Linkwitz Transform & DSP Biquad Calculator
Enter the measured sealed-box frequency and Q, choose the target alignment, and get DSP biquad coefficients with the headroom and power cost shown beside them.
Maximum filter boost (dB)10.21
- Low-frequency boost (dB)
- 10.21
- Maximum-boost frequency (Hz)
- 0
- Recommended preamp reduction (dB)
- 10.21
- Peak power multiplier
- 10.5
- Maximum reference power before boost (W)
- 47.63
- Target -3 dB frequency (Hz)
- 25
- Standard coefficients, H(z) denominator uses 1 + a1 z^-1 + a2 z^-2
- b0=1.0009619171, b1=-1.9953593427, b2=0.9944320434, a1=-1.9953713094, a2=0.9953819939
- miniDSP coefficient block, feedback signs inverted
- b0=1.0009619171, b1=-1.9953593427, b2=0.9944320434, a1=1.9953713094, a2=-0.9953819939
Use these coefficients only for a sealed-box response that behaves like a second-order high-pass system.
How to use this calculator
- Measure or simulate the sealed-box f0 and Q0 before applying other equalization.
- Enter the target frequency and Q you want the transform to create.
- Use the DSP internal sample rate, not just the incoming audio stream rate.
- Reserve at least the recommended preamp reduction before pasting the coefficients into the DSP.
- Verify the response and excursion at low level before increasing playback volume.
What the Linkwitz transform calculates
A Linkwitz transform replaces one ideal sealed-box second-order high-pass response with another. It is useful when the measured box has a higher cutoff or a different Q than the alignment you want, and the loudspeaker, amplifier, and excursion limits can support the extra low-frequency demand.
The analog transform is:
H(s) = (s^2 + (w0 / Q0)s + w0^2) / (s^2 + (wp / Qp)s + wp^2)
Here f0 and Q0 describe the existing sealed system, fp and Qp describe the target system, and w = 2*pi*f. This calculator turns that transform into a digital biquad with a prewarped bilinear transform. It sets K0 = tan(pi*f0/Fs) and Kp = tan(pi*fp/Fs), then uses:
N0 = 1 + K0/Q0 + K0^2, N1 = 2(K0^2 - 1), N2 = 1 - K0/Q0 + K0^2
D0 = 1 + Kp/Qp + Kp^2, D1 = 2(Kp^2 - 1), D2 = 1 - Kp/Qp + Kp^2
The standard direct-form coefficients are b0=N0/D0, b1=N1/D0, b2=N2/D0, a1=D1/D0, and a2=D2/D0 for H(z)=(b0+b1z^-1+b2z^-2)/(1+a1z^-1+a2z^-2). miniDSP biquad entry uses the same b coefficients but reverses the signs of a1 and a2, so the calculator prints both conventions.
Boost, headroom, and the target -3 dB point
The low-frequency boost is the DC gain, 20*log10((K0/Kp)^2). The maximum boost is found by evaluating the digital biquad magnitude from DC to Nyquist, rather than assuming DC is always the peak. That matters with high-Q targets, where the largest gain can occur near the target resonance.
The recommended preamp reduction equals the maximum positive boost. A 10 dB boost needs about ten times the power at the boosted frequency, so the calculator also reports 10^(boost/10) and divides the entered amplifier power by that multiplier. This is only a headroom estimate; it does not prove the driver can stay below Xmax.
The target frequency is not always the target -3 dB frequency. For the target second-order high-pass response, the calculator solves x = (-A + sqrt(A^2 + 4))/2, where A = 2 - 1/Qp^2, then returns f3 = fp*sqrt(x). When Qp is close to 0.707, f3 is nearly the same as fp; other Q values shift it.
What this calculator leaves out
The math assumes a sealed enclosure that behaves like a linear second-order high-pass system. It does not apply to ported boxes, passive radiators, transmission lines, open baffles, or systems whose low-frequency response is dominated by room correction or protection filters. It also leaves out room gain, boundary reinforcement, leakage, voice-coil heating, thermal compression, cone excursion, driver distortion, and any separate high-pass filter used to protect the driver below the operating band.
Worked example
Suppose the measured sealed box is f0 = 45 Hz with Q0 = 0.900, and the target is fp = 25 Hz with Qp = 0.707 at a 48,000 Hz DSP sample rate. The prewarped calculation gives standard coefficients b0=1.0009619171, b1=-1.9953593427, b2=0.9944320434, a1=-1.9953713094, and a2=0.9953819939.
The DC boost is about 10.21 dB, and the maximum filter boost is also about 10.21 dB at DC. Reserving that headroom means a 500 W amplifier has about 47.63 W of flat reference power before the boosted band reaches the same peak demand. With Qp = 0.707, the target -3 dB frequency is about 25.00 Hz.
Common questions
What does a Linkwitz transform do?
A Linkwitz transform changes the low-frequency alignment of an ideal sealed-box loudspeaker. It cancels the measured second-order high-pass response and replaces it with a target second-order high-pass response, usually to extend bass or change Q.
How do I measure f0 and Q0 with Room EQ Wizard?
Measure the driver in its finished sealed enclosure at low signal level, with other EQ and filters bypassed if possible. Fit the low-frequency response or impedance-derived resonance to find the system resonance and total Q. Use the in-box measured values rather than free-air driver data.
Why is target frequency not always the -3 dB frequency?
The characteristic frequency of a second-order high-pass section equals the -3 dB point only for a Butterworth alignment near Q 0.707. Lower Q moves the -3 dB point upward, and higher Q can create response peaking before the response falls.
Can a Linkwitz transform be used with a ported subwoofer?
No. A ported, passive-radiator, transmission-line, or open-baffle system is not the same second-order sealed-box system assumed by this transform. Applying these coefficients to that kind of alignment can remove damping, overdrive the driver, or produce an unstable practical result.
Which sample rate should I use for miniDSP?
Use the sample rate at which the miniDSP processes the biquad internally. That may be fixed by the hardware or plugin and may not match the source file or input stream sample rate. Using the wrong rate shifts the implemented frequencies.
Why are miniDSP a1 and a2 signs reversed?
The standard transfer function writes the denominator as 1 plus a1 z^-1 plus a2 z^-2. miniDSP asks for feedback coefficients in the opposite sign convention, so the same stable filter is entered with a1 and a2 negated.