Numerical Fourier transforms of matrix?

Hi

I was tryaing for the numerical fourier tranform of the function u(z,t) (real space) where u and z as per the attachments:

The fourier transform u~(q,t) (reciprocal space) is recognised as; u~(q,t) = (I/L)*Integration(dz u(z,t)exp(-iqz)

how can I take care of this?

Thank you in advance!!

Answer 1

[Star Strider]

I have no idea what you want. I would not suggest integrating the individual sine and cosone coefficients using numerical integration. The Fast Fourier Transform calculates the coefficients much more efficiently.

This calculates and plots the fast Fourier transform (fft) of that signal —

T1 = readtable('https://www.mathworks.com/matlabcentral/answers/uploaded_files/987480/data.txt');
T1.Properties.VariableNames = {'Signal','Time'}

T1 = 1024×2 table
      Signal         Time   
    __________    __________

    3.0967e-06     3.418e-09
    3.0967e-06    6.8359e-09
    3.0967e-06    1.0254e-08
    3.0967e-06    1.3672e-08
    3.0967e-06     1.709e-08
    3.0967e-06    2.0508e-08
    3.0967e-06    2.3926e-08
    3.0967e-06    2.7344e-08
    3.0967e-06    3.0762e-08
    3.0967e-06     3.418e-08
    3.0967e-06    3.7598e-08
    3.0967e-06    4.1016e-08
    3.0967e-06    4.4434e-08
    3.0967e-06    4.7852e-08
    3.0967e-06     5.127e-08
    3.0967e-06    5.4687e-08

figure
plot(T1.Time, T1.Signal)
grid
xlabel('Time')
ylabel('Amplitude')

L = size(T1,1);
Ts = mean(diff(T1.Time));                                               % Sampling Interval
Fs = 1/Ts;                                                              % Sampling Frequency
Fn = Fs/2;                                                              % Nyquist Frequency
FTSignal = fft(T1.Signal-mean(T1.Signal))/L;                            % Fourier Transform (Subtract Mean To Emphasize Peaks)
Fv = linspace(0, 1, fix(L/2)+1)*Fn;                                     % Frequency Vector
Iv = 1:numel(Fv);                                                       % Index Vector

figure
plot(Fv, abs(FTSignal(Iv))*2)
grid
xlabel('Frequency')
ylabel('Amplitude')
xlim([0 5]*1E+7)

The coefficients of the cosine (real) terms are the real parts of ‘FTSignal’ and the imaginary parts are the coefficients of the sine terms for each frequency in the ‘Fv’ vector.

.

[Somnath Kale]

@Star Strider here I have attached the detailed equation. will you pleaee elaborate this little more. I understood the use of fft but unable to gain how to set the limits. In the attached data r = var1 and u(r) = var2.

Thank you very much for your help!

[Star Strider]

The ‘q’ values are apparently the frequencies, however beyond that, I am lost.

Reiterating my previous code —

T1 = readtable('https://www.mathworks.com/matlabcentral/answers/uploaded_files/987480/data.txt');
T1.Properties.VariableNames = {'Signal','Time'}

T1 = 1024×2 table
      Signal         Time   
    __________    __________

    3.0967e-06     3.418e-09
    3.0967e-06    6.8359e-09
    3.0967e-06    1.0254e-08
    3.0967e-06    1.3672e-08
    3.0967e-06     1.709e-08
    3.0967e-06    2.0508e-08
    3.0967e-06    2.3926e-08
    3.0967e-06    2.7344e-08
    3.0967e-06    3.0762e-08
    3.0967e-06     3.418e-08
    3.0967e-06    3.7598e-08
    3.0967e-06    4.1016e-08
    3.0967e-06    4.4434e-08
    3.0967e-06    4.7852e-08
    3.0967e-06     5.127e-08
    3.0967e-06    5.4687e-08

figure
plot(T1.Time, T1.Signal)
grid
xlabel('Time')
ylabel('Amplitude')

L = size(T1,1);
Ts = mean(diff(T1.Time));                                               % Sampling Interval
Fs = 1/Ts;                                                              % Sampling Frequency
Fn = Fs/2;                                                              % Nyquist Frequency
FTSignal = fft(T1.Signal-mean(T1.Signal))/L;                            % Fourier Transform (Subtract Mean To Emphasize Peaks)
Fv = linspace(0, 1, fix(L/2)+1)*Fn;                                     % Frequency Vector
Iv = 1:numel(Fv);                                                       % Index Vector

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

q_max = Fs/3E-9                                 % Maximum Frequency (Guess) = 2*pi / 3 nm

q_max = 9.7524e+16

q_min = Fs/3300E-9                              % Minimum Frequency (Guess) = 2*pi / 3300 nm

q_min = 8.8658e+13

In my code, corresponds to the sampling frequency ‘Fs’ (the Nyquist frequency ‘Fn’ would be π).

No time or amplitude units are provided, so I assume they are as stated, and not multiplied by some unknown factor to change the data in the file.

The maximum frequency in the Fourier transforrm plot (the Nyquist frequency or π) is 1.46E-8 (cycles/(independent variable units)), so the limits make no sense in the context of the data provided. If they did, constraining the Fourier transform plot to show those limits would be straightforward using the xlim function.

.

[Somnath Kale]

abs(FTSignal(Iv))*2

@Star Strider why this multiplyer 2 is there?

and

Fv = linspace(0, 1, fix(L/2)+1)*Fn;

if write this as: Fv = linspace(q_min, q_max, L);with which im getting this type of out put (off corse i have plotted in loglog scale)

will it make any differance? if yes what?

[Star Strider]

First, the multiplication by 2 results from the fft function calculating a 2-sided Fourier transform and dividing the energy of the time domain signal at every frequency evenly between each half. The multiplication normalises the amplitude to approximate the time domain amplitude. (This also makes it easier to check the results.)

Second, that still does not make sense in the context of the frequencies as stated for ‘q_min’ and ‘q_max’ that are several orders-of-magnitude higher than the frequencies calculated in my code. So that diecrepancy needs to be resolved first. Just now, those do not make any sense in the context of the data provided.

[Somnath Kale]

Thank you very much!

[Star Strider]

As always, my pleasure!