Introduction
Active crossovers are often used in professional sound because they reduce the amount of power wasted in passive crossover components.
Active crossovers can also be very useful in a home system because the drivers each are driven by their own amplifier. A clipping amp will only affect the sound of the driver it is connected to, and (as the folklore goes) the amplifiers may be chosen to reflect the power requirements of music.
The Question
A common question is: "what power is needed for each frequency range?"
I once noted when playing with a spectrum analyzer that some types of relentless heavy metal have nearly a pink noise type spectrum, sometimes excepting the very top octave. So my usual (untested) thought on this question is that you should have approximately equal power per octave. So I decided to do a calculation this way, then test it with some software experiments.
The Pink Noise Guess Method
Here is how I would go about starting to figure this out. In music, the average power goes down as frequency goes up, but the peak power doesn't change nearly as much.
Pink noise has equal power per octave. Pink noise has a bit more HF on average than music, at least above 5k or so. This means that if you take 20-10k as 9 octaves (just ignore top octave for the calculation), if you need 9W RMS for a satisfactory average SPL from a full range speaker system with a given sensitivity, you essentially need 9W/9octaves, or 1W/octave. So for a woofer from 20-320 you need 4W, for a mid from 300-3k you need 3W, and for 3k-10k you need another 2W.
Then you tack on crest factor and just multiply power by 10 (or 15, or 20) and get 40, 30 and 20W amplifiers.
If a component (driver) of that system is less sensitive by 10dB, you need 10dB (or 10x) more power for that component... If more sensitive you need less power for that component.
So this method gives reasonable sounding numbers, but how close are they?
Music Spectrum
I recall reading of an experiment many years ago on Art Ludwig's wonderful Sound Page. The specific section I remember reading (almost 20 years ago!) is this. Here he does a little processing in Cool Edit and Matlab to determine average and peak sound pressure levels in the bands he was using for his crossovers 0-300Hz, 300-3000Hz and 3000-22000Hz (assuming CD source.) You could call this three roughly equal "decades" - a decade is a little more than three octaves - and so they should have roughly the same amount of power required if music is like pink noise.
In the three tracks he used, he found average power in the three bands to be roughly 66% in the low range, 30% in the middle and 4% in the high. He found peak power to be a maximum of 90% in the low range, 60% in the middle and 15% in the high. The percentage given is the percentage of power full range. Something seems amiss in that the peak powers he cites are less than the average powers for the bass and midrange in the jazz and classical tracks.
He does show that the peak/average ratio in the low, mid and high ranges are about 1.5, 1.5 and 4 times average for the talking heads track, and the jazz track shows the mid and high range peak/average at 3.6 and 10.
My Experiment
I did my own check (only on peak power levels) by processing roughly 30-60 second sections of some files with high and lowpass filters in Audacity
Rock1 - Rush - Fountain of Lamneth
Rock2 - Toadies - Possum Kingdom
Jazz1 - Flim and BB's - Tricycle
I Normalized files so peak levels were 0dB fullrange. I then processed them with 24dB/octave high and lowpass filters at 300 and 3000 to generate three bands, as in Art Ludwig's experiments. I then found the peak levels in the resulting filtered programs
0-300Hz Lowpass
Rock1 -3.5dB or 44%
Rock2 -1.5dB or 71%
Jazz1 -5.2dB or 30%
300-3k Bandpass
Rock1 -2.6dB or 55%
Rock2 -1.5dB or 71%
Jazz1 -3.4dB or 46%
3k-22k highpass
Rock1 -2.9dB or 51%
Rock2 -5.5dB or 28%
Jazz1 -2.1dB or 62%
The numbers mean that a signal with the applied filter was down that many dB (peak, not RMS) from the fullrange signal. The filtered signal needed to be amplified by that much to reach 0dB.
The number in percent is the percent of full range signal peak power needed in that particular bandwidth. So if a single amp needed to be 100 watts peak to drive the full range signal, it would need to be 62Watts to run the 3k-22k band without clipping for the jazz track, but only 28W for the Rock2 track. For the tracks I used, you can get away with a slightly smaller amp for each band than for the full range, but not much smaller at all.
This means the equal power per octave method as stated above underpredicted peak power by about half in each bandwidth. It should have been about 1/3 of the fullrange power for each band, but it was closer to 2/3. So if you use the equal power per octave rule, it would be a good idea to use a higher safety factor. More band-limited data is needed to test this rough method.
So the amount of peak power needed in each bandwidth is very program dependent.... Art Ludwig's experiments seem to indicate that the peak/average ratio is higher in the midrange, and much higher in the treble.
Many amplifiers are built with undersized output devices, heatsinks and power supplies and their RMS outputs are derated by 3 or more dB from the maximum short term power they can generate. This is known in the business as dynamic headroom. It would seem that this type of amplifier would be more useful in the high and midrange bands than in the bass.