Phase, Time, and Summation at the Crossover
Matched slopes only set the loudness envelope — whether the drivers add or cancel is down to phase. And because delay is a phase ramp, the moment you “fix the timing” the phase you just dialled in rotates straight back out.
Matched slopes aren’t enough — phase at the crossover decides
When you tune a two-way you usually chase the woofer and tweeter slopes until they meet neatly at the crossover, then expect them to sum flat. They won’t, not automatically. Matched slopes only say how loud each driver is at each frequency — they say nothing about what you get when you add the two, because the sum is a vector addition, not an addition of loudness numbers. The thing that decides it is the relative phase of the two drivers at the crossover.
Chart 1. Two drivers with identical LR4 slopes, crossed at fc=2.5 kHz — same polarity sums to a flat 0 dB, but flip the polarity (180° apart) and the sum drops into a deep null at fc, with the slopes completely unchanged.
That “slopes are matched but the midrange went thin and hollow” complaint you hear so often is usually exactly this null — two drivers equally loud at the crossover, walking out of phase, cancelling each other.
At the crossover both drivers are equally loud, so it’s all phase
At the crossover frequency each driver is attenuated by the same amount (LR comes down −6 dB). Adding two equal-magnitude signals as vectors: in phase (0°) they reinforce by +6 dB over one driver, landing back at a flat 0 dB; 180° out and they cancel to nothing; anything in between gives a value somewhere down the middle.
And the relative phase the drivers should have at the crossover isn’t the same for every topology — it depends on the filter order:
| crossover | level at fc | relative phase | what to do |
|---|---|---|---|
| LR4 (4th order) | −6 dB | 0° (in phase) | same polarity, sums flat |
| LR2 (2nd order) | −6 dB | 180° | invert the tweeter |
| 1st order | −3 dB | 90° | same polarity, flat + transient perfect |
Easy rule of thumb: LR4 same-polarity drops right in, while LR2 gives you a full null if you forget to invert the tweeter — the classic trap for anyone moving from LR4 to LR2.
The phase you hear comes from three places
The phase your ear gets from each driver isn’t just the crossover filter’s phase. It’s three things added together, and the one people forget most often is the third.
The first two are minimum-phase (locked to the magnitude response), but the third — the drivers not sitting in the same plane — is pure delay. It’s a phase that ramps straight up with frequency, and it’s the culprit behind “I tuned it flat at one frequency and it went wrong everywhere else.”
Drivers out of plane = delay = a phase ramp
The woofer and tweeter don’t have their acoustic centres in the same plane, so sound from the deeper one reaches your ear a touch later. That lateness turns into a phase that ramps with frequency, and when it adds onto the crossover phase you carefully dialled in, the sum that used to be flat tilts and dips.
Chart 2. The same same-polarity LR4 pair, just with 117 µs of offset added — the sum tilts and dips by as much as ~−5.2 dB around ~2.9 kHz, with slopes and polarity all correct. This is why drivers that aren’t planar wreck the crossover.
Lobing — two sources sitting in different places
It doesn’t end on axis, because the woofer and tweeter live at different spots on the baffle (separated vertically). Move your ear up or down and the distance to each driver is no longer equal, so an extra geometric phase difference creeps in and the sum forms lobes — the null runs away with the angle.
Chart 3. The sum as you move vertically — flat on axis, but at +15° a null of −7.5 dB appears at ~2.8 kHz, and at +30° it deepens and slides down to ~2.2 kHz. That’s lobing: move your head, the crossover region is a different sound.
Flat on axis doesn’t mean flat at every angle
When you tune for flat you usually sit on one spot on axis, but it’s the whole radiated lobe that decides what you hear in a real room. LR4 has the nice property that its main lobe is symmetric about the design axis (up and down behave the same). Odd-order crossovers, or drivers with an offset, tilt the lobe — the axis where everything sums best slides off your ear height entirely.
Lobing is interference between two sources in different places. The larger the spacing (d) relative to the wavelength at fc, the narrower the lobe and the sharper the null. Placing the drivers close (or a d’Appolito MTM arrangement) and choosing the crossover order is how you control the lobe shape directly — a different job from dragging the on-axis curve flat.
Smear — the magnitude is flat but time isn’t
Now suppose you tune until both magnitude and phase are flat at the listening spot. The frequency response is dead flat — still not done, because group delay, the delay of each band of frequencies, may not be flat. Different frequencies arrive at the ear at slightly different times, so the transient (the leading edge) is stretched and blurred = smear.
