O'Reilly logo

Digital Filters Design for Signal and Image Processing by Mohamed Najim

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

8.5. Filtering in the frequency domain

When the filter properties are specified in the frequency domain, or when the impulse responses have a very large support, it is advantageous to carry out the filtering operation in the Fourier domain by using equation (8.5). Changing the representation domain is done using the 2-D discrete Fourier transform. Consequently, the input/output relation, in the spatial domain, is no longer described by a simple discrete convolution equation, but by a circular convolution equation. We will look at the effects of this phenomenon at the end of this section.

8.5.1. 2-D discrete Fourier transform (DFT)

Let us look at a sequence of two indices {xk,l}. To simplify our presentation, we assume that the two indices have the same variation domain, from 0 to N−1. The transformed sequence images is given by:

images

It is clear that the transform sequence is periodic, of period N for each index.

Equation (8.36) also shows that the transformation is separable, being constituted of a discrete 1-D Fourier transform (DFT) operating row by row, followed with a DFT column by column.

As in the 1-D context, the 2-D discrete Fourier transform is inversible:

images

As in the 1-D context, we ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required