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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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The Discrete Fourier Transform (DFT) decomposes any signal into four orthogonal signal components

Let $F=\frac{1}{\sqrt{n}}(w^{kl})_{k,l=0}^{n-1}$ be the discrete Fourier matrix of size $n$ where $w=\exp^{-\frac{2\pi i}{n}}$. It is a well-known that $F_n^4 = I_n$ where $I_n$ represents the ...
ABB's user avatar
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Are complex exponentials the only eigenfunction for arbitrary LTI systems?

After reading a few posts, like this. I know that arbitrary LTI systems always have complex exponential eigenfunctions. And that for specific LTI systems you can also have other types of ...
roobee's user avatar
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Practical Implications of DFT Eigenvector Formulations

Different sets of eigenvectors for the Discrete Fourier Transform (DFT) are well-established. Are there any potential practical implications associated with deriving explicit formulations for these ...
ABB's user avatar
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Smallest Eigenvalue in the Derivation of the MUSIC Algorithm

I am seeking clarification on a particular step in the derivation of the MUSIC algorithm as presented in a specific paper. Here, there is an intermediate step I cannot follow and I would appreciate ...
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What is the possible application of eigenvalues?

I am a PhD in mathematics. Recently, we made an attempt to compute the eigenvalues of non-normalized discrete sine and cosine transforms. Surprisingly, the issue regarding three particular types, DCT-...
ABB's user avatar
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Direct and numerically robust conversion from zero-pole to state-space representation

Note: this question was initially asked in a different community. Encouraged by the comments, I decided to cross-post here too. Given (z,p,k) my goal is to convert to a state-space representation (A, ...
DaveC's user avatar
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Why do they say that complex exponentials are eigenfunctions of LTI systems, when there are still transient responses?

Let $$\dot{x} = Ax+Bu$$ $$y = Cx + Du$$ be a linear ODE with $x(0)=0$. Here, I am assume $A$ is invertible. As you can see, the relation $$H:u(.) \mapsto y(.),$$ where $(u(.),y(.))$ is a solution to ...
Spencer Kraisler's user avatar
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1 answer
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The benefit of Eigendecomposition of DCT and DST

I am Ph.D in pure mathematics and interested in signal processing. Theoretically, any illustration of the eigendecomposition of the discrete trigonometric transforms (DTTs) is worthwhile. Q. What real ...
ABB's user avatar
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When is the Fourier transform of a periodic discrete signal $\mathcal{F}x[k]$ the same as $x[k]$ up to a diagonal matrix

I am looking for all pairs $(x[n],q)$ where $x[n]$ is a periodic discrete signal with period $N$ and $q$ is a rational number satisfying the following identity: $$\mathcal{F}x[k]=e^{i(q-\frac{\pi k}{...
ABB's user avatar
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Subspace Methods - Eigenvalues of the Signal Subspace

Subspace frequency estimation methods like MUSIC or ESPRIT decompose the signal correlation matrix into a signal and a noise subspace. Assume the signal model is given by $$\boldsymbol{s} = \...
Lukas's user avatar
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Subspace decomposition

While reading the paper "Perturbation analysis for subspace decomposition with applications in subspace-based algorithms" by Zhengyuan Xu, I came across the decomposition technique called ...
Neuling's user avatar
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2 answers
213 views

Why do we need to estimate eigenvalues?

I am not working in signal processing field, but recently I happen to read a paper which estimates source numbers using Gerschgorin radii, and I feel kind of confused about why we need to estimate ...
WBR's user avatar
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What is the relation between eigenvalues and state-space response in control systems?

I understand the mathematics behind it but I want to know what happens physically in a real-life system. How do the eigenvalues come into the picture from a non-mathematical (physical) point of view? ...
Sagnik's user avatar
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3 answers
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Are all exponential functions eigensignals of LTI systems?

I know that complex exponential functions are eigensignals to LTI systems. Do these include real exponential functions? E.g. $2^t, e^t, ...$ Thanks for the help!
Phobos's user avatar
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MUSIC algorithm terminology

Looking for clarification on the notation described in the MUSIC algorithm in Ralph Schmidt's IEEE paper$^{[1]}$. The data model is: $$X = AF + W$$ Schmidt defines the following: ...
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Efficient way to calculate eigen-values of a 2x2 matrix

I'm working on an algorithm that works based on finding the eigen-values of a 2x2 complex covariance matrix. However, from a fixed-point implementation point of view, finding eigen-value of a matrix ...
vmontazeri's user avatar
1 vote
1 answer
255 views

SVD vs matched filter

Reading about singular value decomposition (SVD) in the context of signal processing applications, one can separate the signal from the noise into orthogonal subspaces. On the surface this sounds like ...
BigBrownBear00's user avatar
3 votes
3 answers
1k views

