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I'd like to know where a crossover fits in a PA system. I'm using a basic pa system like below.
[singer]->[microphone]->[mixer]->[?]->[speakers].
The ? means do I connect the mixer to the power amplifier before the crossover then to the speakers or do I connect from the mixer to crossover then to the power amplifier and speakers...?!
There are several variations and it is not possible to answer this as a simple, generic question.
Most speakers have built-in crossovers for 2-way or 3-way (or even 4-way) systems. And, of course, self-powered speakers have the amplifier(s) and crossover built-in.
The primary place you would need a separate crossover would be where you are using a sub-woofer, and you must split the sub-low frequencies into the subwoofer, and the rest of the spectrum into the main speaker(s).
There are two basic forms of crossovers. The kind implemented for line-level signals splits the audio BEFORE it goes into the power amplifiers. You need a power amplifier for each part of the spectrum that is split out by the crossover.
And then there are speaker-level crossovers. You rarely see that kind as a separate unit. They are virtually always built-in to speaker systems.
The specific answer to your question depends on exactly what gear you are using. It is not possible to give an answer that covers all cases.
