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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 ...
Royi's user avatar
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8 votes
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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 $...
Fat32's user avatar
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7 votes
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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} {...
Royi's user avatar
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7 votes
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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 $ \...
Royi's user avatar
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7 votes
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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 ...
Olli Niemitalo's user avatar
6 votes
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Proof of complex conjugate symmetry property of DFT

Hint: According to Euler's formula we have $$e^{-j2\pi k}=\cos(2\pi k)-j\sin(2\pi k)=\ldots$$
Matt L.'s user avatar
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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 ...
Laurent Duval's user avatar
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}\...
Matt L.'s user avatar
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6 votes
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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 ...
Bob's user avatar
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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: $$e^x = 1 \iff x=0$$ ...
Tendero's user avatar
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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 ...
Paul Irofti's user avatar
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)).
Mark's user avatar
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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} ...
Marcus Müller's user avatar
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 ...
Laurent Duval's user avatar
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 ...
Thomas Arildsen's user avatar
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,...
MimSaad's user avatar
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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
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 ...
Rainer P.'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
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3 votes
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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
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3 votes
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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 $I$, ...
Laurent Duval's user avatar
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 ...
Thomas Arildsen's user avatar
3 votes
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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 ...
LJSilver's user avatar
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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()): ...
Royi's user avatar
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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 ...
Laurent Duval's user avatar
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) = ...
Peter K.'s user avatar
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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 ...
Royi's user avatar
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3 votes
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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 ...
Royi's user avatar
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3 votes
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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)...
Fat32's user avatar
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