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Questions tagged [covariance]

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1answer
40 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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0answers
64 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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1answer
37 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
30 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
83 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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1answer
85 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
56 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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1answer
76 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
385 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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2answers
780 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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0answers
48 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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0answers
79 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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0answers
161 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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0answers
68 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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2answers
58 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
212 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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2answers
142 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
141 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 ...
0
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1answer
746 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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0answers
142 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
267 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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0answers
327 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
169 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, ...
3
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1answer
194 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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0answers
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 = \...
0
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1answer
83 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 ...
0
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1answer
212 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 ...
1
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1answer
239 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 ...
0
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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 (...
0
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1answer
36 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 ...
0
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1answer
295 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
949 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 ...
1
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1answer
336 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
103 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 ...
2
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1answer
107 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 ( ...
2
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3answers
1k 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
127 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
531 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 ...
2
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1answer
97 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
177 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 $\...
2
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1answer
2k 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 ...
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0answers
276 views

Problem calculating the average power of a vector?

I am calculating the average power of a vector. I would like to compare the final expression with the simulation. However, they are not equal. Please help me to point out which steps are wrong. Thank ...
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0answers
57 views

Solving an array signal processing estimation problem based on the Rayleigh quotient

The Rayleigh quotient for a covariance matrix $\mathbf{C}$ and a non-zero steering vector $\mathbf{a}$ is given by $$ R(\mathbf{C},\mathbf{a}) := \frac{\mathbf{a}^H\mathbf{C}\mathbf{a}}{\mathbf{a}^H\...
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1answer
479 views

Karhunen loeve transform question

I have read some about Karhunen-Loeve Transform (KLT) and its application to the field of seismic data processing. The method as I understand it based on decomposing the data (actually mostly used in ...
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0answers
507 views

Constant amplitude, uniform phase - what's the distribution of the complex signal then?

The well-known relationships for zero-mean circularly-symmetric complex Gaussian $z = a + jb = |z| \exp(j\varphi)$ signals are the amplitudes $|z| = \sqrt{a^2 + b^2}$ are Rayleigh-distributed the ...
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2answers
374 views

How to Extract Nonorthogonal PCA Principal Components

My problem is essentially a 'blind source separation' problem. I have 3 non-orthogonal sources (or basis functions) and $N$ random linear combinations (mixes) of said sources. My problem is to obtain ...
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1answer
597 views

Purpose of eigenspace of covariance matrix of a blob?

Given a blob of an image (representing an object), according to Wikipedia, we can compute the co-variance matrix using the image moments. I understand that the eigenvectors of that matrix can be used ...