### 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

- periods under the rectangular window,
- periods under the Hamming window,
- periods under the Blackman window,
- periods under the Blackman-Harris -term window,

*effective length*is only 2 periods, as illustrated in Figures 5.13(a) through 5.13(c).

**Next Section:**

Tighter Bounds for Minimum Window Length

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Simple Sufficient Condition for Peak Resolution