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Formula of phase correlation

If I have two 2D signals, and one is the shift of another. I can propose such schema for define offset via continious Fourier Transform: $$f_2(x,y)=f_1(x-x_0,y-y_0)$$ Then $$Ff_2(s_1,s_2)=e^{-2\pi ...
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How would I draw (sketch) the result of DT convolution sum

This picture shows the result of a convolution sum. PICTURE IS HERE! The question is how can I draw $y[n]$. I would appreciate any hints to start
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51 views

Why is the fundamental period $T_0$ of the complex exponential $e^{i\omega_0t}$, $T_0 = \frac{2 \pi}{|\omega_0|}$?

Assuming that $\omega_0 \in \mathbb{R}, t \in \mathbb{R}, T \in \mathbb{R}$. I realize that in order for $e^{i\omega_0t}$ to be perioric, it must be true that $e^{i\omega_0(t + T)} = e^{i\omega_0t}$ ...
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15 views

“Regressing out” nuissance covariates in fMRI data. [on hold]

How can I use the general linear model to remove a noisy signal from another signal of which it is part? In fMRI data, time series representing physiological noise can be included as regressors ...
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1answer
33 views

Summation of $A\cos (\omega n+\phi)$ [on hold]

I'm trying to evaluate the following summation: My original problem is $$\lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^N \left|A \cos(\omega n+\phi)\right|^2$$ Now I'm stuck at calculating the ...
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41 views

When is a mapping the proximity operator of some convex function?

Sorry for cross-posting from MO. It's been a few days and the question hasn't received any attention there. So, is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which ...
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16 views

Implementation of blind deconvolution for signal r(n) = h(n) * s(n) + a(n) in Matlab [closed]

r(n) is the recorded speech h(n) is impulse response of room acoustics s(n) is desired speech signal a(n) is noise from microphone I understand in order to find the desired speech signal, s(n), ...
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22 views

Cross-correlation, Fourier transform and Laplace transform: measure of how much signal are alike?

I'm studying electrical engineering and use correlation, Fourier transform and Laplace transform a lot. I know how and when to use them, however, the interpretation I've seen in the lectures still ...
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1answer
19 views

Simultaneous Diagonalization of A and B via $\Sigma = A^{-1}B$

I am reading the paper "A Generalized Subspace Approach for Enhancing Speech Corrupted by Colored Noise" by Yi Hu and Philipos C. Loizou. In the paper, they claim that given two matrices $R_{n}$ and ...
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13 views

Continuous time fourier transform existance proof explanation

The continuous time fourier transform,$$X(jw) = \int_{-\infty}^{\infty}x(t)e^{-jwt}\mathrm{d}t$$ During a lecture a few months ago in my signals and systems class, the professor showed when the CTFT ...
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1answer
27 views

Fourier Transform of u(-2-t)

I'm trying to find the Fourier Transform of x(t) = u(-2-t) Here's what I've tried: Can anyone tell me what I'm doing wrong? Thanks in advance. EDIT: Forgot to mention that u(t) is the unit step ...
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20 views

Some questions about Hilbert transform

I have some questions about Hilbert transform when I read Real Analysis: In Stein "Real Analysis" p.220, the Hilbert transform is defined by $P=\frac{I+iH}{2}$, where $P$ is an orthogonal projection ...
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89 views

Sampling a sinusoidal signal

Consider the signal $g(t)=\cos(2\pi \lambda t+\phi)$ that is sampled with a frequency $\tau$. Let $g_k$ denote the values of $g$ at the times $t_k=\frac{k}{\tau}$, $k \in \mathbb{N}$. (a) Show that ...
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Estimating pseudo-periodicity of signals

I have pressure data which are measured at a given point in a standing wave. These data(signals) are 'almost' sinusoidal in nature. Each cycle may slightly vary from the original signal i.e the ...
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1answer
27 views

LTI system with sinc input and unit impulse output?

I have a few "conceptual" questions given to me in preparation for a signals and systems exam, and I can't seem to grasp this one. Does there exist a linear time-invariant (LTI) system S such that: ...
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2answers
35 views

Understanding Fourier Transforms

I'm trying to understand Fourier Transforms, so I thought I'd try to explain the following in an english sentence, but I can't. If I bury myself in equations, I can trick myself into understanding ...
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1answer
58 views

Geometric explanation of a methodology in the article about Image Denoising

In article Ghimpeteanu G., et al. - A Decomposition Framework for Image Denoising Algorithms, I found as below: Let $\displaystyle I : \Omega \subset R^2\mapsto R$ be a gray-level image, and $(x, ...
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13 views

