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Questions tagged [covariance]

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In EKF should Kalman Gain converge to a specific value?

I have implemented an EKF using the standard predict and update equations in order to perform state estimation of a vehicle with multiple sensors. The model has process covariance matrix $Q$ and each ...
useeeeer132's user avatar
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1 answer
59 views

Intuitive understanding of the relationship between the variance of a Wiener process and the variance of white noise

The Wiener process $W(t)$ is said to be the integral of white noise $\xi(t)$ from $0$ to $t$. That is, $$W(t) = \int_{0}^{t} \xi(t')dt'$$ However, the variance of $\xi(t)$ is infinite, but the ...
Mashe Burnedead's user avatar
1 vote
0 answers
40 views

Fastest way to track dominant eigenvector of recursively updated correlation matrix

Consider a real vector time series: $\mathbf{x}_k\in{\mathbb{R}}^{N \times 1}$ where $k$ is the sample index. An associated correlation matrix is updated recursively as: $${\bf R}_k = \alpha {\bf R}_{...
rhz's user avatar
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3 votes
1 answer
254 views

What's the exact definition of the power spectral density function?

I learned from my signal processing course that PSD is the Fourier transform of the autocorrelation function. $$\mathscr{F}\Big\{\mathrm{E}\big[x(t)x(t+\tau)\big]\Big\}$$ Today I took my statistic ...
Xiangyu Cui's user avatar
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39 views

Recursive estimation of signal variance of strongly colored noise

I have a quasi stationary but time varying stochastic signal of which I would like to recursively (adapting to its time dependence) estimate the variance. I believe the signal could be modeled like ...
Ladsnilinn's user avatar
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23 views

Confusion with Complex Gaussian process with Auto-covariance

I have a complex sequence $z(t)$ in time which I know to be a Gaussian process. I read that the complex Gaussian process is not only characterized by the covariance, but also the pseudo-covariance ...
CfourPiO's user avatar
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1 answer
84 views

Why is the total noise variance less than the sum of individual noise variances?

I have three random variables: $Y$: my data $Y_n$: my data corrupted by additive white Gaussian noise (AWGN) $Y_{nc}$: my noisy data corrupted by a non-linear transformation $\mathcal{C}$. I have ...
graille's user avatar
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Finding correlation coefficient of two dependent random variables

I know when there are two independent random variables X and Y, the correlation coefficient is But what if the two random variables are dependent, for example, Y-X and X+Y, how can I represent them ...
wannastudycommunication's user avatar
1 vote
1 answer
183 views

Initial Process Covariance in 1-D Kalman Filter

Having a bit of confusion about what the initial process covariance (P) should be. Assume a 1-D tracking problem where I am measuring the distance/position of a static object. Would P not just be ...
6900HS's user avatar
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1 answer
342 views

Intuition for $\mathbf{P} = \mathbf{0}$ in steady-state when $\mathbf{Q} = \mathbf{0}$ (Kalman filter)

Consider the following discrete-time system: \begin{equation} \mathbf{x}(k+1) = \mathbf{A}_d \mathbf{x}(k) + \mathbf{B}_d \mathbf{u}(k) \end{equation} \begin{equation} y(k) = \mathbf{C}_d \mathbf{x}(k)...
Gab's user avatar
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Related to wireless channel variance?

I am reading wireless research papers and found that mostly the wireless channel is considered to be Zero mean circularly symmetric Complex Gaussian (ZMCSCG) with variance 1 or 2 or 5 or 10 etc. My ...
chaaru's user avatar
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Mann-Kendall Trend Test

I'm using the pyMannKendall package in Python to detect the presence of slope of any given waveform. More details of the package here: https://github.com/mmhs013/pyMannKendall I'm not exactly able to ...
EnigmAI's user avatar
  • 111
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61 views

Optimal weighting in case of unknown noise variances

Let's consider $n$ multiple measurements $x_i[j]$ of the same desired signal $s[j]$ but with different uncorrelated zero-mean additive noise $n_i[j]$ with different variances $\sigma_i^2$: $$ x_1[j] = ...
Alister Trabattoni's user avatar
3 votes
2 answers
108 views

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 ...
Lukas's user avatar
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1 answer
893 views

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 ...
SuperCodeBrah's user avatar
3 votes
1 answer
521 views

How to determine covariance matrices $\mathbf P$, $\mathbf Q$, and $\mathbf R$ in Extended Kalman Filter

I am implementing an Extended Kalman-Filter and an Unscented Kalman-Filter for state and parameter estimation of a conveyer belt system. The problem is that I don't really know how to determine the ...
Tristan's user avatar
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1 answer
766 views

Ideally Lowpass-filtered White Gaussian Noise: question about a derivation of variance and covariance

Consider white gaussian noise $\omega(t)$ that has passed through and ideal lowpass filter with bandwidth $B$. This filter has the impulse response $$h(t) = 2B\text{sinc}(2Bt)$$ Sampling the filter ...
DancingIceCream's user avatar
5 votes
2 answers
2k views

Kalman Filter: Why do we decrease the state uncertainty regardless of the current measurement?

