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is a branch of algebra, concerning linear nature of objects: vector or vector spaces, linear transformations, systems of linear equations, quadratic and bi-linear forms, among the main tools used in linear algebra is the determinants of the matrix pair. The theory of invariants and tensor calculus ...

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Two different definitions of coherence parameter

When we have the measurement basis $\varphi_{m \times n}$ and the sparse basis $\psi_{n \times n}$ the coherence parameter is defined as follows, $\mu_1(\varphi , \psi) = \sqrt{n}\max_{j,i}\frac{|\...
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1answer
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DFT as an Orthogonal Basis Change

In one of the homeworks that I am dealing with for Linear Systems course, I have encountered with such a statement: Consider $\mathbb{C}^N$ the vector space of N dimensional complex vectors. We can ...
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2answers
60 views

Conversion from stationarity to non-stationarity

Is there any way to convert a non-stationary signal to a stationary one, perform operations on it meant for a stationary signal and then convert it back to the non-stationary one?
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16 views

Wiener Filter: spacing of the data

I read in an article that for the discrete version of Wiener filter as proposed by Levinson, the data can be arbitrarily spaced. What is implied by this? I believe that the spacing does matter to ...
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1answer
99 views

How to Check Separability of 2D Signal / Matrix

suppose x(n1,n2) = {1 ,n1=0,n2=0 ; 2 ,n1=1,n2=0 ; 3 ,n1=0,n2=1 ; 6 ,n1=1,n2=1 } then, how do i prove it is separable.
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20 views

Blind Signal Separation for Sparse Signals / Sources

Assume we have $N$ measurements $z_1, ..., z_N \in \mathbb{R}^{n_z} $ that generated by $$ z_i = M v_i + e_i $$ where $v_i \in \mathbb{R}^{n_v}$, $n_v < n_z$ and $e_i$ an error sampled from ...
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1answer
47 views

How do I apply a binary mask and STFT to produce an audio file?

So here's the idea: you can generate a spectrogram from an audio file using shorttime Fourier transform (stft). Then some people have generated something called a "binary mask" to generate different ...
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26 views

Blind Signal Separation for Sparse Signals

Assume we have $N$ measurements $z_1, z_2, \dots, z_N \in \mathbb{R}^{n_z}$ that are generated by $$ z_i = M v_i + e_i $$ where $v_i \in \mathbb{R}^{n_v}$, $n_v < n_z$ and $e_i \in \mathbb{R}^{...
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1answer
33 views

How Does Mean Centering Affect the Result of Using SVD to Compress Images?

I have been learning about using the Singular Value Decomposition to find low rank approximations to matrices. I had an image which I converted to a matrix. I regarded each row of the matrix as a '...
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2answers
57 views

What is the relation between kernel functions, kernels used in convolution and null spaces of a matrix?

I have recently started learning about machine learning and have come across kernels and null spaces. I understand that null space is the set of all vectors that satisfy the equation A.v = 0 (Where A ...
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1answer
35 views

How Can PCA Be Used in Image Analysis [closed]

I am still a not how PCA can be used in image analysis and where is it is mostly used. For example how can PCA be used in order to differentiate between different faces? Can you please mention other ...
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65 views

Multilateration of Sound in 3D Space

TL:DR - How can you find the 3D coordinates of a emitter than transmits an impulse signal? STORY: I'm working on something to improve my bird-watching. I've got a camera that can take pictures of ...
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3answers
94 views

Least Squares with blocks/updates

I have a continuous-time system that I want to fit via least squares. I just send $N$ digital samples $x[n]$ through the system and receive (via analog signal chain, ADC etc) $N$ digital samples $y[n]$...
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0answers
29 views

Required number of measrments for signal recovery in a compressed sensing MMV problem?

