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Maximum likelihood (ML) estimator Here will be derived a maximum-likelihood estimator of the power of the clean signal, but it doesn't seem to be improving things in terms of root mean square error, for any SNR, compared to spectral power subtraction. Introduction Let's introduce the normalized clean amplitude $a$ and normalized noisy magnitude $m$ ...


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Kalman filters are useful when your input signal consists of noisy observations of some linear dynamical system's state. Given a series of observations of the system state, the Kalman filter aims to recursively provide better and better estimates of the underlying system's state. In order to apply it successfully, you need to have a model for the dynamics of ...


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as the first answer (with the most votes) says, the kalman filter is better in Any case when signal is changing. Notice the problem statement These use the algorithm to estimate some constant voltage. How could using a Kalman filter for this be better than just keeping a running average? Are these examples just oversimplified use cases of the filter? using a ...


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*STOP! If you only want a hint and not the complete solution please see Stanley P.'s or Peter K.'s answers. * Since you do not specify if there is model for the temperature evolving over time $n$, I will derive an estimator which is a combination of $Y_1$ and $Y_2$ for each fixed $n$. Let $\alpha \in (0,1)$ and suppose the estimate of $X$ can be written as $...


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First of all let us assure that a Kalman filter (estimator) does not only remove Gaussian noise, but can remove (with certain success) any other type of noise as long as it's designed accordingly. However, what lies at the heart of the standard Kalman filter is the linear estimator; and that linear estimator will be the optimum minimum mean square error ...


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I will tell you something, even if it is differntiable, use Unscented Kalman Filter for any non linear case. This flavor of Kalman Filter, based on the Unscented Transform, is almost always superior to the Extended Kalman due to its properties. The main reason is it is able to better predict the mean and variance (Which all Kalman Filter needs) of the ...


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Assumption of a Gaussian process allows us to obtain optimality. This uses the facts A linear( or better affine) map takes a Gaussian random variable and maps to another Gaussian random variable. A linear combination of two jointly Gaussian random variables is again a Gaussian random variable. So we don't have to track the mean and the variance. If we ...


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Another take: The Kalman Filter lets you add more information about how the system you're filtering works. In other words, you can use a signal model to improve the output of the filter. Sure, a moving average filter can give very good results when you're expecting a close-to-constant output. But as soon as the signal you're modelling is dynamic (think ...


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I won't add any equations, I will just add some intuition. I will also limit my self for Additive Gaussian White Noise. Now, in that case the Kalman filter can written as a Least Squares problem to solve. I'd say even more, the Kalman Filter is linear, if you have the samples up to certain time $ T $, you can write the Kalman filter as weighted sum of all ...


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The covariance matrix of a Kalman filter is a function of the $ Q $ and $ R $ matrices of the model. If you use a model where $ R $ and $ Q $ are time invariant or known in prior then the calculation of the covariance matrix $ P $ can be done offline and isn't a function of the measurements. In some cases, advanced implementations of Kalman Filter estimate ...


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If I understood it correctly, you have something that is moving and you can observe the speed and this speed is noisy. From your measurements, you observe 2 kinds of variations.\ Variations caused by the noise Variations because the object is truly changing the speed (e.g. turning) The reason your Kalman gain goes to zero is that you have implicitly ...


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A lot has been said already, allow me to add some comments: Kalman filters are an application of Bayesian probability theory, which means that "a priori information" or "prior uncertainty" can (and must) be specified. As I understand, this is not the case with traditional least-squares fitting. While observations (data) can be weighted with probabilities in ...


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Update: I'm sorry to have to say that testing shows the following argument seems to break down under heavy noise. This is not what I expected, so I have definitely learned something new. My prior testing had all been in the high SNR range as my focus has been on finding exact solutions in the noiseless case. Olli, If your goal is to find the parameters ...


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I suggest this reference regarding the comparison between least-squares and Kalman filters : Fundamentals of Kalman Filtering: A Practical Approach by P. Zarchan & H. Mussof Especially Chapter 3 (Recursive Least-Squares Filtering) and Chapter 4 (Polynomial Kalman Filters). In Chapter 4, the authors show that the discrete (time) n-th order polynomial ...


