Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 19

Sometimes in acoustics and vibration we encounter signals which are pure tones (or very nearly so), e.g. the 120 Hz hum from an electric motor. In the case of a pure tone, the time history of the signal is simple harmonic and could be represented by the waveform x(t) = A sin (2πf₁ t) in Figure 1.2a. The pure tone can be represented in the frequency domain by a spike of height A at frequency f₁, Hz (see Figure 1.2a).

Figure 1.2 Time and frequency domain representations of (a) pure tone; (b) complex tone; and (c) nonperiodic deterministic signal.

More often, sound signals are encountered which are periodic, but not simple harmonic. These are known as complex tones. Such sound signals are produced by most musical instruments (both wind and string). They can also be produced mechanically or electronically (a square wave is an example of a periodic signal or complex tone). The broken line plotted in Figure 1.2b is an example of a complex tone which is made up by the superposition (addition) of two simple harmonic signals, x(t) = A sin (2πf₁ t) + B sin (2πf₃ t). Note in this case we have chosen f₃ = 3f₁. The signal A sin (2πf₁ t) is known as the fundamental (or first harmonic) and B sin (2πf₃ t) is the third harmonic. In this particular case the second harmonic and the fourth and higher harmonics are completely absent from the complex tone x(t). The frequency domain representation of the complex tone is also given in Figure 1.2b.

In fact, Fourier [11] showed in 1822 that any periodic signal may be analyzed as a combination of sinusoids:

(1.1)

or in complex notation:

(1.2)

where ω = 2πf; f is the fundamental frequency; T = 1/f = 2π/ω, is the period of the signal; j = , and A_n and B_n are the Fourier coefficients calculated from [4, 6, 8]

(1.3a)

(1.3b)

The sine and cosine terms in Eq. (1.1) can have values of the subscript n equal to 1, 2, 3, …, ∞. Hence, the signal x(t) will be made up of a fundamental frequency ω and multiples, 2, 3, 4, …, ∞ times greater. The A₀/2 term represents the D.C. (direct current) component (if present). The nth term of the Fourier series is called the nth harmonic of x(t). The amplitude of the nth harmonic is

(1.4)

and its square, , is sometimes called energy of the nth harmonic. Thus, the graph of the sequence is called the energy spectrum of x(t) and shows the amplitudes of the harmonics.