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1answer
47 views

Can any piecewise function be represented as a traditional equation?

In "Fundamentals of Electrical Engineering" we learned about piecewise functions for the "unit-step" and "ramp" which are represented by $f(x)= \begin{cases}0, & \text{if }x< 0 \\ 1, & ...
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28 views

Asymptotic result on quadratic variation of a semi-martingale linear functional estimator

In the same context of this previous question. Consider $$ \mathcal E^{(n)}_t := \sqrt{n}(\widehat\Lambda_n(\phi)_t - \Lambda(\phi)_t )$$ I desire to prove that $$ \left \langle \mathcal ...
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1answer
30 views

Find if the system $(x(t-1))^2 + x(t) +(x(t+1))^2 = y(t)$ is invertible

If there wasn't the $x(t)$ term, I could use $x(t) = x$ and $x(t) = -x$ to disprove invertibility, but I can't think of two functions that give the same $y(t)$ in this case. When I tried proving ...
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84 views

Symbolic math engines barf on this ostensibly tractable integral.

$$\frac14 \int_{-M\pi}^{N\pi - s} \cos(tu/M) \cos((t+s)u/M)(1-\cos(t/M))(1-\cos((t+s)/N))\space \mathrm d t$$ with integer $u$. Alpha runs out of time. Maxima gives a tremendous result that can ...
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1answer
34 views

Arcsine for a value

I want to exactly determine the arcsine (sine inverse) for a value. Say I take $\sin$ for $60000$, which is approximately $-0.866$. I want to get back $60000$ from this. Taking a sine inverse will not ...
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2answers
35 views

Very short theory question signals?

My teacher asked us this question yesterday in the lecture but it didn't make any sense to me. He asked: What do the coefficients of the exponential Fourier series represent? Also, what's the ...
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2answers
44 views

What calculation remains constant for discretely sampled points of a sinusoid on a window of 1/4th its period?

I have a univariate time series that consists of discretely sampled (equally spaced) points of a sinusoid. If you have a window that slides over these points (like this animation) with a length of ...
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1answer
30 views

2-D Fourier Transform of complex exponential with 2-D quadratic phase

I've been looking around to see if there is either an exact transform pair or an approximation to either of the following but have not been able to find anything: $$ \mathcal{F}_{xy}\left( ...
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42 views

Detecting the respiratory rate of a breating lung.

I am currently working with some data-sets that represents the movements of a beating heart and breathing lungs. The data-sets are represented as a collection of floats that range from 47 to 51. We ...
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1answer
15 views

Prove that the function in the domain of Z is a high pass filter

I need prove that the function \begin{equation} L(z) = 1-z^{-1} \end{equation} is a high pass filter, but I have not much understanding of the $z$ transform and what really the $z$ domain is. So how ...
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37 views

How does SVD work?

Trying to find information, and, no-one seems to know the answers. I have a time-series, represented by $T = [0, 1, 1, 0, \ldots, n]$ the time series is then transformed into the Spectral results: ...
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2answers
52 views

Reducing a two term signal to one term

I am trying to solve a phasor addition problem, reducing from its original form to $$X(t) = A \cos(\omega_0 * t + \phi)$$ The original equation is : $$X(t) = 2 \sin(\omega_0 * t + 45) + ...
2
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1answer
82 views

Need to learn wavelet, suggest steps and resources

I am looking for a good introduction to wavelets and wavelet transforms. that covers the following: Basics Vector Spaces – Properties– Dot Product – Basis – Dimension, Orthogonality and ...
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0answers
36 views

how to calculate inverse Fourier transform of given absolute function?

how to calculate inverse Fourier transform of abs(omega)? i have tried by it's standard formula and its properties but didn't get the answer please help... I used duality property also but its not ...
2
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1answer
94 views

Kalman Filter Process Noise Covariance

I want to model the movement of a car on a straight 300m road in order to apply Kalman filter on some noisy discrete data and get an estimate of the position of the car. In a Kalman filter the matrix ...
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0answers
51 views

Fourier Transform of a Gaussian Signal?

As far as I know this is the formula for FT : On this question on part b) I fint on the answer the part with e^-jwt is changed with cos(wt) I have no idea how cos(wt) came in ... would you please ...
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1answer
41 views

Frequency response of Continous-time system

Not sure where to start on this one: $$H(s)={(s-j\omega_0)(s+j\omega_0)\over(s+\omega_0\cos\theta+j\omega_0\sin\theta)\left(s+\omega_0\cos\theta-j\omega_0\sin\theta\right)}$$ Sketch the frequency ...
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0answers
66 views

Artifacts and low frequencies FFT.

