Practically speaking, if one is interested in a frequency band well-separated from the line noise (say, for example, the 10-20 Hz band, with 60 Hz line noise), would it be advisable to notch out this line noise before bandpass filtering to the desired frequency range? It seems that, given of course an acceptable filter rolloff, it would be better not to perform an additional filtering step if it were not necessary; however, I'm wondering whether this may lead to some unforeseen problem.
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$\begingroup$ If the stop band of your bandpass filter is sufficiently attenuated, then the notch filter won't be required. You'll have to determine the filter settings based on the maximum line noise that you observe by varying the experimental setup (distance between transmitter and receiver in case of wireless transmission). A stop-band attenuation of 20dB is usually sufficient. If you are implementing the filter in an embedded platform for real-time applications and need a low order filter, try an order 6 butterworth filter. $\endgroup$– NaveenCommented Mar 31, 2015 at 16:54
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A general rule of thumb is to filter as little as necessary, because every filter distorts your desired signal, even if just a little bit. Also: a notch filter is advised in a situation where narrow-band interference is present within your signal's band. The notch filter will reject the interference while having minimal effect (ideally) in your signal.
In the problem you describe, my preferred approach would be to go with a low-pass filter only, and increase its order to achieve sufficient rejection of the 60 Hz noise.
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$\begingroup$ I agree with this fully, and appreciate hearing your advice. $\endgroup$ Commented Apr 2, 2015 at 3:15