Fourier Transform
The Fourier and inverse Fourier transforms convert a time-series into a power spectrum and viseversa. These are well-known transformations that are employed for many applications including finding and characterizing periodicities in time-series analysis and regression; fast convolution and de-convolution; transfer function modeling in systems analysis; solving systems of differential equations; Bayesian analysis using characteristic functions; and signal filtering.
The discrete Fourier transform involves a time domain (corresponding to an index) and a frequency
domain (corresponding to a frequency index). The time points are equally spaced at internals
of Δt, and the frequency points are equally spaced at ΔF . Both index have npoints. The intervals spacings are related as
[math]\displaystyle{ \Delta t =\frac{1}{n \Delta t} }[/math]
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