The received wisdom on ASR seems to be that you should always have some form of a cross-over between your mains and your subs. However, is this really always true?
Shall we first see why we use different size drivers and crossover/filters between the amplifier and the drive units?
An electromagnetic drive unit is a high-pass filter. It has a resonance frequency (fc) and its output will slope off at 12dB octave below the -3dB point (f3). In order to have the full audio spectrum you have to make the f3 at or above 20Hz. This requires a heavy cone and a very powerful magnet. A heavy cone cannot produce high frequencies well. You have to split the frequency to multiple bands and use two or more drive units each optimised to a limited frequency range. It can be seen by entering the parameters of the common materials and the available manufacturing techniques that a 2-way speaker with a cross over frequency of between 1-3kHz is the most optimised solution.
In order to make the low-frequency driver work up to 1-3kHz the cone size is limited. This in turns limits how low the f3 can go and/or how loud the speaker can go. However, as the musical range of recorded music is mostly above 40Hz. A speaker with an f3 of around 80Hz, placed against a wall (bookshelf) or on a corner can produce that 40Hz at an acceptable level, allowing us to hear almost the entire acoustic instrument range.
If you are happy with the above then a small 2-way speaker will satisfy you. If you are not you have two options: a 3-way speaker or the addition of a subwoofer. In the 3-way speaker the low frequency driver (woofer) and the enclosure is made larger and a second crossover is used between it and the midrange driver. However, various other factors comes into play and the sound quality gets deteriorated when the enclosure gets larger and the crossovers gets complicated. These makes a 3-way speaker much more expensive than an equivalent quality 2-way speaker. That is why using small 2-way speaker and subwoofers is sometimes preferred.
As the 2-way speaker will be high-pass filter working around 80Hz with a slope of 12dB/oct why not use a subwoofer which is optimised to work as low as 20Hz, with its own amplifier feeding the signal via low-pass active filter is that matches the 2-way speaker's inherent filter is easy? The answer to that is, why not indeed? The subwoofer and the 2-way match perfectly as a symmetrical crossover is formed. However, there is a catch: As the input signal is increased the voice coil can move outside the magnetic field. This will put it outside it's operating zone and the sound quality is dramatically lowered. This can happen even at low levels if a source with high dynamic range is played. You will have distorted peaks.
This is why it is advisable to feed a 2-way speakers via a high-pass filter. If you use a 2nd order filter set at the f3 of the 2-way speaker you can form a perfect 4th octave filter. This is where my old friend Linkwitz RIP came in, inventing the proper way to cross two drive units back in the 70s (Linkwitz-Riley filter). The subwoofer's low-pass filter should then be changed to a 4th order filter so that the cross-over is symmetrical and the 2-way speaker's and the subwoofer's outputs blend seamlessly. (Side note: I have never seen a 4th order filter on a sub, has anyone?)
There should be no argument on above. It is simple physics and elementary electronics. Richard Small wrote his thesis on how loudspeakers work half a century ago, in 1972. Unfortunately the Hi-Fi industry is still producing devices that are plainly breaking the rules and make us question the otherwise obvious.
This is a great forum. If I managed to add something to it I will be happy.