The Fourier Transform is a valuable instrument to analyze non-periodic functions. In this way, it complements the Fourier Series, which allows analyzing systems where periodic functions are involved.
That is, through the Fourier Series we can represent a periodic signal in terms of its sinusoidal components, each component with a particular frequency. The Fourier Transform allows you to do the same with non-periodic signals.
Fourier reasoned that an aperiodic signal can be considered as a periodic signal with an infinite period. More precisely, in the Fourier Series representation of a periodic signal, as the period increases, the fundamental frequency decreases and the harmonically related components become closer to the frequency. As the period becomes infinite, the frequency components form a continuum and the sum of the Fourier series becomes an integral.
Let f be a real function defined in the continuous domain, say f(t) defined in the t domain. Then, The Fourier Transform (FT) is defined as:
It is said that a signal f(t) has a Fourier Transform if the integral of equation (1) converges (that is, it exists). The integral converges if f(t) “behaves well” and is fully integrable; this last condition means that:
All real signals behave well, and therefore satisfy the previous condition. That is, most of the real signals have FT. However, the following is an example of a signal that does not have FT:
The signal of equation (3) is well known as a CD signal or constant signal. And it has no FT because it is not a real signal, that is, no signal that is different from zero all the time can be physically possible. If we substitute this signal in equation (1) we could verify that this integral does not converge just by observing that the area under the constant signal is infinite, so that integral does not have a finite value. Later, however, we will show that a constant signal does have FT in a generalized sense.
The Fourier Transform Pair
We can define two integrals called the Fourier Transform pair:
For the TF of f(t) to exist, it must be fulfilled that:
F(ω) is the transform of the spectrum of f(t). From here we see that f(t) is being analyzed in a finite number of frequency components with infinitesimal amplitude equal to:
Fourier Transform Considerations
1. In general F(ω) is a complex function, which transforms a given signal into its exponential components;
2. F(ω) is called the Direct Fourier Transform of f(t), and represents the relative amplitudes of several frequency components, so F(ω) is the representation of f(t) in the frequency domain:
3. The time representation of f(t) specifies a function at each time value, while F(ω) specifies the relative amplitudes of the frequency components of the signal, for each frequency value.
4. Thus, F(ω) is a complex function with the following form
F(ω) is a complex function that can be represented graphically by the magnitude and phase Θ(ω) versus frequency. In this way, the graph of is called Continuous Spectrum of Amplitude of f(t), and the graph of Θ(ω) is called Continuous Spectrum of Phase of f(t). The spectrum is said to be a continuous spectrum, since both the amplitude and the phase of F(ω) are continuous functions of the frequency ω. This graphic representation of both spectra is known as the Frequency Spectrum. Note the difference between this continuous spectrum and the discrete spectrum generated by the Fourier Series
5. In many cases F(ω) is real or imaginary pure. Therefore, only one graph is needed since:
Fourier Transform Properties
The relationship between a signal and its Fourier Transform will be denoted as follows:
The following is a summary of the most prominent properties of the TF:
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