Spectral Peak Q Factors

• The sounds produced by impulsively excited objects, such as marimba bars or plucked strings, can often be well approximated as consisting of a set of independent exponentially decaying sinusoidal components (or vibrational modes), each of the form of:
    $$x_n(t) = e^{-\alpha_n t} A_n \cos(2 \pi f_n t + \phi_n),$$ (13)
  where $\alpha_n$ is a decay constant, $A_n$ is the amplitude, $f_n$ is the sinusoidal frequency and $\phi_n$ is a phase offset.

• Each decaying sinusoidal component will produce a peak, or resonance, in the spectrum of such sounds and the frequency bandwidths of the corresponding spectral peaks will be directly proportional to the sinusoidal decay constants.

• The decay constant can be expressed in terms of the quality factor or Q factor of a mode or resonance as $\alpha = 2 \pi f_n / 2 Q$.

• The Q factor is approximately the number of oscillations required for a freely oscillating system's energy to drop by the factor $e^{-2\pi}$ from its original energy and is commonly defined as $Q = f_n / \Delta f_n$, where the bandwidth $\Delta f_n$ is the range of frequencies (in Hertz) for which the spectral 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 then estimating the peak bandwidth at a level -3 dB from its maximum.

• A higher value of Q (a smaller $\Delta f_n$) corresponds to a lower sinuosoidal decay constant, which corresponds to a more slowly decaying sinusoidal component.

• The Matlab script parafit.m demonstrates parabolic interpolation and estimation of the -3dB bandwidth.