Time and frequency representation
Fourier Transformation

The Fourier transform, named by Joseph Fourier, is important in mathematics, engineering, and the physical sciences.

In simple way we can say that Fourier transformation represents mathematical function of time as a function of frequency. This function is known as frequency spectrum.

Very important note, Fourier Transformation is used only for non-periodic analog signals. In case of periodic analog signal Fourier series are used.

Let's say you have a function f(t) that maps some time value t to some value f(t).

Now, we try to approximate f as the sum of simple harmonic oscillations, i.e. sine waves of certain frequencies ω. Of course, there are some frequencies that fit well to f and some that approximate it less well. Thus we need some value f(ω) that tells us how much of a given oscillation with frequency ω is present in the approximation of f.

Take for example the function (black line) from here:

Two harmonic frequencies forming signal

which is defined as f(t)=sin(t)+0.13sin(3t). The oscillation (dotted line) with ω=1 has the biggest impact on the result, so let's say F(1)=1. The other wave (ω = 3, dashed line) has at least some impact, but its amplitude is much smaller. Thus we say F(3)=0.13. Other frequencies may not be present in the approximation at all, thus we would write F(ω)=0 for these.

Now if we knew F(ω) not only for some but all possible frequencies ω, we could perfectly approximate our function f(t). And that's what the continuous Fourier transform does.

It takes some function f(t) of time and returns some other function F(ω)=FT(f), it's Fourier transformation, that describes how much of any given frequency is present in f. It's just another representation of f(t), of equal information but with a completely different domain. Often though, problems can be solved much easier in this other representation (which is like finding the appropriate coordinate system).

But given a Fourier transform, we can integrate over all frequencies, put together the weighted sine waves and get our f again, which we call inverse Fourier transform IFT.

Most importantly, the Fourier transform has many nice mathematical properties (i.e. convolution is just multiplication). It's often much easier to work with the Fourier transforms than with the function itself. So we transform, have an easy job with filtering, transforming and manipulating sine waves and transform back after all.

Let's say we want to do some noise reduction on a digital image. Rather than manipulating a function image:Pixel→Brightness, we transform the whole thing and work with F(image):Frequency→Amplitude. Those party of high frequency that cause the noise can simply be cut off – F(image)(ω)=0,ω>...Hz.

The Fourier transform (usually known as forward transform) is defined as:

and inverse Fourier transformation is defined:

Fourier function F(ω) is frequency representation of signal f(t), also called spectral function. Spectral function depends on real variable ω so it can be defined as:

where is absolute value and is phase spectrum.

In every formula j is defined as . The complex exponential is the heart of the transform. A complex exponential is simply a complex number where both the real and imaginary parts are sinusoids. The exact relation is called Euler's formula ejφ = cosφ + jsinφ, which leads to the famous (and beautiful) identity . Complex exponentials are much easier to manipulate than trigonometric functions, and they provide a compact notation for dealing with sinusoids of arbitrary phase, which form the basis of the Fourier transform.

Complex exponentials (or sinus and cosines) are periodic functions, and the set of complex exponentials is complete and orthogonal. Thus the Fourier transform can represent any piecewise continuous function and minimizes the least-square error between the function and its representation.

There exist other complete and orthogonal sets of periodic functions; for example, Walsh functions (square waves) are useful for digital electronics (more about it in chapter 8).

Why do we always encounter complex exponentials when solving physical problems? Why are monochromatic waves sinusoidal, and not periodic trains of square waves or triangular waves? The reason is that the derivatives of complex exponentials are just rescaled complex exponentials. In other words, the complex exponentials are the functions of the differential operator. Most physical systems obey linear differential equations. Thus an analog electronic filter will convert a sine wave into another sine wave having the same frequency (but not necessarily the same amplitude and phase), while a filtered square wave will not be a square wave. This property of complex exponentials makes the Fourier transform uniquely useful in fields ranging from radio propagation to quantum mechanics.

Fourier transformation is defined also for two-dimensional signals and mathematically is defined as:

and inverse transformation is defined: