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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 $ 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),

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

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.
\subfigure[Length $M=100$\ rectangular window]{
\epsfxsize =\twidth
\epsfysize =1.5in \epsfbox{eps/EffectiveLength1.eps}
\subfigure[Length $M=200$\ Hamming window]{
\epsfxsize =\twidth
\epsfysize =1.5in \epsfbox{eps/EffectiveLength2.eps}
\subfigure[Length $M=300$\ Blackman window]{
\epsfxsize =\twidth
\epsfysize =1.5in \epsfbox{eps/EffectiveLength3.eps}

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University