Resonance Filters

• A second-order IIR filter ( $y[n] = x[n] - a_1 y[n-1] - a_2 y[n-2]$) can be designed with a single peak in its frequency magnitude response using coefficients of the form:
  $$\begin{eqnarray*} a_1 &=& -2 r \cos( 2 \pi f_c T_s ) \\ a_2 &=& r^2, \end{eqnarray*}$$

  where $f_c$ is the center frequency of the resonance peak, $T_s = 1 / f_s$ is the sample period, and $0 < r < 1.0.$ The closer $r$ is to 1.0, the narrower the bandwidth of the resonance peak.

• As a further refinement, we can design the resonance filter such that the resonance peak always has a gain of 1.0 by specifing numerator (or feedforward) coefficients as:
  $$\begin{eqnarray*} b_0 &=& (1 - r^2) / 2 \\ b_1 &=& 0 \\ b_2 &=& -b_0. \end{eqnarray*}$$

  in a difference equation of the form $y[n] = b_0 x[n] + b_1 x[n-1] + b_2 x[n-2] - a_1 y[n-1] - a_2 y[n-2]$.

• An example digital resonance filter magnitude response is shown in Fig. 10. The coefficients were determined as indicated above with a 5000 Hz center frequency.

  Figure 10: Magnitude frequency response of resonance filter with $f_c = 5000$ Hz.

  
  

• The reson˜ MSP object implements a second-order digital resonance filter with parameters for gain, center frequency, and Q.

• The quality factor, Q = center frequency / 3dB bandwidth, will typically range between 1-100.