Subwoofer time alignment: why the distance formula is not enough

Ask how to align a subwoofer and the answer comes back immediately: measure the distance to the sub and to the main speakers, subtract, divide by the speed of sound, dial that many milliseconds into whichever is closer. It is the right answer to a slightly different question, and it is the reason so many systems still show a dip at the crossover after being “aligned”.

The arithmetic is not wrong. The model behind it is incomplete. Distance sets a propagation delay, but it is not the only thing delaying one source relative to the other, and the correction you apply is not the same kind of quantity as the error you are fixing. Three separate controls are involved, they do different things, and only one of them is a distance.

The calculator below solves for the setting that sums flatly across two octaves either side of the crossover, and reports what the system does with nothing corrected at all.

Three controls that are not interchangeable

The confusion starts because all three change the phase of one source relative to the other, and phase is the only thing that decides whether two sources sum or cancel. But they change it in different ways, and that difference is what makes one of them right for a particular error.

• Distance is not a control at all — it is the physical cause. Sound covers a metre in about 2.91 ms at 20 °C, so a sub half a metre nearer arrives 1.46 ms early, which is a 37° phase lead at 70 Hz.
• Delay is the correction. It produces a phase shift proportional to frequency: 90° where the delay is a quarter period, 180° where it is half, 360° where it is a whole one. Because the proportion moves with frequency, a fixed delay corrects exactly one frequency and nothing else.
• Polarity is 180° at every frequency at once. It cannot be traded against a delay, and it cannot be emulated by one. Its weakness is that 180° is all it has: it fixes a half-turn error and nothing else.

At an 80 Hz corner one period is 12.5 ms, which is 4.29 m of air. Half a period is 6.25 ms, or 2.145 m. That is the exchange rate between the three controls: getting the polarity wrong is equivalent to moving the subwoofer 2.1 metres, and no amount of care with distances will compensate for it.

The crossover is a fourth delay, and it is the big one

The mechanism almost never gets mentioned. The low-pass on the sub and the high-pass on the main speaker are filters, and every filter has a group delay in its transition band. That delay is a real delay of the signal, indistinguishable from any other.

For a Linkwitz-Riley 4th-order pair at 80 Hz it is 5.627 ms in each branch. That is 1.93 m of air — larger than the distance error most people are trying to correct. Both branches share it, so it cancels in the difference and the distance formula appears to work. It cancels only because the two branches are the same filter. The moment they differ, this 1.93 m stops being common to both and becomes part of the error.

Worse, it is not constant. The same LR4 branch has 7.503 ms of group delay at 60 Hz and 4.502 ms at 100 Hz, scaling as 1/f. A fixed delay cannot follow that. It cancels the difference at the corner — which is what a crossover is designed to do — and lags behind everywhere else, which is the residue you see as a tilt rather than a dip.

Why Linkwitz-Riley is the alignment that behaves

The crossover's phase contribution depends entirely on the shape of the two branches at the corner. There are four arrangements worth knowing, and the calculator will show each of them:

Pair at the corner	Level per branch	Phase difference	Sum at the corner
Butterworth 2nd order, both	−3.01 dB	180° apart	full cancellation
Butterworth 4th order, both	−3.01 dB	in phase	+3.01 dB bump
Linkwitz-Riley 2nd order, both	−6.02 dB	180° apart	full cancellation
Linkwitz-Riley 4th order, both	−6.02 dB	in phase	flat, 0 dB

Only the last row is flat, and it is flat for a reason worth stating. A 4th-order Linkwitz-Riley response is two Butterworth sections in cascade, which lands each branch exactly 6.02 dB down at the corner and exactly in phase with the other. Two equal in-phase halves, each 6.02 dB down, sum to unity: nothing is left over, so nothing has to be corrected, and the distance formula is the whole answer.

The 2nd-order row above it is the exception that proves the rule. There each branch is a single section rather than a cascade, so the two land 90° one way and 90° the other — 180° apart — and no delay can put them back together across the band. That pair is fixed by polarity, which is the subject of a section below.

Butterworth is not in that position. Each branch is only 3.01 dB down and the two are in phase, so they sum to +3.01 dB — the crossover bump that gave Butterworth its reputation. That bump is in the filters, not in the geometry, and moving the subwoofer cannot remove it: the best delay the calculator can find takes the spread from 2.98 dB to 2.04 dB by moving the excess energy away from the corner, which is a choice about where the error goes rather than a fix.

When the two branches differ, no delay is enough

Pair an LR4 low-pass with a Butterworth 2nd-order high-pass — a common situation, since a small satellite is often specified differently from the sub — and the branches are 90° apart at the corner instead of 0°. At 80 Hz, 90° is 3.125 ms, so the distance formula now needs a 3.125 ms correction on top of whatever the geometry asked for.

Applying it does fix the corner. What it does not do is flatten the region: the phase-matched setting leaves 2.41 dB of ripple across two octaves, and the flattest setting the calculator can find — which reverses the subwoofer polarity as well as moving the delay — still leaves 1.54 dB. The residual is not a misalignment. It is the mismatch between a 24 dB/octave slope and a 12 dB/octave one, which no time correction can reconcile. The honest reading of that number is that the filter pair, not the alignment, is what needs changing.

