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Well, there's no such thing as 'normal ADC'....ADC0804 doesn't directly allow negative voltages at its input (although you can use a rectifier at the input to measure the negative voltages as well .. and use some zero-crossing detection mechanism to inform ur microcontroller/microprocessor about negative and positive voltages).
Still, there are some other ICs that do support negative voltages directly.
Or you can simply use a level-shifter/clamper to shift your signal to ground level but this requires you to know about the peaks of your measure signal or you can use a clipper.
This method has one major drawback, it can only be used to capture the waveform but can't be used to determine between positive or negative sides unless you know some characteristics of the wave, like some sort of symmetry like in AC signals etc.
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