Discrete Fourier Transform (DFT)

Time series data can be decomposed into components of different sine and cosine waves of different frequencies, where frequency is an inverse measure of the time period of the wave; a high frequency wave corresponds to small period whereas a low frequency corresponds to high period. The spectral density function computed as a function of frequency quantifies the relative contributions/weights of different frequencies of sine (or cosine) waves to the time series under study. A spatial data can similarly be quantified by strengths of different sine and cosine waves of different wavenumbers. A wavenumber can be thought of as a `spatial frequency’, or the number of times that a pattern is repeated in a unit of spatial length

When we think of spatial data, periodicity is visualized as wavelength, which is inversely related to wavenumber (i.e., a small wavenumber corresponds to a large wavelength and conversely). A spatial spectral density function is typically plotted as a function of wavenumbers and it quantifies the relative contributions/weights of different wavenumbers of sine (or cosine) waves to the spatial data under study.

Spatial spectral reddening is formally captured by the Discrete Fourier Transform (DFT) that decomposes spatial variation into a summation of periodic sine and cosine function  as:

(1)   \begin{eqnarray*} \hat{z}[p,q] & = & \frac{1}{MN} \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} (z[m,n]-\bar{z}) e^{-i\,2\pi(mp/M+nq/N) } \\ & = & \frac{1}{MN} \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} (z[m,n]-\bar{z}) \Big(\cos\big(2\pi (\frac{mp}{M}+\frac{nq}{N})\big) - i \sin\big(2\pi(\frac{mp}{M}+\frac{nq}{N})\big) \Big)\\ & = & a[p,q] - i\,b[p,q] \end{eqnarray*}

where i is the imaginary number defined such that i^2=-1, a[p,q] and b[p,q] are the real and imaginary parts of DFT \hat{z}[p,q], and symbols p and q correspond to wavenumbers in x– and y-dimensions. DFT, in general, can be a complex number, and therefore we plot the power spectrum I[p,q] = |\hat{z}[p,q]|, where |\hat{z}| is the modulus of the complex number \hat{z}.

Spatial correlation and DFT are related to one another under certain conditions (referred to as the Weiner-Kinchin theorem). An increase in correlation length as computed from the spatial correlation function and spectral reddening i.e., increased contribution of small wavenumbers or large wavelength to the spatial fluctuations are equivalent quantification of the same underlying dynamics. However, depending on the nature of the dataset, spatial resolution, etc, one quantification may appear more informative than the other.

 

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