# Modelling cycles trough Fourier Analyss/Synthesis on financial time series

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 Hi everyone!I am trying to develop modeling in Mathematica that presuppose the use of Fourier analysis and synthesis applied to financial historical series (eg stocks, financial indices, currencies, and so on). So I wanted to ask you if anyone knows any textbooks, papers, or any other references on these topics for my purposes (even better if with examples of applications on Mathematica).The purpose of my research is to apply Fourier analysis and synthesis on financial time series (as already mentioned), in order to extrapolate from them the relative market cycles, both to be able to carry out appropriate analyzes and to be able to make forecast analyzes. The output should be to obtain a graph similar to the one attached, where above there is a "candlestick" graph and below the sinusoidal trend of the market cycle reconstructed through the application of Fourier. This kind of analysis would also make it possible to create a sort of trading system to identify entry points (sell/buy) to the market, exploiting the phase-lag of the various cycles, and their crossover (the idea in fact is to exploit cycles with different periods- eg. daily, weekly, monthly, ergo with amplitude and different frequency) This in turn implicitly requires a strategy to optimize the cycle obtained as a function of the volatility of the underlying chart, as well as the optimal choice of parameters to be used in the Fourier transform (and this is another point I wanted to ask).Furthermore, another question (secondary, but equally important for me to understand) is that by doing some research, I could see how the Hilbert-Huang transform could be a more efficient strategy in this sense, since financial time series are non-series. stationary, so I was also wondering if it was appropriate to model everything using this type of transform. Many thanks in advance.  Attachments: Answer