Properties of Fourier Transforms

•  Importance of phase and magnitude
•  Circularly symmetric signals
•  Examples of 2D signals and transforms

There are a variety of properties associated with the Fourier transform and the inverse Fourier transform. The following are some of the most relevant for digital image processing.

 * The Fourier transform is, in general, a complex function of the real frequency variables. As such the transform con be written in terms of its magnitude and phase.

* A 2D signal can also be complex and thus written in terms of its magnitude and phase.

* If a 2D signal is real, then the Fourier transform has certain symmetries.

The symbol (*) indicates complex conjugation. For real signals equation leads directly to,

* If a 2D signal is real and even, then the Fourier transform is real and even

* The Fourier and the inverse Fourier transforms are linear operations

where a and b are 2D signals(images) and and are arbitrary, complex constants.

* The Fourier transform in discrete space, ,is periodic in both and .Both periods are

integers