Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 22

Equation (1.1) is known as a Fourier series and can only be applied to periodic signals. Very often a sound signal is not a pure or a complex tone but is impulsive in time. Such a signal might be caused in practice by an impact, explosion, sonic boom, or the damped vibration of a mass‐spring system (see Chapter 2 of this book) as shown in Figure 1.2c. Although we cannot find a Fourier series representation of the wave in Figure 1.2c because it is nonperiodic (it does not repeat itself), we can find a Fourier spectrum representation since it is a deterministic signal (i.e. it can be predicted in time). The mathematical arguments become more complicated [1, 7, 9] and will be omitted here. Briefly, the Fourier spectrum may be obtained by assuming that the period of the motion, T, becomes infinite. Then since f = ω/2π = 1/T, the fundamental frequency approaches zero and Eq. (1.2) passes from a summation of harmonics to an integral:

(1.5)

Just as C_n was complex in Eq. (1.2), X(ω) is complex in Eq. (1.5), having both a magnitude and a phase. The Fourier spectrum (magnitude), ∣ X(ω)∣, of the wave in Figure 1.2c is plotted to the right of Figure 1.2c. ∣ X(ω)∣may be thought of as the amplitude of the time signal at each value of frequency ω.