# Properties of the Fourier Transform

One-dimensional Fourier Transformation is defined as follows:

(1)

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:

(2)

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

1) Linearity:

(3)

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

2) Shifting:

(4)

(5)

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:

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

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:

(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 -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:

(11)