Resonance Filters

A second-order IIR filter given by the difference equation can be designed with a single peak in its frequency magnitude response using coefficients of the form:

$\displaystyle a_1$ $\textstyle =$ $\displaystyle -2 r \cos( 2 \pi f_c T_s )$

$\displaystyle a_2$ $\textstyle =$ $\displaystyle r^2,$

where is the center frequency of the resonance peak, is the sample period, and The closer 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:

$\displaystyle b_0$ $\textstyle =$ $\displaystyle (1 - r^2) / 2$

$\displaystyle b_1$ $\textstyle =$ $\displaystyle 0$

$\displaystyle b_2$ $\textstyle =$ $\displaystyle -b_0.$

in a difference equation of the form
$\displaystyle y[n] = b_0 x[n] + b_1 x[n-1] + b_2 x[n-2] - a_1 y[n-1] - a_2 y[n-2].$ (1)
An example digital resonance filter magnitude response is shown in Fig. 13. The coefficients were determined as indicated above with a 5000 Hz center frequency.

Figure 13: Magnitude frequency response of resonance filter with Hz.
$\begin{figure}\begin{center} \epsfig{file=figures/resonance.eps, width=3.5in} \end{center} \end{figure}$
The reson~ Pd 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.
The quality factor is related to the parameter above as $r = e^{-\pi T_s f_c / Q}$ .

	$\displaystyle a_1$	$\textstyle =$	$\displaystyle -2 r \cos( 2 \pi f_c T_s )$
	$\displaystyle a_2$	$\textstyle =$	$\displaystyle r^2,$

$\displaystyle b_0$	$\textstyle =$	$\displaystyle (1 - r^2) / 2$
$\displaystyle b_1$	$\textstyle =$	$\displaystyle 0$
$\displaystyle b_2$	$\textstyle =$	$\displaystyle -b_0.$