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8 votes
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Intuitive explanation of subspace

TL;DR: Subspaces are low-dimensional, linear portions of the entire signal space that are expected to contain (or be close to) a large part of the observable and useful signals or transformations ...
Laurent Duval's user avatar
7 votes

Intuitive explanation of subspace

Subspaces are a Linear Algebra concepts. The best representative example I can think of is the relationship of the XY plane to XYZ space, The former is a subspace of the latter. Any vector in the ...
Cedron Dawg's user avatar
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5 votes
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What are the eigenvalues of the 8 point DFT matrix?

The eigenvalues belong to the same set of quartic roots of unity verifying $\lambda^4=1$, whatever the order of the DFT. Indeed, this is a consequence of Fourier conjugation (adapted from Fourier ...
Laurent Duval's user avatar
4 votes
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MUSIC algorithm terminology

Yeah, notation is not ideal. It is not - he assumes that each of the $M$ antenna elements is connected to its own RF chain, i.e., there are also $M$ receivers available. If you have fewer receivers ...
Florian's user avatar
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4 votes
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SVD vs matched filter

That's not true, it's not better. The thing is: the matched filter just implements the projection in the signal vector space, onto the signal vector itself (or a multiple thereof). (You'll find ...
Marcus Müller's user avatar
4 votes

Intuitive explanation of subspace

A subspace is just a vector space that's included in a bigger vector space. Separating a random signal space into two statistically uncorrelated subspaces, a desired signal space and a noise space, ...
Fat32's user avatar
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4 votes
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Zero forcing vs matched filtering vs LMMSE

Given no formal system model in the question, I will outline in words what each does and the relation between them. Matched Filter: The MF maximizes SNR when the signal is in additive Gaussian noise....
Engineer's user avatar
  • 3,042
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
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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 ...
TimWescott's user avatar
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4 votes
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Why do they say that complex exponentials are eigenfunctions of LTI systems, when there are still transient responses?

Note that you only get a transient if you switch the system (or the input) on at a certain finite point in time. If the input $x(t)=e^{s_0t}$ has existed forever, the output is given by the ...
Matt L.'s user avatar
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4 votes
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Smallest Eigenvalue in the Derivation of the MUSIC Algorithm

This is a simple consequence of previous insights. As was observed that $APA^H$ is positive semi-definite, this means that also $S-\lambda S_0$ has to be the same. Now if $\lambda$ is not the smallest ...
Lutz Lehmann's user avatar
  • 1,215
3 votes

Smallest Eigenvalue in the Derivation of the MUSIC Algorithm

This is something that I studied doing my PhD research on a competing algorithm. The number of eigenvalues corresponds to the number of sensors, the number of "large" eigenvalues ...
JosephDoggie's user avatar
3 votes
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Subspace decomposition

You can rewrite the equation in a block matrix form: \begin{eqnarray}X &=& U_s \Lambda_s V_s^H + U_n \Lambda_n V_n^H \\ &=& \left[ \begin{matrix} U_s & U_n\end{matrix} \right] \...
David's user avatar
  • 2,891
3 votes

Are all exponential functions eigensignals of LTI systems?

Complex exponentials are eigenfunctions of LTI systems because they are eigenfunctions of the convolution operator: $$\begin{align}e^{j\omega_0t}\star h(t)&=\int_{-\infty}^{\infty}h(\tau)e^{j\...
Matt L.'s user avatar
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3 votes
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Question about eigendecomposition, signal subspace and their properties

My first swing at the answer had some very incorrect claims. I do not have access to the article, so I am inferring some things from the portion posted in the question. NOTA BENE: My arguments ...
Joe Mack's user avatar
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3 votes
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Mistake in Python implementation of Pisarenko's harmonic decomposition

By plotting all of the eigenvectors and not including their eigenvalues, you are trying to do MUSIC, but it won't work as I'll explain. MUSIC (and Pisarenko's by extension) solve the equation \begin{...
Baddioes's user avatar
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3 votes
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Computing modern spectral estimation techniques with FFTs

The idea that this can be done is based on the idea that the Fourier transform is an orthobasis expansion. See more details on this answer. The idea that we pull from this though is that the Fourier ...
Baddioes's user avatar
  • 1,784
2 votes

What Is the Difference Between PCA and Karhunen Loeve (KL) Transform?

