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 ...


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

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} + {\...


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}\...


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 $$\mathbf{R}_i = E\left[\textbf{u}_i^*\textbf{u}_i \right]$$ ...


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

You can't really have a covariance of a matrix. What you can have, is a covariance matrix of a set of vectors. So, if you think of the rows of your matrix A as two vectors in 3D: [2 3 4] and [5 5 6], then the covariance matrix of this set of two vectors is C = A' * A (A transpose times A). Note that if you shuffle the rows of A in a different order, C ...


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

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 ...


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

What you want to do is dimension reduction. The most basic, yet very powerful and commonly used, technique to do it is principal component analysis (PCA). PCA operates on the covariance matrix. You can look it up on Wikipedia for a throughout tutorial. PCA decomposes your data $y$, having dimension $d$, into $d$ principal components. The first principal ...


2

Variance is defined as $V(x)=\frac{\sum_{i=1}^n(x_i-\mu)^2}{n}$. Just in case for you, mean $\mu$ is defined as $\mu=\frac{\sum_{i=1}^nx}{n}$. Covariance between two random variables $x$ and $y$ (or columns of a matrix) is defined as $Cov(x,y)=\frac{\sum_{i=1}^n[(x_i-\mu_x)(y_i-\mu_y)]}{n}$ and $Cov(x,x)=V(x)$. The term covariance matrix may be misleading ...


2

I would think that it is indeed the "normal" variance of the image. You look at the distribution of the pixel values (i. e. gray levels), and compute its variance.


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

Pay attention that for Scalar Random Process the Power Spectrum Density is non negative. Namely, let $ y \left[ n \right] \in \mathbb{R} $ 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 Denisty is: $$ {S}_{y, ...


2

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}} = \...


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

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

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 ...


1

Unfortunately, in Classical Kalman filter applications you need to have a guess of your state transition model governing the states. Classical choices for such models for time-varying systems are the random walk model, or its generalization, a first-order Markov model. If do not have a clue about which state model you can assume, you can try to extract a ...


1

I'd recommend looking into the relation of correlation and covariance; the one is just the "bias-corrected" version of the other. Then, use the Wiener-Khinchin┬╣ theorem: If, and only if, the signal is weak-sense stationary, the Fourier transform of the autocorrelation of that signal is the same as Expectation value of the magnitude-squared Fourier transform ...


1

To make the Online variance algorithm work for vectors, change one line: M2 += np.outer( delta, delta2 ) # deltas 3-vecs, M2 3 x 3 # outer( x, y ) is pairs x[i] * y[j] -- # [[ x0 y0, x0 y1, x0 y2 ], # [ x1 y0, x1 y1, x1 y2 ], # [ x2 y0, x2 y1, x2 y2 ]] A small python class for this is under gist.github.com/denis-bz . It can covariance points with ...


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