How to do a Amplitude Adjusted Fourier Transform on time series?

I need to use MATLAB (R2022b) to surrogate my data (1x500 excel) using Amplitude Adjusted Fourier Transform.

Any advice how to do that?

Answer 1

[Jaimin]

Hi @Mir Sahand

To surrogate your data using the Amplitude Adjusted Fourier Transform (AAFT) in MATLAB, you will need to perform the AAFT algorithm, which involves randomizing the phases of the Fourier transform of your data and then adjusting the amplitude.

For understanding kindly refer following code.

% Step 1: Import Data
filename = 'your_data.xlsx'; % Replace with your file name
data = readtable(filename);

% Ensure data is a row vector
if size(data, 1) > 1
    data = data';
end

% Step 2: AAFT Algorithm
% Sort the original data
sortedData = sort(data);

% Randomize the original data
randomizedData = data(randperm(length(data)));

randomizedFFT = fft(randomizedData);

% Randomize the phases
randomizedPhases = angle(randomizedFFT);
amplitudes = abs(randomizedFFT);
randomizedFFT = amplitudes .* exp(1i * randomizedPhases);

% Inverse Fourier Transform
surrogateData = ifft(randomizedFFT, 'symmetric');

% 2.6: Adjust the amplitude by sorting
[~, sortIndex] = sort(surrogateData);
surrogateData(sortIndex) = sortedData;

For more information kindly refer following MathWorks documentation.

readtable: https://www.mathworks.com/help/releases/R2022b/matlab/ref/readtable.html

fft: https://www.mathworks.com/help/releases/R2022b/matlab/ref/fft.html

ifft: https://www.mathworks.com/help/releases/R2022b/matlab/ref/ifft.html

[Paul]

@Jaimin

Do you have source for this code/algorithm? I'm curious about a few things.

rng(100);
% Step 1: Import Data
data = 1:10;
data = data(randperm(10));

% Step 2: AAFT Algorithm
% Sort the original data
sortedData = sort(data);
% Randomize the original data
randomizedData = data(randperm(length(data)));
randomizedFFT = fft(randomizedData);
temp = randomizedFFT;

Nothing is being randomized here. All this code does is recreate the randomizedFFT (to withing numerical precision).

% Randomize the phases
randomizedPhases = angle(randomizedFFT);
amplitudes = abs(randomizedFFT);
randomizedFFT = amplitudes .* exp(1i * randomizedPhases);
norm(temp-randomizedFFT)

ans = 8.2441e-15

% Inverse Fourier Transform
surrogateData = ifft(randomizedFFT, 'symmetric');
% 2.6: Adjust the amplitude by sorting
[~, sortIndex] = sort(surrogateData);
surrogateData(sortIndex) = sortedData;

Consequently, surrogateData is identical to randomizedData.

randomizedData

randomizedData = 1×10

2 7 9 6 3 4 10 8 5 1

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surrogateData

surrogateData = 1×10

2 7 9 6 3 4 10 8 5 1

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It's unclear as to why any more code is needed after randomizedData is computed.