Thursday, March 10, 2011

Physical all-pass network cannot be canceled by another all-pass

From AES paper by Moller (2007) in JAES:

"The all-pass sections described so far have zeros in the right half of the complex s plane, and poles in the left half-plane.  The phase is negative, and the impulse response is causal.  In principle, all-pass sections can also have zeros in the left half-plane, and corresponding poles in the right half plane.  In that case the phase is positive and the impulse response is noncausal.  Such noncausal all-pass sections do not exist in physical systems, but it is relevant to consider them because they may occur as a result of calculations, such as when a transfer function that includes a causal all-pass section is inverted with the aim of equalizing it."

So, this states exactly what I have long figured.  If you have a network that introduces all-pass response (such as Linkwitz-Riley crossover), you cannot cancel that out with another physical network.  You can only cancel it with a calculation, such as DSP.  Another twist is that the impulse response of an all-pass section is infinitely long.  Any implementation of the inverted version has to be truncated in time and delayed.

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