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) | $$B{W}_{\text{Hz}}=\left\{\begin{array}{cc}\frac{{f}_{0}\sqrt{{A}_{0}}}{{Q}_{x}}\text{,}\hfill & {A}_{0}\ge 1\hfill \\ \frac{{f}_{0}}{\sqrt{{A}_{0}}{Q}_{x}}\text{,}\hfill & {A}_{0}<1\hfill \end{array}\right. $$ |

where Q_{x} is yet to be determined. We also know that BW_{Hz} = f_{0} / Q_{b} for all A_{0}, so we can substitute that relationship into (27) above, giving:

(28) | $${Q}_{b}=\left\{\begin{array}{cc}{Q}_{x}/\sqrt{{A}_{0}}\text{,}\hfill & {A}_{0}\ge 1\hfill \\ {Q}_{x}\sqrt{{A}_{0}}\text{,}\hfill & {A}_{0}<1\hfill \end{array}\right. $$ |

Comparing (28) with (10) shows that Q_{x} = Q_{m}. Since Q_{m} 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 (Q_{b}) in terms of the MSO Q (Q_{m}) as follows:

(29) | $${Q}_{b}=\left\{\begin{array}{cc}{Q}_{m}/\sqrt{{A}_{0}}\text{,}\hfill & {A}_{0}\ge 1\hfill \\ {Q}_{m}\sqrt{{A}_{0}}\text{,}\hfill & {A}_{0}<1\hfill \end{array}\right. $$ |

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

Likewise, we can express the MSO Q (Q_{m}) in terms of the Bristow-Johnson Q (Q_{b}) as follows:

(30) | $${Q}_{m}=\left\{\begin{array}{cc}{Q}_{b}\sqrt{{A}_{0}}\text{,}\hfill & {A}_{0}\ge 1\hfill \\ {Q}_{b}/\sqrt{{A}_{0}}\text{,}\hfill & {A}_{0}<1\hfill \end{array}\right. $$ |

The half-gain bandwidth BW_{Hz} can be expressed in terms of the MSO Q as:

(31) | $$B{W}_{\text{Hz}}=\left\{\begin{array}{cc}\frac{{f}_{0}\sqrt{{A}_{0}}}{{Q}_{m}}\text{,}\hfill & {A}_{0}\ge 1\hfill \\ \frac{{f}_{0}}{\sqrt{{A}_{0}}{Q}_{m}}\text{,}\hfill & {A}_{0}<1\hfill \end{array}\right. $$ |

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

(32) | $${BW}_{\text{Hz}}={f}_{0}/{Q}_{b}$$ |

for all A_{0}. 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.

- The REW generic PEQ uses the Bristow-Johnson Q convention.
- MSO uses the same Q convention as the Hypex PSC2.400 amplifier and the Behringer DCX2496.
- For miniDSP devices, the Q convention can be made irrelevant, as long as you use custom biquad export from your equalization software and custom biquad import into the miniDSP.
- When using miniDSP devices
*without*biquad import/export, and instead entering the Q values manually into the device's software user interface, the Q convention being used is the Bristow-Johnson convention, per a post by John Mulcahy. - The Bristow-Johnson Q is always lower than the MSO Q for the same PEQ filter. Consider two PEQ filters having the same numerical value of Q and all other parameters identical. If one is using the Bristow-Johnson Q convention and the other the MSO convention, the PEQ using the Bristow-Johnson convention will have a narrower bandwidth.