Properties of the Fourier Transform
One-dimensional Fourier Transformation is defined as follows:
![\label{Fourier1}
H(k) = \mathcal{F}\left( h(x) \right) = \int \limits_{-\infty}^{\infty} h(x) \mathrm{e}^{-\mathrm{i}2\pi kx} \mathrm{d}x](lib/equation/pictures/babcfc179d90b1e09d35db143cdf021f.png)
We mention here that the domain of this transformation is the set of functions whose absolute value has a finite integral over but hereinafter we will always suppose that the picked function is element of this set as in practice we will deal with finite functions limited in a finite time interval, and therefore their absolute value can always be integrated.
The inverse Fourier transformation is very similar to the transformation itself, they only differ in a complex conjugate:
![\label{Fourier2}
h(x) = \mathcal{F}^{-1}\left( H(k) \right) = \int \limits_{-\infty}^{\infty} H(k) \mathrm{e}^{\mathrm{i}2\pi kx} \mathrm{d}k](lib/equation/pictures/63daba069f695d4bc12fa31836ae6211.png)
The frequently used properties of the Fourier Transform are the followings:
1) Linearity:
![\label{Fourier3}
\mathcal{F}\left( \alpha g(x) + \beta h(x) \right) = \alpha \mathcal{F} \left( g(x) \right) + \beta \mathcal{F} \left( h(x) \right)](lib/equation/pictures/40db7bab8d9213acce999a11b94c7873.png)
for every complex numbers and
. Specifically, a constant phase multiplication is preserved during the Fourier transform.
2) Shifting:
![\label{Fourier4}
\mathcal{F}\left( h(x-x_0) \right) = \mathcal{F} \left( h(x) \right) \mathrm{e}^{-\mathrm{i}2 \pi k x_0} = H(k) \mathrm{e}^{-\mathrm{i}2 \pi k x_0}](lib/equation/pictures/99cbf71c6f644e9a2891084c5f58bab2.png)
![\label{Fourier5}
\mathcal{F}^{-1}\left( H(k-k_0) \right) = \mathcal{F}^{-1} \left( H(k) \right) \mathrm{e}^{\mathrm{i}2 \pi k_0 x} = h(x) \mathrm{e}^{\mathrm{i}2 \pi k_0 x}](lib/equation/pictures/b119bd11952084b36a2e500db05d6d8a.png)
For every real and
. This means that a linear phase ramp in the Fourier space (
-space) results a spatial shift in the image space and vice versa. The shift is proportional to the steepness of the phase ramp.
3) Convolution:
![\label{Fourier6}
\mathcal{F}\left( g(x) \Conv h(x) \right) = \mathcal{F} \left( g(x) \right) \mathcal{F} \left(h(x) \right) = G(k)H(k)](lib/equation/pictures/f9ff807699f51018cf814c509dc1f7c3.png)
![\label{Fourier7}
\mathcal{F}^{-1}\left( G(k)H(k) \right) = \mathcal{F}^{-1} \left( G(k) \right) \Conv \mathcal{F}^{-1} \left( H(k) \right) = g(x) \Conv h(x)](lib/equation/pictures/e2e3dcfc95ba160f2e0c3177af63690f.png)
![\label{Fourier8}
\mathcal{F}\left( g(x) h(x) \right) = \mathcal{F} \left( g(x) \right) \Conv \mathcal{F} \left(h(x) \right) = G(k) \Conv H(k)](lib/equation/pictures/2e52d2af851b62f30bf1dc12c46c7ff8.png)
![\label{Fourier9}
\mathcal{F}^{-1}\left( G(k) \Conv H(k) \right) = \mathcal{F}^{-1} \left( G(k) \right) \mathcal{F}^{-1} \left( H(k) \right) = g(x) h(x)](lib/equation/pictures/db2c47d2a5522761478f52e525d09270.png)
In other words, a multiplication in the -space results as a convolution in the image space with the inverse Fourier transformed version of the multiplying function. In the other direction, a convolution in the
-space means a multiplication in the image space.
4) Symmetry. If is a real function then:
![\label{Fourier10}
H(-k) = \overline{H(k)}](lib/equation/pictures/0f9cb9764fdda6bda4aa6ccb7dfea248.png)
This symmetry is of utmost importance in the so-called partial Fourier imaging where the effective spin density is assumed to be real, and therefore the -space data is redundant as negative
-vector values can be replaced by the complex conjugate of the positive
-vector values.
The extension of the upper properties to higher dimensions can be done quite intuitively. The variables are then vectorial denoted by and
, and the Fourier Transform (for example in 3D) is defined by the following with
equals to their dot product:
![\label{Fourier10}
H(\mathbf{k}) = \int h(\mathbf{r}) \mathrm{e}^{-\mathrm{i}2\pi \mathbf{kr}} \mathrm{d^3r}](lib/equation/pictures/42346c090789213b7bf19323545e4d32.png)