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 ...
- 4,195
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 ...
- 185
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 ...
- 6,178
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 ...
- 19k
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 ...
- 11.7k
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 ...
- 31.6k
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 ...
- 31.6k
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 ...
- 1,976
3
votes
Accepted
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{...
- 87.4k
3
votes
Accepted
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 ...
- 4,557
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 ...
- 1,450
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 ...
- 453
3
votes
Accepted
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 ...
- 25.1k
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:
...
- 423
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 $^...
- 87.4k
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 ...
- 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 ...
- 1,589
3
votes
Accepted
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.
...
- 41.6k
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 ...
- 3,280
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 $\ \dagger$ ...
- 20.1k
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 ...
- 19k
2
votes
Accepted
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 ...
- 87.4k
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 ...
- 708
2
votes
Maximising each element in a matlab array
Try using bsxfun if your version has it.
A = bsxfun(@max, B, C.')
As per their documentation, ...
- 121
2
votes
Maximising each element in a matlab array
If the matrices are not too big, repmat could work:
...
- 31.6k
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 ...
- 2,791
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 ...
- 5,445
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 ...
- 512
2
votes
Accepted
Noise estimation SNR matrix
In your case you probably want to calculate the SNR as mean over standard deviation.
...
- 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 ...
- 598
Only top scored, non community-wiki answers of a minimum length are eligible