Definition of Fourier Transform
Fourier Transform is a very useful mathematical tool, mainly used to provide the frequency content of a function (signal). It can be considered as a limiting case of Fourier Series when the period of the periodic finite duration function limits to infinity.
The Fourier Transform of a function f(t) is given by the equation:
Note that before the transform, the function had as independent variable the variable t. After Fourier Transform, the independent variable of the function changed to the angular frequency w. In other words, Fourier Transform changes the domain of a function, from the domain of time (or whatever the independent variable is) to the domain of frequency w (2πf).
Fourier Transform is noted by a calligraphic F. For the resulting function we use the same letter of the initial function, but in capital. For example to note the Fourier Transform of the function f(x), we write it as follows:
Note again that before the transform we have a function of time t and after the transform we have a function of frequency w.
Let see how we can calculate the Fourier Transform of delta function.
Consider f(t) a periodic function (or signal) with period T seconds and angular frequency ω=2π/T rad/s (i.e. T=2 sec. and ω=π rad/s). The Fourier Series of this function in complex exponential form are given by:
And the complex coefficients are given by:
Now consider that f(t) is not periodic, in the sense that it exists between –T/2 and T/2 and it is zero everywhere else.
The complex coefficients can be written as:
And finally for the complex coefficients we have:
The equation of the Fourier Series, substituting the complex coefficients become:
We considered our function non periodic. This means that:
The notation of angular frequency changed to w, to distinguish it from the angular frequency value ω=1/T rad/s of the periodic function.
The spectrum of the Fourier Series was distinct lines. Due to the fact that Δω becomes zero, the spectrum lines of Fourier Transform become infinitely close resulting to a continuous spectrum.
Noticing the formula of forward and inverse Fourier Transform we see that they actually are power series.
Figure 1: Spectrum of Fourier Series and Fourier Transform
Euler’s formula states that:
Integrating the above formula we have:
Also if we calculate the above integration we have:
Equating the previous formulas:
If we divide a function f(x) by cos(wx), we actually calculate how many cosines this function contains. Just like when we divide, for example 12/3, we calculate how many 3s number 12 contains.
So the Fourier Transform answers the question, how many cosines function f(x) contains. These cosines are the frequency components of f(x) and are orthogonal between them. And why cosines are selected and not any other function? Because it doesn’t change its form when it passes through LTI (Linear Time Invariant), and it’s easier to work with.
Fourier Transform is given by the formula:
Laplace Transform is given by the formula:
So Fourier Transform is a special case of Laplace Transform when σ=0.
The relation between them is:
Consider the function:
This function is described by the summation of three orthogonal vectors. Adding these orthogonal vectors gives the whole function.
Similarly, in the Fourier Transform, adding orthogonal signals, called frequency components, gives us the whole function:
In the above formula, the summation is the integration, the magnitude of each orthogonal signal is the F(w)/2π and the orthogonal signal is the exponent which corresponds to cosine function.
Figure 2: Orthogonality in Cartesian system and in Fourier Transform