12

According to your definition of autocorrelation, the autocorrelation is simply the covariance of the two random variables $Z(n)$ and $Z(n+\tau)$. This function is also called autocovariance. As an aside, in signal processing, the autocorrelation is usually defined as $$R_{XX}(t_1,t_2)=E\{X(t_1)X^*(t_2)\}$$ i.e., without subtracting the mean. The ...


7

Let's solve a more general problem (Least Squares with Linear Equality Constraints): $$ \begin{alignat*}{3} \arg \min_{x} & \quad & \frac{1}{2} \left\| A x - b \right\|_{2}^{2} \\ \text{subject to} & \quad & C x = d \end{alignat*} $$ The Lagrangian is given by: $$ L \left( x, \nu \right) = \frac{1}{2} \left\| A x - b \right\|_{2}^{2} + {\...


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

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 $ ...


5

What are reasons to choose for cross-correlation or cross-covariance when comparing signals with non-zero mean? Well, part of the issue is that cross-correlation as defined in your equation: $$(f \star g)[n]\ \stackrel{\mathrm{def}}{=} \sum_{m=-\infty}^{\infty} f^*[m]\ g[m+n].$$ will not exist (or be infinite) if $f$ and $g$ have non-zero mean. So, in ...


5

For discrete data both are the same - Finding set of orthogonal directions which maximizes the Variance (Energy) of data along them. Sometimes those are called the natural axis of the. Since we're dealing with variance it is only natural both are calculated from the covariance matrix of data. You may encounter places where KL might be even defined on the ...


4

For the first case, as you wrote, it means the elements are not correlated. Since this is a Gaussian Random Vector it means the elements are independent. It means that at most only one element of $ \boldsymbol{\mu} $ is not zero. Since if there were more than 1, the matrix $ \boldsymbol{R} $ wasn't diagonal. Update Let's define $ \hat{\boldsymbol{x}} = \...


4

Pay attention that for a Scalar Random Process the Power Spectrum Density is non negative. Namely, let $ y \left[ n \right] \in \mathbb{R} $ be a WSS Random process with its Auto Correlation function given by: $$ {R}_{y, y} \left[ m \right] = \mathbb{E} \left[ y \left[ n \right] y \left[ n - m \right] \right] $$ Then the Power Spectrum Density is: $$ {S}_{y, ...


4

The Covariance Matrix is commonly defined as $$\mathbf Q = E\left[ (\mathbf x -\mathbf\mu_{x})(\mathbf x -\mathbf\mu_{x})^*\right]$$ with $\mu$ denoting the mean value, i.e. $\mu_{x}=E\left[\mathbf x\right]$, and $\mathbf x$ being column vectors. The fact that you define the covariance matrix as $$\mathbf{R}_i = E\left[\textbf{u}_i^*\textbf{u}_i \right]$$ ...


4

For power signals $x(t)$ and $y(t)$, the function $$R_{xy}(\tau)=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}x(t)\bar{y}(t+\tau)dt\tag{1}$$ is the cross-correlation of $x(t)$ and $y(t)$. So the expression you're asking about is the cross-correlation of $x(t)$ and $y(t)$ evaluated at lag $\tau=0$: $$R_{xy}(0)=\lim_{T\rightarrow\infty}\frac{1}{2T}\...


3

A PRN sequence is a Pseudo-Random Noise sequence, often generated by using an Linear Feedback Shift Register (LFSR) with the feedback taps done by using a primitive irreducible polynomial in GF{2}, which is the Golois Field of 2 elements. When a primitive and irreducible polynomial in GF{2} is used, the LFSR will produce a "maximum length sequence", meaning ...


3

MVDR is a narrowband beamformer. For broadband signals it is usually applied for each frequency bin. That means that $\mathbf{R}_{xx}$ is frequency dependent. In other words, for each time you should have $M$ matrices, each one is $3\times 3$. Now, since you usually cannot compute $\mathbf{R}_{xx}$ exactly, you perform covariance estimation $\tilde{\mathbf{...


3

Usually, for power signals, we define the inner product to be \begin{align} \left<x\,,\,y\right> &= \lim_\limits{T\rightarrow \infty} \frac 1 {2T} \int\limits_{-T}^T x(t)\bar y(t)\,dt \end{align} which induces the vector norm \begin{align} ||x||^2 &= \left<x\,,\,x\right>\\ &=\lim_{T\rightarrow \infty} \frac 1 {2T} \int\limits_{-T}^...


3

I don't understand the subscript $n$ notation, however, in the least squares problem that is given by: \begin{equation} {\bf{y}}={\bf{H}}{\theta}+\bf{n}, \end{equation} where ${\bf{n}}\sim\mathcal{N}(\bf{0}, \sigma^2I_N)$ is a zero mean additive white Gaussian noise and $I_N$ is the $N \times N$ identity matrix, the maximum likelihood and the least squares ...