Chart 4. Group delay of the sum (all three magnitude curves are equally flat at 0 dB) — LR4 spikes to 217 µs in the crossover region, while 1st order is dead flat. So the higher the order, the cleaner the slope, but the more smeared the transient.
Why aligning phase then adding time blows it up again
This is the most baffling part when you’re tuning — you adjust phase until the crossover sums flat, then you open up a delay to “time-align” the drivers so they arrive together, and the sound goes wrong again. The answer is in Chart 5 — delay is a phase that ramps in a straight line (Δφ=2πf·Δt). Adding delay doesn’t just shift “time”; it rotates the relative phase at the crossover too. What used to be 0° becomes something else, and the sum that was flat collapses immediately.
Chart 5. Relative phase between the two drivers — the flat line is the tuned-in case (0° at fc). Add 150 µs of delay and the line tips into a ramp, swinging through −135° at fc and crossing −180° (null) at some frequency. Time and phase are the same knob.
At a single crossover you cannot set “time” and “phase” independently — the delay that makes the transients arrive together is the phase ramp that rotates the sum at fc. Real tuning is finding the balance point of both at once, not chasing one knob while assuming the other holds still.
Why you can’t fix them separately: minimum-phase, allpass, linear-phase
The reason they can’t be separated is that drivers and most passive filters are minimum-phase systems — phase is locked to the shape of the magnitude by the Bode relations. Slide an EQ to fix magnitude and phase moves with it; you can’t separate them. Offset/delay, on the other hand, is allpass/linear-phase — flat magnitude but it adds group delay.
Getting all three is possible, but it costs
There are three things you want at the crossover: (1) flat magnitude, no dips or bumps; (2) flat time, every frequency arriving together, clean transients; (3) steep slopes, so each driver doesn’t play far outside its band. The problem is no method gives you all three — you have to choose what to give up. The three routes below are each “you can’t have it all” in a different way.
1st order + planar drivers — the only one that gets truly time-perfect (flat magnitude + flat phase + clean transient, what’s called transient perfect). The cost is the slope is very shallow (6 dB/oct), so each driver plays far outside its band — the tweeter cops bass it can’t take, the woofer reaches up high where it beams, and with both radiating over a wide overlapping range from different spots, lobing comes early. You give up the slope.
DSP FIR linear-phase — digital FIR filters genuinely control magnitude and phase separately (steep slope with flat phase, which analogue can’t do). The cost is latency (a buffer’s worth of delay, several ms — awkward for live or video sync) and pre-ringing (ringing that arrives before the real transient, which doesn’t happen in nature). You give up latency and the naturalness of the leading edge.
LR4 + offset/delay made planar — what most good speakers use: clean steep slopes, in-phase sum, flat on axis, a symmetric vertical lobe. The cost is the residual group delay in the crossover region (that band’s transients are a touch blurred, per Chart 4) — not time-perfect, but everything else is clean. You give up a little time only.
| method | what you get | what you give up |
|---|---|---|
| 1st order + planar drivers | truly time-perfect (transient perfect) | shallow slope, wide overlap, early lobing |
| DSP FIR linear-phase | steep slope + flat phase, controlled separately | latency + pre-ringing |
| LR4 + made planar | steep slope, flat on axis, symmetric lobe | residual group delay in the crossover band |
Go in this order: (1) make it planar — align the acoustic centres first (move the driver or add delay) so Δt_offset is near zero; (2) pick the order and polarity so the relative phase at fc is right; (3) then tidy up magnitude. Don’t swap the order — fix magnitude first and it fools the phase, and you’ll loop forever.
In short: slope, phase, and time are one knot
Slope sets the loudness envelope, relative phase at the crossover decides add-or-cancel, offset/delay ties phase to time, and the driver positions tie all of it to the listening angle through lobing. Three knobs that look separate are really one knot — tuning is untying it all at once, not pulling one thread at a time. Once you see that delay is just a phase ramp, the “aligned the phase, then time made it wrong” mystery stops being a mystery.
References
- aesLinkwitz, S. H. “Active Crossover Networks for Noncoincident Drivers,” JAES 24(1):2–8 (1976).
- aesLipshitz, S. P., Pocock, M. & Vanderkooy, J. “On the Audibility of Midrange Phase Distortion in Audio Systems,” JAES 30(9):580–595 (1982).
- appBohn, D. “Linkwitz-Riley Crossovers: A Primer,” Rane Note 160.
- bookD’Appolito, J. Testing Loudspeakers, Audio Amateur Press 1998 (lobing error / driver geometry).
- bookBlauert, J. Spatial Hearing, MIT Press 1997 (precedence & lobe perception).