9
votes
Under what conditions does DFT(f(x)) = f(DFT(x)) hold?
The fft is an efficient computation of the DFT.
So your question is about the DFT, not the fft.
The DFT of a signal can be ...
- 19.3k
8
votes
Accepted
Group delay of $H(\omega)= 1- re^{j \theta}e^{ - j \omega} $
The book's formula is right.
Let $$H(w) = 1 - r e^{j(\theta - w)} = [1-r \cos(\theta - w)] + j [-r \sin(\theta - w)]$$ Since the group delay $\tau$ is the negative of the derivative of the phase of $...
- 28k
7
votes
Accepted
Accessing Maximum Value from a Singular Value Decomposed Matrix
The SVD Decompose the image into the (One way to look at it) many matrices.
For instance, given an Image $ I $ its SVD is given by:
$$ I = U S {V}^{T} = \sum_{i=1}^{\textrm{rank}(I)} {s}_{i} {u}_{i} {...
- 19.3k
7
votes
Accepted
Sequential Form of the Least Squares Estimator for Linear Least Squares Model
Slope from all samples obtained
To summarize the question's problem, you want to calculate the slope based on all samples obtained thus far, and as new samples are obtained, update the slope without ...
- 13.5k
6
votes
Accepted
Proof of complex conjugate symmetry property of DFT
Hint:
According to Euler's formula we have
$$e^{-j2\pi k}=\cos(2\pi k)-j\sin(2\pi k)=\ldots$$
- 88.9k
6
votes
Proof of complex conjugate symmetry property of DFT
Did you ever wonder about where $\pi $ came from? Watch out...
Let us first draw this weird function complex exponential $e^{-2j\pi t}$ for several discrete values of $t\in[0,10]$ (the little blue ...
- 31.7k
6
votes
Group delay of $H(\omega)= 1- re^{j \theta}e^{ - j \omega} $
This is a slightly tedious but nevertheless straightforward exercise in computing the derivative of a function:
$$\begin{align}\tau(\omega)&=-\frac{d\phi(\omega)}{d\omega}=-\frac{d}{d\omega}\...
- 88.9k
6
votes
Accepted
How to Check Separability of 2D Filter / Signal / Matrix
Nilesh Padhi, Welcome to the DSP Community.
The classic definition of separable means the data (2D) given by $ X \in \mathbb{R}^{m \times n} $ can be written as:
$$ X = \sigma u {v}^{T} $$
Where $ \...
- 19.3k
6
votes
Accepted
Decomposing Sobel Filter
The decomposition of a separable filter is not unique,
since $u v' = (u a) (v a^{-1})'$
The solution you are expecting is found by using a $-\sqrt{2}$ factor, namely ...
- 1,911
5
votes
Proof of complex conjugate symmetry property of DFT
Remember that $e^z$ has a very different meaning than $e^x$ (taking $z\in\mathbb{C}$ and $x\in\mathbb{R}$).
If the exponent was real, then, as you state in your question:
$$e^x = 1 \iff x=0$$
...
- 4,980
5
votes
Alternative to Orthogonal Matching Pursuit (OMP) Algorithm
The main advantage of OMP is that the residual is orthogonal to the current solution.
Let's say you select all $k$ columns from $A$ (also called atoms) at once and let us also presume that $A$ is an ...
- 209
5
votes
Under what conditions does DFT(f(x)) = f(DFT(x)) hold?
One (almost trivial) function is the ifft.
So fft(ifft(x))=ifft(fft(x)).
- 357
4
votes
Accessing Maximum Value from a Singular Value Decomposed Matrix
[EDIT: some code made available] A common framework for (multivariate) image processing is to suppose that its useful features (edges, textures, spectral correlation) contain redundancy, while the ...
- 31.7k
4
votes
The mathematical interpretation of DFT
It's not valid.
If any matrix $W$ is invertible (such as the DFT matrix is), then there's the inverse $W^{-1}$ with
$$\begin{align}
W^{-1}W &=I\\
&\implies\\
X &= Wx \\
&\iff\\
W^{-1} ...
- 29.6k
4
votes
Alternative to Orthogonal Matching Pursuit (OMP) Algorithm
What you propose is actually being used in other algorithms. Your proposal corresponds to the first step of iterative hard thresholding. After the first step, the residual is updated, correlation ...
- 1,322
4
votes
What are the practical constraints on designing Sensing matrix in compressed Sensing?
Checking for RIP of a matrix is an NP-Hard problem which means it is not computationally feasible to accomplish. RIP is used in matrix design mostly in theoretical aspects. Stealing @David 's comments,...
- 1,976
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 ...
- 31.7k
4
votes
Solving Inverse Problem of Multiple Pulses Over Multiple Channels with Convolution Kernel and Cross Channel Mix
Crosstalk between channels is small and well-conditioned, at least in the example you provided. Your matrix A has a condition number of 1.85, which is really good. It means you can invert it and it ...
- 531
3
votes
What are the practical constraints on designing Sensing matrix in compressed Sensing?
When you say "practical constraints on designing the sensing matrix", it depends on whether you mean realising the sensing matrix in actual hardware. In that case, physical constraints probably ...
- 1,322
3
votes
Accepted
How to find out if a transform matrix is separable?
I admit I did not really thought about it before. I hope my notations won't be too sloppy.
I assume that given an operator matrix $A(u,v)$, you can apply this operator as a transform on an image $I$, ...
- 31.7k
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]$$
Multiplying ...
- 453
3
votes
Accepted
Showing a system is always controllable?
There's a very simple way to check controllability, indeed if you define the reachability matrix
$$
R = \begin{pmatrix}B & AB & \dots & A^{n-1}B\end{pmatrix}
$$
then the reachable subspace ...
- 768
3
votes
Generate the Matrix Form of 2D Convolution Kernel
I created a function to create a Matrix for Image Filtering (Similar ideas to MATLAB's imfilter()):
...
- 19.3k
3
votes
Accepted
Sensing matrix for compressed sensing
A sensing matrix maps input vector to measurement vector through linear wighted summation of input. What makes a specific matrix good, is application dependent. Now, both distributions more or less ...
- 1,976
3
votes
Group delay of $H(\omega)= 1- re^{j \theta}e^{ - j \omega} $
Using the logarithmic derivative of the transfer function, as detailed in Julius O. Smith's Numerical Computation of Group Delay, the following computations seem to involve a little less of ...
- 31.7k
3
votes
Check whether a given equation is linear
To write what Laurent says in the comments a bit more fully, what you want to show is that if
$$f_1(x, y) = 56g_1(x,y)+93g_1(x−1,y)+92g_1(x+1, y)−57g_1(x, y−1)+555g_1(x, y+1) $$
and
$$f_2(x, y) = ...
- 25.2k
3
votes
Sequential Form of the Least Squares Estimator for Linear Least Squares Model
There are really great answers.
I will try to give the Sequential Least Squares approach which generalizes to any Linear Model.
Sequential Least Squares Model
We're after solving the Linear Least ...
- 19.3k
3
votes
Accepted
Resources on Solving Convex Optimization Problems in the Compressed Sensing Field
There are few options:
Stephen Boyd, Lieven Vandenberghe - Convex Optimization.
This is the classic in this field. Very well written book.
Also have a look on other papers of Boyd on similar subjects ...
- 19.3k
3
votes
Accepted
Using Linear Algebra for DSP
Yes indeed modern signal processing uses matrix methods of linear algebra (or Linear System Theory more correctly), in addition to the classical calculus and harmonic analysis.
Linear algebra (matrix)...
- 28k
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