See: Jan J. Gerbrands, On the relationships between SVD, KLT and PCA, Pattern Recognition,Volume 14, Issues 1–6, 1981, Pages 375-381, ISSN 0031-3203,https://doi.org/10.1016/0031-3203(81)90082-0. (...
GreyHound's user avatar
2 votes
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What Is the Difference Between PCA and Karhunen Loeve (KL) Transform?

For discrete data both are the same - Finding set of orthogonal directions which maximizes the Variance (Energy) of data along them. Sometimes those are called the natural axis of the data (Inferred ...
Royi's user avatar
  • 20.5k
2 votes
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Eigenvalues of FIR Convolution Matrix

For any convolution matrix (even truncated), your diagonal entries would be $h_0$; hence, the characteristic polynomial, setting $b_0 = h_0-\lambda$, \begin{align}|H-\lambda I| = \begin{vmatrix} b_0 &...
Marcus Müller's user avatar
2 votes

Problem with Covariance matrix using diagonal loading involved in calculation of eigenvalues

You can write $$ R=YY^H $$ where $Y$ is a matrix of size $N\times N_f$ and $N$ is the dimension of $y_k$. $Y$ contains all the measured $y_k$ as its columns. Then, the rank of $R$ is upper bounded ...
Maximilian Matthé's user avatar
2 votes

Proving that a product of matrices invertible

First, this question is probably better for MATH.SE, but I'll give it a shot. It's been a long long time since I did this stuff. If $N > M$: 1) $C^*$ has rank M. 2) $R_x^{-1}$ has rank M. (or ...
Cedron Dawg's user avatar
  • 7,600
2 votes

Real time signal processing use cases for eigenvalues of symmetric matrices

OK, you know that complex sinusoids are eigenfunctions (eigenvectors) of linear time-invariant (LTI) systems (which are linear operators). Guess what you see when you look at a spectrum: it's the ...
Marcus Müller's user avatar
2 votes
Accepted

Are complex exponentials the only eigenfunction for arbitrary LTI systems?

This is not a rigorous proof but just to make it plausible that complex exponentials are the only eigenfunctions for general LTI systems. We know that for LTI systems the input-output relation is ...
Matt L.'s user avatar
  • 92.5k
2 votes

Computing modern spectral estimation techniques with FFTs

For $$V(\omega) := v^H e(\omega)=\sum_{n=0}^{N-1}v_n^*e^{jn\omega},$$ if you choose to evaluate the $V(\omega)$ at $\omega=k\frac{2\pi}{N}$ $$V(k)=v^H e(k\frac{2\pi}{N})=\sum_{n=0}^{N-1}v_n^*e^{jnk\...
AlexTP's user avatar
  • 6,725
1 vote

Practical Implications of DFT Eigenvector Formulations

The unitary scaling convention for the DFT is identical in scaling with its inverse and preserves energy across the transform or inverse transform: $$ \begin{align} \mathcal{DFT}\Big\{x[n]\Big\} & ...
robert bristow-johnson's user avatar
1 vote
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The benefit of Eigendecomposition of DCT and DST

Okay, here is a question: We know that if we define the DFT as: $$\begin{align} X[k] &= \mathcal{DFT} \Big\{ x[n] \Big\} \\ &\triangleq \frac{1}{\sqrt{N}} \sum\limits_{n=0}^{N-1} x[n] \,...
robert bristow-johnson's user avatar
1 vote

When is the Fourier transform of a periodic discrete signal $\mathcal{F}x[k]$ the same as $x[k]$ up to a diagonal matrix

tl;dnr version: No nonzero vector can satisfy the requirement stated in the body of this question. The rest of this answer is a long-winded proof of the assertion above. The Discrete Fourier Transform ...
Dilip Sarwate's user avatar
1 vote

When is the Fourier transform of a periodic discrete signal $\mathcal{F}x[k]$ the same as $x[k]$ up to a diagonal matrix

Let $\newcommand{\F}{\mathbf{F}_{{}_N}} \F$ be the (unitary) DFT-Matrix of size $N$. Let $\newcommand{\x}{\mathbf x}\x$ be the vector $\x=(x[0],\ldots x[N-1])$. Your equation says: \begin{align}\...
Marcus Müller's user avatar

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