The trick that looks clever and is not

An LR2 pair is 180° apart at the corner, so the branches cancel. The correct fix is to reverse the polarity of one driver, which is 180° at every frequency and makes the pair flat across the whole band.

There is a tempting alternative: leave the wiring alone and delay one source by half a period, 6.25 ms at 80 Hz. It works at the corner — half a period is exactly 180° there — and it produces a flat sum at 80 Hz. At 160 Hz the same 6.25 ms is a full period, which is 360°, which is no phase shift at all, so the 180° error comes back undiminished. Measured across the region the half-period delay gives this:

Frequency	40 Hz	60 Hz	80 Hz	100 Hz	120 Hz	160 Hz
Sum with a 6.25 ms delay instead of inversion	−1.67 dB	−0.63 dB	0.00 dB	−0.65 dB	−2.41 dB	−4.44 dB
Sum with the polarity reversed	0.00 dB	0.00 dB	0.00 dB	0.00 dB	0.00 dB	0.00 dB

A 4.44 dB dip an octave above the crossover, from a setting that measures perfectly at the crossover. That is the failure mode this article exists to describe: a correction that is exact at the one frequency you checked and wrong everywhere else. It is also why an alignment done by ear at a single tone can still sound wrong on music.

A delay and a polarity inversion are not two ways of doing the same thing. They are a frequency-dependent shift and a frequency-independent one, and the error you are correcting is one or the other. An odd number of poles of phase error — a 180°, a 540° — belongs to polarity. A timing error that grows with frequency belongs to delay.

Setting the polarity by measurement

The numbers above suggest a test that needs no instrument beyond a level meter, and it is the one worth trusting because it uses the system's own crossover rather than a model of it.

Play a sine at the crossover frequency and measure at the listening position. Note the level. Reverse the polarity of the subwoofer and measure again. For a matched Linkwitz-Riley pair the two settings differ by the numbers in the table above: one is 0 dB and the other is a null tens of dB deep, because the branches that were summing now cancel. Keep the louder of the two. For an LR2 pair the same test gives the same rule and the opposite answer — the louder setting is the inverted one — which is exactly why the polarity should be set by measurement rather than by a convention remembered from a different system.

Two cautions. First, do the polarity test with the geometry already corrected, so the branches are otherwise aligned and the null is real; a mismatched distance can move the null away from the frequency you are measuring. Second, use the crossover frequency itself. The difference between the two settings is largest there — the full depth of the null — and it shrinks as you move away in either direction: a third of an octave out it is under 6 dB, and two octaves out it is a tenth of a decibel, at which point the two branches have stopped overlapping and there is nothing left to decide.

Check the arrival, then stop trusting the arithmetic

Everything above is a model of two second-order sections and a delay, which is the right model for setting up the controls and the wrong model for predicting what a room will do. Once the delay and the polarity are set, measure the actual system with a swept sine or a noise stimulus and look at the summed response across the crossover region, not at the corner alone. A single-frequency check cannot see a 4.44 dB error an octave away, and neither can a single-frequency alignment procedure.

If the measured response shows a broad dip rather than a sharp one, the delay is wrong or the crossover frequency is. If it shows a sharp null, the polarity is wrong. If it shows a tilt across the whole region with the corner itself roughly right, the two branches differ in order and no alignment will fix it.

Five mistakes that survive review

1. Correcting the distance and nothing else. For a matched pair that is the whole job and it works. For a mismatched pair the distance is one of two terms and the smaller one.
2. Using a delay where polarity is the right tool. Half a period is a 180° rotation at exactly one frequency. It measures perfectly there and leaves 4.4 dB of dip an octave up.
3. Assuming a mixed crossover can be aligned. A 24 dB/octave low-pass and a 12 dB/octave high-pass leave 1.5 dB of ripple at best. Changing the slopes is the fix; time correction is not.
4. Forgetting the electronics in the sub's own path. A plate amplifier with its own DSP, or a wireless link, adds latency that no distance measurement knows about. It belongs in the same total as the geometry, which is why the calculator has a field for it.
5. Trusting the arithmetic for the temperature. 2.91 ms per metre becomes 2.87 ms at 30 °C, a 1.7% change. That is second-order against a 5.63 ms group delay and irrelevant next to a polarity error — worth knowing so it can be dismissed deliberately rather than overlooked.

Summary

Three quantities decide whether two sources sum flatly. Distance is the cause, delay is the correction for anything that grows with frequency, and polarity is the correction for 180°. A crossover adds its own delay to both branches, which cancels when they are the same shape and does not when they differ, and it is usually larger than the distance error being corrected. Set the polarity by measurement at the crossover frequency, set the delay from the geometry plus any known latency, and then measure the whole region rather than the corner.

The calculator is an engineering aid rather than a measurement; see the disclaimer, and the tool index for the rest of the series.