This section provides a brief overview of crossover performance in the form of graphs of frequency, phase, and square wave response. As discussed in the section on sound demos, I have test files if you would like to do your own listening tests. Arny Kruger has also posted files that simulate the effect of a Linkwitz-Riley crossover.
Crossover design is considered in much more detail in another section (blue highlighted words link to the Glossary, References, to another web page, or, if followed by a size in kb, link to a figure). In the real world, a crossover is electronically imperfect, and it interacts with the drivers There are a large number of crossover types. To keep things reasonable, here we consider only a 3-way system, and two crossover designs: a 1st-order Butterworth, and a 4th-order Linkwitz-Riley. I personally think these two are the best choices.
The results given here mathematically simulate a perfect crossover mated with perfect drivers. This isolates the effects, if any, of the crossover design itself. I consider time alignment, in which the drivers are arrayed such that the sound from each driver arrives at ones ears at the correct time, as part of the crossover design. Constructing a time aligned system is also discussed in a separate section
The crossover frequencies are 300 Hz, and 3000 Hz. The time-aligned 1st order Butterworth is described immediately below; then the effect of removing time alignment from this design is discussed; and then the Linkwitz-Riley design is presented. Finally I give my own impressions of listening to the demos.
Time-aligned 1st-Order Butterworth
A perfect 1st-order Butterworth is truly perfect. The frequency response curve [33.4 kb] is flat for both coherent and incoherent addition. Coherent addition from the 3-drivers is what one gets at the "sweet spot" with time-alignment. Well off-axis the addition tends to be incoherent; since the wall reflections contribute to the total sound, it is nice to have a flat incoherent response as well. The big problem with a 1st-order design is that the out-of-band response rolls off at only 6 dB per octave, which can be problematic for real drivers. The phase response curve [33 kb] is also flat, meaning there is no group delay. Note that curves for the individual drivers do have delay, which is inherent in any filter, but the total coherent sum of the three drivers does not have delay. Finally the square-wave response [46.1 kb] is also perfect. The responses of the woofer, midrange, and tweeter add to provide a perfect result (the ripple is due to numerically cutting off frequencies above 50kHz). Technical note: the curves in this paragraph are for a "tree topology A", whereas the remaining plots in this section are for "tree topology B"; see the crossover design section.
1st-Order Butterworth, no Time-alignment
To achieve time alignment, drivers have to be at a different distance from ones ears. Here we place all of the drivers at the same distance from the ears. Using the midrange as a reference, and values I measured for my own system, this introduces a delay of .16 milliseconds for the woofer, and an advance of .092 milliseconds for the tweeter. This has a very significant effect on the frequency response curve [45kb]. The incoherent addition is not affected, by definition, but the coherent addition develops a big notch.(In the initial edition of this section the sign of the delay was reversed, which causes less of an effect) The phase response curve is affected quite dramatically, and the square-wave response gets pretty ugly as well.
"Time-aligned" 4th-Order Linkwitz-Riley
A 4th-order crossover has the advantage that the out-of-band response rolls off at 24 dB per octave - so drivers don't have to struggle to reproduce frequencies over a huge range. A 4th-order Butterworth has a 3 dB bump in the coherent response at the crossover frequencies; Linkwitz and Riley came up with a fix for this, as shown in the frequency response curve [53.9 kb] for their design. The incoherent response does have a 3 dB dip at the crossover frequencies. All crossovers of order greater than one unavoidably have significant group delay, which is the case here as shown by the phase response curve [52.1 kb]. This makes the use of the term "time-aligned" a bit inappropriate in this case, since the crossover itself messes up the time-alignment. However, it still helps to space the speakers at the same distances as for the time-aligned 1st-order Butterworth, and that is done here. The square wave response [47.0 kb] is still ugly. Note that the 24 dB per octave response has essentially eliminated the contribution from the woofer to the 1 kHz square wave.
4th-Order Linkwitz-Riley, no Time-alignment
Without time alignment the 4th order Linkwitz-Riley frequency response has a bump of just over 1dB around 1.3 kHz and a dip of about 5 dB at 3.5 kHz. There is very little other effect on amplitude.
Listening to (and looking at) the Demos
After extensive ABX tests I am now convinced that I cannot hear a difference between the 1st-order Butterworth and 4th-order Linkwitz-Riley time-aligned designs. Although I had heard that filter group delay is inaudible, I was skeptical. Granted, square-waves don't occur in real music, but I am really amazed that such an ugly response isn't audible. Furthermore, using Cool Edit 96 you can zoom in on the actual music wave forms and compare the two files. An example of this shows very significant differences [41.4 kb] in the two waveforms. The top two curves are the right and left channels for one design, and the bottom two the right and left channel of the second design (I forget which design is which). Despite this, they sound the same! (at least to me). I also tried this with a 2nd-order Linkwitz-Riley, with the same result. The time-alignment however is another matter. In this case with the 1st order Butterworth I hear a very distinct difference. Whew! For a minute there I was worried that all the effort I put into time-alignment was a waste.
So the bottom line is that an obvious change in the frequency response curve is quite audible, but an equally obvious change in the time domain response can be quiet inaudible.
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