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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 imaginary parts must be uncorrelated (this is equivalent to independence if they are assumed jointly Gaussian) and identically distributed with zero mean. Thus,...

13

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 want to look up "Pisarenko's Method", "Prony's Method" and read about related problems such as ESPRIT (Roy, R.; Kailath, T. (1989). "Esprit - Estimation Of ...

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I've never seen the word "Formula" with "AMDF". My understanding of the definition of AMDF is $$Q_x[k,n_0] \triangleq \frac{1}{N} \sum\limits_{n=0}^{N-1} \Big| x[n+n_0] - x[n+n_0+k] \Big|$$ $n_0$ is the neighborhood of interest in $x[n]$. Note that you are summing up only non-negative terms. So $Q_x[k,n_0] \ge 0$. We call "$k$&...

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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 evaluated vs. Additive White Gaussian Noise (AWGN) with high level of noise real world images are mostly with moderate level of noise with Mixed Poisson Gaussian ...

8

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 signal itself. The obvious method here would be using DFT as the Maximum Likelihood Estimator of such case. It will probably be able to generate the best results in ...

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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 going through all the samples again. On the page you cite is the equation for calculation of the slope $m_n$ that together with $b_n$ minimizes the sum of ...

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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 $m$ ...

7

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 hypotheses. ML does not. This set of probabilities, known as "a priori" probabilities or simply "priors", is often known imperfectly, but even rough approximations ...

7

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 model could be also written as $a \sin \left( \omega t + \psi \right)$. Then the model can be: $${a}_{k} \sin \left( {\omega}_{k} {t}_{k} + \psi \right) = {... 7 This isn't quite what you're asking, because it neglects the amplitude, A, but it's a relatively straightforward example of application of an extended Kalman filter to the frequency tracking problem. See section 1.2 of this PDF, that I wrote some time ago. I'd also recommend starting with B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall, ... 7 For classic Kalman Filter, where  {Q}_{k} = Q  and  {R}_{k} = R , namely the process noise covariance and the measurement noise covariance (I'm using Wikipedia - Kalman Filter notations) the Posterior Covariance  {P}_{k}  is a deterministic matrix independent of the measurements themselves. Since your code set the measurement and the process noise to ... 6 The Wiener Filter can also be derived by another (Easier) way. Let's assume the following model:$$ y = h \ast x + n $$Namely the data is a result of a linear combination (Convolution) of  x  with Additive Noise. If we assume the noise model is Gaussian and our data is also formed by a Gaussian distribution then we should try to minimize (MAP ... 6 In ideal world you'd have the correct model and use it. In your case, the model isn't perfect. Yet the steps you're suggesting are based on a knowledge you have about the process - which you should incorporate into your process equation using your dynamic model matrix: The classic and correct way given F matrix is built correctly according to your knowledge.... 6 Given measurements$$\begin{align} Z_1 &= x + N_1\\ Z_2 &= x + N_2 \end{align}$$where N_1 and N_2 are independent zero-mean Gaussian random variables with variances \sigma_1^2 and \sigma_2^2 respectively, it can be shown that the minimum-mean-square-error (MMSE) estimate of x in terms of Z_1 and Z_2 is a linear function of Z_1 and ... 6 I think I have the solution. I'd be happy to hear others' thought. Defining  F \left(r, v, a, {T}_{tth} \right) = r + v {T}_{tth} + \frac{a {{T}_{tth}}^{2}}{2}  which is the implicit function which connects all variables. Since we're dealing with non linear function the variance is given by:$$ var \left( {T}_{tth} \right) = J P {J}^{T} $$Where  P  ... 6 Samples of colored noise (taken at different times) generally are correlated random variables because the autocorrelation function of the noise process is not a delta function as it is in the case of white noise. Thus, if we assume a zero-mean process (noise is generally assumed to be regardless of its color), then the covariance of two signals separated in ... 6 Given  \left\{ x \left[ n \right] \right\}_{n \in M}  where  M  is the set of indices given for the samples of  x \left[ n \right] . The trivial solution (Which it would be great to have a faster more efficient solution is what I'm looking for) would be:$$ \arg \min_{y} \frac{1}{2} \left\| \hat{F}^{T} y - x \right\|_{2}^{2} $$Where  \hat{F}  is ... 6 You want a method that removes noise while preserving edges. This cannot be achieved well by linear filtering, as you noticed yourself. I know of two approaches that might work well for your problem. The first is median filtering, where samples inside a window are replaced by their median. The following plot shows the result of median filtering with a window ... 6 So this is just the start of an answer. I'll have to keep updating it as I go. The first attempt is to say that the quantities you are interested in are the location of the center of the four LEDs, and the roll, pitch, and yaw (rotation angles) of the LEDs. That means your Kalman FIlter state will be:$$ \mathbf{x}_k = \left[x_k\ y_k\ \alpha_k\ \beta_k\ \...

