hacking a ADC: adding a digital aux input

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If by ADC you mean DAC, then you usually need a Digital Audio Interface Receiver chip to convert S/PDIF to the format the chip likes. If it really is an ADC, then a Digital Audio Interface Transmitter.

Current production chips are AFAIK all surface mount, but there appears to be some stock of DIP versions of older chips on eBay at a reasonable price. (For a while, the DIP versions were fetching scalper prices on eBay and elsewhere.)

Sometimes you can find readymade boards with the interface chip mounted and a Toslink transmitter or something (as appropriate). They'd be intended for adding S/PDIF outputs to digital music hardware that lacked it (MiniDisc, satellite radio). The DIT4096 was one popular chip. S/PDIF output extension for Portable MiniDisc Devices

Or if you want a Digital Receiver using the DIR9001:
DIR9001 SPDIF decoder
 
I do mean ADC. This is what I'm trying to do, just for fun: take the coax digital audio out of my computer and connect it to the output of the ADC on the car stereo, thereby creating a "digital in" on my car stereo. I realize that the DIY digital in would play all the time no matter which source I had selected (USB, Tuner, CD, AUX, etc) so I'd have to be careful to only have something connected to digital in when there is silence on the stereo (set source to AUX or something similar)

With that said, if I'm using the coax digital output from the computer do I still need a Digital Audio Interface Transmitter?
 
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I think the main problem will be the clock domains.

Your car stereo has a clock oscillator, which is used to clock the ADC. The PC also has a clock generator, which will determine the timing of the digital output signal, e.g. on an SPDIF output.
The two clocks will be asynchronous, so the relative timing will vary over time.

There are solutions to this problem, but it does make it more complicated.

1. Use a sample rate converter, e.g. CS8420 or CS8422 from Cirrus Logic. They also have SPDIF input.

2. Use an asynchronous USB interface, e.g.
http://www.diyaudio.com/forums/digital-source/188902-xmos-based-asynchronous-usb-i2s-interface.html
This requires that a suitable clock is available from the car stereo. And the exact timing on the I2S interface is probably still a challenge.

3. Synchronize the PC clock and the car stereo clocks. This is not something I would recommend, since it could become quite difficult!
 
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