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You need a dynamic system to use a Kalman Filter. I would suggest $$y = \sum\limits_{i =0}^n a_i\, x^i $$ $$a[k+1] = a[k] + w $$ $$ cov(w) = Q$$ Measurement: $$z = \sum\limits_{i =0}^n a_i\, x^i = y$$ So instead of using $x$ as states, introduce the coefficient ($a$) as states


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In ideal world you'd have the correct model and use it. In your case, the model isn't perfect. Yet the steps you're suggesting are based on a knowledge you have about the process - which you should incorporate into your process equation using your dynamic model matrix: The classic and correct way given F matrix is built correctly according to your knowledge....


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Given the Covariance Matrix $ {P}_{k - 1 \mid k - 1} $ then: $$ {P}_{k \mid k} = {F}_{k} {P}_{k - 1 \mid k - 1} {F}_{k}^{T} + {Q}_{k} $$ Where $ {F}_{k} $ is the Model Matrix at iteration $ k $ and $ {Q}_{k} $ is the Process Noise Covariance Matrix at iteration $ k $.


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The Kalman Filter properties allows is to be the best linear estimator (What you call removing noise) for any White Noise (Not only Gaussian White Noise). The idea of Kalman Filter is estimating the Mean and Covariance of the State Vector at each iteration. Since optimal linear estimator (For the MMSE criteria) are based on the Mean and Covariance, as long ...


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The covariance decreases to a steady state regardless of how much error I introduce into the measurement. Yes, as @Drazick notes, if the $Q$ and $R$ matrices are time invariant, then the $P$ matrix will converge to a steady state that does not depend on the data (measurements). The variance for x and y are exactly the same even though I introduce more ...


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Remarks The MMSE is Bayesian Framework. Namely it should be employed between random variables which have joint distribution. In your case it seems $ x \left[ n \right] $ is a deterministic parameter hence Parameter Estimation framework should be employed. Parameter Estimation In the case above it seems the Maximum Likelihood Estimation fits. Since the data ...


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I'm going to assume that we have no information about how $X[n]$ varies with time, so we can just do one-at-time estimation of $X[n]$ using $Y_1[n]$ and $Y_2[n]$. One way to get an estimate from $Y_1$ and $Y_2$ is to average the measurements: $$ \hat{X}[n] = \frac{Y_1[n] + Y_2[n]}{2} $$ Using the fact that the sum of two independent Gaussian variables, $A$...


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I will use Wikipedia notations - Kalman Filter. In most models the state transition model matrix $ F $ depends on the interval parameter $ T $. The same goes for the Process Noise Covarinace Matrix $ Q $. For instance, for the constant velocity model: $$ F = \begin{bmatrix} 1 & T \\ 0 & 1 \end{bmatrix}, \; Q = \begin{bmatrix} \frac{ {T}^4 }{4} & ...


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Update If I understood your model, you have a model of Constant Velocity in 2D (Cartesian Coordinate System). While your measurement are in Polar Coordinate System. Pay attention that your measurement function is: $$ h \left( x, y, {v}_{x}, {v}_{y} \right) = \begin{bmatrix} \sqrt{ {x}^{2} + {y}^{2} } \\ {\tan}^{-1} \left( \frac{y}{x} \right ) \end{bmatrix} $...


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Anuar Y, Welcome to the DSP community. What you're talking about is called smoothing. Let me explain, assume we have samples $ {\left\{ x \left[ n \right] \right\}}_{n = 0}^{N - 1} $ and we want to build estimator for $ x \left[ k \right] $ which we will define as $ \hat{x} \left[ k \right] $. Now, we have 3 types of estimation: The case $ k > N - 1 $ ...


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An interesting approximative solution of the maximum likelihood (ML) estimation problem is obtained by using the asymptotic formula $$I_0(x)\approx \frac{e^x}{\sqrt{2\pi x}},\qquad x\gg 1\tag{1}$$ Using the notation and formulas from Olli's answer, the optimum ML estimate of the normalized clean signal amplitude satisfies $$\hat{a}=m\frac{I_1(2m\hat{a})}{...


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Scale-invariant minimum mean square error (MMSE) improper uniform prior estimators of transformed amplitude This answer presents a family scale-invariant estimators, parameterized by a single parameter which controls both the Bayesian prior distribution of amplitude and the transformation of amplitude to another scale. The estimators are minimum mean square ...


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KF is actually a mixture of a deterministic state propagator and a statistical estimator. Despite it's name including the term filter, Kalman filter is not a simple frequency selective one. It's indeed a statistical recursive estimator of a state of a (linear) dynamic system. Yet on a broader sense it's called as a filter as it will separate a desired ...


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