Linkwitz explains why I had to delay the signal to the subwoofers to get time alignment with the panels. At the end of this incredibly informative page on Crossovers.
I don't think I like his solution better than mine (I delay the woofer channel instead of phase shifting the midrange channel) but his analysis is crystal clear despite being totally counter-intuitive.
One normally doesn't think of woofers as being a high-pass device, especially subwoofers. They are supposed to go lower than we can hear after all. But lower than we can hear, is still infinitely higher than DC. At some point above DC the woofer HAS TO cut off (except for fan-type subwoofers, and they require massive amounts of power and cannot be made cheaply).
AND, here's the rub. The lower the low cutoff, the greater the phase lead! So, woofers will naturally lead higher frequency drivers, since the former have lower low cutoff.
My solution of delaying the subwoofer channel can work perfectly at a single frequency, but not other frequencies because the problem may not be that the woofer is simply delayed...it has a highpass phase lead, which means the delay varies with frequency. In that sense, Linkwitz' solution is better, it brings the woofer and midrange into alignment at all frequencies. Mine might be better because I'm not introducing more allpass behavior in the midrange.
With a reasonably steep crossover, I don't believe the mismatch in allpass behavior is such a problem generally (at least in digitally crossed over systems like mine that can easily apply delay instead), and especially in my system where the "midrange" panels can go nearly as low as the subwoofers. However, it's worth keeping in mind. It's an incredibly important insight. It seems to suggest that my friend who was often criticizing speakers with recessed woofers especially as not being "time aligned" may have been all wrong. Also explains why one of the least costly "transient correct" speakers of all time, the Spica's, have deeply recessed woofer. I vaguely recall, back in the day, that a defender of Spica or some such mentioned the woofer lead issue, but I never understood it before at all. I thought it was the crossover that was causing the woofer to "lead," and in that case the distances still needed to be equal so the crossover would work correctly. But the problem is, it's not the crossover that causes the woofer to lead, it's the difference between the woofer and the midrange drivers in themselves, their differing low frequency cutoffs. Ideally both should have identical low frequency delays, but they don't.
However it still troubles me that this "lead" might not be applicable entirely to transients. That's where my intuitions regarding how filters work seriously breaks down.
It seems to me that information cannot travel through a system in negative time, and that information does not travel through woofers hugely faster than midranges. Any kind of "phase lead" must represent some kind of information loss, like the front part getting lopped off. But in saying that, I still can't see how this would apply to the woofer/midrange situation.
Perhaps it's because I'm not thinking of the information loss to the midrange--it is losing the lowest frequency information (because of it's own highpass behavior at least) relative to the woofer. The lopping off of this low frequency information is possibly causing it to be delayed MORE. The woofer is being delayed LESS because it is not losing as much low frequency information.
I don't really know how to understand this yet, but it does seem that to work properly in a system with a crossover, the woofer must be delayed even if this means it doesn't start contributing information to the listening position until a later time than the midrange. And that is what one sees in a decently engineered but nevertheless allpass speaker impulse...in which the high response starts and rolls into the lower responses. But how would it work with transient perfection? Perhaps the greater phase correction is needed for the woofer itself than the crossovers.
It seems that electrical engineering has many different abstractions, and issues arise with mixing the abstractions the wrong ways.
Although we are talking about the low pass function of a woofer (Kellogg-Rice dynamic woofer), but actually, as far as the abstract high pass function nature of it, we could just as well be talking about a capacitor in a series-capacitor followed by shunt resistor circuit (following some AC generator), which is another high pass function if we idealize it slightly.
In that circuit, current must lead voltage in the capacitor. For abstraction purposes, we assume the resistor has no parasitic capacitances and inductances, and as such it samples the voltage at its terminal instantaneously. (Somehow when people talk about the capacitor in the voltage leading thing, they rarely talk about the load.)
But what does this mean for information traveling from the generator to the load? I'm thinking out loud here.
At the moment the generator applies a voltage, that voltage appears across the capacitor and the load. But the capacitor is having none of it. Instead, it all appears across the load. As current flows through the load, it flows into the capacitor and therefore a voltage appears across the capacitor AFTER it has appeared across the load. The capacitor is soaking up the signal AFTER it has started. So in the long run, DC or a Step would be quenched all into the capacitor and none into the load.
