Filter Recommendation for Smoothing a Specific Signal

Question

I have data like the following:

x = [0,0,1,1.98999999999978,2.99000000000069,3.98000000000047,14.0500000000002,15.0599999999995,15.8700000000008,15.8700000000008,16.9399999999996,17.8900000000003,18.8800000000001,29.7900000000000,30.8299999999999,31.8700000000008,32.8699999999999,32.8699999999999,33.8299999999999,34.8599999999997,35.8199999999997,46.0400000000000,46.7600000000002,47.8100000000004,48.8000000000002,48.8000000000002,49.7600000000002,50.7500000000009,62.0400000000000,62.7900000000009,63.7900000000000,64.8000000000002,65.8000000000002,65.8000000000002,66.7800000000007,67.7900000000000,78.0900000000002,78.8999999999996,79.9000000000006,80.9000000000006,81.8999999999996,81.8999999999996,82.8899999999994,83.8900000000003,94.0900000000002,94.5500000000002,95.6400000000003,96.5500000000002,96.5500000000002,97.5500000000002,98.5700000000006,109.560000000000,110.510000000000,111.500000000000,111.500000000000,112.500000000001,113.500000000000,113.500000000000,114.520000000000,126.090000000000,126.090000000000,127.050000000000,127.050000000000,128.050000000000,128.050000000000,129.050000000000,130.040000000000,141.540000000000,142.570000000000,142.570000000000,143.530000000000,144.120000000000,144.120000000000,145.310000000000,157.330000000000,158.320000000001,159.320000000000,160.070000000000,160.610000000000,160.610000000000,161.660000000000];
y = [0,0,0.464601238374681,0.929755484255206,1.39546164308871,1.86171861902337,2.79588063231313,3.73223273143554,4.20122730982987,5.14084800633951,6.08263671695609,7.02658457720498,8.44653533211063,9.39584215120541,10.3472769347812,11.3008307275720,12.7351149203275,13.6939269261834,14.6548264413097,16.1000700040090,16.5828518024014,18.0342831256240,18.5191184145376,19.4903194207480,20.4635533783748,20.9509298520746,21.9271960550294,22.9054722725899,23.8857492968480,24.3766352288654,25.8522688397480,25.8522688397480,26.3451347901984,26.8384928487593,27.8266806453806,28.3215080581393,28.8168229284115,29.3126240906732,29.8089103782565,30.3056806233516,30.8029336570103,30.8029336570103,31.3006683091478,31.7988834085470,31.7988834085470,32.2975777828593,32.2975777828593,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,31.7988834085470,31.3006683091478,30.3056806233516,29.3126240906732,27.8266806453806,26.8384928487593,24.8680179010629,22.9054722725899,21.4388111269700,19.0044642269341,17.5499595010577,15.1360561892542,11.3008307275720,8.92092218717511,6.08263671695609,3.26378347149450,1.86171861902337,1.39546164308871,1.39546164308871,1.39546164308871];
plot(x, y);

The actual visualization of the data is as on the left of the image below. However, I want to smooth the data like the one on the right. Any suggestions/thoughts? Thank you.

Answer 1

Here is the easy version

I created this by simply ignoring the X-axis and assuming the data is uniformly sampled.

I think there is a problem with your sampling grid. It's very non-linear with large more or less periodic jumps in it. There are also duplicates in the x-axis that are NOT duplicates in the y-axis, which is physically impossible. That's very unusual and partially non-sensical) for a sampling grid and that's exactly why your data looks so funky.

I'm guessing something went wrong when importing or exporting the sampling grid.

