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Suppose I have a staircase function, which has a periodic increase but no periodic decrease. I've been playing with Fourier transforms recently, and I know one main use is to pick out frequencies because they appear as spikes in the Fourier transform. Unfortunately, that doesn't seem to work so well if there is no periodic decrease corresponding to a periodic increase. For example, below are two graphs: one of a staircase function and the other is its Fourier transform.

Staircase function Staircase function FFT

There are a few spikes in the transformed data, but if I had only that to go on, I would not have been able to guess at the original function. So, that's why I'm asking if there is a different method I can use that's still related to Fourier transforms.

El'endia Starman
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1 Answers1

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If you were to transform the staircase by subtracting $x$ from it, you would find a nice periodic function with which to apply the Fourier transform. My advice would be, assuming you are working with a monotonic time series, detrend your data and take the Fourier transform of the result.

A common method to detect such an upward trend is to take the convolution at time $t$ of $f(x)$ with a Gaussian function, $e^{-x^2/2\sigma^2}$ - i.e. study instead $f - g$ where $$ g(x) = \int f(t) e^{-(x-t)^2/2\sigma^2} dt.$$ This 'smoothed' function $g$ typically contains much of the long-term behavior. See http://en.wikipedia.org/wiki/Trend_estimation for more ideas.

Titus
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