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Note that for a small sampling interval $T$, $\big(d[k+1]-d[k]\big)/T$ is a good approximation for the velocity. So if you fit $au[k]+b$ to a given set of measurements $v[k]$, it is valid to conclude $$d[k+1]=d[k]+T\big(au[k]+b\big)\tag{1}$$ In the text you refer to they might have normalized $T$, so it changes the units without changing the values of $a$ ...

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Approaches can be any of the following: Model the noise that you expect, for ex: gaussian (least squares in Minimum variance unbiased estimator for a linear signal model in presence of gaussian noise). Based on this model try and estimate the noise variance, the regularization term should be close to noise varainace. Deploy machine learning techniques based ...

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As mentioned in the comment, I modified the code given here and was able to adapt the LMS filter with error tapering to zero. The only assumption I made is that (since I am not an audio expert and do not know how the channel from speaker to the microphone would look like), I assumed a 10 tap channel with only first 3 non-zero values (multi-path reflection ...

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The reason x[n] must be white is because the solution will effectively spectrally weight the channel response based on the amount of energy present in each spectral frequency location. A white noise source provides equal weight to all frequencies. If energy is not present in any particular frequency bin, a proper solution cannot be found for that frequency. ...

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The book is correct, there is no discrepancy. When we reverse a system in time, only the time-variable will get negated and not the shift. Time-reversal does not mean that the whole argument of $x[n]$ gets negated. Take example of a sequence : $x[n] = {\hat{0},1,2,3,4,5}$. Shift this by 2 samples, so $x_{shift}[n] = {\hat{0},0,0,1,2,3,4,5}$. Now, reverse ...

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One option from the realms of adaptive filtering is the Least Mean Squares (LMS) filter depicted below: The idea is you take the output of the unknown system, compare it with the output of your adaptive filter and minimize the difference by tweaking the filter coefficients, using a LMS algorithm. When the error $e(n)$ is zero (or more often, lower than a ...

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