Questions tagged [covariance]

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Sample with zero mean and its covariance matrix

From a covariance matrix how we can get the sample has zero mean ? Suppose a covariance matrix is given and we have to find out if the original matrix is zero mean sample . How can we found it?
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30 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.
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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 ...
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35 views

Retrieve covariance matrix from array of fft

I have 10000 power spectral density vectors of 64 frequency bins each. My input is therefore are 10000x64 matrix of real values. Those are obtain from fft of time series of temporal signal I don't ...
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93 views

Kalman Filter State Covariance Matrix

If I have a discrete time process model of the form: $$x_{k+1} = x_{k} + v_{k}\cos(\theta_{k})dt$$ $$y_{k+1} = y_{k} + v_{k}\sin(\theta_{k})dt$$ $$v_{k+1} = v_{k} $$ $$\theta_{k+1} = \theta_{k}$$ ...
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1answer
32 views

covariance block of MUSIC Algorithm

I am implementing MUSIC algorithm in verilog and need to implement the blocks in following order input signal --> co-variance/correlation matrix--> EVD --> peak search But I am confused because ...
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1answer
53 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 ...
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188 views

About the variance of the white noise process

I saw such an example in a book. Assume that the $w(t)$ is a white Gaussian noise with zero-mean and power spectrum density $N_0/2$. Now, consider the sample function: $$n(t)=\sqrt{2\over T}\int_{...
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50 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 \...
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1answer
39 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, -...
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1answer
125 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-...
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99 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 ...
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2answers
61 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 ...
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82 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 ...
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1answer
514 views

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 ...
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1k views

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

I have written the following code to smooth an (almost) linear function: ...
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58 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 ...
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88 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 ...
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205 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 ...
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83 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 ...
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112 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 ...
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1answer
251 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 ...
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143 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 ...
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1answer
182 views

How to estimate covariance matrix using Fourier representation?

So, I have multidimensional time-series $X \in R^{(d \times T)}$, and I want to determine the covariance matrix of that signal in a specific frequency band. I might filter the signal to that specific ...
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1answer
868 views

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} \...
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161 views

Deterministic method to compute “Process noise covariance matrix, Q” for a Kalman filter when parameter variations of the model is known apriori

I am implementing a Kalman filter (for a linear ODE system for now). My model represents a physical device that has 6 "parameters", i.e. those values of the device do not evolve over time (within a ...
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2answers
1k views

Kalman Filtering with Unknown State Transition Matrix

I'm currently studying the use of Kalman filters for estimating linear systems. My current State Transition Matrix (STM) is the identity since so far I've been dealing with non time-varying systems. ...
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1answer
276 views

Auto-covariance of the product of deterministic and wide-sense stationary signal

Anybody give me an advice how to find the auto-covariance of the product of deterministic and wide-sense stationary signal. I couldn't find how to solve this, I have looked and searched the internet, ...
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351 views

Noise covariance matrix

I am attempting to implement a generalized least squares estimator that uses a noise covariance matrix from measured data. The basic model is as follows $$\hat{c}_{GLS} = \left(\bf{A}^T\bf{\Sigma}^{-...
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1answer
188 views

Covariance matrix of an adaptive filter input

I run many times in equations containing the trace of covariance matrix of an adaptive filter input. But it is not really clear what it is. For example in this paper the input covariance matrix is $$...
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2answers
7k views

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, ...
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199 views

Help in understanding from book expression of variance of an estimator : PRBS vs real valued input

The question is based on Book : Fundamentals of Statistical Signal Processing by Steven Kay, Chapter 4 : Eq(4.21). The expression for the variance of the estimated coefficients when the input is PRN ...
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51 views

Variance and Co-variance of a Linear Forecast

Consider a linear forecasting problem where all shocks $\{\epsilon_i\}_1^n$ are independently distributed with $\epsilon_i\sim N(0,\sigma_i^2)$ for all $i$. Suppose you want to forecast $\theta = \...
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1answer
89 views

