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I am using a Recursive least square adaptive filter to process electromyography signals and it is working decently so far. I decided to implement an LMS adaptive filter as a noise cancellation, so that I can compare the results, however, going through the matlab documentation for the LMS filter and seeing the LeakingFactor $$0<\alpha<1$$ and the ForgettingFactor of the RLS filter$$0<\lambda<1$$ I am now confused if there is an actual difference between those two parameters or is the difference only in the names?

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The play similar role in those algorithms - the ability to forget the past and adapt to current reality.

In the LMS, the classic implementation has $ \alpha = 1 $.
Namely the optimal weights at any point are function of all inputs.

The Leakage factor allows to weigh the past differently in a damped manner which over times means the far past has practically no significance on the current result.
The other side means the ability to integrate data against noise is decreased which is the same balance the user stands when selecting the parameters of the RLS.

Namely, as usual, the balance between high bandwidth filter which adapts fast yet is sensitive to noise or the low bandwidth filter which is slow yet if the data is stationary can handle high energy noise.

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