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173 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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2answers
146 views

Applying a Kalman filter to a WiFi power signal

I have created an app that uses the power of a WiFi signal to determine distance to the WiFi access point. Problem with that power reading is that it is not very stable. I have been looking into ...
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1answer
94 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 ...
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1k 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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144 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
186 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
224 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
135 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
175 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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46 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
647 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
96 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 ...
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98 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 ...
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1answer
67 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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0answers
47 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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101 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
54 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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150 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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61 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
46 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
548 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, $$ a_{k} = \frac{1}{T}\int_{0}^{T} x(t)e^{-jkw_{0}t}dt $$ ...
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2answers
84 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
27 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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160 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 ...
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45 views

Relationship between 2 sinusoidal signal data sets?

I'm trying to relate a near shore tidal signal (point A) to 3 points along a long model boundary (points B C D). I want to possibly have a relationship between B C D with which we can convert A ...
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2answers
190 views

Fourier Series Coefficient of a given signal

$$ {\rm x}\left(t\right) = \sum_{k = -\infty}^{\infty}\left[\delta\left(t-\dfrac{k}{3}\right) + \delta\left(t-\dfrac{2k}{3}\right)\right] $$ I need to find the Fourier series coefficient of x(t). I ...
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1answer
26 views

How do you compute the Fourier Transform of this Unit-Impulse Function?

I have been given this problem from a textbook (not homework, trying to study for an exam. The goal is to find the Fourier transform of this function. $\sum_{k=0}^\infty a^k*\delta(t-kT), |a|<1$ ...
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130 views

Fourier Transform over function depend on time and frequency

In my task I need to perform Inverse Fourier Transform from spectrum that depend on time and frequency arguments simultaneously. E.g., I have a discrete spectrum of some function $S(t, f)$ with $2N$ ...
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1answer
50 views

Omitting part of Frequency domain, Fourier Transform, Image Processing

In my Image and Signal Processing lecture, the Professor said that if every other column of the frequency domain of an image is zeroed out, then the reconstructed image is aliased. (along the x-axis) ...
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1answer
194 views

Why arctan equal to -90 degrees?

Can somebody show me why $$-\arctan\left(\frac{2\pi}{1-\cos(2\pi)}\right)$$ equals to $-90^\circ$ degrees? Thanks.
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1answer
76 views

symmetric window for discrete windowed Fourier transform

The discrete windowed Fourier transform of a signal $f$ of period $N$ is given by $$ Sf[m,l]=\sum_{n=0}^{N-1}f[n]g[n-m]\exp\left(\frac{-i2\pi l n}{N}\right). $$ Why is it that the window $g$ must be ...
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71 views

How to prove Fourier inverse transform worked?

$$g(t)=\int\limits_{-\infty}^{\infty}g(f)e^{i\omega t}df$$ $g(t)$ is a function of time, $g(f)$ is a function of frequency, $e^{i\omega t}$ represent wave, and $\omega = 2\pi f$, the angular ...
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63 views

Recovery of Bandlimited Signals

Let $\Omega > 0$ and denote by $\mathcal{B}_\Omega$ the subspace of $L^2(\Bbb R)$ consisting of signals that are bandlimited to $(-\Omega, \Omega)$. Denote $\mathcal{L}_{\Omega} : L^2(\Bbb R) ...
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168 views

Q: Calculating Fourier Coefficients and Inverse Fourier Transform

Let $\Omega >0$ and $x \in \mathcal{B}_{\Omega/2}$ is continuous. Define $\hat{y}(\omega) = \sum_{n \in \Bbb Z} \hat{x}(\omega - n\Omega)$. If $\hat{y}$ is expressed as \begin{equation} ...
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1answer
33 views

Elimination of complex variable in integral

I have the equation: $$\frac{1}{\tau}\intop_{0}^{\tau}A\sin\left(\Omega t\right)\cdot A\sin\left(\Omega\left(t-\lambda\right)\right)\mathrm{d}t$$ for which the attempted solution is to convert the ...
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1answer
108 views

