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 .^5.9 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.^5.10

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 $\Delta f = f_1$ . To resolve these harmonics in a spectrum analysis, we require, adapting (4.8),

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

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 , for periodic signals with period samples, is $\zbox {M \geq KP}$ , where is the main-lobe width in bins (when critically sampled) given in Table 4.2. 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,

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 4.19(a) through 4.19(c).

**Figure 4.19:** 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.
$\begin{figure}\centering \subfigure[Length $M=100$\ rectangular window]{ \eps... ... =\twidth \epsfysize =1.5in \epsfbox{eps/EffectiveLength3.eps} }\end{figure}$

Previous: A Simple Sufficient Condition for Peak Resolution
Next: Tighter Bounds for Minimum Window Length

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Spectral Audio Signal Processing
    Spectrum Analysis of Sinusoids
       Resolving Sinusoids
          Periodic Signals