For multiple measurement vector (MMV) problem $Y=AX$ where $A$ is $m \times n$ sensing matrix and $X$ is $n \times L$ matrix haveing K non zero rows. What are the necessary conditions on the ...
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1answer
199 views

Signals and systems book with linear algebra approach

I am currently taking a course on discrete-time signals and systems. I am using B. P. Lathi's Signal Processing and Linear Systems, which I don't like at all, as it doesn't draw any paralells to ...
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42 views

linear equation system with partially unknown mixing matrix

I have a system of two linear equations with one column of observation matrix missing $$ \begin{bmatrix} y_{1} \\ y_{2} \\ \vdots \\ y_{n} \end{bmatrix} = \begin{bmatrix} u_{1} & s_{1} \\ u_{2} &...
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1answer
48 views

Tikhonov Regularization Alternate Formulation

2.2 Tikhonov Reqularization Tikhonov regularization, named for Russian mathematician Andrey Tikhonov, attempts to fix the issue that arises when the least squares method is used with an ill-posed ...
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1answer
202 views

3D world homography matrix calculation

I have a matrix in (x,y,z) coordinates and also i have a transformed version like (x_new,y_new,z_new). I need to calculate a transformation matrix that can transform (x,y,z) to (x_new,y_new,z_new). ...
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1answer
23 views

Are there analogues to orthogonal transformations in non-orientable surfaces?

I am working on extremely large, symmetric matrices of counts, and attempting to identify patterns/shapes within them. Wavelets are a popular tool in image processing, and have some nice statistical ...
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1answer
112 views

Time domain basis

I have some troubles with understanding time domain, not on the intuitive level, but in math terms. For example I have a vector signal $$ x = [x_0,x_1,x_2,...,x_{N-1}]$$ I understand that generally ...
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1answer
89 views

Underdetermined deconvolution of windowed output

Consider a discrete 'blurred' output $h[t]$ given by the convolution of filter $f[t]$ and signal $g[t]$. This question considers recovering $g[t]$ from a window (subset) of $h[t]$. This causes the ...
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1answer
42 views

Can linear transforms of vector space be seen as LTI systems? [duplicate]

Linear transforms of vector spaces has the linearity (i.e., homogeneity and superimposability), which is shared by LTI system. There are also shared concepts such as eigenvectors/eigenvalue... I am ...
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0answers
115 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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3answers
164 views

Group delay of $H(\omega)= 1- re^{j \theta}e^{ - j \omega} $

I'm studying chapter 5 of Discrete-Time Signal Processing 3rd edition by Alan Oppenheim and I'm having serious difficulties understanding how he obtained equation 5.57. For those who don't have this ...
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36 views

Matrix data transmission over fading channel

I have to send a data matrix $A$ over a Rayleigh fading channel $h$. $$y=hA+n$$ where $A$ is data matrix, $h$ is channel vector, and $n$ is receiver noise. The problem is that the data matrix $A$ is ...
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6answers
261 views

The Least Norm Solution of Under Determined Linear System

Suppose I have a matrix $$A= \begin{pmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 1& 0\\ \end{pmatrix} $$ where the variables are channel information like assume $X_1$, $X_2$, $X_3$ ...
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0answers
56 views

Linear Systems, Sparse Solutions, and $4 \times 4$ Sudoku Algorithm [closed]

I am unable to understand the paper Linear Systems, Sparse Solutions, and Sudoku. I have to form a $4 \times 4$ Sudoku using the algorithm in this paper. Can somebody please provide me the algorithm ...
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1answer
32 views

Help with Determinants [closed]

Okay so i understand the first det, which is why i got the correct solution. However for the second det, im not sure how to solve it since because of the different coefficents (0,2,4) and i tried of ...
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2answers
211 views

Deriving Frequency Response for 2-pole Zero-Delay Feedback State Variable Filter

I have an existing zero-delay feedback (ZDF) 2-pole state variable filter implementation (along the lines of the theory presented in VA Filter Design by V. Zavalishin), and I wish to determine the ...
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1answer
82 views

Restriction of Fourier Transform

I am currently reading Candes et. al.'s 2006 paper[1] on recovery of sparse signals from incomplete frequency samples. I am having trouble figuring out what is the form of the Fourier transform ...
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1answer
582 views

Sensing matrix for compressed sensing

What are the differences between random binary sensing matrix  and random Gaussian sensing matrix? What the advantages and disadvantages of each matrix? How can I choose the suitable matrix for a ...
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3answers
3k views

Proof of complex conjugate symmetry property of DFT

According to the Proof : \begin{align} X_n &= \sum_{k=0}^{N-1}x_ke^{-j\frac{2\pi k n}{N}}\\ X_{N-n} &= \sum_{k=0}^{N-1}x_ke^{-j\frac{2\pi k (N-n)}{N}}\\ &=\sum_{k=0}^{N-1}x_k e^{-j 2\pi ...
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2answers
199 views

What are the practical constraints on designing Sensing matrix in compressed Sensing?

