Linkwitz Transform

In this application note, we show you how to use the miniDSP custom biquad programming feature to implement a Linkwitz transform.

What's a Linkwitz transform?

A loudspeaker driver in a sealed enclosure has a mathematical description that determines its low-frequency response. At some point, the response of the speaker starts rolling off and reduces in level at 12 dB per octave. (Each time the frequency halves, the output level drops by 12 dB.) The shape of this curve can be characterized by two parameters: tuning frequency F, and quality factor Q.

A Linkwitz transform is a mathematical operation that changes the effective F and Q to different values. Typically, this is used to lower F to get more low bass output, or to lower the Q to make the box behave like a larger box. You can implement this transform in any miniDSP plugin that supports the custom biquad programming feature, such as the Advanced 2-way crossover, Advanced 4-way crossover, the 4x10 and 10x10 plugins, and so on. Check the miniDSP plugins page to find suitable plugins.

Note that the Linkwitz transform only works properly for sealed boxes. It's not suitable for ported boxes or for open-baffle speakers.

1. Download the biquad programing spreadsheet

Download the community-contributed biquad programming spreadsheet from the following link, and select the "LT" tab.

2. Determine your speaker's F and Q

If you have built the speakers yourself, you may already know these parameters from your box design phase. Otherwise, you will need to determine them by performing a nearfield measurement of the woofer - that is, with the microphone placed close to the cone, or about 5 cm (2"). The curves below illustrate a set of curves with a F of 80 Hz and Q of 0.5, 0.7, 1.0, and 1.4 (green, red, purple, black).

Various values of Q for f = 80 Hz

With a measurement taken, you will need to guess at F and Q by comparing the measured response with the curves above. Enter your guesses at F and Q into the spreadsheet and compare the curve displayed in blue with your measurement. If needed, adjust F and Q in the spreadsheet until the curve displayed in blue is reasonably close.

It's not super-critical to get F and Q exactly right the first time around. After applying the transform (steps 3 to 5 below), you can adjust and try again if the result is not what you want.

3. Calculate the biquad parameters

In the spreadsheet, enter the box F and Q as f(0) and Q(0), and the desired F and Q as f(p) and Q(p). For example:

Linkwitz Transform parameters

The spreadsheet will display the original response, the desired or target response, and the equalization curve:

Linkwitz Transform curves

You will note at this point that you will need to be realistic about the target F and Q as typically the transform will result in significant boost at low frequencies. Set the target F and Q so that the amount of boost is compatible with available amplifier power and the woofer's excursion limits.

4. Enter the parameters

Scroll down on the spreadsheet to find the biquad coefficients section:

Linkwitz Transform coefficientsto your measurement

In the miniDSP plugin, set a filter to Advanced mode and enter the five parameters a1, a2, b0, b1, and b2, as shown below. (Note that the order in the spreadsheet is different than the order shown by default in the miniDSP plugin. Just copy each number and paste into the correct location.) Then press the Process button. The frequency response graph will update to show the transform.

Linkwitz transform loaded into miniDSP

5. Verify your results

Run the nearfield measurement of the woofer again. If all is well, you will see the target response! If the response is not what is desired, fine-tune your F and Q settings and try again. Here is the before and after response of the example speaker, as measured with Room EQ Wizard:

Linkwitz transform, before and after

Have fun, and don't forget to ask on the miniDSP forum if you have questions!