2001 BMW 325i After market Radio

First you have to figure out if there's a remote amplifier from the factory. If there is, you'll need to find that and cut the wires there and run them to the radio. If there's no amplifier, the speaker wires will be in the dash behind the radio. You'll have to look there too to see if someone cut the original plugs off, if they left them there and spliced into the wires, or if they left them unbutchered and just ran new wires to new speakers. If the wires in the dash are still intact and the connectors aren't cut off, seriously consider finding an original radio, especially if there's a remote amp involved. There can be over a dozen factory speakers. You have to use all of them to get the proper tone response because they used separate bass, mid, and treble speakers in each corner, and tying three speakers together reduces the resistance to a much lower value than radios are meant to run. You run the risk of damaging the output circuits in the aftermarket radio, or at the very least the sound will not be clear.

BMW is one of the manufacturers that won't allow us to buy radio service manuals so I can't tell you if the amplifier is necessary. With Chryslers, for example, all radios run speakers directly, and all the same radios can use an amplifier. The amp is just for tone conditioning for the shape of the body. It doesn't increase volume or power. When GM and Ford radios use a remote amplifier, they have to be used with it or the sound will be too low to hear clearly. If an amp is used, an amp MUST be used, and if it isn't used, it can't be used. I don't know if your car is like a Chrysler system or a GM / Ford system. If your aftermarket radio has line output jacks, that is low-level output that has to go through an amp, and that might work if you have a factory amp. Doing it that way would be much easier than trying to jump dozens of wires together to bypass the amp.

Dandy. So what's your plan of attack? Your aftermarket radio will most likely have eight speaker wires. If you extend them and run them to where the amp was, you can splice them there to feed the speakers. I can get a diagram to show the wire colors but there are too many speakers in the car to connect them all. You might get away with running two speakers on one pair of wires from the radio but more than that will start to sound distorted if the radio can't drive that many.