Representing a set of data points as a Fourier Series

Hi, I wanted to represent a set of discrete data points as a fourier series. Since I don't have a function I was wondering how to get the fourier coefficients. I tried researching and I think I should use fft(), but I dont know how exactly and how I would go about getting the coefficents from there as I am new to the concept of Fourier Series and am having trouble understandin them. The data points are [x,y]

[1 13.38

2 15.33

3 18.11

4 21.15

5 25.74

6 27.89

7 29.18

8 29.74

9 29.32

10 26.01

11 19.79

12 17.24].

Any help would be greatly appreciated.

Answer 1

[Star Strider]

Try this:

data = [1  13.38
        2 15.33
        3 18.11
        4 21.15
        5 25.74
        6 27.89
        7 29.18
        8 29.74
        9 29.32
        10 26.01
        11 19.79
        12 17.24];

t = data(:,1);
s = data(:,2);

Ts = mean(diff(t));                                 % Sampling Interval
Fs = 1/Ts;                                          % Sampling Frequency
Fn = Fs/2;                                          % Nyquist Frequency
L = numel(t);                                       % Signal Length

smean = mean(s);                                    % Mean Amplitude
FTs = fft(s - smean)/L;                             % Fourier Transform
Fv = linspace(0, 1, fix(L/2)+1)*Fn;                 % Frequency Vector
Iv = 1:numel(Fv);                                   % Index Vector

figure
plot(Fv, abs(FTs(Iv))*2)
grid
xlabel('Frequency')
ylabel('Amplitude')

The coefficients are ‘FTs’. Note that this calculates a two-sided Foiurier transform, however only a one-sided Fourier transform is plotted.

Subtracting the mean of the signal in the fft call makes the peaks other than the zero-frequency peak more easily visible. This signal has a mean value of 22.74, so the zero-frequency peak would hide the other peaks if the zero-frequency peak was included in the plot.

[Samuel James]

Thank you for your answer, although I'm afraid this isnt quite what I'm looking for. It seems I did a poor job explaining my problem (my mistake). I somehow forgot to point this out, but this is a time series. I just wanted to model this time series using a fourier series, not to get a frequency amplitude graph. I also forgot to point out that my goal is to get the coefficents an and bn of the fourier series, the problem being that I dont have a function . As I mentioned earlier, I initially thought of using fft() to get the fourier coefficients but was unable to do so. Sorry for the confusion, I got very little sleep last night.

[Star Strider]

You do not need a function. The Fourier transform decomposes the original signal into sine and cosine elements of it, so the reconstruction of it (using the inverse Fourier transform) will re-create the original signal.

The cosine coefficients are the real parts of ‘FTs’ and the sine coefficients are the imaginary parts of ‘FTs’.

The fundamental frequency is ‘Fs’ and the relevant harmonics are integer multiples of it, corresponding to the elements of ‘FTs’.

[Samuel James]

Ok, thank you so much

[Star Strider]

As always, my pleasure!

[omar ali]

This code has been very useful to me. I would be grateful for how a plot of phase angle vs frequency could be plotted for this case?

[Star Strider]

@omar ali —

figure

subplot(2,1,1)

plot(Fv, abs(FTs(Iv))*2)

grid

ylabel('Amplitude')

subplot(2,1,2)

plot(Fv, rad2deg(angle(FTs(Iv))))

grid

xlabel('Frequency')

ylabel('Phase (°)')

.