Intuitive explanation of subspace

There are many techniques in signal processing that use eigen analysis (MUSIC, SVD, eigen decomposition, etc) that result in signal and noise subspaces.The mathematical definitions for signal ...
BigBrownBear00's user avatar
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1 answer
232 views

Question about eigendecomposition, signal subspace and their properties

Lately, I am readig a paper titled A Subspace Method for Estimating Sensor Gains and Phases. In it, it is mentioned: There are $m$ sensors in the array, $n$ known narrowband far-field signals($m \...
tyrela's user avatar
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Real time signal processing use cases for eigenvalues of symmetric matrices

I realize that this might be somewhat of an unusual and specific question. I know that the eigenvalues of symmetric matrices are used in a number of ways in scientific computing, such as for finding ...
user3120921's user avatar
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1 answer
2k views

Zero forcing vs matched filtering vs LMMSE

In what scenarios would you choose each of Zero forcing, LMMSE and matched filtering receivers: Possible points to consider are: Receiver SINR, High Interference levels, Low interference levels, ...
Dsp guy sam's user avatar
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3 votes
2 answers
102 views

Proving that a product of matrices invertible

Given $R_x$ a Positive Definite (PD) covariance matrix of size $M\times M$ and $C$ a full rank $M \times N$ matrix, I want to prove that $C^* R_x^{-1} C$ is invertible to derive the Linearly ...
Oriol B's user avatar
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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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How to estimate noise by eigendecomposition of the variance covariance matrix?

I'm new here so I will try to be as clear as possible. I am trying to apply some techniques from signal processing framework to denoise financial time series. I would like to know if what I am ...
Chaos's user avatar
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Is there a situation when maximizing SNR doesn't also maximize probability of detection?

The problem I'm working on is signal detection of a radar signal, but I suspect this problem shows up in many different branches of signal processing. Background: From [1], the factor $I_f$, by ...
Robert L.'s user avatar
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1 answer
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What are some efficient algorithms for real-time calculation of eigenvalues?

I need to calculate eigenvalues of a square matrix for a real-time implementation using microcontrollers or dSPACE. Obviously, I can find the eigenvalues of a 2by2 matrix analytically. I am wondering ...
Yasi's user avatar
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10 votes
2 answers
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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
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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
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1 answer
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Eigenvalues of FIR Convolution Matrix

Are the eigenvalues of the FIR convolution matrix the zeros of the corresponding FIR filter? Suppose I have an FIR filter $H(z) = h_{0} + h_{1}z^{-1} + h_{2}z^{-2}$. I want to implement it using a ...
Some_Guy_2018's user avatar
3 votes
1 answer
2k views

What are the eigenvalues of the 8 point DFT matrix?

I know that the eigenvalues for 4 point DFT matrix can be found from $F_4^4=I$. Is this also valid for 8, 16 and higher orders? For example with 8 points, will it be $F_8^8=I$ ? If not, how can I ...
Mostafa's user avatar
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1 answer
2k views

Angle of arrival (AOA) estimation using FMCW radar using MUSIC algorithm

I am working with FMCW phased array radar with only upchirps. The range doppler matrix is obtained using the two dimensional fast Fourier transform on multiple chirps. I want to obtain the angular ...
Zeeshan's user avatar
  • 386
3 votes
1 answer
720 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 ...
Ricardo García's user avatar
5 votes
2 answers
1k views

MUSIC algorithm derivation

Setup Suppose we have a complex $L\times 1$ signal $\mathbf{x}$ with two tones at (unknown) frequencies and phases defined as: $$ x_n = A_1 e^{j \omega_1n + \varphi_1} + A_2 e^{j \omega_2n + \...
Atul Ingle's user avatar
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6 votes
1 answer
2k views

Eigen Values and Eigen Vectors in Image Processing

I have been reading about eigen values and eigen vectors but I haven't been able to find any decent explanation relating their application in Image Processing / Computer Vision. For example, How can ...
silver surfer's user avatar
14 votes
4 answers
5k views

Are complex exponentials the only eigenfunctions of LTI systems?

Is there an example of an eigenfunction of a linear time invariant (LTI) system that is not a complex exponential? Justin Romberg's Eigenfunctions of LTI Systems says such eigenfuctions do exist, but ...
Vinod's user avatar
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8 votes
3 answers
2k views

Eigenvalues and Eigenvectors of a 3D Image Laplacian

I need to compute the eigenvalues and eigenvectors of a 3D image Laplacian. I'm trying to evaluate the heat kernel on the 3D uniform grid (the uniform structure generated by the voxelized image) at ...
Federico's user avatar
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