Learning point spread function image processing

Given a set of images, that are blurred by Gaussian point spread function, how can I learn the parameters of the PSF, i.e. standard deviation of the Gaussian kernel. One way that I can think of is to ...
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1answer
14 views

Converting from complex to sinusoidal form and vise versa [closed]

I'm having some trouble understanding this type of transformation. The materials provided by my professor doesn't even mention the method that is being used to switch from complex to sinusoidal and ...
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24 views

Computing Hilbert transform and envelope of a function

The following is a function with $\alpha$ being a real constant $$f(t) = \frac{\sin(\alpha t)}{\alpha t}.$$ Determine the analytic signal $f_a (t),$ Hilbert transform $\hat{f}(t),$ and the envelope ...
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26 views

Hilbert transform analytic signal frequency range

For the real signal $f(t),$ show that if it is band-limited to the range $$\nu_0 - \frac{1}{2} \alpha \leq \nu \leq \nu_0 + \frac{1}{2} \alpha$$ (where $\nu_0 >\frac{1}{2} \alpha >0$), then the ...
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0answers
15 views

Significance of the complex conjugation symmetry of the DFT for real-valued input

For real-valued input $\mathbf{x} = (x_0, ..., x_{N-1})$ and its discrete Fourier transform (DFT) $\mathbf{X} = \mathcal{F}(\mathbf{x})$ we have that $$X_{N-k} = X_k^*$$ where * denotes complex ...
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1answer
18 views

How can I analyse signal with discrete wavelet transform?

With CWT it's clear enough. We have function of two variables which are scale and translation. But what about DWT? Here is Matlab code: ...
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2answers
27 views

Optimal estimation of the fusion of two measurements

Suppose I have a sensor measuring a quantity $\text R$. For example the sensor could be a radar estimating the range of a target. We can write: $$R(t)=r(t)+\nu_0(t)$$ where $r(t)$ is the real range ...
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7 views

How to solve a difference equation with an input?

How do you solve the difference equation (initial conditions are given) $$y(k)+ay(k-1)+by(k-2)=cx(k-1)+dx(k-2)$$ where the input $x(k)=\theta(k)$ (the unit step function). I know that the general ...
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25 views

Solving traveling wave usin the shooting method

The spatially-dependent Hodgkin-Huxley equation for a cylindrical dendrite or unmyelinated axon: where $\frac{a}{2\rho}\frac{\partial^2V}{\partial x^2}$ is a diffusion term $a$ is the fiber radius, ...
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2answers
21 views

Fourier series: can a function be odd and have a dc component?

Long story short: fourier series is taken in two subjects (for now). One doc says that the dc component is 0 if the function is odd. The other says that odd and even has no effect on the dc ...
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19 views

Using Linear Kalman Filters with a Nonlinear System?

Can you answer these questions I have about using linear Kalman filters and extended Kalman filters with a nonlinear system? 1. Does using a linear Kalman filter mean that I must have a ...
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0answers
19 views

What is Z-tranform of signum function?

If Z-transform of x(k) is X(z), then what will be the Z-transform of sign(x(k))? Furthermore, what will be the Z transform of sign(x(k-1))?
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1answer
20 views

Autocorrelation of heaviside functions

I'm trying to find the expression that describes the auto-correlation $r_{xx}(\tau)$ of two heaviside functions $u(t)$. I was told that the result must be $1/2$, which makes total sense, as the power ...
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1answer
26 views

Instant frequency of sine sweep function?

Firstly, I'm not a mathematician, I'm an engineer, so you can freely make fun of the question. I have the following counter-intuitive behaviour in a sweep function. I have a sweep sine function ...
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25 views

Power of a signature (sum of squares divided by number of elements)

I need to find some literature to study the theory of an exercise I am working on (it is from a course in digital image processing and pattern recognition). I have an $n\times n$ matrix, I have to ...
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34 views

Estimating a sparse vector: Mean squared error when support known

I was reading this paper ("How well can we estimate a sparse vector?" by Candès and Davenport, link: http://arxiv.org/pdf/1104.5246v5.pdf). They consider the problem of estimating a $k$-sparse vector ...
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1answer
30 views

CTFS: What happened in this integral?

Specifically, what happened in the last line to obtain the answer? It seems like they ignored the exponential term $e^{-jw_{o}kt}$?
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1answer
25 views

Help in understanding a coding technique based on inverse mapping of a dynamical system

Based on paper titled : Simultaneous Arithmetic Coding and Encryption Using Chaotic Maps by Kwok-Wo Wong et. al The Authors use a non-linear dynamical system for generating keys to be used in ...
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2answers
42 views

$\cos(2\pi f nT +2\pi N_n) =\ cos(2\pi f nT)$ and more, why?

