Parametric Equalizer "Q" Definitions and Bandwidth (page 3)

Applying the Results of the PEQ Bandwidth Calculation

The equalizer section of the Room EQ Wizard documentation has lots of useful bandwidth data on various DSP devices, much of it derived from measured data of actual units. For instance, the Behringer DCX2496 is stated in the REW documentation to have a half-gain bandwidth as follows.

"Bandwidth = sqrt(gain)*centre frequency/Q"

Taken in a purely literal sense, this statement would imply that the DCX2496 does not have the desirable symmetry property discussed earlier in this article. However, John Mulcahy verified that the DCX2496 does have this symmetry property, and when the gain is less than 1, its reciprocal must be used in the above formula in place of the gain. We can state this more formally as:

(27) B W Hz = f 0 A 0 Q x , A 0 1 f 0 A 0 Q x , A 0 < 1

where Qx is yet to be determined. We also know that BWHz = f0 / Qb for all A0, so we can substitute that relationship into (27) above, giving:

(28) Q b = Q x / A 0 , A 0 1 Q x A 0 , A 0 < 1

Comparing (28) with (10) shows that Qx = Qm. Since Qm is the MSO Q, this says the Q convention of the Behringer DCX2496 is the same as that of MSO.

Summary

To summarize, we can express the Bristow-Johnson Q (Qb) in terms of the MSO Q (Qm) as follows:

(29) Q b = Q m / A 0 , A 0 1 Q m A 0 , A 0 < 1

The Bristow-Johnson Q is therefore always lower than the MSO Q for the same PEQ filter.

Likewise, we can express the MSO Q (Qm) in terms of the Bristow-Johnson Q (Qb) as follows:

(30) Q m = Q b A 0 , A 0 1 Q b / A 0 , A 0 < 1

The half-gain bandwidth BWHz can be expressed in terms of the MSO Q as:

(31) B W Hz = f 0 A 0 Q m , A 0 1 f 0 A 0 Q m , A 0 < 1

and the half-gain bandwidth can also be expressed in terms of the Bristow-Johnson Q as:

(32) B W Hz = f 0 / Q b

for all A0. The half-gain bandwidth of a PEQ is its center frequency divided by its Bristow-Johnson Q.

Other pertinent details can be summarized as follows.