8
votes
Accepted
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 ...
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 ...
5
votes
Accepted
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 ...
4
votes
Accepted
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 ...
4
votes
Accepted
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 ...
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, ...
4
votes
Accepted
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....
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 ...
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 ...
4
votes
Accepted
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 ...
4
votes
Accepted
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 ...
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 ...
3
votes
Accepted
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]
\...
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:
$$\begin{align}e^{j\omega_0t}\star h(t)&=\int_{-\infty}^{\infty}h(\tau)e^{j\...
3
votes
Accepted
Question about eigendecomposition, signal subspace and their properties
My first swing at the answer had some very incorrect claims.
I do not have access to the article, so I am inferring some things from the portion posted in the question.
NOTA BENE: My arguments ...
3
votes
Accepted
Mistake in Python implementation of Pisarenko's harmonic decomposition
By plotting all of the eigenvectors and not including their eigenvalues, you are trying to do MUSIC, but it won't work as I'll explain.
MUSIC (and Pisarenko's by extension) solve the equation
\begin{...
3
votes
Accepted
Computing modern spectral estimation techniques with FFTs
The idea that this can be done is based on the idea that the Fourier transform is an orthobasis expansion. See more details on this answer. The idea that we pull from this though is that the Fourier ...
2
votes
What Is the Difference Between PCA and Karhunen Loeve (KL) Transform?
See:
Jan J. Gerbrands,
On the relationships between SVD, KLT and PCA,
Pattern Recognition,Volume 14, Issues 1–6, 1981, Pages 375-381,
ISSN 0031-3203,https://doi.org/10.1016/0031-3203(81)90082-0.
(...
2
votes
Accepted
What Is the Difference Between PCA and Karhunen Loeve (KL) Transform?
For discrete data both are the same - Finding set of orthogonal directions which maximizes the Variance (Energy) of data along them. Sometimes those are called the natural axis of the data (Inferred ...
2
votes
Accepted
Eigenvalues of FIR Convolution Matrix
For any convolution matrix (even truncated), your diagonal entries would be $h_0$; hence, the characteristic polynomial, setting $b_0 = h_0-\lambda$,
\begin{align}|H-\lambda I| = \begin{vmatrix} b_0 &...
2
votes
Problem with Covariance matrix using diagonal loading involved in calculation of eigenvalues
You can write
$$ R=YY^H $$
where $Y$ is a matrix of size $N\times N_f$ and $N$ is the dimension of $y_k$. $Y$ contains all the measured $y_k$ as its columns.
Then, the rank of $R$ is upper bounded ...
2
votes
Proving that a product of matrices invertible
First, this question is probably better for MATH.SE, but I'll give it a shot.
It's been a long long time since I did this stuff.
If $N > M$:
1) $C^*$ has rank M.
2) $R_x^{-1}$ has rank M. (or ...
2
votes
Real time signal processing use cases for eigenvalues of symmetric matrices
OK, you know that complex sinusoids are eigenfunctions (eigenvectors) of linear time-invariant (LTI) systems (which are linear operators). Guess what you see when you look at a spectrum: it's the ...
2
votes
Accepted
Are complex exponentials the only eigenfunction for arbitrary LTI systems?
This is not a rigorous proof but just to make it plausible that complex exponentials are the only eigenfunctions for general LTI systems.
We know that for LTI systems the input-output relation is ...
2
votes
Computing modern spectral estimation techniques with FFTs
For $$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\...
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:
$$ \begin{align}
\mathcal{DFT}\Big\{x[n]\Big\} & ...
1
vote
Accepted
The benefit of Eigendecomposition of DCT and DST
Okay, here is a question:
We know that if we define the DFT as:
$$\begin{align}
X[k] &= \mathcal{DFT} \Big\{ x[n] \Big\} \\
&\triangleq \frac{1}{\sqrt{N}} \sum\limits_{n=0}^{N-1} x[n] \,...
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 ...
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}\...
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