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I am reading a paper in which they filter EEG signal using a notch filter to remove the power supply component of 50 Hz, then a high pass filter at 0.3 Hz to remove artifacts resulting from breathing, and finally a low pass filter with a cut-off frequency of 40 Hz.

Is there any difference to simply using a band pass filter of 0.3 – 40 Hz?

Additionally, why the notch at 50 Hz when everything 40+ Hz is thrown away, anyway?

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Is there any difference to simply using a band pass filter of 0.3 – 40 Hz?

No - but some filter design tools might not provide an option to design a band-pass filter with a unity gain "plateau" over a range of frequencies (instead, the response of the band-pass filter will be a bump at a given frequency).

why the notch at 50 Hz when everything 40+ Hz is thrown away, anyway?

50 Hz is dangerously close to 40 Hz, and might still be in the transition band of the filter. The 40 Hz filter would have to be very steep to entirely attenuate the 50 Hz. It would be much easier to remove the 50 Hz component separately with a steep notch filter.

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You are right that this is a sub-optimal approach. I agree with pichenettes answer that it is mainly a question of having a sufficiently flexible filter design tool. With such a tool, even the notch at 50Hz can be incorporated into one single design specification with the advantage that the overall filter probably performs better for a given filter complexity.

If a linear-phase FIR filter is used, then all design specifications (lower band-edge, upper band-edge, notch frequency, etc. ) can be formulated as a linear programming problem. This paper is an example of such a method based on linear programming.

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