In a typical compressed sensing scenario, $y=Ax$, where $x$ is a sparse signal and $A$ is the sensing matrix. To reconstruct the sparse signal $x$ from $y$, $A$ should posses the Restricted Isometry ...
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2answers
416 views

Alternative to Orthogonal Matching Pursuit (OMP) Algorithm

In the Compressed Sensing context, assume there is a signal $ x \in {\mathbb{R}}^{n} $ which is $ k $ sparse. Namely its Pseudo $ {\ell}_{0} $ Norm is $ {\left\| x \right\|}_{0} = k $ (The signal has ...
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1answer
1k views

How to find out if a transform matrix is separable?

In image processing, when we have a series of basis images, how could we know if the transformation is separable or not? For example, I know that following bases are separable and transformation can ...
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0answers
51 views

MIL given for RLS equations vs the Woodbury derivation [closed]

Can any one help me in deriving the matrix inversion lemma rule for RLS algorithm? I don't know how to start with. Many books have just stated but they haven't derived it.
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79 views

Loss of precision in matrix operations

I am implementing Gaussian Elimination method on PC (x86_64) windows 8, using C language. Data is float. A small value will start vanishing as the algorithm proceed. That is why partial pivoting is ...
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1answer
149 views

Finding a good inverse for an ill-conditioned matrix transformation

I have a time-series observation dataset that has been distorted. I want to recover the best approximation of the original signal as possible. Disclaimer:: I know only the basics of linear algebra, so ...
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165 views

Wavelet domain Gaussian processes

Maraun et al, "Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significant testing" (2007) defines a class of non-stationary GPs that can be specified by multipliers in ...
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2answers
147 views

The mathematical interpretation of DFT [closed]

We have the DFT(matrix form) $X = Wx$ ($W$ is the Fourier basis matrix, $x$ is the original signal in time domain, $X$ is in the frequency domain). In mathematics, $x$ represents the coordinates of $...
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1answer
103 views

Geometric explanation of a methodology in the article about Image Denoising

In article Ghimpeteanu G., et al. - A Decomposition Framework for Image Denoising Algorithms, I found as below: Let $\displaystyle I : \Omega \subset \mathbb{R}^2\mapsto \mathbb{R}$ be a gray-level ...
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2answers
62 views

Proving conditions for controllability

Let's say I have the following LTI system: $$\dot{x}(t) = \mathbf{A} x(t) + \mathbf{B} u(t)$$ I need to somehow show the following is true or false (proof): This system is controllable if and only ...
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1answer
61 views

Showing a system is always controllable?

I need to show that the following system is always controllable: \begin{align}A &= \begin{bmatrix} -\alpha_1I_{k\times k}& -\alpha_2I_{k\times k}& \cdots &-\alpha_{n-1}I_{k\times k}&...
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32 views

Eigenvalues of correlation matrix which have the form of an harmonic function [duplicate]

As a continuation to this question, I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\frac{|\phi_1-\phi_2|}{2}} &...
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2answers
96 views

How to make the $\ell_2$ norm of all columns and rows of an $n \times n$ matrix equal to $\sqrt{n}$?

I have an $n \times n$ matrix and I would like its columns and rows to have $\ell_2$ norm equal to $\sqrt{n}$. Is this possible?
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28 views

How to reduce polynomial?

I have some 4 polynomials like this. these equation is made by polyfit() from MATLAB tool. ...
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0answers
110 views

In which way do image transformations change camera extrinsics?

Given a pinhole camera model, we have an image and according camera intrinsics K and extrinsic parameters R. When we scale the ...
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1answer
111 views

New Horizons at Pluto

I'm teaching an undergraduate linear algebra course, and looking for applications. The New Horizons spacecraft is approaching Pluto, and returning images I find fascinating: http://pluto.jhuapl.edu/...
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2answers
202 views

Improvement of Minimum description length (MDL) estimate

I earnestly request apology if this question is inappropriate for the forum. The question has two parts one technical and the other is not technical. I would appreciate any response. Let me consider ...
5
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1answer
335 views

Weighted Nuclear Norm Minimization for Image Denoising

Recently, I saw new published papers like Shuhang Gu, Lei Zhang, Wangmeng Zuo, Xiangchu Feng, Weighted Nuclear Norm Minimization with Application to Image Demonising [pdf]. about denoising images ...