotMarkers -> Graphics@{Disk[{0, 0}, Scal[email protected]]},
      AspectRatio -> 1/10, ImageSize -> Full]

The next step is to make the filter and then apply it to the data. I start by defining the sample rate of the data (equals 1/increment). Then I note the start and stop frequencies you require. These have to be expressed as a ratio of the sample rate and we need to express this as a fraction of 2 Pi. I have only done the high-pass filter here, but the low-pass filter is a similar procedure.

sr = 1/inc;
{f1, f2} = {0.07, 0.15};
filtHP = ToDiscreteTimeModel[
   ButterworthFilterModel[{"Highpass", 2, f1/sr 2 \[Pi]}],
   sr, Method -> { "BilinearTransform", "CriticalFrequency" -> f1/sr}];
d1HP = RecurrenceFilter[filtHP, d1[[All, 2]]];
Show[
 ListLinePlot[Transpose[{d1[[All, 1]], d1HP}], PlotStyle -> Green,
 PlotRange -> {All, All}],
 plotd1, AspectRatio -> 1/10, ImageSize -> Full,
 PlotRange -> {All, All}]

Note that the start falls from a larger value. It takes a few samples to take out the mean.

The Butterworth filter has variants of lowpass, highpass and bandpass, etc. I suggest you experiment with those. I think you are taking the correct approach to check as you progress. There is another step which involves pre-warping frequencies link here (it is only necessary if you are dealing with filter values that approach half the sample rate).

Happy new year!

Hope that helps.