In what sense, then, is the signal "leading" into the resistor? In the sense that the maximum rate of rise for the load is at 0 degrees, whereas the maximum rise in the AC signal itself is at 45 degrees. As soon as the input signal has started, the rate of change in the load begins falling, as would happen at 45 degrees.
The voltage at the load...which is the output of the low pass filter, is therefore 45 degrees ahead of the input signal.
However, none of this means that start time of signal is changed. It simply starts at 45 degrees rather than 0. One may be tempted to draw sine waves like arches but they aren't. The first 45 degrees is a gradually inflecting upwards, starting like nothing. OTOH, the output of a high pass filter starts at maximum upwards movement.
Now, for a larger capacitor this leading time is longer. So the high pass filter retains the maximum upwards characteristic longer. I still find it baffling that this means the larger capacitor "leads" more. It seems to me more that it holds on to the lead longer.
CIVIL: Capacitor: I leads V, and by 90 degrees, Inductor: Lags.
The Capacitor is maximally charging at 0 degrees into the input signal, which means the current is at maximum at that point, which means the voltage across the load leads the AC input signal by 90 degrees, because the voltage across a resistive load has the same phase as the current flowing through it, and because the current flowing through the capacitor must be the same as the current flowing through the load.
This is not to say it's starting any earlier in response to any signal, but that it's starting at the 90 degree point instead of zero for AC waveforms.
I'm still not clear why the lower high pass filter would lead more. However, if the lower high pass filter leads by 90 degrees of it's cutoff frequency, that would be "more leading" time. I just don't see how it's cutoff frequency comes into play here. It should be just 90 degrees. The capacitor is maximally charging at 0 degrees, i.e. when the signal starts.
And even then, it's still even more baffling how the lower high pass filter leads more then the higher high pass.
The "more leading time" seems to be the correct reason and answer. The leading DOES have to do with the cutoff of the RC circuit. It leads by 90 degrees of that cutoff frequency, even if that frequency is no where in the input, that same effective leading is applied to everything along with the associated cutoff. It's weird but I think I'm beginning to get it.
One again this has little or nothing to do with how fast the input starts, but the phase angles are going to contaminate the step response unless the leading is accounted for...this tends to mean delaying the "more leading" woofer...and it does seem desirable that it should be delayed, and also so that the impulse response "rolls into " the bass, rather than the bass in any way contaminating the leading edge of the impulse. In fact that contamination is what happens is the woofer is not delayed, as the speaker with the lower highpass cutoff is going to have pre-mature phase angle from a dirac impulse--actually beginning to tend downward as the midrange is still going up, causing cancellation of the leading edge.
The exception would be designing each range of speaker for identical highpass response. Some very quirky designs do that, not that I'm endorsing them in total, but it's easy to understand the desirability of it now.
The woofer's natural highpass behavior causes a phase lead which is probably far from zero at the crossover point and therefore affects the addition of the woofer and midrange outputs. This can be corrected by placing a first order allpass in the midrange channel which simulates the highpass phase shift of the woofer.
I don't think I like his solution better than mine (I delay the woofer channel instead of phase shifting the midrange channel) but his analysis is crystal clear despite being totally counter-intuitive.
One normally doesn't think of woofers as being a high-pass device, especially subwoofers. They are supposed to go lower than we can hear after all. But lower than we can hear, is still infinitely higher than DC. At some point above DC the woofer HAS TO cut off (except for fan-type subwoofers, and they require massive amounts of power and cannot be made cheaply).
AND, here's the rub. The lower the low cutoff, the greater the phase lead! So, woofers will naturally lead higher frequency drivers, since the former have lower low cutoff.
My solution of delaying the subwoofer channel can work perfectly at a single frequency, but not other frequencies because the problem may not be that the woofer is simply delayed...it has a highpass phase lead, which means the delay varies with frequency. In that sense, Linkwitz' solution is better, it brings the woofer and midrange into alignment at all frequencies. Mine might be better because I'm not introducing more allpass behavior in the midrange.
