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I run bandpass filtering via the butter() in matlab to apply butterworth bandpass filtering to find out the signals are centered at zero after the filter. Would this always happen for any kind of low/high-pass filtering and why?

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High pass filtering means that low-frequencies, including DC, are removed. So if your signal has a DC offset (is not zero mean), then this will be removed by a highpass filter or a bandpass filter that does not include 0 in the passband.

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