Hello,

one of the ways to cope with such data sets is to resample the data (which is to apply some kind of interpolation algorithm). In your case, a 1 min time base would apply - fitting the 4 and the 5 min intervals perfectly.

Once you have the data stream with a value for every minute, it is easy to apply the standard Fourier algorithms.

Regards

The alternative idea could be to rewrite the Fourier expansion so as to take into account a variable step (e.g., by replacing integrals with a variable step rectangle formula). This eliminates the complexity of the interpolation phase but is not expected to be implementable as a fast algorithm.

Hi,

I would prefer to solve this problem by writing a code which can do the integration for non-uniform time steps using trapezium rule or simpsons rule and thus obtain the fourier coefficients.

This problem is know as the irregular sampling problem. Different methods have been devised to solve it. You can find an introduction here: https://www.math.ucdavis.edu/~strohmer/research/sampling/irsampl.html. And a comparison of different methods here: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=113853.

An iterative algorithm is usually used. First, as suggested by others, you can use some kind of interpolation and make a preliminary fitting of the Fourier coefficients. However, the residuals obtained from this process still contain information on the spectrum of your time series. Then you can iteratively obtain the Fourier coefficients of the residual time series and use them to update the original ones and improve your estimation.