Signal Processing Stack Exchange
Signal Processing Stack Exchange

Questions tagged [covariance]

81 questions
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What Does Make an Error Surface Convex? Is It Determined by the Covarinace Matrix or the Hessian?

I am currently learning about least-squares (and other) estimations for regression, and from what I am also reading in some adaptive algorithm literatures, often times the phrase "... and since the error surface is convex..." appears and any depth…
Spacey
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Covariance vs Autocorrelation

I'm trying to figure out if there is a direct relationship between these concepts. Strictly from the definitions, they appear to be different concepts in general. The more I think about it, however, the more I think they are very similar. Let $X,Y$…
rbell
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Question on covariance matrix of 2 spatial signals

Every time I think I have understood the covariance matrix, someone else comes up wih a different formulation. I am currently reading this paper: J. Benesty, "Adaptive eigenvalue decomposition algorithm for passive acoustic source localization", J.…
Spacey
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What Is the Difference Between PCA and Karhunen Loeve (KL Transform)?

I have been reading about Karhunen-Loeve or also known as KL transform and I see that when it is used to reduce dimension the procedure is identical to PCA, that is, for both methods the covariance matrix of the data is constructed and then the…
Roger Figueroa Quintero
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Kalman Filter State Covariance Matrix for Non Constant Process Noise Matrix in PyKalman

I'm experimenting with the pykalman Python library to learn about Kalman Filters. In the code below, I'm generating a random walk where each step is the last step multiplied by 1 plus some noise: 1 + std_dev. In the first block, the covariance…
SuperCodeBrah
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Difference between $\mathbb{E}[\mathbf{x} \mathbf{x}^{\rm{H}}]$ and $\mathbb{E}[(\mathbf{x}-\boldsymbol{\mu}) (\mathbf{x}-\boldsymbol{\mu})^{\rm{H}}]$

Let us have a random vector $\mathbf{x} \sim \mathcal{CN} (\boldsymbol{\mu}, \boldsymbol{\Sigma})$ with $\boldsymbol{\mu} \neq \mathbf{0}$. What can we say about the relationship between the elements of $\mathbf{x}$ in the following two separate…
TheDon
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Variance of an Implicit Function of Kalman State Vector

Given a state vector, $ x $, given by $ x = {[r, v, a]}^{T} $ (Range, Velocity, Acceleration) the Time to Hit is the the time which holds the following: $$ r + v {T}_{tth} + \frac{a {T}_{tth}^{2}}{2} = 0 $$ Now, given the State Vector covariance $ P…
Royi
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Is there a way to reduce the covariance matrix of several source signals to the dominant source signal?

The problem I have can be seen in the context of DoA estimation or blind source signal separation and similar fields, where several source signals are observed by several antennas (or by an antenna array, respectively). The basic model is something…
Michael
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Quadratic Programming with Linear Equality Constraints

I need to solve an equality constrained minimization problem as give below $$\min_{\textbf{w}} \mathbf{w}^TR\mathbf{w} $$ such that $$X\mathbf{w} = \mathbf{1}$$ where $R\in \mathbb{R}^{n\times n}$ is covariance matrix (hence positive…
user5045
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Why does diagonal loading of a covariance matrix make an adaptive beamformer more robust in the case of a perturbed array?

It has been shown that 'diagonal loading' a covariance matrix derived for an adaptive beamformer can improve robustness of the beamformer when the antenna array is perturbed, albeit at the expense of background noise power. The question is why does…
Spacey
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Matched filter for "amplitude SNR" vs power SNR

I am working on an application for which a matched filter seems like the right concept. In the derivation for the matched filter which I went through (here), they define the SNR as the ratio of signal power to noise power as the basis for the…
ChateauDu
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Show That the Power Spectrum Density Matrix Is Positive Semi Definite (PSD) Matrix

Given a Wide Sense Stationary Multi Variate (Vector) Random Process $ \boldsymbol{x} \left[ n \right] $ it Auto Covariance Matrix Function is given by: $$ {R}_{x, x} \left[ m \right] = \mathbb{E} \left[ \boldsymbol{x} \left[ n \right]…
Royi
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Cross-correlation or cross-covariance of non-zero mean signals

Cross-correlation for uniformly sampled signals is defined as [1] $$(f \star g)[n]\ \stackrel{\mathrm{def}}{=} \sum_{m=-\infty}^{\infty} f^*[m]\ g[m+n].$$ Cross-covariance for wide-sense stationary (WSS) signals is defined identically [2]. This…
Erik
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What is a covariance matrix?

Suppose you have k samples from each of the N elements of a uniform linear array (ULA) of sensors: What is the physical meaning of a covariance matrix? How do you form a covariance matrix with the samples? How do you decide how many samples you…
random_dsp_guy
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Generalized correlation coefficients

Assume there are given two Gaussian random vectors $\boldsymbol{x}$ and $\boldsymbol{y}$ of equal length $N$ with corresponding means $\boldsymbol{\mu}_x$, $\boldsymbol{\mu}_y$ and covariance matrices $\boldsymbol{C}_{xx}$, $\boldsymbol{C}_{yy}$ as…
Lukas
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