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The space domain

In this module, we will extend the concept of the Fourier transform from the time domain into the space domain. In making this extension, we will encountersome significant additional complexity. For example, while time is one-dimensional, space is three-dimensional. While you can only move forward andbackwards in time, you can move up, down, forward, backward, and from side to side in space.

(In order to keep the complexity of this module in check, we will assume that space is only two-dimensional, allowing movement up, down, andfrom side to side only. This will serve us well later for such tasks as image processing. Three-dimensional Fourier transforms are beyond the scopeof this module.)

It is also possible and very common to combine time domain signal processing with space domain signal processing. However, that also is beyond the scope ofthis module.

Time and space are analogous

We will consider the space domain to be analogous to the time domain, with the stipulation that the space domain has two dimensions. The unit of measure inthe time domain is usually seconds, or some derivative thereof. The unit of measure in space is usually meters, or some derivative thereof.

As with the time domain, we will assume that all space domain surfaces are purely real (as opposed to being complex) . This will allow us to simplify our computations when performing the 2D Fourier transform to transformour data from the space domain into the wavenumber domain.

(I will point out that from a practical viewpoint this assumption is much more limiting in the space domain than in the time domain. Complexspace domain functions are quite common in such areas as antenna array processing.)

Frequency and wavenumber are analogous

We will consider the wavenumber domain to be analogous to the frequency domain. The unit of measure in the frequency domain is cycle per second, or somederivative thereof. The unit of measure in the wavenumber domain is cycles per meter or some derivative thereof.

Period and wavelength are analogous

The reciprocal of the typical unit of measure in the frequency domain is seconds per cycle, commonly referred to as the period. The reciprocal of thetypical unit of measure in the wavenumber domain is meters per cycle, commonly referred to as the wavelength.

Some real world examples

With all of this as background, I will begin by discussing some real world engineering problems for which the solution lies in an understanding of thewavenumber domain. I will use these examples to show some of the practical uses of 2D Fourier transforms.

Following that (in Part 2 of this series) , I will present and explain a class that you can copy and use to perform 2D Fourier transforms. Then I will presentand explain a program that exercises and tests the 2D Fourier transform class for some common 3D surfaces.

A commercial radio station

Assume that you have just acquired an FCC license to build and operate a new commercial radio station in a small town in west Texas. As is frequently thecase in west Texas, your town is situated at the intersection of two highways. One highway runs northeast and southwest. The other highway runs northwest andsouthwest. The two highways are generally perpendicular to one another. Like many highways in west Texas, each of these highways is straight as an arrow withvery few curves.

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Source:  OpenStax, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38
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