14 votes
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What would be the variance for complex number?

I will focus on the reason of the factor $1/2$ and leave aside the estimation things. The exact understanding should be : if a scalar Gaussian random variable (rv) is circular symmetric, its real and ...
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13 votes

How to learn MUSIC algorithm?

Read the original paper: Schmidt, R. O. "Multiple Emitter Location and Signal Parameter Estimation." IEEE Transactions on Antennas and Propagation. Vol. AP-34, March, 1986, pp. 276–280 You may also ...
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12 votes
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What is an AMDF?

I've never seen the word "Formula" with "AMDF". My understanding of the definition of AMDF is $$ Q_x[k,n_0] \triangleq \frac{1}{N} \sum\limits_{n=0}^{N-1} \Big| x[n+n_0] - x[n+n_0+...
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12 votes
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Which Noise Reduction Algorithms Are Used in Commercial RAW Image Processors?

Common Approaches for Commercial Denoisers Commercial denoisers are different than what you'd see on most papers. While on papers the results are mostly using objective metrics (PSNR / SSIM) and are ...
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11 votes
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What sensors can be fused using the Kalman Filter framework

Remark: I will answer this using the Linear framework of the Kalman Filter but the idea is the same. The Kalman Filter basically propagate and fuses Gaussian Distributions in order to calculate the ...
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9 votes
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Estimate and Track the Amplitude, Frequency and Phase of a Sine Signal Using a Kalman Filter

We can build a non linear dynamic model in order to estimate the parameters of a sine signal. Let's model the signal as $ a \sin \left( \phi \right) $ where $ \phi $ is the instantaneous phase. So the ...
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9 votes

Simple and Effective Method to Estimate the Frequency of a Single Sine Signal in White Noise

I assume the model to be: $$ x \left[ n \right] = \sin \left[ 2 \pi \frac{f}{ {f}_{s} } n + \phi \right] + w \left[ n \right] $$ Where $ w \left[ n \right] $ is white noise uncorrelated with the ...
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9 votes

A Machine Learning Based Algorithm as an Alternative to the Matched Filter

Sure, you can learn the matched filter, as convolution with a filter is just a function applied to a signal, and e.g. Neural Networks (through the universal approximation theorem) are good function ...
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8 votes

Understanding the Difference Between MAP Estimation and ML Estimation

Maximium A Posteriori (MAP) and Maximum Likelihood (ML) are both approaches for making decisions from some observation or evidence. MAP takes into account the prior probability of the considered ...
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8 votes

Estimation of Amplitude, Frequency and Phase of Linear Combination of Harmonic Signal Beyond the Leakage Resolution of DFT

Solving the Linear Combination of Real Harmonic Signal The data model is given by: $$ x \left( t \right) = \sum_{i = 1}^{M} {a}_{i} \sin \left( 2 \pi {f}_{i} t + {\phi}_{i} \right) + n \left( t \right)...
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7 votes

How Is the Formula for the Wiener Deconvolution Derived?

The Wiener Filter can also be derived by another (Easier) way. Let's assume the following model: $$ y = h \ast x + n $$ Namely the data is a result of a linear combination (Convolution) of $ x $ ...
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7 votes
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Concept About Estimated Standard Deviation

Given data $ { \left\{ {x}_{i} \right\} }_{i = 1}^{N} $ the Empirical STD of the data is well defined: $$ STD = \sqrt{ \frac{1}{N - 1} \sum_{i = 1}^{N} { \left( {x}_{i} - \bar{x} \right) }^{2} } $$ ...
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7 votes
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Sequential Form of the Least Squares Estimator for Linear Least Squares Model

Slope from all samples obtained To summarize the question's problem, you want to calculate the slope based on all samples obtained thus far, and as new samples are obtained, update the slope without ...
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7 votes

Estimators for improved spectral subtraction of noise

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, ...
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7 votes
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Signal Estimation after detection Part 2

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 ...
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7 votes

Kalman Filter on Sinusoidal Signal

This isn't quite what you're asking, because it neglects the amplitude, $A$, but it's a relatively straightforward example of application of an extended Kalman filter to the frequency tracking problem....
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7 votes
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Kalman Filter on Sinusoidal Signal

I'm copying my answer to Estimate and Track the Amplitude, Frequency and Phase of a Sine Signal Using a Kalman Filter which solves a more general problem with example code: We can build a non linear ...
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7 votes
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Kalman Filter State Covariance Matrix for Non Constant Process Noise Matrix in PyKalman

For classic Kalman Filter, where $ {Q}_{k} = Q $ and $ {R}_{k} = R $, namely the process noise covariance and the measurement noise covariance (I'm using Wikipedia - Kalman Filter notations) the ...
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7 votes
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A Machine Learning Based Algorithm as an Alternative to the Matched Filter

The idea is to have a simple experiment to see if we can get, for a known signal, a better results than the Matched Filter for time delay estimation. Experiment Objective Generate, using ML (DL) a ...
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6 votes

minimizing the mean squared error?

Given measurements $$\begin{align} Z_1 &= x + N_1\\ Z_2 &= x + N_2 \end{align}$$ where $N_1$ and $N_2$ are independent zero-mean Gaussian random variables with variances $\sigma_1^2$ and $\...
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6 votes
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Maximum Likelihood Estimation in Presence of Colored Noise

Samples of colored noise (taken at different times) generally are correlated random variables because the autocorrelation function of the noise process is not a delta function as it is in the case of ...
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6 votes
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Variance of an Implicit Function of Kalman State Vector

I think I have the solution. I'd be happy to hear others' thought. Defining $ F \left(r, v, a, {T}_{tth} \right) = r + v {T}_{tth} + \frac{a {{T}_{tth}}^{2}}{2} $ which is the implicit function which ...
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6 votes
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Kalman Filter - Optimal Way to Handle "Derived" Measurements?

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 ...
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6 votes
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Estimate the Discrete Fourier Transform / Series of a Signal with Missing Samples

Given $ \left\{ x \left[ n \right] \right\}_{n \in M} $ where $ M $ is the set of indices given for the samples of $ x \left[ n \right] $. The trivial solution (Which it would be great to have a ...
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6 votes
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Duration of unknown rectangular pulse with additive white Gaussian noise

You want a method that removes noise while preserving edges. This cannot be achieved well by linear filtering, as you noticed yourself. I know of two approaches that might work well for your problem. ...
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6 votes
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Subsample Time Delay Estimation

Even though the signals are sampled you can get accuracy which is well above the accuracy offered by the samples as long as you sample using Nyquist. Actually, Using the Matched Filter you can achieve ...
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6 votes

Kalman Filter for estimating position with nonconstant velocity & acceleration

So this is just the start of an answer. I'll have to keep updating it as I go. The first attempt is to say that the quantities you are interested in are the location of the center of the four LEDs, ...
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6 votes
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Tracking a Sine Wave with Kalman Filter - How to Account for Offset (DC Signal)?

Well, in continuous time, a sinusoid with a bias can be seen as the output of the linear system \begin{align*} \begin{bmatrix}\dot x_1\\\dot x_2\\\dot x_3\end{bmatrix} &= \begin{bmatrix}0 & 1 ...
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6 votes
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Why use parametric based estimation methods - confusion regarding terms

Hi: I'll try to answer as briefly as possible and only with respect to statistics. not dsp. In statistics, if you have a nice pdf such as the normal distribution, then maximizing the likelihood is ...
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6 votes
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Extended Kalman Filter (EKF) for Non Linear (Coordinate Conversion - Polar to Cartesian) Measurements and Linear Predictions

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 ...
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