Bi-quadratic Resonant Filter

• For a biquad resonant (bandpass) filter,
  $$H(z) = g \frac{1 + \beta_1 z^{-1}+ \beta_2 z^{-2}} {1 + a_1 z^{-1}+ a_2 z^{-2}},$$
  the filter coefficients are given by
  $$\beta_1 = 0, \quad \beta_2 = -R$$
  and
  $$a_1 = -2R\cos(2\pi f_c T), \quad a_2 = R^2,$$
  where $f_c$ is the resonant (or center) frequency, and $R$, is set according to the desired bandwidth of the resonator
  $$R = e^{-\pi B_wT},$$
  where $B_w$ is the bandwidth at -3dB in Hz given by
  $$B_w = \frac{f_c}{Q},$$
  and $T$ is the sampling period.

• This function is often called bi-quadratic or simply a biquad because both the numerator and denominator of it's transfer function are quadratic polynomials.