在使用MATLAB将傅立叶级数拟合到离散数据之前，如何定义周期？

Question

I'm using MATLAB's fit function:

fourier_series=(x,y,'fourier8');

to fit an 8th order Fourier series to a set of discrete data (x,y). I need the period of the Fourier series to be 2*pi. However I can't work out how to fix this so that when I call the function it fits the series to my required period. Any suggestions would be greatly appreciated. Thanks.

Background to problem: I am analysing video capture data of cyclist's pedalling which outputs as a cloud of joint positions in 3D space over time. The joint positions change slightly every pedal stroke. I am therefore wishing to fit Fourier series' to these joint positions and joint angles as a function of crank arm angle to find the cyclist's "average" position. The period of the Fourier series' therefore need to be constrained to be 2*pi as the "average" positions must return to the same location when the crank arm angle is zero (ie top dead centre, TDC) and the crank arm angle is 2*pi (ie TDC after one crank arm rotation).

Currently MATLAB is selecting the period to be slightly greater than 2*pi which means that when I use the Fourier series' to calculate the cyclist's position, the cyclist's position changes for the same crank arm angle on consecutive pedal strokes.

Answer 1

The best way to force the fit function on a certain period is to resort to a custom equation model, via fittype . Another option (that will throw a warning) is to fix the lower and upper bounds of the parameter w to the same value, and select as solution method LinearLeastSquares .

A cleaner solution is obtained by observing that, since you already know the period the fitting problem is linear in the parameters, and so you can resort to the linear least-squares method. I'll show hereafter an example of this approach.

%// Build a simple time series with period 2*pi.
t = (0:0.01:4*2*pi)';
y = sawtooth(t);
T = 2*pi;

%// Compute the angular speed and the azimuth.
Omega = 2*pi/T;
alpha = Omega*t;

%// Build the regressor matrix, with n harmonics.
n = 8;
A = ones(length(t), 2*n+1);
for i = 1:n
    A(:,[2*i 2*i+1]) = [cos(i*alpha) sin(i*alpha)];
end

%// Solve the linear system.
%// The parameters are sorted as:
%// p = [y0 a1 b1 a2 b2 ...]'
%// being y0 the average of y, a_i the terms multiplying the sines
%// and b_i the terms multiplying the cosines.
p = A\y;

%// Evaluate the Fourier series.
yApprox = A*p;

%// Compare the solution with the data.
figure();
hold on;
plot(t, y, 'b');
plot(t, yApprox, 'r');