Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 21

We are required to represent x(t) = At over the interval 0 ≤ t ≤ 1, T = 1, and the fundamental frequency is ω = 2π/T = 2π. Then, we determine the corresponding Fourier coefficients using Eqs. (1.3a) and (1.3b)

Therefore, substituting for the Fourier coefficients in Eq. (1.1) we get

The energy spectrum of x(t) is shown in Figure 1.4. For this case the frequency spectrum is discrete, described by the Fourier coefficients (Eq. (1.4)). Note that for a Fourier series with only sine terms, as in Example 1.1, the amplitude of the nth harmonic is C_n = |B_n|. The energy spectrum has spikes at multiples of the fundamental frequency (harmonics) with a height equal to the value of . Thus, the spectrum is a series of spikes at the frequencies ω = 2πf, and overtones 2ω = 2 × 2πf, 3ω = 3 × 2πf, … with amplitudes (A/π)², (A/2π)², (A/3π)², … respectively.

Figure 1.3 Periodic sound signal.

Figure 1.4 Frequency spectrum of the periodic signal of Figure 1.3.