I was curious about crossover filter design, so I did some reading on Linkwitz-Riley filters. Seems to me that the general idea is that if you add HP and LP filters and they are properly designed, you get an all-pass filter, which made sense on its face. However, I cannot seem to work this out mathematically. Using an LR 2, my understanding is that the filter transfer functions are simple biquads with a Q of 0.5.
$H_{LP}\left ( s \right )=\frac{\omega _{0}^{2}}{\omega _{0}^{2} + \frac{\omega _{0}}{Q}s + s^{2}} $
$H_{HP}\left ( s \right )=\frac{s^{2}}{\omega _{0}^{2} + \frac{\omega _{0}}{Q}s + s^{2}} $
And if they're summed you'd get:
$H_{LP+HP}\left ( s \right )=\frac{\omega _{0}^{2}+s^{2}}{\omega _{0}^{2} + \frac{\omega _{0}}{Q}s + s^{2}} $
Which is not an all pass filter unless Q is infinite. Am I totally missing something here?