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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 & 0\\-\omega^2&0&0\\0 &0 &0\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\\ y &= \begin{bmatrix}1&0&1\end{bmatrix} \...

6

You have a set of message set $m_i$, $0 \le i \le N-1$. (For example, QPSK will be $N=4$). For the transmitted message $m_i$, the corresponding symbol vector is $\textbf{x}_i$, and the received symbol vector is $\textbf{y} = \textbf{x} + \textbf{w}$, where $\textbf{w}$ is the AWGN at the receiver. The above is a simplified baseband model assuming a simple ...

6

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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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 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 ...

5

Here's the way I think about a discrete Wiener Filter Consider a sequence of observations $\mathbf{y} \in \Re^n$ Form a matrix from the input $\mathbf{x} \in \Re^{n+r-1}$ by shifting columns one sample each: $$X= \begin{bmatrix} x_1 & x_2 & ... & x_r \\ x_2 & x_3 & & x_{r+1} \\ x_3 & x_4 & & x_{r+2} \\ ... & &... 5 The usual way to estimate the amplitude of a particular frequency is to use the Goertzel algorithm. There is a good write-up by Rick Lyons here. Even though Rick's writeup is about single tone detection, it can be applied when multiple tones are present, too. 5 The Hermitian symmetry is used to obtain a real-valued time-domain signal. It is a special case of OFDM called discrete multitone (DMT). It exploits a property of the discrete Fourier transform (DFT), namely that the DFT of a real-valued signal has Hermitian symmetry. The motivation is usually the channel: if the signal shall be transmitted over a low-pass ... 5 1) It depends on the application, but in general, both. In some cases we can whiten the noise, say in the following situation:$$Y = x+ \eta\text{ where }\eta \sim N(0, \Sigma)$$where we want to estimate x \in \mathcal{A} for some set \mathcal{A} from Y. In this case, we multiply both sides by the inverse of the square root of \Sigma, and we have ... 5 Given data  { \left\{ {x}_{i} \right\} }_{i = 1}^{N}  the Empirical STD of the data is well defined:$$ STD = \sqrt{ \frac{1}{N - 1} \sum_{i = 1}^{N} { \left( {x}_{i} - \bar{x} \right) }^{2} } $$Where  \bar{x}  is the empirical mean of the data given by:$$ \bar{x} = \frac{1}{N} \sum_{i = 1}^{N} {x}_{i}  Now, if there's a model on the data (Such ...

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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 the CRLB (Cramer Rao Lower Bound) for Delay Estimation (Easy to derive for white noise). If you calculate the CRLB for Time Delay Estimation you'll see it ...

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To answer your final question: Why is it that the estimation community ignores the use of a discrete lowpass filter for estimation? As far as I can tell it's the best approach to estimate the signal above. That's because you're feeding them the wrong signal model. TL;DR: Use the right tool for the job! Gory Details Any time you start in with the ...

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