I'm talking about designing "second-order" filters. The goal is to achieve "flat" frequency response. We have a couple alignments to implement. Let us pick two alignments for comparison: Butterworth and Linkwitz-Riley characteristics.
Imagine the "FLAT" FR curve, depicted on attached, the difference we can see on it is the "overlap" region.
Which one do you prefer between large and small overlap, and why?
Note:
Red curves = Linkwitz-Riley (large overlap)
Green curves = Butterworth (small overlap)
Imagine the "FLAT" FR curve, depicted on attached, the difference we can see on it is the "overlap" region.
Which one do you prefer between large and small overlap, and why?
Note:
Red curves = Linkwitz-Riley (large overlap)
Green curves = Butterworth (small overlap)
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One can imagine it, but your depicted curves are not those of classic second order LR (-6dB at the crossover point), which sum flat, and the BW (-3dB at the crossover point) which sums +3dB at the crossover point:
Jim Geisler designed and built analog four-way active crossovers (Mill City Systems AS410) that could (sort of) "morph" between the two.
Back in 1979, few of us users knew the difference between the BW & LR, we conferred over what to call the control knob, and settled on "DB DOWN AT CROSSOVER POINT".
At the 12 o'clock position, -6dB ("LR"), 5 o'clock -3dB ("BW"), 7 o'clock position not sure, but probably -9dB.
I was surprised to find the photo on line, few were made.
Sold mine (the first pair built) shortly after finding how much better the BSS (Brooke Siren Systems) using LR fourth order crossovers sounded, minimizing the overlap area, each output in phase through the crossover region.
The early BSS crossover points were fixed, to change them required exchanging lots of components on removable cards.
Even though the BSS LR crossovers in theory would sum flat, still had to cut the three crossover frequency points on a 1/3 octave EQ another -3dB to make a flat acoustic response with the speaker systems we were using. We used standard ISO 1/3 octave points back then, parametric EQ's were not too well accepted 😉
Since each second order filter introduces a 180 degree phase shift, the low-mid and high polarity should have been reversed from the low and high mid, but polarity was a crap-shoot back then, the XLR pin 1 was always shield, but the 2 or 3 being +/- convention had not been settled for microphones, amplifiers, and anything in between.
JBL still was still using the opposite speaker polarity from most everyone else on a good portion of their drivers..
I think ANSI/EIA RS-221A (1979)-Issued, Abbrev. Title: “Polarity of Broadcast Microphones" started alerting us they decided on pin 2+, pin 3-, the opposite of what about half of us thought was the better choice.
Many products using XLR connectors are still Pin 3+...
In retrospect, I don't know what polarity the outputs were on the Mill City Systems AS410.
"Textbook" filters seldom perform like they look when combined with real drivers, but in general I prefer the flat response of the LR.
That said, for a HP filter used to limit excursion below Fb (box tuning) on ported speakers, I prefer the BW's response.
Anyway, whatever IIR filters, EQ, polarity (and delay..) it takes to get a smooth phase transition between the acoustic crossover points are the ones I prefer.
Art
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IIRC the 2nd order Butterworth can sum flat when averaging the angles at which the sound is projected. For a typical 2-way, where the tweeter and woofer are offset, constructive interference where the phases are perfectly aligned is likely to be at some odd angle rather than straight ahead as with a well aligned Linkwitz Riley.
In any case, a Bessel type filter seems to define the limit of how 'steep' a pole-based filter (with no zeroes or pre-ringing) can be without artifacts like overshoot or ringing in the time response of the filter. A downside of a Bessel is that it's hard to make matching high-pass and low-pass sections that sum flat, so a L.R. is a nice compromise.
For a 2nd order Butterworth, the overshoot probably doesn't matter, the impulse response goes slightly negative before settling down to zero.
However, for higher order filters, the impulse response can start to 'bounce' back and forth, after the initial spike, leading to a perceptible drone or whine at the cut-off frequency. It's as though our ears don't really hear the absolute balance of the frequency response with the help of a bubble level, but the sharp edge between a perfectly flat plateau and a cliff-like drop-off can sound like an unwanted peak.
In any case, a Bessel type filter seems to define the limit of how 'steep' a pole-based filter (with no zeroes or pre-ringing) can be without artifacts like overshoot or ringing in the time response of the filter. A downside of a Bessel is that it's hard to make matching high-pass and low-pass sections that sum flat, so a L.R. is a nice compromise.
For a 2nd order Butterworth, the overshoot probably doesn't matter, the impulse response goes slightly negative before settling down to zero.
However, for higher order filters, the impulse response can start to 'bounce' back and forth, after the initial spike, leading to a perceptible drone or whine at the cut-off frequency. It's as though our ears don't really hear the absolute balance of the frequency response with the help of a bubble level, but the sharp edge between a perfectly flat plateau and a cliff-like drop-off can sound like an unwanted peak.