Interpolate non-uniform signals

I have a 12 hour signal X, that was recorded at 1Hz.

However, some samples were missed and I don't have all 43200 samples (the 12h period in seconds [1Hz]). Furthermore, the sampling rate does not seem to be constant, meaning that besides missed samples, the interval between the ones I do have might be slightly above or below 1s.

Together with my signal of interest (X) I also recorded the timestamps (t) at which each sample was recorded (lengths of X and t are equal).

I need to resample/interpolate my signal sucha that I have 43200 samples.

Any suggestion how I could accomplish this?

Thanks in advance!

[Jan]

Of couse with an interpolation. If you show us, in which format your data are availble, it is easy to post the corresponding interp1 command.

Do you assume, that the data are smooth? Would you prefer a fair linear interpolation?

[MDias]

Hi Jan,

Here's one of the datasets (att.)

In the first column are the timestamps (in seconds) and in the second column the signal. Note the number of samples vs. the expected 43200.

The data is not expected to be smooth. More than having the "actual data" I need to preserve the order and spacing of the samples I do have. In other words, I'm perfectly happy with filling the missing values with NaN as long as the samples I do have are in their "correct place".

Cheers

[John D'Errico]

Magic would be easy, if we only had a magic wand. :)

t = data(:,1);
X = data(:,2);
t(1:10)
ans =
                         0
         0.750389099121094
          1.74427032470703
          2.78888702392578
          3.73203277587891
          4.77415466308594
          5.85184478759766
          6.77722930908203
          7.83953094482422
          8.87712097167969
          
diff(t(1:10))
ans =
         0.750389099121094
         0.993881225585938
          1.04461669921875
         0.943145751953125
          1.04212188720703
          1.07769012451172
         0.925384521484375
          1.06230163574219
          1.03759002685547      

How wild is the spacing?

max(diff(t))
ans =
          1.14429473876953
min(diff(t))
ans =
         0.750389099121094    

So not too bad.

Your data really is unequally spaced. You want to interpolate this onto a uniform spacing, perhaps at every 1 unit of time? Before we go any further, lets plot it. ALWAYS PLOT EVERYTHING. And then look carefully at the plot.

plot(t,X,'.')

Sigh. You want to interpolate that????? Wildly oscillatory swings, what appear to be effectively random spikes and outliers. As well, a system that appears to oscillate between two distinct levels for a large part of the plot.

Can we make it easier, and just pick random numbers for a result? :)

What would a meaningfully interpolated plot look like here? If you were able to draw a good curve through that mess, would it be smooth? In that case, you do NOT want to interpolate, but you want to do smoothing. So what would you think is a reasonable result from that data?

[MDias]

Hi John,

My signal corresponds to the position of an object. The object sometimes jumps arround and there are omisions due to detection issues. This is data for one block but the actual recording lasts many days. The interpolated data is not really what I care about and filling missing values with NaN is perfectly fine.

Cheers

[MDias]

@John D'Errico I just saw your edit.

The swings are what I care about (transitions between 300 and the last bit at 100 for example). But for the next steps I do need each 12h block to be sampled at 1Hz (43200 samples).

Cheers

Answer 1

[Jan]

John D'Erricos warning is important: The data do not look like an interpolation is fair. But if you have a good reason to do this, this is how it works:

data2 = interp1(data(:, 1), data(:, 2), 0:(12 * 3600 - 1));

A comparison:

figure;
plot(data(:, 1), data(:, 2), 'ro');
hold on;
plot(0:43199, data2, '-');

You see the typical effect of removed spikes for a linear interpolation.

[MDias]

Thanks! I'll test it

[John D'Errico]

You definitely never want to use a spline interpolant with data like this. But even a linear interpolant may be too much, as the linear interpolant tends to average neighboring points together, but somewhat randomly.

plot(t,X,'.')

[min(t),max(t)]
ans =
                         0          43198.1130371094
                         
that = 0:1:max(t);
Xhat0 = interp1(t,X,that,'nearest');
Xhat1 = interp1(t,X,that,'linear');

plot(that,Xhat0,'.')

plot(that,Xhat1,'.')

So the nearest neighbor interpolant retains much more of the character of the original plot, showing the dual nature of the series, where it seems to oscillate between two distinct levels.

Once the data is onto a uniform spacing, now some variety of Savitsky-Golay filter (with a wide window), or perhaps a wide window median filter might be appropriate.

Answer 2

[Star Strider]

Try something like this, using the Signal Processing Toolbox resample function —

LD = load('example_data.mat');
data = LD.data;
t = data(:,1);
s = data(:,2);

Fs = 1;                                             % Sampling Frequency (Hz)
[sr,tr] = resample(s,t,1);                          % Resample At Uniform Sampling Frequency Of 1 Hz, Return Interpolated Signal (‘sr’) and Time (‘tr’) Vectors

figure
plot(t, s)
hold on
plot(tr,sr)
hold off
grid
legend('Original','Resampled', 'Location','best')

.

[MDias]

Thanks! I'll give it a try.

Cheers

[Star Strider]

My pleasure!

The resample function uses a common technique to interpolate unevenly-sampled signals to a uniform sampling frequency, since this is required by all digital signal processing procedures that I am aware of. The function uses an anti-aliasing filter to prevent aliased signals from appearing in the resampled vectors, a common problem using simple interpolation.