I am working on analyzing a time signal and want to preform a FFT. However I run in to some artifacts at low frequencies. I have managed to reproduce the behavior in a test signal. Given by $S(t) = ...
0
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1answer
44 views

what is the period of this sinusoidal function funtion

How can we find the fundamental period of this sinusoidal discrete function $x(n) = 10\cos{(\frac{4n\pi}{31}+\frac{\pi}{5}})$ I tried using the formulae $\frac{2π}{\omega}$ and got the answer 31/2, ...
2
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2answers
76 views

Z-Transform, Transfer Function, Poles & Zeros

I've been working on a question that I'm now stuck on. I need to: Determine the transfer function and poles-zeros of: $y[n]=0.5y[n-1]-0.25y[n-2]+x[n]$ So far I've carried out a z-transform in ...
2
votes
1answer
97 views

Bandpass filter with Fourier and inverse.

My understanding of signals is limited. I did a signal processing subject in engineering, but I can't say I got much from it. For me, the subject wasn't taught with enough 'real world' explanation - ...
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1answer
35 views

Z-transform: Convolution <=> Multiplication giving strange results

I try two slightly different routes to an answer, and get two different answers. (This is from a past exam paper.) Find h[n] if $ H(z) = \frac{1}{(1-az^{-1})^2} $ First try: $ H(z) = H_1(z)\cdot ...
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0answers
29 views

Error bounds in representing a vector using a truncated Moore-Penrose biorthogonal basis

I was reading and trying to reproduce the results in the arXiv preprint of Periodic Gabor Functions with Biorthogonal Exchange: A Highly Accurate and Efficient Method for Signal Compression by Asaf ...
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1answer
54 views

Given a Poisson-noisy signal, what is the noise distribution of its Fourier transform?

Disclaimer: I'm not a mathematician, but here's my attempt at a mathy version of my question Start with a noiseless, discretely sampled expected signal $I(x_n)$. Construct a Poisson-noisy measurement ...
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1answer
45 views

Fourier analysis of an exponential function review

I am working through and reviewing some of the examples presented on Fourier analysis from a Modern Digital and Analog Communication Systems book. In one of the examples, the author goes through the ...
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1answer
81 views

calculating an incoherence property

With respect to Matrix Completion and Compressive Sampling (CS) I'm trying to understand how to calculate an incoherence property μ between two bases Φ and Ψ. Getting this incoherence is important ...
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42 views

Compressive Sensing - Incoherence Property

Compressive Sensing is built on 2 properties: 1) the sparsity of the representation basis relative to the sampling basis and 2) the incoherence between the singular vectors from each of the 2 bases in ...
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0answers
46 views

Gradient of a complex quadratic form

I have to compute the gradient of the following expression: $$ \nabla_\overline{h} \left( h^H R h - h^H s\right) $$ where the overline means "conjugate of" and $^H$ means conjugate transpose (or ...
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1answer
51 views

fitting the content of an elliptical region in an image into the equivalent stretched circle

Assuming that I know the ellipse parameters (the major/minor axis, theta and the centre), and I could get the equivalent circle of this ellipse (I mean the equivalent x and y positions of the pixels ...
0
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1answer
150 views

Prove that Unit Impulse Function Integral is equal to one? [duplicate]

Unit impulse function is one of the special functions which is widely used in the field of signal processing. It has nice properties that helps in some situations specially its sifting property. But ...
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49 views

Continous wavelet transform and shannon Entropy.

Note: I have asked the same question on signal processing forum,but didn't get any answer. so it might be more like a math or physics question. Hope you don't consider it as cross-post. I am trying to ...
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1answer
53 views

circle affine transformation

I am trying to convert a region of pixels surrounded by an ellipse into an equivalent stretched circle.This basically means affine transformation. Assume I want to extract the region in light blue ...
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1answer
84 views

Convolution of indicator function with itself

A paragraph in Mallat's "A wavelet tour of signal processing" says: Spline Dyadic Wavelets A box spline of degree $m$ is a translation of $m+1$ convolutions of $\mathbf{1}_{[0,1]}$ with itself. ...
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1answer
68 views

Finding period of a periodic function

I am having some trouble finding the period of this function: $$W(\omega) = \frac{\sin[(2N +1)\omega \Delta t / 2]}{(2N + 1)\sin[\omega \Delta t /2]}$$ Here $N$ is an integer, $\omega$ is angular ...
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2answers
96 views

Why is the DTFT (Discrete Time Fourier Transform) unique to each input?