3

It's the key point of array signal processing, I suppose. Say $x$ is the input vector of $[N,1]$ dimension collected from $N$ array sensors. $x(k)$ is its realization at the $k$ moment of time. By its definition covariance matrix (sometimes it's called autocorrelation matrix): $R = E[x\cdot x^H]$ , where $E[]$ is expectation operator and $x^H$ is Hermitian ...


3

It depends on what you mean by SNR. It's a common joke in the DSP community to spell it out as "something to noise ratio", referring to the fact that there is no unique definition of SNR, so the term by itself means nothing. Define it yourself and use it appropriately. What's common is to define it as ${\rm SNR} = \frac{P_{\rm s}}{P_{\rm n}}$ where $P_{\rm ...


3

This question belongs more on stats.SE (where many similar questions have been thrashed out in detail) but nonetheless here goes. Let's take the simplest case of $N=1$. Just because $X$ and $Y$ are Gaussian random variables, it is not necessarily the case that $X$ and $Y$ have a jointly Gaussian distribution. See, for example, this answer on stats.SE for ...


2

There is something that is not clear of what you have done with the data, and that is who do you form the random vectors to perform de SVD (or EVD) on the covariance matrix. 1 -The KLT can be succesfully used on a one dimensional signal (only one Geophone), taking frames of $M$ samples and estimating a covariance matrix from it, and the performing ...


2

This answer is not fundamentally different from the others; it's more of a complement and addendum. If $n(t)$ is zero-mean white Gaussian noise, then its variance is actually infinite; its power spectral density is constant and often denoted $\mathcal P_n(f)=N_0/2$. If this noise is input into a filter with impulse response $h(t)$, then the power spectral ...


2

In general, the equations from your first code block work equally for real and complex values. Note that the ()' operation is the Hermitian Conjugate (i.e. transpose + conjugate). If you are in the real domain, it only becomes transpose (because conjugation on reals doesn't change them). Hence, you would not need to change anything in the code. Regarding ...


2

Your question (as expanded in the comments) is asking if we start with $$ x(t) = x_{\rm true}(t) + n_1(t) $$ and filtering it using a filter $H(\omega)$ to get $$ x_{\rm hp}(t) = h(t) * x(t) = x_{\rm true}(t) + n_2(t) $$ where the variance of $n_1$ is $v_1$, then what is $v_2$, the variance of $n_2$? That seems ill-posed because the filter $H$ will change $...


2

Hi: In order to estimate the variance, you need to have an underlying model for your signal. So, suppose that the model is $y_{t+1} = y_t + \epsilon_t$ $~\forall ~ t = 1,\ldots n $. assuming that $E(\epsilon_t) = 0$ and $var(\epsilon_t) = \sigma^2$. In this case, you would difference your data in order to get estimates of $\epsilon_{t}$ at each time $t$, ...


2

Short answer: just use $\sigma = 10^{-8}$. Covariance matrices have eigenvalues $\geq 0$ (theoretically), so $Ri + 10^{-8} \, I$ will have eigenvalues $\geq 10^{-8}$, safely non-singular. A longer answer: split Covar = S + N, "signal" + "noise", by eigenvalues or by SVD, Singular-value decomposition aka PCA, Principal component analysis. This has several ...


2

One must be careful when asking questions about the relationships between the elements of a complex random vector. The short answer to your question is that you cannot say much for either cases simply by considering the covariance (or correlation) matrix. Actually, the covariance (correlation) matrix is not enough to capture all the relationships that ...


2

As mentioned in another answer, the variance of the measurement noise is a property of the model which is baked in to the Kalman filter. I think you're running into a different problem, which is that your model is different from reality. It sounds like you're modeling the system assuming that all the microphones are 'live'. If that's the case, then the ...


2

Assume $Y = g(X)$ be the function of RV $X$, then by using the following $$E\{ g(X) \} = \int g(x) f_X(x) dx $$ variance of $Y$ can be computed without the computation of pdf $f_Y(y)$ as: $$ \begin{align} \text{Var(Y)} &= E\{ (Y-\mu_Y)^2 \} = E\{ Y^2 \} - (\mu_Y)^2 \\ & = E\{ g^2(X) \} - E\{ g(X) \}^2 \\ & = \int g^2(x) f_x(x) dx - \left(\...


2

The covariance matrix is given by $$C_{X,Y}=\begin{bmatrix}E(XX)& E(XY) \\ E(YX )& E(YY) \end{bmatrix}$$ This can be written as below: $$C_{X,Y}=\begin{bmatrix}E(R^2cos^2(\Theta) )& E(Rcos(\Theta)Rsin(\Theta)) \\ E(Rsin(\Theta)Rcos(\Theta) )& E(R^2sin^2(\Theta)) \end{bmatrix}$$ Since $R$ and $\Theta$ are independent the expectation will ...


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