I'm struggling with fully understanding the concept behind a Kalman filter. For the sake of simplicity, let's ingore the input variable $u$ and assume constant process $Q$ and measurement noise $R$. ...
user57647's user avatar
0 votes
1 answer
40 views

Random function covariance

I was studying signal processing and I was frequently asked to verify if a certain covariance is possible for a given random function. I tried to check by verifying the property to prove it: ${γ}_{xx}(...
LeA's user avatar
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1 answer
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Variance Due to white noise input

I have the problem below. It sounds simple but for some reason I have been stuck on it for a long time and don't know what am doing wrong. Am trying to solve this using correlations. So we all know ...
JordenSH's user avatar
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1 vote
0 answers
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Colored noise and adjusting covariance matrix with change of frequency

Assume a signal sampled at 2Hz. Assume it is a colored noise with known PSD. The measurement equation to be used in a Kalman filter requires a covariance matrix. If I want to sample at 9Hz (for ...
baptiste's user avatar
4 votes
2 answers
207 views

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 ...
ChateauDu's user avatar
6 votes
2 answers
827 views

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 ...
user5045's user avatar
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1 vote
2 answers
2k views

Covariance Matrix Polar to Cartesian

Let $R$ and $\Theta$ be zero mean independent Gaussian random variables with variances $\sigma_R^2$ and $\sigma_\Theta^2$ respectively. Hence their covariance matrix would be diagonal give by $$C_{R,\...
user5045's user avatar
  • 331
1 vote
1 answer
39 views

Variance of circular vector correlation

Say I have two zero-mean vectors, with a size of N and the translation between them is k. Say the image signal std is $\sigma_s$ and the noise std is given by $\sigma_n$. What is the variance of the ...
Gideon Genadi Kogan's user avatar
1 vote
0 answers
154 views

Whitening Filter for Noise Estimation

The signal I'm working on includes some noise which is I assume Gaussian white noise with zero mean and an arbitrary variance. So, it is like; ...
kubicwerke's user avatar
2 votes
0 answers
145 views

Finding the error in the total integrated intensity of a fitted 2D Gaussian

I have been trying to fit signals to a 2D Gaussian function, and while I have bene able to use sciKit-image's curve_fit function to find the covariance matrix for ...
TheEponymousProgrammer's user avatar
0 votes
1 answer
363 views

Problem implementing KL expansion with square-exponential kernel: output just looks like Gaussian

I am trying to implement the Karhunen Loeve expansion for a 1-D Gaussian random field with a square-exponential kernel. Specifically, I know that a Gaussian process has a KL expansion $\hat{U}=\sum_{...
George's user avatar
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3 votes
1 answer
4k views

Generate signals with a particular variance and SNR

Consider a system model of the form: $y_n = ax_n + v_n$ where $x_n$ is the input that is corrupted by $v_n$ which is an Additive White Gaussian Noise of zero-mean and variance 1 for $n = 1,2,...,N$ ...
Sm1's user avatar
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0 answers
125 views

Proof of weak stationary random process autocovariance always goes to zero?

Professor told me that if a random process is weak stationary, and it does not feature any periodic component, then its autocovariance always goes to zero. I can intuitively understand it, however, ...
Juà's user avatar
  • 21
2 votes
1 answer
682 views

Variance of function of random variable

Is their an easier way to find variance of function of random variable? Till now what I am doing is first find probability density function of (function of random variable) then integrate over range.
Buzz bee's user avatar
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3 votes
0 answers
402 views

Role of Riccati Equations in Kalman Filter Design

I am working on a Kalman Filter (KF) design problem and I am struggling to understand the role of the Riccati equations in the design process of a KF. Some sources discuss the importance of Riccati ...
Simon Diemert's user avatar
3 votes
1 answer
815 views

Kalman Filter State Covariance Matrix

If I have a discrete time process model of the form: $$x_{k+1} = x_{k} + v_{k}\cos(\theta_{k})dt$$ $$y_{k+1} = y_{k} + v_{k}\sin(\theta_{k})dt$$ $$v_{k+1} = v_{k} $$ $$\theta_{k+1} = \theta_{k}$$ ...
indigoblue's user avatar
3 votes
1 answer
155 views

Covariance block of MUSIC Algorithm

I am implementing MUSIC algorithm in Verilog, and I need to implement the blocks in following order $$\text{Input signal}{\longrightarrow}\text{Co-variance/Correlation matrix}{\longrightarrow}\text{...
uzmeed's user avatar
  • 31
0 votes
1 answer
312 views

Why discretize a continuous transition matrix in Kalman Filter?

