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Could you help me? How can I understand the function of the filter coefficients practically? In the simple case, it is the impulse response of the LTI system. but how do they work?

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    $\begingroup$ Filter and LTI system theory can probably not be answered briefly in an answer here. As a first step, you could read about FIR filters, which are related to a moving average. In this case, the filter coefficients are the impulse response of the LTI system that you mention. Once you get the basic idea, continue with more involved FIR and then IIR filters. In general, filters work by adding delayed and scaled versions of the input and output signals. This book provides a good and intuitive introduction to these topics. $\endgroup$ – applesoup Feb 9 '18 at 8:50
  • $\begingroup$ When discussing the benefits of adaptive filters over non-adaptive filters, it is not the values of the coefficients themselves that are a disadvantage. The disadvantage is that the coefficients are fixed and, as a consequence, so is the filtering behavior. In certain applications it is useful to adapt the filter behavior to the input signal. An example is an adaptive notch filter for hum reduction: as the hum frequency may change slightly around the center mains power frequency (50 Hz or 60 Hz), an adaptive notch filter allows for tracking the frequency changes. $\endgroup$ – applesoup Feb 9 '18 at 9:24
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    $\begingroup$ Really, your questions make no sense. If you need to learn about adaptive filters, learn the basics of filtering first. There's no shortcut here. Read a book. $\endgroup$ – Marcus Müller Feb 9 '18 at 9:51
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    $\begingroup$ Or this: ccrma.stanford.edu/~jos/filters $\endgroup$ – Hilmar Feb 9 '18 at 13:44
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There is more than one way of looking at the coefficient values. An insightful answer is necessarily subjective so , filter coefficients $a_0,a_1,a_2\dots a_{N-2} a_{N-1}$correspond to the N coefficients of a polynomial of a single variable $s$ (or $z$ or also in $z^{-1}$).

$$ f(s)=a_0s^0+a_1 s^1+ s^2 a_2 + \dots + a_{N-2}s^{N-2} + a_{N-1} a^{N-1}=\sum_{i=0}^{N-1}a_i s^i $$ which can be equivalently expressed in terms of it's roots $$ f(s)=(s-r_0)(s-r_1)\dots (s-r_{N-2})(s-r_{N-1})=\prod_{i=0}^{N-1}(s-r_i) $$

An LTI filter is generally expressed as a ratio of two polynomials. One can equivalently, as shown, express a polynomial in terms of products of $(s-r_i)$ where the $r_i$ correspond to zeros and poles of the transfer function depending if it is a numerator or denominator root.

If an input $s=e^{j \omega t}$ is applied to the filter, under some requirements of existence and stability which depends on the location of the roots, a shifted and scaled complex function of the same frequency will be output : $\hat{f}_{out}(s)=Ae^{j \omega t+\theta }$

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