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I'm having some confusion learning about the LMS Adaptive Filter. I understand that the whole model of adaptive filters relies on the fact that we give it a reference signal to which it keeps comparing the input * filter with and the filter coefficients keep changing until the error between input and reference is zero.

Let's say I have a telephone conversation with a time-varying sinusoid added on top that changes frequencies every so often. How exactly do I give it a reference signal? I mean, if I had a reference signal of what I wanted the input to become, I would have just used it? And similarily if I had a reference signal for my unwanted sinusoid, I could have just subtracted it from my original sound.

What am I missing here?

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The LMS and many of the variants of Adaptive Filters (In the Linear System context) work in the following settings (Intuitive):

  1. You have access to 2 signals.
  2. One signal is the result of the other one when a Linear System is applied.

This sounds really limiting, yet in practice it is powerful and flexible.

In the settings you mentioned the most known and similar problem is the Echo Cancellation model.

Pay attention that your model can't be formed as a Linear System. Namely the connection between the clean signal and the corrupted signal can't be described using Linear System.

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  • $\begingroup$ I should have probably added, my unwanted sinusoid is modelled as a broadband signal and my voice is narrowband. I think I might be able to use a delayed version of my input as the reference. $\endgroup$ – AlfroJang80 Nov 18 '18 at 13:04
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In interference cancellation problems, the reference signal is processed by an adaptive filter. Afterwards, the primary signal is subtracted from the filter output, and the error is used to adapt the filter, for example using the LMS algorithm. LMS-based algorithms have the advantage that they are model independent so no statistical assumptions (e.g. stationarity) have to be made.

Using the primary input, in this case your undistorted signal, and the reference, the input plus sinusoidal distortion, you are able to estimate a filter which reduces the difference between clean and noisy signal. This estimate changes over time.

This approach has the advantage of adaption over a linear filtering approach such as a fixed frequency notch filter. Your particular problem is discussed in detail as an application for adaptive noise cancelling in the book Adaptive Filter Theory by Simon Haykin.

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