Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 23

So far we have discussed periodic and nonperiodic signals. In many practical cases the sound or vibration signal is not deterministic (i.e. it cannot be predicted) and it is random in time (see Figure 1.5). For a random signal, x(t), mathematical descriptions become difficult since we have to use statistical theory [1, 7, 9]. Theoretically, for random signals the Fourier transform X(ω) does not exist unless we consider only a finite sample length of the random signal, for example, of duration τ in the range 0 < t < τ. Then the Fourier transform is

(1.6)

where X(ω,τ) is the finite Fourier transform of x(t). Note that X(ω) is defined for both positive and negative frequencies. In the real world x(t) must be a real function, which implies that the complex conjugate of X must satisfy X(−ω) = X^*(ω); i.e. X(ω) exhibits conjugate symmetry. Finite Fourier transforms can easily be calculated with special analog‐to‐digital computers (see Section 1.5).

Figure 1.5 Random noise signal.