Stanley P. Lipshitz, John Vanderkooy.
"A family of linear-phase crossover networks of high slope derived by time delay".
Presented at the 69th AES Convention 1981 May 12-15 Los Angeles.
http://www.aes.org/e-lib/browse.cfm?elib=11953Digital audio was not reigning yet in 1981. Pure, clean delays remained unfeasible. Such communication remained ignored for years.
Please try the following .fsm :
- Bandsplitter - Lipshitz-Vanderkooy - 8 bands (full) (650 pix).jpg (73.96 KiB) Viewed 47042 times
Try it for yourself. All 8 outputs are mostly phase-synchronous in the transition bands. The summed output is rigorously phase-linear. A square-wave in, leads to a reconstructed rigorously exact square-wave out. Quite unbelievable, isn't?
Latency is not zero. I think that a zero latency is physically impossible.
Latency depends mainly on the 1st crossover frequency, here 42 Hz, obliging a 356 samples latency.
Latency also depends on the 2nd, 3rd and 4th crossover frequency, here 96 Hz, 219 Hz and 503 Hz, obliging a 156 + 68 + 29 samples latency.
Consequently, the total latency is 609 samples, which is approx 14 millisecond when Fs = 44,100 Hz.
In case you want latency to drop below 10 millisecond, you need to set the 1st crossover frequency to a higher frequency, say 60 Hz instead of 42 Hz.
The Lipshitz-Vanderkooy band-splitter I am suggesting is consuming less CPU% than a 4th-order Linkwitz-Riley band-splitter that's incapable of delivering a phase-linear summed output. In case you want to insert a phase correction before a Linkwitz-Riley band-splitter, you will require a quite long FIR filter that's consuming a lot of CPU%, and anyway, for such long FIR filter to reasonably flatten the phase response between 50 Hz to 16 kHz, such long FIR filter will introduce at least 20 millisecond of latency.
Clearly, the Lipshitz-Vanderkooy band-splitter I've designed, is the best you can find (because it is transient perfect by nature), it is the most efficient (because of not requiring a phase correction), and it is exhibiting the shortest latency in the context of phase-linear band-splitters.
In case you dig into Stanley P. Lipshitz & John Vanderkooy "A family of linear-phase crossover networks of high slope derived by time delay", presented at the 69th AES Convention 1981 May 12-15 Los Angeles, you will discover that lowpass filtering by a double 2nd-order Butterworth, is
not what Stanley P. Lipshitz & John Vanderkooy advocated for. Like everybody, they were in search of the maximal highpass
monotonic slope that was attainable with a subtractive approach. They managed to find the 3rd, 4th, 5th etc. Butterworth lowpass filtering solution, that are all leading to a pseudo 3rd-order highpass slope, qualified by P. Lipshitz & John Vanderkooy as "
high slope derived from time delay".
Unfortunately P. Lipshitz & John Vanderkooy dropped (or did not find) the "slightly oversized delay, double 2nd-order Butterworth" solution that I am applying here, that's leading to a pseudo 4th-order highpass slope. It should be named
SOCDS-DB2.
Slightly
Over
Compensated
Delay
Subtractive -
Double
Butterwoth
2nd-order.
John Kreskovsky paper dating back from 2002 got largely under-valuated. Possibly because few people had access to Digital Signal Processing. I am talking about John Kreskovsky paper entitled “A New High Slope, Linear Phase Crossover Using the Subtractive Delayed Approach”, Dec. 2002. If I remember, John Kreskovsky was the first to popularize the
SOCDS-DB2 approach.
The slightly over-compensated delay subtracting scheme that's applied on a double 2nd-order Butterworth lowpass filter is probably the wisest choice, best balanced choice. And, as cherry on the pie, it is requiring approx. the same computational power than the ubiquitous 4th-order Linkwitz-Riley.
Unfortunately, below -35 dB, the SOCDS-DB2 attenuation curves appear to be
non-monotonic. A Linkwitz-Riley crossover is slightly better on such aspect (because it is perfectly monotonic, but nobody expects perfection).
Unfortunately, in the transition band, the lowpass and highpass are
not perfectly synchronous. A Linkwitz-Riley crossover is slightly better on such aspect (because it is perfectly phase-synchronous, but nobody expects perfection).
John Kreskovsky was not 100% clear about this. It didn't helped the SOCDS-DB2 approach to become understood, and mainstream. Meanwhile, the SOCDS-DB2 approach is transient-perfect, which means that a square-wave in, gets out as square-wave. And there, the Linkwitz-Riley crossover is disastrous. We are not talking about the Linkwitz-Riley being "slightly better". The Linkwitz-Riley crossover is a plain disaster on such aspect, contrary to the SOCDS-DB2 that's perfect on such aspect.
Please let me know if the SOCDS-DB2 band-splitter I've designed, helps improving the subjective quality of multi-band dynamics processors (expanders, compressors, limiters). And speakers crossovers also.
Have a nice day