loudspeaker X-over setup

Part1:  Rattle up your theory.

Some 150 years ago a couple of brilliant guys like Fourier, Hertz and Helmholtz proved that every periodic signal (like music) can be seen as a combination of sine waves:

So, when we do understand the behaviour of sine waves in our system we do understand our system.
Two parameters of sine waves are quite obvious, amplitude for loudness and wavelength for pitch.
Taken in to account that both these are perceived in a logarithmic manner we often use magnitude in dB's and frequency in Hz for these two parameters.

Ok, so far everybody with only a wee bit of audio knowledge will know this and work with these two concepts on a daily base.

Now for a more tricky concept.

Actually music consisting of only 1 sine wave at the time can be a bit boring ;)
So normally when we would decompose our music into sine waves we would see really a lot of them.
At this point we also have to include the way they are related in time, in the analytic description of our systems.
Phase expresses this in a manner of how these frequencies are related to each other, rather then absolutely, in time.
Certainly we could express this relation in much more intuitive units like seconds (or ms). (actually we sometimes do and call this groupdelay)
But 1ms of time shift on wave of 10kHz has a completely different impact (in terms of phase) than the same time shift on a wave of 100 Hz.
So expressing this time difference in phase degrees will work better in our case.

Now we all know these pictures and the concept of describing a system in terms of frequency response:

More highs, more lows and for the more literate we also have mids... ;)
By now we should know that this is only half the story.
If we do a frequency analyses like the above (also called a bode plot) we should  also include the phase behaviour.

So for correctly describing how system X responses to a input one should see this kind of pictures:

Don't think this will be a nice sounding loudspeaker system, but hey..

Time (delay) and phase are intertwined and there is really a lot of confusion out there.
Let's agree to some conventions:
  • For now lets define phase difference as every time related difference between the sine components of our signals where one of them will be set to zero-time or 0' degrees shift. (usually the highest frequency).
  • Furthermore: if a group of these sine components are delayed by a equal amount of time I will not regard this as a phase difference! (for example in a two way system in which both sources have a physical distance)

next part2: FFT analyses