%% data
x = [0,0,1,1.98999999999978,2.99000000000069,3.98000000000047,14.0500000000002,15.0599999999995,15.8700000000008,15.8700000000008,16.9399999999996,17.8900000000003,18.8800000000001,29.7900000000000,30.8299999999999,31.8700000000008,32.8699999999999,32.8699999999999,33.8299999999999,34.8599999999997,35.8199999999997,46.0400000000000,46.7600000000002,47.8100000000004,48.8000000000002,48.8000000000002,49.7600000000002,50.7500000000009,62.0400000000000,62.7900000000009,63.7900000000000,64.8000000000002,65.8000000000002,65.8000000000002,66.7800000000007,67.7900000000000,78.0900000000002,78.8999999999996,79.9000000000006,80.9000000000006,81.8999999999996,81.8999999999996,82.8899999999994,83.8900000000003,94.0900000000002,94.5500000000002,95.6400000000003,96.5500000000002,96.5500000000002,97.5500000000002,98.5700000000006,109.560000000000,110.510000000000,111.500000000000,111.500000000000,112.500000000001,113.500000000000,113.500000000000,114.520000000000,126.090000000000,126.090000000000,127.050000000000,127.050000000000,128.050000000000,128.050000000000,129.050000000000,130.040000000000,141.540000000000,142.570000000000,142.570000000000,143.530000000000,144.120000000000,144.120000000000,145.310000000000,157.330000000000,158.320000000001,159.320000000000,160.070000000000,160.610000000000,160.610000000000,161.660000000000];
y = [0,0,0.464601238374681,0.929755484255206,1.39546164308871,1.86171861902337,2.79588063231313,3.73223273143554,4.20122730982987,5.14084800633951,6.08263671695609,7.02658457720498,8.44653533211063,9.39584215120541,10.3472769347812,11.3008307275720,12.7351149203275,13.6939269261834,14.6548264413097,16.1000700040090,16.5828518024014,18.0342831256240,18.5191184145376,19.4903194207480,20.4635533783748,20.9509298520746,21.9271960550294,22.9054722725899,23.8857492968480,24.3766352288654,25.8522688397480,25.8522688397480,26.3451347901984,26.8384928487593,27.8266806453806,28.3215080581393,28.8168229284115,29.3126240906732,29.8089103782565,30.3056806233516,30.8029336570103,30.8029336570103,31.3006683091478,31.7988834085470,31.7988834085470,32.2975777828593,32.2975777828593,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,32.7967502586087,31.7988834085470,31.3006683091478,30.3056806233516,29.3126240906732,27.8266806453806,26.8384928487593,24.8680179010629,22.9054722725899,21.4388111269700,19.0044642269341,17.5499595010577,15.1360561892542,11.3008307275720,8.92092218717511,6.08263671695609,3.26378347149450,1.86171861902337,1.39546164308871,1.39546164308871,1.39546164308871];
clf
plot(x, y);
%% Try 1
hold on
n = length(x);
x1 = linspace(0,x(end),n);
plot(x1,y,'Linewidth',2);
legend('original','smoothed');
grid('on')

Answer 2

Here are a few options:

1. Lowpass Filtering:

2. Smoothing:

3. Savitzky-Golay Filtering:

Here is the overall comparison:

Note that these are representative plots. You can tune the filtering parameters to get better results. Or you could concatenate the results of two operations to get a better overall result.

Here is the code:

clear; close all; clc;
x_lpf = lowpass(x,0.03); x_smooth = smoothdata(x); x_sg = sgolayfilt(x,3,9);
figure(1) plot(x, y, 'r','LineWidth',2); hold on plot(x_lpf,y, 'y','LineWidth',2); hold on plot(x_smooth,y, 'g','LineWidth',2); hold on plot(x_sg,y, 'b','LineWidth',2); legend('Original','Lowpass Filtered', 'Smoothing','S-G Filtered') legend('boxoff') box off set(gcf, "color", "w");
figure(2) plot(x, y, 'r','LineWidth',2); hold on plot(x_lpf,y, 'y','LineWidth',2); legend('Original','Lowpass Filtered') legend('boxoff') box off set(gcf, "color", "w");
figure(3) plot(x, y, 'r','LineWidth',2); hold on plot(x_smooth,y, 'g','LineWidth',2); legend('Original','Smoothing') legend('boxoff') box off set(gcf, "color", "w");
figure(4) plot(x, y, 'r','LineWidth',2); hold on plot(x_sg,y, 'b','LineWidth',2); legend('Original','S-G Filtered') legend('boxoff') box off set(gcf, "color", "w");