I am studying signals and systems and this came up? Could someone explain why is $\cos(2\pi f nT +2\pi N_n)$ equal to $\ cos(2\pi f nT)$? The book says: "because $N_n$ is an integer" I am ...
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2answers
38 views

Prove Differentiator is Linear and Time-Invariant

The differentiator gives an output equal to the derivative of its input. Show that the differentiator is a linear time invariant system. Consider the input $f(t)=\sin(t^2).$ Attempt For ...
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1answer
19 views

Quick Fourier Series help?

I was given a graph (shown above) and was asked to represent this as a Fourier Series. I was able to solve $a_0$ with no problem. However, when I was integrating for $a_n$ and $b_n$, I was having a ...
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1answer
24 views

Signal whose Laplace transform contains derived Dirac-deltas: How do I find the inverse transform?

I must reconstruct the input signal to a system, knowing the output signal and the system transfer function. At the end, I found that the Laplace-Transform of the input signal is something like: $$ ...
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23 views

Find the inverse z transform of $H(z)=\frac{0.2685}{1-0.146z^{-1}}-\frac{0.2685}{1-6.8493z^{-1}}$

I need to find the inverse z transform of: $H(z)=\frac{0.2685}{1-0.146z^{-1}}-\frac{0.2685}{1-6.8493z^{-1}}$ My initial attempt gave: $h(n)=0.2685(0.146)^nu(n)+0.2685(6.8493)^nu(-n-1)$ by using the ...
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0answers
9 views

Calculate system output of 2nd Order discrete LTI with cosine input

Consider this time-discrete LTI: $$ H(z) = \frac{z^{-1} - 0.25z^{-2}}{1 - 0.5z^{-1} +0.4z^{-2}}$$ $$ = -0.625 + \frac{0.7907 e^{j1.1645}}{1-0.6324e^{-j1.1645}z^{-1}} + \frac{0.7907 ...
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2answers
46 views

How to show that the signal $x_n = A\cos(\omega n)$ can be fully predicted by a system with two weights $w_1,w_2$

I am trying to solve the following exercise: Show that the signal $x_n = A\cos(\omega n)$ can be fully predicted by a system with two weights $w_1,w_2$ (i.e. $x_n = w_1 x_{n-1} + w_2 x_{n-2}$). ...
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0answers
28 views

Polar form of the Fourier transform of $\sin(t)$

I'm studying signal processing, and I came across the Fourier transform of sin(t). It ends up being a purely imaginary (dirac delta) impulse pair. But when considering the frequency domain ...
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1answer
21 views

What is the Permutation Matrix in FFT DFT Factorization?

Given: $$F_N = \frac{1}{\sqrt{N}} \begin{bmatrix} 1&1&1&1&\cdots &1 \\ 1&\omega&\omega^2&\omega^3&\cdots&\omega^{N-1} \\ ...
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1answer
27 views

Why is the following system is not time invariant?

The system is as follows: $y[n] = x[2n]$ Shouldnt the system be time invariant because $y[n-n_0] = x[2n-2n_0]$ and $T(x[n-n_0]) = x[2n-2n_0]$ These are both equal, therefore why is the system not ...
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2answers
31 views

Find an function that oscillates between a given upper and lower envelope

Suppose I'm given two real, continuous functions $f(x)$ and $g(x)$ such that $f(x)\ge g(x)$ for all real $x$. I'd like to determine an oscillating function $h(x)$ that has $f(x)$ as its upper-envelope ...
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39 views

Prove that taking the inverse Fourier transform of frequency returns time.

If we evaluate the inverse Fourier transform of X(w) how do we know we get x(t) back? Link to X(w) and x(t) equations I know that integrating in the frequency domain results in getting information ...
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25 views

Creating a balanced bidirectional pulse pair, also called the Lilly Wave and change it's sample amounts in octave / matlab

I was told this should be in the mathematical stackexchange Original question link below ...
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1answer
43 views

Shape of Impulse Responses of $ARMA(p,q)$ Processes

Suppose that $x_t$ is an $ARMA(p,q)$ stochastic process, $$ \phi(L)x_t = \theta(L)\varepsilon_t ,$$ where $\varepsilon_t \sim N(0,\sigma^2)$, and $\phi(L)$ and $\theta(L)$ are lag-polynomials given ...
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19 views

I downloaded the SPARCO package (MATLAB)and when setting it up a problem has occured

I downloaded SPARCO1.2 (MATLAB) from http://www.cs.ubc.ca/labs/scl/sparco/ and when I use the command 'sparcoSetup' it says everything is successful. But then when I use the command 'checkProblems' ...