LowpassFilter Mathematica

Question

I have this example signal:

w1 = 3; (* rad/s *)
w2 = 50; (* rad/s *)
w3 = 4;  (* rad/s *)
signal[t_] = Sin[w1*t] + Sin[w2*t] - Sin[w3*t]

I would filtrer this signal with ideal Low Pass filter (w3=50 rad/s) using command LowpassFiltersuch that the filtered signal is:

signalF[t_] = Sin[w1*t] - Sin[w3*t]

I tried so:

data = Table[signal[t], {t, 0, 5, 0.01}];
wc=50;
n=200; (*What does it mean specifically this parameter?*)
ListLinePlot[LowpassFilter[data, 50, 200, DirichletWindow]]

but I got the same starting signal. What am I doing wrong?

Answer 1

w1 = 3; w2 = 50; w3 = 4; 
signal = Sin[w1*t] + Sin[w2*t] - Sin[w3*t];
data = Table[signal, {t, 0, 5, 1/100}] // N;

Edit

To play with the cut frequency I have made this edit.

  Manipulate[
 Show[
  Plot[signal, {t, 0, 5}], 
  ListLinePlot[LowpassFilter[data, w], DataRange -> {0, 5}, PlotStyle -> Darker@Red]],
 {w, 0.5, 0.05, Appearance -> "Open"}]

I prefer filter with well-known transfer function, e.g. Butterworth filter. One can adjust the passband and stopband frequencies and the attenuations. Here, I choose a loss of 30 dB at the frequency of w2 = 50.

filter = ButterworthFilterModel[{"Lowpass", {w3, w2}, {0.1, 30}}, s];
out[t_] = OutputResponse[filter, signal, {t, 0, 6}];
ϕ = Arg[filter[N[I*w3]]]/w3; (*phase shift*)
Plot[{signal, out[t - ϕ]}, {t, 0, 5}]

Answer 2

I will restrict my answer to a class of signals given in the example. The signal is a Sine series, and I am assuming that w1, w2 and w3 are integers. Therefore you can extract the coeffients of the series:

coeffs = Table[FourierSinCoefficient[signal[t], t, n], {n, 50}]

with $n$ an integer that depends on your highest frequency.

Build your low pass filter by selecting the coeffs within your low frequency range:

LowPassFilter[coeffs_, n_] := Take[coeffs, n]

and reconstruct your filtered signal:

lowcoeffs = LowPassFilter[coeffs, 4];
funcs = Table[Sin[n t], {n, Length[lowcoeffs]}];
smoothsignal[t_] := LowPassFilter[coeffs, 4].funcs;

The resulting plot:

Plot[{smoothsignal[t], signal[t]}, {t, 0, 5}, PlotStyle -> {Red, Blue}]