Parametric Equalizer "Q" Definitions and Bandwidth

Introduction

One of the most useful types of electrical filters for audio equalization (EQ) is the parametric EQ (PEQ). Robert Bristow-Johnson ("RBJ") has written the most thorough discussion I've read of the math related to the audio PEQ. To get the most of this article, you'll need to read two articles by RBJ. The most important of these is a PDF article titled The Equivalence of Various Methods of Computing Biquad Coefficients for Audio Parametric Equalizers. In that article, he derives a number of important results we'll look at shortly. However, there's only a brief mention of filter Q in that article. On his web page titled Cookbook formulae for audio EQ biquad filter coefficients, he introduces a new definition for the Q of PEQ filters (which he calls "peaking EQ" filters). His definition of Q is different from the classical definition found in electrical engineering texts. We'll discuss different conventions for defining Q, and their relationship to one another. The classical definition of Q will be discussed first.

The Classical Definition of Q

The classical definition of Q (which RBJ calls the "EE Q") is sometimes called the "pole Q". I'll refer to it as Q_p here. We'll look at a PEQ filter having a center frequency f₀ and a gain A₀ at the center frequency. The transfer function H(s) of this PEQ filter in terms of Q_p (the pole Q) can be written as:

(1)	$H (s) = \frac{s^{2} + \frac{A_{0} ω_{0}}{Q_{p}} s + ω_{0}^{2}}{s^{2} + \frac{ω_{0}}{Q_{p}} s + ω_{0}^{2}}$

where ω₀ = 2πf₀ and the gain in dB at the center frequency is A₀(dB) = 20 * log₁₀(A₀).

Symmetry is a Traditional Requirement for Audio PEQ

From the point of view of the user of such a PEQ, a desirable property would be that two filters having identical Q and center frequency, but one with a boost of "x" dB and another with a cut of "x" dB at the center frequency would combine to be perfectly flat. That is, if the former filter had a transfer function H₁(s) and the latter a transfer function H₂(s), we want H₂(s) = 1 / H₁(s). The filter as defined above in equation (1) does not have this symmetry property.

Modifying the Definition of Q for Symmetry

One popular way of defining Q to obtain the symmetry mentioned above is described in the PDF manual of the Hypex PSC2.400 amplifier and a PDF whitepaper by THAT Corp. The revised definition for Q is given by equations 1 and 2 in the latter paper. In the Hypex amplifier manual, the revised Q definition is described as follows.

"Dip/peak filter. For peaks, Q is defined by the poles. For dips, Q is defined by the zeros. Thus the same filter with opposite gains will cancel."

This is the definition of Q that's used by the Multi-Sub Optimizer software (MSO), so I'll call it Q_m, the "MSO Q". To see how it's used, I'll also introduce Q_z, the "zero Q" as hinted in the quote above from the Hypex manual. Using this notation, we can write the transfer function H(s) of equation (1) above as follows.

(2)	$H (s) = \frac{s^{2} + \frac{ω_{0}}{Q_{z}} s + ω_{0}^{2}}{s^{2} + \frac{ω_{0}}{Q_{p}} s + ω_{0}^{2}}$

Inspection of (1) and (2) shows that:

(3)	$A_{0} = Q_{p} / Q_{z}$

In MSO, the gain value A₀ and Q value Q_m are chosen, and Q_p and Q_z are computed as follows.

(4)	$Q_{p} = {\begin{matrix} Q_{m}, & A_{0} \geq 1 \\ {A_{0} Q}_{m}, & A_{0} < 1 \end{matrix}$

and

(5)	$Q_{z} = {\begin{matrix} Q_{m} / A_{0}, & A_{0} \geq 1 \\ Q_{m}, & A_{0} < 1 \end{matrix}$

It's easily seen that (4) and (5) satisfy (3) for all values of A₀. This approach also achieves the desired symmetry property. This isn't the only way to achieve the symmetry property though. We'll see how RBJ does it next.

RBJ's Definition of Q

In his Cookbook formulae for audio EQ biquad filter coefficients, RBJ expresses the PEQ transfer function H(s) in normalized form in the "peaking EQ" section. This form can be denormalized to a center frequency ω₀ in radians/sec as follows.

(6)	$H (s) = \frac{s^{2} + \frac{A_{b} ω_{0}}{Q_{b}} s + ω_{0}^{2}}{s^{2} + \frac{ω_{0}}{A_{b} Q_{b}} s + ω_{0}^{2}}$

I've called his Q definition Q_b, the "Bristow-Johnson Q", adding the "b" subscript to distinguish it from the other Q definitions discussed here. At the center frequency, the gain value, expressed herein as A₀, is easily shown to be:

(7)	$A_{0} = A_{b}^{2}$

From (2) and (6), we get:

(8)	$Q_{p} = Q_{b} A_{b} = Q_{b} \sqrt{A_{0}}$

and

(9)	$Q_{z} = Q_{b} / A_{b} = Q_{b} / \sqrt{A_{0}}$

From (4), (5), (8) and (9), we get:

(10)	$Q_{b} = {\begin{matrix} Q_{m} / \sqrt{A_{0}}, & A_{0} \geq 1 \\ Q_{m} \sqrt{A_{0}}, & A_{0} < 1 \end{matrix}$

Equation (10) expresses the Bristow-Johnson Q in terms of the MSO Q (Q_m) and the center-frequency gain A₀. Next, we'll look at the bandwidth.