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Properties of the Fourier Transform

One-dimensional Fourier Transformation is defined as follows:

H(k) = \mathcal{F}\left( h(x) \right) = \int \limits_{-\infty}^{\infty} h(x) \mathrm{e}^{-\mathrm{i}2\pi kx} \mathrm{d}x(1)

We mention here that the domain of this transformation is the set of functions whose absolute value has a finite integral over $\mathbb{R}$ 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:

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(2)

The frequently used properties of the Fourier Transform are the followings:

1) Linearity:

\mathcal{F}\left( \alpha g(x) + \beta h(x) \right) = \alpha \mathcal{F} \left( g(x) \right) + \beta \mathcal{F} \left( h(x) \right)(3)

for every complex numbers $\alpha$ and $\beta$. Specifically, a constant phase multiplication is preserved during the Fourier transform.

2) Shifting:

\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}(4)


\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}(5)

For every real $k_0$ and $x_0$. This means that a linear phase ramp in the Fourier space ($k$-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:

\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)(6)
\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)(7)
\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)(8)
\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)(9)

In other words, a multiplication in the $k$-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 $k$-space means a multiplication in the image space.

4) Symmetry. If $h(x)$ is a real function then:

H(-k) = \overline{H(k)}(10)

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 $k$-space data is redundant as negative $k$-vector values can be replaced by the complex conjugate of the positive $k$-vector values.


The extension of the upper properties to higher dimensions can be done quite intuitively. The variables are then vectorial denoted by $\mathbf{r}$ and $\mathbf{k}$, and the Fourier Transform (for example in 3D) is defined by the following with $\mathbf{kr}$ equals to their dot product:

H(\mathbf{k}) = \int h(\mathbf{r}) \mathrm{e}^{-\mathrm{i}2\pi \mathbf{kr}} \mathrm{d^3r}(11)

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