- If we record the decaying sound of a string or other impulsively excited object, we can interpret spectral magnitude peaks as natural modes of vibration, which can each be represented as exponentially decaying sinusoids defined by:

(17) *A*is the amplitude, is the natural radian frequency and is a phase offset. - The decay constant can be expressed in terms of the quality factor or Q factor of a mode or resonance as
.
- The Q factor is approximately the number of oscillations required for a freely oscillating system's energy to drop by the factor from its original energy and is commonly defined as
, where the bandwidth (or ) is the range of frequencies (in radians or Hertz) for which the energy is at least half its peak amplitude.
- Thus, we can estimate the Q factor of a resonance from frequency magnitude data by finding the center frequency of a peak and estimating the peak bandwidth at a level -3 dB from its maximum.
- If we wish to model a resonance with a second-order digital filter, we can relate the Q factor to the pole radius
as:

(18) *T*is the sample period.

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