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In short, yes: the key phrase that you want to search on is optimal smoothing. You didn't ask how to do it, and the "how" is chapter-length, in a good book about Kalman filtering. So I'm not going to try to describe that. In general hand-wavy terms, though, optimal smoothing takes into account that if you know both past and future values of a ...


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Don't worry too much about defining these terms too precisely, because they are used in many contexts with slightly different meanings. In very general terms, "estimation" is the calculation of a signal parameter, for example the phase, the mean, the PSD, etc. In other words, you have a signal, possibly noisy or distorted, and you want to find ...


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In that range it is guaranteed to converge. It doesn't mean it will necesseraly won't converge for higher values. If you want deeper understanding you can read about the step size in Convex Optimization context where there the step size related to the Lipschitz Constant of the function (Which matches the eigen value for Quadratic functions). If you share ...


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You specify that $T$ is known and invertible, so you know $X_1, X_2$, and then it's really trivial: $y[3]=[X_1,X_2]*[h_1, h_2]$, and $y[9]=[X_7, X_8] *[h_1,h_2]$; write that down as matrix system $$\begin{pmatrix}y[3]\\y[9]\end{pmatrix}=\begin{pmatrix}X_1&X_2\\X_7&X_8\end{pmatrix}\begin{pmatrix}h_2\\h_1\end{pmatrix}$$ and solve it.


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After signal detection, how to estimate the clean signal $s(t)$? Matched filtering is used to detect the presence of a known signal in noise. There is no estimation part when you are talking about a matched filter. The estimate part comes after you have done the matched filter and need to estimate the symbols. It looks like you are talking about a ...


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It's important to point out here that Kay is talking about classical estimation, which means that the parameters to be estimated are unknown but deterministic. Hence, the term $E[\hat{\theta}]-\theta$ is deterministic, and, consequently, the last term in your equation becomes $$\begin{align} E \left[\left( \hat{\theta} - E[\hat{\theta}] \right) \left( E[\hat{...


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HINT: $$ E\left[\hat \theta - E\left(\hat \theta \right)\right] = E\left(\hat \theta\right) - E\left(\hat \theta\right) $$


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