Periodic Signals

Many signals are periodic in nature, such as short segments of most tonal musical instruments and speech. The sinusoidal components in a periodic signal are constrained to be harmonic, that is, occurring at frequencies that are an integer multiple of the fundamental frequency $f_1$ .^6.6 Physically, any ``driven oscillator,'' such as bowed-string instruments, brasses, woodwinds, flutes, etc., is usually quite periodic in normal steady-state operation, and therefore generates harmonic overtones in steady state. Freely vibrating resonators, on the other hand, such as plucked strings, gongs, and ``tonal percussion'' instruments, are not generally periodic.^6.7

Consider a periodic signal with fundamental frequency $f_1$ Hz. Then the harmonic components occur at integer multiples of $f_1$ , and so they are spaced in frequency by $\Delta f = f_1$ . To resolve these harmonics in a spectrum analysis, we require, adapting (5.27),

$$\zbox {M \geq K \frac{f_s}{f_1}}$$ (6.28)

Note that $f_s/f_1 \isdef P$ is the fundamental period of the signal in samples. Thus, another way of stating our simple, sufficient resolution requirement on window length $M$ , for periodic signals with period $P$ samples, is $\zbox {M \geq KP}$ , where $K$ is the main-lobe width in bins (when critically sampled) given in Table 5.2. Chapter 3 discusses other window types and their characteristics.

Specifically, resolving the harmonics of a periodic signal with period $P$ samples is assured if we have at least

• $2$ periods under the rectangular window,
• $4$ periods under the Hamming window,
• $6$ periods under the Blackman window,
• $2L$ periods under the Blackman-Harris $L$ -term window,

and so on, according to the simple, sufficient criterion of separating the main lobes out to their first zero-crossings. These different lengths can all be regarded as the same ``effective length'' (two periods) for each window type. Thus, for example, when the Blackman window is 6 periods long, its effective length is only 2 periods, as illustrated in Figures 5.13(a) through 5.13(c).

Figure 5.13: Three different window types applied to the same sinusoidal signal, where the window lengths were chosen to provide approximately the same ``resolving power'' for two sinusoids closely spaced in frequency. The nominal effective length in all cases is two sinusoidal periods.