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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 .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 Hz. Then the harmonic components occur at integer multiples of , and so they are spaced in frequency by . To resolve these harmonics in a spectrum analysis, we require, adapting (5.27),

 (6.28)

Note that is the fundamental period of the signal in samples. Thus, another way of stating our simple, sufficient resolution requirement on window length , for periodic signals with period samples, is , where 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 samples is assured if we have at least

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).

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