Decomposing the time series of wind turbulance using Fourier series, and computing the spectra.

I have wind turbulence data for every 30 minutes. I need to compute spectra using Fourier series and to get the graph in the image below. How can I do it with using MATLAB. In the attachment file, I have the u,v and w component of the wind turbulence. Thanks for answers.

Answer 1

[Star Strider]

Try this —

T1 = readtable('https://www.mathworks.com/matlabcentral/answers/uploaded_files/1125355/data-MOOCTurbulenceDataSeries.xls')

T1 = 36000×4 table
      u          v           w        time 
    ______    ________    ________    _____

    8.1492     -1.0046    -0.38644        0
    8.1792     -1.0381    -0.36686     1800
    8.1242    -0.96182    -0.32607     3600
    8.2941    -0.96584    -0.38851     5400
    8.2917     -1.0513    -0.41848     7200
    8.2179     -1.1291    -0.37742     9000
    8.2462     -1.0865    -0.34782    10800
     8.263     -1.0178    -0.34806    12600
    8.2572     -1.0048    -0.36798    14400
    8.2972    -0.98813    -0.29854    16200
     8.228    -0.97129    -0.33756    18000
    8.1946    -0.94709    -0.33708    19800
    8.2038     -1.0585    -0.32721    21600
    8.1683     -1.0659     -0.3467    23400
     8.337     -1.1015    -0.35912    25200
    8.3213     -1.1163    -0.28888    27000

u = T1.u;
v = T1.v;
w = T1.w;
t = T1.time/60;                                     % Guessing Here (Units = Minutes)

figure
plot(t, [w v u])
grid
legend('u','v','w', 'Location','best')
xlabel('Time (Minutes)')
ylabel('Amplitude (Units?)')

Fs = 1/(t(2)-t(1))/60;                              % Sampling Frequency (Hz)  (Cy/Min * Min/Sec)
Fn = Fs/2;                                          % Nyquist Frequency
L = size(T1,1);
NFFT = 2^nextpow2(L);                               % FOr Efficiency
FTuvw = fft([u v w],NFFT)/L;                        % Fourier Transform
Fv = linspace(0, 1, NFFT/2+1)*Fn;                   % Frequency Vector
Iv = 1:numel(Fv);                                   % Index Vector

figure
plot(Fv, abs(FTuvw(Iv,:))*2)
grid
% xlim([1E-4 10])
Ax = gca;
Ax.XScale = 'log';
Ax.YScale = 'log';
legend('u','v','w', 'Location','best')
xlabel('Frequency (Hz)')
ylabel('Magnitude')

Scale the time (‘t’ vector) correctly in that assignment. I believe I did it correctly, however the frequencies do not match with the posted plot, being several orders-of-magnitude smaller. Set the xlim values correctly in the Fourier Transform plot to show whatever the region of interest is (possibly Hz to Hz).

.

[E. Cheynet]

To further smooth the spectrum, you can use Welch's algorithm with multiple overlapping segments and use bin-averaging over logarithmically-spaced bins. See e.g. https://se.mathworks.com/matlabcentral/fileexchange/73584-averaging-noisy-data-into-bins