How to calculate double Fourier coefficients A_kl and B_kl from W(x,y) when the W(x,y) was not defined in mathematical way?

As the title, if I have a data x,y,z which is axial length, circumferential length and deviation, respectively. And I would like to use double Fourier series to approximate the random field W(x,y) which is

So it means that I can use the Fourier coefficients A_kl and B_kl to describe the deviation pattern W(x,y)

But the problem is that the W(x,y) is a random data and it was not defined in mathematical way. Then how can I calculate the A_kl and B_kl from unknow W(x,y)?

I tried to use curve fitting to fit the surface but it doesn't give me satisfying result. (Black dots are my xyz data)

Interpolation fitting gives a good agreement with my scatter but I cannot get the mathematical definition of the fitting curve.

So my problem is: How to get the Fourier coefficients from arbitrary W(x,y) which was not defined? If it must be defined as a function W(x,y), then which method should I use? Hope that someone could give me an idea, thank u so much.

Answer 1

[Drishti]

Hi 서해 신,

To calculate the fourier coefficients from an arbitarary function ( W(x, y) ), it needs to be defined over a specified domain or can be approximated using the data points.

To approximate the function ( W(x, y) ) using a double Fourier series, we determine the coefficients ( A_{kl} ) and ( B_{kl} ) through integration.

You can refer to the below given code snippet for calculating the coefficients:

% Define the function W(x, y), taking example function 
W = @(x, y) sin(x/10) .* cos(y/10); 

% Initialize coefficients
Akl = zeros(n1+1, n2+1);
Bkl = zeros(n1+1, n2+1);

% Calculate coefficients Akl and Bkl
for k = 0:n1
    for l = 0:n2
        integrandA = @(x, y) W(x, y) .* cos(k*pi*x/L) .* cos(l*pi*y/R);
        integrandB = @(x, y) W(x, y) .* cos(k*pi*x/L) .* sin(l*pi*y/R);
        
        Akl(k+1, l+1) = (alpha/(2*pi*R*L)) * integral2(integrandA, 0, L, 0, 2*pi*R);
        Bkl(k+1, l+1) = (alpha/(2*pi*R*L)) * integral2(integrandB, 0, L, 0, 2*pi*R);
    end
end

% Define the approximation function
f_F = @(x, y) t * sum(sum(...
    cos((0:n1)'*pi*x/L) .* (Akl .* cos((0:n2)*pi*y/(2*pi*R)) + Bkl .* sin((0:n2)*pi*y/(2*pi*R))) ...
    ));

I hope this helps in getting started.