Fourier’s Smart Tool: Altering a Problem to be Easily Solved

       Linear transforms, especially Fourier and Laplace transforms are widely used in solving problems in science and engineering, The Fourier transform is used in linear systems analysis, antenna studies, optics, random process modeling, probability theory, quantum physics, and boundary value problems and has been very successfully applied to restoration of astronomical data. The Fourier transform, a pervasive and versatile tool is used in many fields of science as a mathematical or physical tool to alter a problem into one that can be more easily solved. Some scientists understand Fourier theory as a physical phenomenon not simply as a mathematical tool. In some branches of science the Fourier transform of one function may yield another physical function.

       The Fourier transform, in essence, decomposes or separates a waveform or function into sinusoids of different frequency which sum to the original waveform. It identifies or distinguishes the different frequency sinusoids and their respective amplitudes. The Fourier transform of f(x) is defined as:

Since the Fourier transform F(s) is a frequency domain representation of a function f(x) the s characterizes the frequency of the decomposed co-sinusoids and sinusoids and is equal to the number of cycles per unit of x. If a function or waveform is not periodic, then the Fourier transform of the function will be a continuous function of frequency.

Fourier Transform Properties

       It is often useful to think of functions and their transforms as occupying two domains. These domains are referred to as the upper and the lower domains in older texts as if functions circulated at ground level and their transforms in the underworld. They are also referred to as the function and transform domains, but in most physics applications they are called the time and frequency domains respectively.

       Operations performed in one domain have corresponding operations in the other. For example, as will be shown below, the convolution operation in the time domain becomes a multiplication operation in the frequency domain, that is f(x)Äg(x) = F(s).G(s).The reverse is also true, F(s)ÄG(s) = f(x).g(x). Such theorems allow one to move between domains so that operations can be performed where they are easiest or most advantageous.

 

Scaling Properties        From time scaling property, it is an evident that if the width of a function is decreased while its height is kept constant, then its Fourier transform becomes wider and shorter. If its width is increased, its transform becomes narrower and taller
Shifting Properties        This time shifting property states that the Fourier transform of a shifted function is just the transform of the un-shifted function multiplied by an exponential factor having a linear phase.
Convolution Theorem        As previously mentioned, the Fourier transform of a product is given by the convolution of the individual transforms, and so it works reversely. Fourier transform of a convolution is given by product of the individual transforms.
Correlation Theorem        The correlation integral, like the convolution integral is important in theoretical and practical applications. This is the statement of the correlation theorem. If f(x) and g(x) are the same function. The integral above is normally called the autocorrelation function and the cross-correlation if they differ.