Interpolation of circular data

Hello everybody out there using MATLAB,

Is there a built-in function for interpolating circular data, i.e. a timeseries of angles (in degree) between 0° and 360° to other timepoints.

I've carefully read the documentation for the interp1 function, which doesn't really mention this case, although this post is mentioning it.

This post is mentioning the interpft function, but I'm not sure whether this would help in my case.

David Long (2021). Interpolate angles (https://www.mathworks.com/matlabcentral/fileexchange/65289-interpolate-angles), MATLAB Central File Exchange. Retrieved September 1, 2021. is showing a way using the unwrap function, which I don't quite understand.

Answer 1

[Bjorn Gustavsson]

To my understanding simply extend the input arrays such that you have a couple of points from the first few samples added onto the end adding 2*pi to those angles and a few points from the end of your arrays at the beginning subtracting 2*pi from those:

theta = linspace(0,2*pi-0.001,31);
f =  12*cos(theta) + 5*sin(3*theta).^2;
Theta = [theta(end-3:end)-2*pi,theta,theta(1:3)+2*pi];
F1 = [f(end-3:end),f,f(1:3)];
THETA = linspace(0,2*pi,301);
F2 = 12*cos(THETA) + 5*sin(3*THETA).^2;
Fi = interp1(Theta,F1,THETA,'pchip');

That way you force the interpolation to be periodic, you only need to braket your theta and f array with three (possibly 2 but I'm too lazy to check) points to get the cubic and spline interpolation-methods to give you a periodic interpolation.

HTH

[Nicolas Bourbaki]

Thanks a lot for your help. I'm not quite sure whether I understand your suggestion correctly. The problem is, that the angles are not ordered. There might be a sequence of angles somewhere in my vector, where the angle changes from say 358° to 1° and `interp1` would yield a value of 179.5° instead of 359°. (A solution in radians would of course also be fine.)

[Bjorn Gustavsson]

Oh, I see, I misunderstood your problem completely. Here's my second attempt at understanding clear plain English...

t = 1:151;
theta = cumsum(rand(size(t))/2);
thetai = unwrap(atan2(interp1(1:51,sin(theta),ti),interp1(1:51,cos(theta),ti)));

This will give you a thetai without the branch-cut at 180,-180 (or 360,0).

HTH

[Nicolas Bourbaki]

Thanks, using `unwrap` was what is needed in this case.