Kullback-Leibler Distance of Spectral Data

I am currently reading through Music Structure and Analysis from Acoustic Signals and am having some difficulty in understanding how the modified Kullback-Leibler distance is calculated. (I am just ...
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1answer
235 views

Covariance matrix $P$ and $Q$ for complex valued data

The equations for the Kalman filtering are IM = H*X; IS = (R + H*P*H'); K = P*H'/IS; X = X + K * (y-IM); P = P - K*IS*K'; The covariance matrix in my ...
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1answer
270 views

Problem with Covariance matrix using diagonal loading involved in calculation of eigenvalues

I have the following problem : I'm calculating the sample covariance matrix in the frequency domain ( $y_{k}$ is the FFT of a time domain $k_{th}$ symbol vector signal , basically a simulated ...
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1answer
58 views

Should you include division by coefficient of variation in WV reconstruction for its normed 1D signal?

I make a new ECG time series from WV spectrum of original signal and its L2 energy normalisation. I am thinking if the reconstruction step benefits from covariance at each time point. I take later a ...
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1answer
2k views

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 (...
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1answer
37 views

Testing for changes in auto-covariance

I am working with uniformly-spaced time series data where I am interested in knowing whether there are changes in temporal auto-covariance. The mean can be assumed constant. Visually, there are no ...
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1answer
334 views

Determining the covariance of point clouds in real-time

So basically I have a set of multidimensional data that I need to determine the covariance of between dimensions in real-time. Each point that comes in is a vector. I have gotten the mean and variance ...
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1answer
1k views

Calculating covariance matrix for MVDR beamforming

I am trying to calculate the covariance matrix that is required for the calculation of an MVDR beamformer. I am getting confused as to how to calculate it. I have an array of 3 microphones each with a ...
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1answer
364 views

Relation between Covariance matrix & Energy of a random signal

Let's say I have the below random signal: $ Y[n] = [y(n), y(n-1), y(n-2), \ldots, y(1)] $ I have two random variables now: The first one $X_1 $ which express the maximum eigenvalue of the covariance ...
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1answer
104 views

Best way to find an object in a picture

I'm trying to find an object in a picture, my solution is to take the picture and a photo of the object and find the maximum of the mutual covariance, this is my ...
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1answer
116 views

Can I model process noise as a known “error” in my dynamics while designing a Kalman Filter?

Consider I am modelling the dynamics of a robot and using a Kalman filter to obtain estimates of some state. I have certain terms in my equation which correspond to data not accessible to this robot ( ...
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3answers
2k views

Variance of a filtered signal

I'm using a very simple 1st order Butterworth Filter shown in Matlab code: order = 1; cutOff = 0.1; [b, a] = butter(order, (2*cutOff)/SampleRate, 'high'); So ...
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2answers
128 views

PSD and $\lim_{T\rightarrow \infty} \frac 1 {2T} \int_{-T}^T x(t)\bar y(t)\,dt$

From Wikipedia, I taken a definition of power spectral density: For continued signals that describe, for example, stationary physical processes, it makes more sense to define a power spectral ...
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1answer
579 views

power spectral density plots

Hi all, How do we interpret a power spectral density plot. I have used modified covariance and burg methods and plotted using MATLAB. What does the peaks at some frequencies suggest? and what is the ...
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1answer
104 views

What is the Technique to Find Variance of Estimation Error

Given an $n$-vector $y$ (responses) and a design matrix $X$, I wish to fit them with a simple linear regression model $$y=X\beta+e,$$ or, $y_t = x_t'\beta_0 + e_t$ where $e\sim\mathcal{N}(0, \sigma^...
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0answers
183 views

Distribution of a signal covariance matrix

A common estimation problem in signal processing assumes the following signal model \begin{equation} \mathbf{r} = \sum_{i=1}^{Q}\alpha_i\mathbf{s}\left(w_i\right)+\mathbf{n} \end{equation} where $\...
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1answer
3k views

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 ...