### 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

**Previous Section:**

Simple Sufficient Condition for Peak Resolution