2.1.3. Spectra of Three Excitations

The spectrum of *δ*(*t*) below means that *δ*(*t*) contains the equal frequency components for *ω* ∈ (0, ∞).

$$\int\_{-\infty}^{\infty} \delta(t) e^{-i\omega t} dt = 1. \tag{22}$$

The spectrum of *u*(*t*) is in the form

$$\int\_{-\infty}^{\infty} u(t)e^{-i\omega t}dt = \pi\delta(\omega) + \frac{1}{i\omega}.\tag{23}$$

The Fourier transform of cos *<sup>ω</sup>*1*t* is given by

$$\int\_{-\infty}^{\infty} \cos \omega\_1 t e^{-i\omega t} dt = \pi [\delta(\omega + \omega\_1) + \delta(\omega - \omega\_1)].\tag{24}$$

Three functions or signals above, namely, *δ*(*t*), *u*(*t*), and sinusoidal functions, are essential to the excitation forms in oscillations. However, their spectra do not exist in the domain of ordinary functions but they exist in the domain of generalized functions. Due to the importance of generalized functions in oscillations, for example, *δ*(*t*) and *u*(*t*), either theory or technology of oscillations nowadays is in the domain of generalized functions. In the domain of generalized functions, any function is differentiable of any times. The Fourier transform of any function exists (Gelfand and Vilenkin [67], Griffel [68]).
