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2D Convolution as a Doubly Block Circulant Matrix Operating on a Vector

The point is that circular convolution of two 1-D discrete signals can be expressed as the product of a circulant matrix and the vector representation of the other signal. The circulant matrix is a ...
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8 votes
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Circular Convolution Matrix of $ {H}^{H} {H} $

If $ H $ is a matrix form of Circular Convolution then it is a Circulant Matrix. Being a Circulant Matrix means it can be diagonalized by the Fourier Matrix $ {F} $: $$ H = {F}^{H} D F $$ Where the ...
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6 votes
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Least Angle Regression (LARS) without Matrix Inversion

If you want to solve for single value of $ \lambda $ in the model: $$ \arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{1} $$ Then you can use Coordinate ...
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5 votes
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Deriving the Matrix Inversion Lemma for RLS Equations vs the Woodbury Derivation

It is not clear what are you asking but I will try answer both things. Deriving the Matrix Inversion Lemma The Matrix Inversion Lemma goes as: $$ {\left( A + U C V \right)}^{-1} = {A}^{-1} - {A}^{-...
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5 votes
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Is There a Way to Perform a 2D Image Rotation by Matrix Multiplication?

Let's say you have an $ M \times N $ image. If you turn it into a vector in $ {R}^{\left( M N \right)} $ and create a matrix which is $ \left(M N \right) \times \left( M N \right) $ by ...
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5 votes

The Least Norm Solution of Under Determined Linear System

Least Squares solution is always well defined for Linear System of Equations. In your case, which is under determined it means there are many solutions to the Linear Equations. The Least Squares ...
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4 votes

Apply Low Pass (Smoothing) Filter on a Set of Matrices and Reject Outliers

I would build a tensor of those matrices and use low rank or some thresholding methods on it. You may have a look for Tensor SVD.
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4 votes
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Apply Low Pass (Smoothing) Filter on a Set of Matrices and Reject Outliers

The method used was a parametric representation of the matrix $ H $. For intuition, think of rotation matrix in $ \mathbb{R}^{2} $, namely $ H \in \mathbb{R}^{2 \times 2} $. Since we have series of ...
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4 votes
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What is a covariance matrix?

It's the key point of array signal processing, I suppose. Say $x$ is the input vector of $[N,1]$ dimension collected from $N$ array sensors. $x(k)$ is its realization at the $k$ moment of time. By its ...
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4 votes
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The inverse of an orthogonal matrix is its transpose

An orthogonal matrix has orthogal columns, i.e. the scalar product of two different columns is zero (the case $i\neq j$). For the case $i=j$ you have $a_i^Ta_i=\|a_i\|^2>0$. So, all you can say ...
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4 votes
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Why do we need to estimate eigenvalues?

It seems to me that we can directly calculate ... which can be done in a few second in matlab. Who says that Matlab is calculating it directly, or that it isn't using Gershgorin's circle theorem in ...
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Why do we need to estimate eigenvalues?

We do use eigenvalues, because they behave like invariants in linear systems (invariant inputs are well connected to outputs) and generally finding invariants of unknown or model systems provides a ...
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4 votes

Optimization of square matrix multiplied with another matrix to have the final result a unitary matrix

Could it be that you are indeed looking for the closest orthogonal matrix $Y$? Then, there is a solution which involves computing the square root of $ D^TD$ . If $E=(D^TD)^{1/2}$ were invertible, the ...
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3 votes
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Why is this matrix invertible in the Kalman gain?

Note that $\mathbf{P} _{k\mid k-1}$, just like $\mathbf{R}_k$, is also a covariance matrix, and for this reason it is (at least) positve semi-definite, i.e., $\mathbf{y}^T\mathbf{P}_{k\mid k-1}\mathbf{...
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3 votes
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involutory transformations - why are they not so much used in signal processing?

