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I feel this is a basic question but I found no answer for it so far.

Let's take a random sound card that lives within a PC. From the connections at the end of the datasheet I assume that it is powered by 5 V and uses the computer's ground. Maybe it is a faulty assumption, but by some small experiments with my laptop it seems that the audio input and output ground is connected to the overall ground of the computer. So the sound card does not have any fixed negative voltage to refer to.

The waveform of the input and audio should have negative and positive regions. How do the ADC and DAC handle that? If there is any external circuitry that shifts the voltage it doesn't seem so.

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Audio inputs are usually capacitively coupled. This means low frequencies are attenuated and DC, the lowest frequency of all, is blocked entirely. Usually audio circuits are designed such that low means something around the lower limit of human hearing, about 20 Hz.

A typical audio input stage might look like this:

schematic

simulate this circuit – Schematic created using CircuitLab

C1 serves to block any DC. The parallel combination of R1 and R2 set the input impedance. You can also view this arrangement as a simple low-pass RC filter, with the RC time constant, and thus the cutoff frequency of the filter, defined by R1||R2 and C1.

R1 and R2 meanwhile create a voltage divider that set the DC level as seen by the ADC or whatever follows. Usually R1 and R2 will be equal so that the DC level is midway between the supply rails, to allow for maximum voltage swing in either direction.

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Computer power supplies do have multiple voltages, including negative ones, so if some amplifier needs dual rails, these are available.

A single-supply pre-amp input is biased to some positive middle-of-the-range voltage, and the input signal is applied to that node via a coupling capacitor which passes AC, while blocking the DC component, preventing that input's bias voltage from driving DC through the source. (Additionally, a DC return resistor is often provided on the outside of that capacitor in case the output stage of the previous device needs its load to be a complete DC circuit.)

If the input line goes right to some op-amp chip without a coupling capacitor, that chip must be handling the negative voltage somehow. If there is no negative power rail, then the circuit topology is doing it. The Texas Instruments document Single-supply Op-amp Design has a lot of information about this. Look at the Boundary Conditions paragraph right on the first page:

Use of a single-supply limits the output voltage range to the positive supply voltage. This limitation precludes negative output voltages when the circuit has a positive supply voltage, but it does not preclude negative input voltages. As long as the voltage on the op amp input leads does not become negative, the circuit can handle negative input voltages. [Emphasis mine]

It is possible to design a very simple input stage with a single discrete component that handles voltages more negative than the negative supply to the device. The obvious example are JFET transistors (and, similarly, depletion mode MOSFETS). The pinchoff gate voltage of a n-channel JFET is below the source voltage. You can bias a JFET input at zero, and it will nevertheless handle negative swings without clipping them off. Look this very simple pre-amp for a passive pickup in an electric guitar. Note how there is only a single voltage supply, and yet the pickup, which generates both positive and negative voltages, is coupled directly to the gate of the JFET. Only the output is capacitively coupled.

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The soundcard audio input and output stages that I have seen are AC-coupled (capacitor in the line). This is in fact a major problem when you try to use PC + a soundcard as oscilloscope, because the is no easy way to measure the DC level of the input.

The AC-coupling means that the input (and output) stages can have their no-signal (DC) level at any point they choose.

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