The Equivalence of Various Methods of Computing Biquad Coefficients for Audio Parametric Equalizers

Several authors have put forth design procedures for obtaining the coefficient values for second order digital filters to be used as parametric equalizers or "presence" filters. Given the EQ parameters, these differing design methods, having greatly disparate degrees of complexity, compute the filter coefficients. This paper shows that, with the possible exception of Q or bandwidth definition, all methods must be equivalent and the simplest method may be used. A method of computation is presented here in which the bandwidth is specified in octaves. 0 INTRODUCTION Applying a second order IIR filter to the problem of parametric equalization appears to be so natural and straightforward that one might expect the design solution to have been worked out, published, and universally accepted long ago. However, this does not seem to be the case. A few different second order IIR filter topologies have been suggested for use for the parametric equalizer or "presence" filter. These include the Direct Form I (or Direct Form II) (McNally [1], Moorer[2], White [3], Mitra, et.al. [4]) and Allpass (usually using the Lattice or Normalized Ladder forms) with feedforward (harris & Brooking [5], Regalia & Mitra [6], Massie [7]). Although they have differing round-off noise, limit cycle, and saturation characteristics, all can be simplified to have the same form of transfer function as the Direct Form I filter having 5 independent coefficients. This means that any scheme for computing coefficients (given the input parameters: boost/cut frequency, boost/cut gain, and Q or bandwidth) for one form can be compared to and adapted to another form if desired. This means, specifically, that the scheme used in Moorer's "Manifold Joys. " paper [2] (for 5 coefficient Direct Form I) can be used in the Normalized Ladder instead of the method proposed by Massie/Mitra. Or vice versa. Moorer's method is much more complex than the methods proposed by Massie, Mitra, or White but seems to deal with the definition and implementation of the EQ bandwidth more satisfactorily than the latter papers. This paper compares the different methods, the claims made therein, and the input parameter definitions assumed. While the emphasis is on determining the standard biquadratic transfer function coefficients from which the Direct Forms I or II derives its coefficients directly, some 2 attention will be given to the nearly trivial problem of converting to the coefficients in the Lattice or Normalized Ladder forms in case that may be the desired implementation. 1 FOUR EQUATIONS, FIVE UNKNOWNS The five coefficient biquadratic discrete-time filter shown in the Direct Form I in Fig. 1 has the transfer function shown below. H z ( ) = b0 + b1z −1 + b2z −2 1+ a1z −1 + a2z −2 (1) If the filter is designed to be a parametric equalizer (or “presence” filter) boosting (or cutting) the gain for a given frequency, the magnitude frequency response may appear as one of the plots in Fig. 2. Immediately, there are four requirements, expressed as four equations that can be postulated: The magnitude gain at DC and at the Nyquist frequency must be zero dB. H e j 0 ( ) = H 1 ( ) = 1 (2) H e jπ ( ) = H −1 ( ) = 1 (3) ∑