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11 votes
Accepted

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 ...
msm's user avatar
  • 4,295
4 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 ...
Luke Hutchison's user avatar
4 votes
Accepted

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 ...
Maximilian Matthé's user avatar
4 votes
Accepted

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 ...
Royi's user avatar
  • 19.7k
4 votes
Accepted

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 ...
TimWescott's user avatar
  • 12.8k
4 votes

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 ...
Laurent Duval's user avatar
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 ...
Laurent Duval's user avatar
3 votes
Accepted

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 ...
MimSaad's user avatar
  • 1,976
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{...
Matt L.'s user avatar
  • 90.4k
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 ...
Jazzmaniac's user avatar
  • 4,583
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 ...
ThP's user avatar
  • 1,460
3 votes

Ifft through Matrix multiplication

You can alternatively create a DFT matrix in matlab using this code: exp(-1j*2*pi* ((0:N-1)/N).' * (0:N-1)) And the IDFT matrix thus: ...
kippertoffee's user avatar
3 votes
Accepted

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 $^...
Matt L.'s user avatar
  • 90.4k
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 ...
jbondu's user avatar
  • 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 ...
V.V.T's user avatar
  • 1,739
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. ...
Hilmar's user avatar
  • 45.4k
3 votes

What is the complexity of big-$O$ $O(N \times \mathrm{log}_2(N))$ vs real operations

Big O abstracts away knowledge about multiplies and adds and complex math, and focuses on how (whatever operations) scale when you increase N. For the case of FFTs, the core operation that motivate ...
Knut Inge's user avatar
  • 3,434
3 votes
Accepted

Understanding y=Hx+n equation in detail?

The result $E[nn^\dagger] = I_r$ comes from writing out explicitly the diagonal, and the off-diagonal terms in the $r\times r$ matrix $nn^\dagger$, paying special attention to that $\ \dagger$ ...
Dilip Sarwate's user avatar
3 votes
Accepted

To find the unitary matrix which is the null of the results of multiplication with another matrix

Here is an attempt, tell me what I misunderstood. You say that $F\times F^{H}$ (which is of dimension $m\times m$ ) is unitary, which implies it is invertible. Therefore it means that $F$ is of full ...
NokiYola's user avatar
  • 507
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 ...
ssk08's user avatar
  • 708
2 votes
Accepted

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 ...
Royi's user avatar
  • 19.7k
2 votes

Deriving the Matrix Inversion Lemma for RLS Equations vs the Woodbury Derivation

I'm not sure if the OP was looking for a proof or derivation. In my mind a derivation is bit different than what Royi provided. I have looked for but never seen a derivation of the various versions of ...
David's user avatar
  • 2,871
2 votes
Accepted

How can I get the uncertainties for peaks on an image?

One way is to simply model each peak with a Gaussian, with mean $\mu_i$ and variance $\sigma_i$. In fact what you mean by uncertainty corresponds to the variance. You can iteratively fit Gaussians ...
Tolga Birdal's user avatar
  • 5,465
2 votes

The Least Norm Solution of Under Determined Linear System

It means you will have a non-unique solution or redundancy. In your formulation, $X_4$ is completely free and $X_3$ is an offset parameter. You can deflate your matrix and obtain the full rank part to ...
percusse's user avatar
  • 522
2 votes
Accepted

Noise estimation SNR matrix

In your case you probably want to calculate the SNR as mean over standard deviation. ...
Gelliant's user avatar
  • 136
2 votes

How to calculate the Diagonal loading factor evaluate calculate the inversion of a covariance matrix

Short answer: just use $\sigma = 10^{-8}$. Covariance matrices have eigenvalues $\geq 0$ (theoretically), so $Ri + 10^{-8} \, I$ will have eigenvalues $\geq 10^{-8}$, safely non-singular. A longer ...
denis's user avatar
  • 608
2 votes

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

A kernel in the context of digital signal processing refers to the impulse response $h[n]$ of a filter. Particularly for finite length FIR kernels, filtering is carried out by the convolution operator;...
Fat32's user avatar
  • 28.3k
2 votes
Accepted

In OFDM, does $N N^H$ equal $I$?

You're conflating two different definitions of orthogonal. Orthogonal in the OFDM sense means that the subcarriers are mutually orthogonal to one another from the perspective of a correlator-based ...
Jason R's user avatar
  • 24.6k
2 votes
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What is about the circular convolution in OFDM

Assuming you have $N$ symbols to transmit encoded in block $k$, \begin{equation} s(k) = \begin{bmatrix} s_1(k) \\ \vdots \\ s_N(k) \end{bmatrix} \end{equation} Performing $N-$FFT at the ...
Ahmad Bazzi's user avatar
2 votes

Image processing - Why is sum of values of a blurring filter = 1?

Blurring an image means reducing its high frequencies while retaining the rest. Usually it's done with a low-pass filter with a cutoff frequency of $\omega_c$. A standard low-pass filter would have a ...
Fat32's user avatar
  • 28.3k

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