You don't chose transforms by whether they are involutions or not. If invertibility is of interest, any simple form of inverse is sufficient. Useful transforms reveal structure of some sort or ...
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3 votes

Calculating covariance matrix for MVDR beamforming

MVDR is a narrowband beamformer. For broadband signals it is usually applied for each frequency bin. That means that $\mathbf{R}_{xx}$ is frequency dependent. In other words, for each time you should ...
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  • 1,440
3 votes

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

HINT If we have the diagonal matrix: $$ D = \left[\begin{array}{cccc} d_1&0&0&0\\ 0&d_2&0&0\\ 0&0&\ddots&0\\ 0&0&0&d_n \end{array}\right]$$ Multiplying ...
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3 votes
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What does "kernel based" mean?

In general, a kernel is a function that acts as a parameter to some algorithm. Filtering: For example, it's possible to call the impulse response of a filter $h[n]$ a kernel, so that it is the ...
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3 votes
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Sensing matrix for compressed sensing

A sensing matrix maps input vector to measurement vector through linear wighted summation of input. What makes a specific matrix good, is application dependent. Now, both distributions more or less ...
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3 votes
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how to set Equalizer's coefficient using generalized eigenvector.

The generalized eigenvalue problem is given by $$Bw=\lambda Cw\tag{1}$$ where $\lambda$ is the generalized eigenvalue of the matrices $B$ and $C$. Multiplying $(1)$ from the left with $w^H$ (with $^...
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3 votes

On the simplification using trigonometric functions

If you use the Euler's formula, you can simplify like this: $$ [d]_{k,n} = \frac{\sqrt{2}}{N}\left( \cos{\left[ \frac{(k-1)(2n-1)\pi}{2N} \right]} e^{j\frac{2 \pi nk}{N}} \right) $$ I think we can't ...
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  • 31
3 votes

why use svd() to invert a matrix?

The two methods differ, above all, by their applicability to matrix classes. col (cholesky) decomposes Hermitian, positive-definite rectangular matrices into the ...
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3 votes
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Proving that the uncertainty can not increase during the update step of a Kalman filter - positive semidefiniteness

$Q_t$ is real-valued and positive definite, thus $Q_t^{-1}$ is real-valued and positive definite. Now it's just making up a lemma of the Cholesky decomposition: If $Q_t^{-1}$ is real-valued and ...
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3 votes
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What is the complexity of multiplication a real matrix with real vector

If you multiply an $M \times N$ matrix with an $N \times 1$ vector you get a vector of size $M \times 1$ For the generic case you will need $M \cdot N$ multiplications and $M \cdot (N-1)$ additions. ...
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2 votes

Analytical expression for the eigenvectors of a 3x3 real, symmetric matrix?

There's a newer (2017) closed-form formulation for the eigendecomposition of 2x2 and 3x3 Hermitian matrices here: Charles-Alban Deledalle, Loic Denis, Sonia Tabti, Florence Tupin. Closed-form ...
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2 votes
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Difference Between Correlation / Convolution and Matrix Multiplication

Well I will try to explain. Let us first discission in time domain: 1) Let us say you have two signals, x and y. By Convolution in time domain, you mean that you flip(invert) one of the signals(lets ...
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  • 116
2 votes
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How is the sound converted to matrix in Matlab?

The function audioread doesn't generate any values, it just reads audio samples stored in a file. If you want to generate the sound of a guitar, you need to look ...
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2 votes

5.1 Rear To 5.1 Side mixing matrix

The commonly called 5.1 format uses only surround channels, which are defined as rear/side channels in ITU-R BS 775. The case you want to deal with (turning rear surround channels to side surround ...
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  • 277
2 votes
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Mechanics of a matrix Interleaver

We have different interleaving techniques, and matrix interleaving is one of them. But at the end all of them do one thing: interleaving is a technique to protect against burst errors (no matter how ...
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  • 4,105
2 votes

Why is this matrix invertible in the Kalman gain?

Let me take a stab at it. You agree that $\mathbf{R}_k$ is positive definite. Since it is the variance. Now, $\mathbf{P}_{k|k-1}$ is also positive definite as it is a covariance matrix, as ...
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