With a reasonably steep crossover, I don't believe the mismatch in allpass behavior is such a problem generally (at least in digitally crossed over systems like mine that can easily apply delay instead), and especially in my system where the "midrange" panels can go nearly as low as the subwoofers. However, it's worth keeping in mind. It's an incredibly important insight. It seems to suggest that my friend who was often criticizing speakers with recessed woofers especially as not being "time aligned" may have been all wrong. Also explains why one of the least costly "transient correct" speakers of all time, the Spica's, have deeply recessed woofer. I vaguely recall, back in the day, that a defender of Spica or some such mentioned the woofer lead issue, but I never understood it before at all. I thought it was the crossover that was causing the woofer to "lead," and in that case the distances still needed to be equal so the crossover would work correctly. But the problem is, it's not the crossover that causes the woofer to lead, it's the difference between the woofer and the midrange drivers in themselves, their differing low frequency cutoffs. Ideally both should have identical low frequency delays, but they don't.
However it still troubles me that this "lead" might not be applicable entirely to transients. That's where my intuitions regarding how filters work seriously breaks down.
It seems to me that information cannot travel through a system in negative time, and that information does not travel through woofers hugely faster than midranges. Any kind of "phase lead" must represent some kind of information loss, like the front part getting lopped off. But in saying that, I still can't see how this would apply to the woofer/midrange situation.
Perhaps it's because I'm not thinking of the information loss to the midrange--it is losing the lowest frequency information (because of it's own highpass behavior at least) relative to the woofer. The lopping off of this low frequency information is possibly causing it to be delayed MORE. The woofer is being delayed LESS because it is not losing as much low frequency information.
I don't really know how to understand this yet, but it does seem that to work properly in a system with a crossover, the woofer must be delayed even if this means it doesn't start contributing information to the listening position until a later time than the midrange. And that is what one sees in a decently engineered but nevertheless allpass speaker impulse...in which the high response starts and rolls into the lower responses. But how would it work with transient perfection? Perhaps the greater phase correction is needed for the woofer itself than the crossovers.
It seems that electrical engineering has many different abstractions, and issues arise with mixing the abstractions the wrong ways.
Although we are talking about the low pass function of a woofer (Kellogg-Rice dynamic woofer), but actually, as far as the abstract high pass function nature of it, we could just as well be talking about a capacitor in a series-capacitor followed by shunt resistor circuit (following some AC generator), which is another high pass function if we idealize it slightly.
In that circuit, current must lead voltage in the capacitor. For abstraction purposes, we assume the resistor has no parasitic capacitances and inductances, and as such it samples the voltage at its terminal instantaneously. (Somehow when people talk about the capacitor in the voltage leading thing, they rarely talk about the load.)
But what does this mean for information traveling from the generator to the load? I'm thinking out loud here.
CIVIL: Capacitor: I leads V, and by 90 degrees, Inductor: Lags.
The Capacitor is maximally charging at 0 degrees into the input signal, which means the current is at maximum at that point, which means the voltage across the load leads the AC input signal by 90 degrees, because the voltage across a resistive load has the same phase as the current flowing through it, and because the current flowing through the capacitor must be the same as the current flowing through the load.
This is not to say it's starting any earlier in response to any signal, but that it's starting at the 90 degree point instead of zero for AC waveforms.
I'm still not clear why the lower high pass filter would lead more. However, if the lower high pass filter leads by 90 degrees of it's cutoff frequency, that would be "more leading" time. I just don't see how it's cutoff frequency comes into play here. It should be just 90 degrees. The capacitor is maximally charging at 0 degrees, i.e. when the signal starts.
And even then, it's still even more baffling how the lower high pass filter leads more then the higher high pass.
The "more leading time" seems to be the correct reason and answer. The leading DOES have to do with the cutoff of the RC circuit. It leads by 90 degrees of that cutoff frequency, even if that frequency is no where in the input, that same effective leading is applied to everything along with the associated cutoff. It's weird but I think I'm beginning to get it.
One again this has little or nothing to do with how fast the input starts, but the phase angles are going to contaminate the step response unless the leading is accounted for...this tends to mean delaying the "more leading" woofer...and it does seem desirable that it should be delayed, and also so that the impulse response "rolls into " the bass, rather than the bass in any way contaminating the leading edge of the impulse. In fact that contamination is what happens is the woofer is not delayed, as the speaker with the lower highpass cutoff is going to have pre-mature phase angle from a dirac impulse--actually beginning to tend downward as the midrange is still going up, causing cancellation of the leading edge.
The exception would be designing each range of speaker for identical highpass response. Some very quirky designs do that, not that I'm endorsing them in total, but it's easy to understand the desirability of it now.
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