3-bit sensors — A question about Hamming distance in signals

I came across this question on Willy Wu's riddle site You have two 3-bit sensors, A and B, that measure the same thing, whatever it is -- temperature of the room, radioactivity levels, whatever. ...
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165 views

Differential equations and Kalman filters

I have been told that every differential equation has an associated Kalman filter. How do we get the Kalman filter of a given differential equation. For example let's say we have $$my''+cy'+ky=f(x)$$ ...
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1answer
172 views

How to interpret the results of 2D Fourier Transform on an image?

I have a class where we're studying signals processing (mostly filtering of sounds and images) and while I kind of understand the results of a Fourier Transform for sounds I don't really get the ...
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1answer
98 views

Using the FFT to align two instances of the same signal

I'm working on a program that has a software oscilloscope-like viewer for audio signals. The scope basically takes in blocks of signals at a regular rate and adds them to its existing signal data. ...
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3answers
806 views

Fourier transform of $\left|\frac{\sin x}{x}\right|$

Is there a closed form (possibly, using known special functions) for the Fourier transform of the function $f(x)=\left|\frac{\sin x}{x}\right|$? $\hspace{.7in}$ I tried to find one using ...
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2answers
2k views

How can a unit step function be differentiable??

Recently, I am taking a Signal & System course at my college. In all of the signal & system textbooks I have read, we see that it is written " When we differentiate a Unit Step Function, we ...
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2answers
195 views

Mathematical explanation for image edge detection and denoising

I am trying to understand why the convolution kernel, $$\left[\begin{array}{rrr} -1&-1&-1\\ 2&2&2\\ -1&-1&-1 \end{array}\right]$$ detects the edges in an image. If anyone has a ...
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1answer
205 views

Can the measurement matrix used for compressive sensing be a sparse matrix?

I am interested in analyzing Compressive Mechanism: Utilizing Sparse Representation in Differential Privacy. In my research, the measurement matrix $A\mathbb \in R^{m \times n}$ needs to be sparse. ...
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918 views

DSP, discrete time, how to calculate the frequency

this is from Digital Signal Processing, 4th ed, Sanjit K Mitra, problem 2.39b. The question is: Determine the fundamental period of $x[n] = cos(0.6n\pi + 0.3\pi)$ Since x[n], is in square brackets, ...
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2answers
56 views

Gaussian noise generation with Matlab

I am a Matlab beginner & need to understand what is going on this code. I know it is for Gaussian noise generation, but what do these variables mean mathematically? ...
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1answer
990 views

Regarding $x^2-a^2$ inside the argument of dirac delta

My undergraduate system textbook has this property in the appendix $$\delta(x^2-a^2)=\frac{1}{2|a|}[\delta(x-a)+\delta(x+a)]$$ and I can't seem to derive the result I tried the following: ...
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3k views

Is impulse response always differentiation of unit step response of a system?

I was trying to solve a question in which the transfer function of a system was asked, its unit step response being given: c(t) = 1-10exp(-t) The method that the book followed was to first find out ...
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2answers
201 views

Why are these two delta function equal

In my system textbook it claims that $$\delta(x)=\delta(-x)$$ I understand the proof as follow $$\int_{-\infty}^\infty f(x)\delta(-x)\,dx$$ let $u=-x\,\:,\: du=-dx$ $$\int_{-\infty}^\infty ...
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3answers
2k views

Proofs of dirac delta property

How would I formally prove this property of dirac delta? $$\int \delta(a-x) \delta(x-b) \,dx = \delta(a-b) $$ I attempted to use the definition of a dirac delta $$\int ...
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42 views

understanding bases and frames for Gabor transform

For the 2D discrete Gabor transform, why is it that we cannot use a set of orthonormal basis for its representation, instead we have to use frames for representing it?