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While IIR notch filters are attractive, I need to retain phase linearity at the filter output. I imagine that it's possible to use a standard IIR notch filter: https://www.researchgate.net/publication/224174145_A_Complex_Adaptive_Notch_Filter followed by some form of phase equalization. An alternative would be an FIR filter with conjugate symmetric coefficients. I am unaware of any FIR designs in the literature explicitly for a single complex tone.

Any thoughts on the best way forward?

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  • $\begingroup$ Question what's the phase of a standard IIR notch filter when you're not close to the notch frequency? Answer : Almost 0, hence almost linear. I suspect that a standard IIR notch filter is good enough $\endgroup$
    – Ben
    Jan 30, 2020 at 1:26
  • $\begingroup$ I already designed an adaptive notch filter. I will try to answer tomorrow $\endgroup$
    – Ben
    Jan 30, 2020 at 2:10

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I might consider doing a complex modulation of the frequency you want to notch down (or up) to 0 Hz. Then run your favorite deep DC blocker (plus phase adjustment if needed). Then complex remodulate the result back to the original unshifted spectrum.

DC block: For non real-time, just subtract the entire signal’s average. Or a long (weighted?) moving average if you can tolerate the latency (Rick Lyons has some notes on this technique).

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  • $\begingroup$ That may be OK, but this is supposed to work continually in a closed loop manner. Seems like a lot of downshifting and upshifting of blocks of samples. Maybe there's a way to make that more efficient. Irrespective of that, how would you update/track the time-varying frequency for the notch? I have seen LMS type update equations but don't know where they come from... $\endgroup$
    – rhz
    Sep 4, 2019 at 0:02
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The simplest solution is probably window design method FIR filter. Even though literature mostly only tells about designing lowpass filter, window design method is actually capable of creating any arbitrary frequency response FIR filter. Here is a reference Arbitray Frequency Response.

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  • $\begingroup$ Note that the question is about adaptive notch filters. $\endgroup$
    – Matt L.
    Sep 2, 2019 at 7:54
  • $\begingroup$ @MattL. I thought it is similar. $\endgroup$
    – mfcc64
    Sep 2, 2019 at 15:13

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