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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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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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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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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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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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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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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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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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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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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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*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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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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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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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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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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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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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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Actually the first section of the notes in the link your provided are about the most likely value in the Bayesian Framework. So we have a comparison between the Minimum Mean Square Error (MMSE) Estimator and the Maximum a Posterior Estimator. Both are Bayes Estimator, namely they are a loss function of Posterior Probability: $$ \hat{\theta} = \arg \min_{a} ...


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R is depends on the sensor sensitivity. If this is a real world problem this can be obtained from the manufacturer. If not use the identity matrix multiplied by a scalar that is less than 1. Q is the covariance of the process noise. Again if this is a real world problem this can be obtained in the noise level in the states of the system at steady state. if ...


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Since $ M \in \mathbb{S}^{N}_{++} $ (In other convention $ M \succ 0 $) by Cholesky Decomposition there is a Triangular Matrix $ R \in \mathbb{R}^{N \times N} $ such that $ M = {R}^{T} R $. Using this fact one could prove $ 1 \iff 2 $ as following: $$\begin{align*} \arg \min_{\hat{x}} \mathbb{E} \left[ {\left( \hat{x} - x \right)}^{T} M \left( \hat{x} - x \...


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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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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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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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They used to have different convention for writing this stuff back then. But actually what you saw is really simple. It's all based on the Orthogonal Principle of MMSE. They say in (9) that the Additive Noise is uncorrelated (By defining its Auto Correlation by Delat Function). In (10) they say the optimal estimation of $ x \left( t \right) $ is given by ...


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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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In any Kalman Filter one need to calculate the 1st and 2nd moment of the data under the transformation. The image above taken from The Unscented Kalman Filter for Nonlinear Estimation by Eric A. Wan and Rudolph van der Merwe The problem is that there are some transformations which their linearization (As done in the Extended Kalman Filter - EKF) yield the ...


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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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