As the title implies. I know the DFT of a signal is unique due to the matrix, but can anyone give a solid explanation as to why the DTFT is unique for each signal input? Thanks for your time!
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0answers
37 views

Find the maximum of an integral function with respect to another function

I'm facing this statistical data analysis problem, where I have to maximize a certain statistic in order to find the optimal filtering function. I'm a little bit out of practice with the mathematics ...
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1answer
200 views

Calculate phase and amplitude of a sampled sine wave

I have an electronics project where I sample two sine waves. I would like to know what the amplitude (peak) and difference in phase is. Actually I just need to know the average product of the two ...
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1answer
75 views

How to derive this frequency response?

Given this difference equation $y(k)$ ... $$y(k) = \frac{1}{K^2} \sum_{m = k-K+1}^k \; \sum_{n = m-K+1}^m x(n) - \frac{1}{L^2} \sum_{m = k-L+1}^k \; \sum_{n = m-L+1}^m x(n)$$ ... how does one derive ...
2
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1answer
58 views

Paley Wiener Theorem on sinc function

Use the Paley-Wiener theorem to argue that, although ${\rm sinc}\left(t\right)$ is bandlimited, ${\rm sinc}\left(t^{3}\right)$ is not. Explain how the above result allows reconstruction of some ...
2
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1answer
45 views

Complex Integration of DTFT

Question A discrete-time signal $u \in \mathcal{l}^2(\mathcal{Z})$ has DTFT \begin{equation} \hat{u}(\omega) = \frac{5+3\cos(\omega)}{17+8\cos(\omega)} \end{equation} Use complex integration to find ...
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31 views

Integration containing a complex number

Folks, Can I treat the complex number in the following integral: $$\frac1{2\pi}\int\frac1{(1+jw)^2}dw$$ as a constant and move it outside of the integral, like this: ...
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0answers
47 views

problem integrating a dirac comb

Let: $$h(t)=\frac{\sin(\pi t(2N+1))}{\sin(\pi t)}$$ $$I=\int_\frac{-1}{2}^\frac{1}{2} h(t) dt$$ when $N\rightarrow\infty$ , obviously (with a change of variable $v=\pi t(2N+1)$ ): ...
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1answer
38 views

Understanding a step in the proof of the Inverse Fourier transform theorem

I'm trying to understand the proof of the Inverse Fourier Transform theorem in Stéphane Mallat's "A wavelet tour of signal processing". Near the end of the proof, we have: $ \lim_{\epsilon ...
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53 views

Reconstructing sine wave from samples

Suppose there is a sine wave signal, like the following: $$V(t) = M * sin(\phi_0 + \omega*\Delta t)$$ I can have it sampled and obtain $V_1$, $V_2$ and $V_3$ at $t_1$, $t_2$ and $t_3$ such that ...
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1answer
40 views

Multiple Characteristic Function and the Dirac Comb

Given the impulse train(Dirac comb): $$\Delta_T(t)=\sum_{k\in\mathbb{Z}}\delta(t-kT)$$ where $T$ is the signal period, $\delta(t)$ is the Dirac delta function and $\mathbb{Z}$ is the set of integers ...
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1answer
36 views

CT Fourier Transform

I need to find the Fourier Transform of the given signal below; $$ x(t) = \frac{\sin(\pi t)}{\pi t} \frac{\sin(2\pi t)}{\pi t}.$$ I know that if $ x(t) = \frac{\sin(Wt)}{\pi t} $ , then $ X(w) = ...
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3answers
193 views

Fourier Series coefficients/Trigonometric functions

I need some help about finding the Fourier Series coefficient of the given signal; $$ x(t) = \sin(10\pi t + \frac {\pi}{6} ) $$ I know that, $$ a_{k} = \frac{1}{T}\int_{0}^{T} x(t)e^{-jkw_{0}t}dt $$ ...
3
votes
2answers
38 views

Generating points from a standard Gaussian

I'm new to Gaussian distributions and I'm trying to generate say, $ N$ points from a $ M$ dimensional standard gaussian. What does this mean? How would I do this in matlab?
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2answers
22 views

Data preprocessing

How would you preprocess 2 dimensional data to have 0 mean? Say you have a matrix $M $ that is $p \times q $. Would you calculate the mean of each row, get a vector of length $q $ and subtract each ...
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0answers
75 views

Whitening matrix for Fast ICA

I have a matrix $X $ with dimension say $ m \times n $ with $ m> n $. I am trying to whiten this matrix in matlab by first taking the $C= \operatorname{covariance}(X)$ followed by eigenvalue ...