In the Kalman filter toolbox at http://becs.aalto.fi/en/research/bayes/ekfukf/cwpa_demo.html the example code shows that a function lti_disc is called, which is essentially a matrix exponential ...
babelproofreader's user avatar
-1 votes
1 answer
116 views

About the variance of the coefficient of narrowband noise when using signal-space representation

Assume that n(t) is a white Gaussian noise process with zero-mean and power spectrum density $N_0/2$. By using the signal-space representation, it can be expressed as: $$n(t) =\sum_{j=1}^N n_j \phi_j(...
Berman Song's user avatar
1 vote
2 answers
403 views

Should I pass Kalman Filter absolute or offset-from-mean sensor values?

I'm using Kalman filters to segment the loudness of an acoustic signal from surrounding noise. The problem I've encountered is that muffled or faulty microphones measuring 'silence' (-70dB, -69dB, -...
Petrus Theron's user avatar
0 votes
1 answer
943 views

Covariance matrix associated with random DC level in Gaussian noise

Given a signal $x[n] = A + w[n]$ where $A$ is a Gaussian random variable and $w[n]$ is Gaussian white noise, then the covariance matrix of the signal is given by $[C(\sigma^2_A)]_{ij}=E[x[i-1]x[j-...
ROS's user avatar
  • 105
0 votes
1 answer
265 views

Estimating variance in arbitrary, periodic signal

I have a periodic signal $x[m], m \in [0;M-N+1]$ made of modulated templates $s[n],~ n \in [0;N-1],~ N \ll M = NK$ of finite energy and support (i.e. zero outside of its defined interval, which does ...
KaiserHaz's user avatar
2 votes
2 answers
1k views

Is there a difference what measurement units use in covariance matrix

The R matrix in the Kalman filter contains measurement noise. Diagonal elements of the matrix is the power of standard deviation. Is there a difference what measurement unit to use for standard error ...
Gluttton's user avatar
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0 votes
1 answer
459 views

Why does it mean that the process/signal is not stationary when its variance varied with time? [closed]

Why does it mean that the process/signal is not stationary when its variance varied with time? that is, $VAR[X(t)]= \alpha \times t$,$t$ is time,and $\alpha$ is a constant,then $X(t)$ is not the WSS ...
Shine Sun's user avatar
  • 159
10 votes
2 answers
5k views

What Is the Difference Between PCA and Karhunen Loeve (KL) Transform?

I have been reading about Karhunen-Loeve (KL) transform. 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 ...
Roger Figueroa Quintero's user avatar
0 votes
2 answers
3k views

[Python ]How can I improve my 1D Kalman Filter estimate?

I have written the following code to smooth an (almost) linear function: ...
JimiChango's user avatar
0 votes
0 answers
102 views

Interpretation of Eigen Values of covariance matrix

I am trying to obtain an intuitive understanding of Eigen Values of covariance matrix and have used a few layman terms because I fully do not understand the concept yet. The following is the code ...
Raj's user avatar
  • 279
1 vote
0 answers
132 views

Sensor fusion under unknown correlations: can covariance intersection account for delays?

Of late, there has been some interest in cooperative estimation algorithms in robotics, where the information sources are usually sensors such as cameras. When multiple robots observe surrounding ...
HighVoltage's user avatar
2 votes
0 answers
576 views

Estimate standard deviation of random-walk using Kalman filter

I'm new to Kalman filters so this might be a stupid question. I created a Kalman filter that takes in time series observations and estimates the mean of that time series. This is simply modeling a ...
Grant Bartel's user avatar
0 votes
0 answers
114 views

2-D parameter vector: Cramer Rao lower bound

Given a 2-D parameter vector, $\mathbf{X} = [x_1, x_2]$, let the corresponding $2\times2$ Fisher Information Matrix be $\mathbf{F}$. The Cramer-Rao Lower Bound (CRLB) is the inverse of the FIM. I ...
r2d2's user avatar
  • 97
1 vote
2 answers
1k views

Is it possible to estimate variance of noise for a step answer signal?

I know there is not possible to find the true noise of a measured signal. The only way to "find" the noise is to estimate the noise. Noise has the mean 0, but the variance varies. So assume that we ...
euraad's user avatar
  • 405
0 votes
1 answer
1k views

How to calculate the Diagonal loading factor evaluate calculate the inversion of a covariance matrix

I am programming a Generalised Likelihood Ratio Test (GLRT) detector. When it comes to inverting a covariance matrix $Ri$, I need to do a diagonal loading to fix the problem of sigularity of this ...
Linda bendjama's user avatar
7 votes
2 answers
213 views

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 ...
TheDon's user avatar
  • 171