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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. I plotted 1000 random sorted integers between 0 and 1000 (in blue) and ...
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Geometric explanation of an article methodology

In an article i found as below- Let $I : \Omega \subset R^2\mapsto R$ be a gray-level image, and $(x, y)$ be the standard coordinate system of $R^2$. We denote by $Ix \hspace{.2cm}resp. Iy$ the ...
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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
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Converting from complex to sinusoidal form and vise versa [on hold]

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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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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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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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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17 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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25 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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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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24 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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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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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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18 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
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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
23 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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22 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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31 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
28 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
23 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
37 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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36 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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What is the absolute phase and the relative phase, of a signal? [on hold]

I need to know what is the absolute phase and the relative phase, of a signal? and why these phases are important in the signal processing?
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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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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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25 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
18 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
26 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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21 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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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' ...
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Signal operation, shifting and scaling

so I have a question regarding this continuous time signal: $$y(t) = \int_{-\infty}^t x(2\tau) \, d\tau$$ Now the question was to find if this function was causal, so i proceeded to check the impulse ...
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1answer
21 views

Is this system invertible?

$y(t) = \int\limits_{-\infty}^{\infty} \frac {x(t)^2}{x(t-1)} dt\\$ I was trying to prove or disprove the invertibility of this function. The only thing I could think of was differentiating it. But ...
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How to define this test signal?

I generate any plots in any parts of the interval $[0,100]$ (let it be set $A=V^{n}$ where $n \in \mathbf Z$), I get a test signal. I would like to understand how you can write this mathematically for ...
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37 views

Equation involving fractions of integrals

In the context of a signal processing problem, let's say we have the following angles that are functions of time $\tilde{\tau}\in[0,1]$ $ \phi_i(\tilde{\tau}) = \left\{\begin{array}{ll} ...
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Pearson correlation of neural responses with it's linear estimation

I am trying to anderstand the following fact: Suppose I have a linear estimation of a stimulus: $ \hat{s} = \mathbf{w}^T(\mathbf{r} - \mathbf{f}(s_0)) + s_0$ where $\mathbf{w}$ is a vector of ...
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Fourier Transforms and Sums

Suppose I have the following sum: $$ \sum_{x = -\infty}^{\infty} \int_{-\pi}^{\pi} f(j) \; e^{i j x} dj $$ Assuming that everything is sufficiently smooth and convergent, then exchanging the sum with ...
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39 views

Calculating a function from its auto-correlation

How do I calculate a function if I know its auto-correlation? To be more specific, I have a function of one variable, let's call it $g(x)$, and I know it's the cross-correlation of a function $f(x)$ ...
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44 views

How can we use theory from $L^2(\mathbb{R})$ on a sequence of numbers (discrete signal)

In have problems understanding connection between theory that is done in $L^2(\mathbb{R})$ and its application on discrete signal. look at this paper ...
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1answer
62 views

Fourier transform of a 2D image, and noise cancelation

I'm a statistics grad student, and I just started getting into Digital-Image-Processing (an analogy for processing super-large contingency tables). In the book "Digital Image Processing" by Gonzalez ...
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1answer
28 views

Infinite sum of discrete unit-step signals

Trying to sketch the following signal: $$\sum_{k=-\infty}^\infty (u[k]-u[k-3])(u[n-k]-u[n-k-3])$$ Where $u[n]$ is the unit step signal (the Heaviside function, $1$ when $n\ge 0$ and $0$ otherwise). ...
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22 views

Discrete fourier transform two point signal

I would greatly appreciate some help in explaining how I would go about finding the DFT of the discrete signal x = [1,0] on the form X = [?,?]
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Multi dimensional multiresolution analysis, designing biorthogonal wavelets.

With inspiration from this question, I'm wondering if biorthogonal bases for arbitrary dimensions are possible to construct with the same mechanism. I am thinking a subsampling of a factor of $N$ in ...
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wavlete transform vs (scaled) Gabor transform

I've read about the scaled Gabor transform $$(G_\Psi f)(b,a)(\omega) = \frac{1}{\sqrt{a}} \int_\mathbb{R} f(x)\Psi(\frac{x-b}{a})e^{-i\omega x}dx$$ and the wavlete transform $$(L_\Psi f)(b,a) = ...
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Is $x(t)=\sin(5t/2)+\cos(2t/8)+\sin(3t/6)$ periodic or aperiodic? Find the fundamental period and frequency of the signal.

Is $x(t)=\sin(5t/2)+\cos(2t/8)+\sin(3t/6)$ periodic or aperiodic? $w_1=(5/2)=2.5 \rightarrow T_1 = 2\pi/w_1 = 2\pi/2.5 =2.513$ $w_2=(1/4)=0.25 \rightarrow T_2 = 2\pi/w_2 = 2\pi/0.25=25.13$ ...