A continuous-time resonator is characterized by its natural frequency $\omega_n$ and damping ratio $zeta$ in the differential equation: $ \frac{d^2y(t)}{dt^2}+2\zeta \omega_{n} \frac{dy(t)}{dt} +\omega_{n}^2 y(t) = \omega_{n}^2 x(t) $ where $x(t)$ and $y(t)$ are the continuous-time input and output of the system. The continuous-time transfer function can be computed with the Laplace transform: $ H(s) = \frac{\omega_n^2}{s^2 +2\zeta\omega_n s + \omega_n^2} = \frac{\omega_n^2}{(s-c_1)(s-c_2)}, \mathit{Re}(s) > -\zeta\omega_n $ When $0<\zeta<1$ the system is underdamped and the poles $c_{1}$ and $c_{2}$are complex conjugates. A continuous-time resonator can be used to model a single resonance of the vocal tract, called a [[formant]]. Each formant is defined by a **formant frequency** $F$ and a **formant bandwidth** $B$, both in Hz. ![[formant-frequency-and-bandwidth.png]] The magnitude in [[deciBel Full Scale (dBFS)|dBFS]] of the frequency response of a second-order system with $F=500 Hz$ and $B=100 Hz$. The **formant frequency** corresponds to the undamped resonant frequency (or natural frequency): $ \omega_{n} = 2 \pi F $ The **formant bandwidth** is measured between the cutoff frequencies, most frequently defined as the frequencies at which the frequency response has fallen to half the value at its peak (-6dB). $ B = f_{c_2} - f_{c_1} = \frac{\omega_n \zeta}{\pi} $ Solving for $\zeta$: $ \zeta = \frac{\pi B}{\omega_{n}} $ These continuous-time frequency values can be used to define the position of the poles of the continuous-time transfer function: $ \begin{align} c_{1} &= -\zeta \omega_{n} + \omega_{n}\sqrt{ \zeta^{2}-1 } \\ c_{2} &= -\zeta \omega_{n} - \omega_{n}\sqrt{ \zeta^{2}-1 } \end{align} $