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8
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154 views

Sampling theorem.

Let us consider \begin{equation} \hat{f}(x)=\sum_{n\in \mathbb Z}\left\langle\hat{f},e^{i n x}\right\rangle_{L^2[-\pi,\pi]} e^{i n x} \ \ \ \ \ \ \ \ (1) \end{equation} where $\langle g, ...
4
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106 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 ...
4
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90 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 ...
4
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124 views

Need a fast algorithm of adaptive convolution

Good morrow, gentlemen! I have to apply some kind of adaptive filter to my function $f(x).$ I present each point of my signal as a Gaussian, whose bandwidth depends on its location (not the point of ...
3
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31 views

Looking for Math books recommendations to study Electronics

My background is the very basics, and I mean, literally, I can add, sub,mul,div and a little of algebra (near, nothing) and that's it. As you can see I need the best Total Beginner Book(s) that can ...
3
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0answers
44 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 ...
3
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0answers
46 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 ...
3
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65 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) ...
3
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901 views

Wavelets: Cone Of Influence

While reading this paper I came across the term Cone of Influence which is described as ...
3
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0answers
115 views

Exponentials of chi-squared random variables (and their sums)

Let $X_1,X_2,\ldots,X_n$ be a sequence of i.i.d. chi-squared random variables with $t$ degrees of freedom, i.e. $X_i\sim\chi^2_t$. I am wondering what is known about the distribution of ...
3
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243 views

Relationship between DFT and FFT/DTFT

Question 1: Assume we already know $x(t):[0,2\pi]\to\mathbb{R}$'s Fourier series $x[n]_{n=-\infty}^\infty$. Perform DFT on $x[n]_{n=0}^N$. How is the result connected to $x(t)$? WHY? Question 2: ...
3
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119 views

Analysing an optics model in discrete and continuous forms

A discrete one-dimensional model of optical imaging looks like this: $$I(r) = \sum_i e_i P(r - r_i)$$ Here, the $e_i$ are point light sources at locations $r_i$ in the object and $P$ is a point ...
2
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22 views

Sampling a Chebyshev polynomial with the discrete cosine transform

I have a Chebyshev polynomial $f$ of degree $n$ in point-value form \begin{align} f&=:S = \left( \left( x_i, y_i \right) \right)_{i=0}^n, \tag{1} \\ x_i &= \cos\left( \frac{i \pi}{n} \right), ...
2
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29 views

Generating cross-correlated stochastic processes

I am looking for a robust way to represent and generate multiple stochastic processes that contain time and cross-correlations i.e. I am looking at stochastic processes $X_t^{1}$, $X_t^{2}$, $\ldots$, ...
2
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0answers
15 views

Rate of convergence of a Weyl-Heisenberg (Gabor) frame expansion

If $\{g_{m,n}\}$ is a Gabor frame for $L^2(R)$, with window function $g$, and $f \in L^2(R)$, is there a bound on the approximation error of $f$ using a finite subset of the frame? That is, is ...
2
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29 views

Acceleration/Position signal correction

I have a set of data for a car position, velocity and acceleration. % my data time car_x car_velocity car_acc The problem is that these arrays have error and I ...
2
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14 views

Fourier coefficients for pattern analysis

There are many areas like, gait analysis, where we recognize persons by analyzing their silhouettes taken while they are at different stages of their walking where analysis also carried on in ...
2
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96 views

Properties of eigenvectors of a sample covariance matrix?

My apology if the question is not appropriate. For me Eigenvectors are quite a mystery. Does it have any property that we can relate to the matrix it came from? By property I mean something like the ...
2
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32 views

Regarding the unilateral Laplace transform of LTI systems

Consider an LTI system described by the following differential equation, $$ \sum_{k=0}^{N}a_k\frac{d}{dt^k}y(t) = \sum_{k=0}^{M}b_k\frac{d}{dt^k}x(t) $$ With initial conditions, $$ y(t)|_{t=0}, ...
2
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0answers
94 views

Fourier spectrum reflected across origin and Nyquist frequency

Recently I've been trying to figure out what's the point of negative frequencies produced by the fourier transform. One answer was it's just there to make calculations more elegant. It could be ...
2
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0answers
21 views

How to reconstruct a sparsely sampled multiperiodic function?

I have $m$ oscillators, where $m$ is unknown, with periods $\vec p = p_1, p_2, \ldots, p_{m}$. Each of the oscillators $j$ has associated with it a vector of sine coefficients $\vec A_j$ and angle ...
2
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56 views

Fourier transform on trig wave

Find the fourier transform for signal in this picture (sorry for the bad quality) Could it be done like this? The signal is a sum of two triangular waves that are each delayed. ...
2
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47 views

Why are there so many different symbols to represent the Heaviside (unit step) function

In signal processing, the unit step function is typically written as $u(t)$. In other references though I have seen it represented as $H(t)$ and even $\theta(t)$. The unit impulse is fairly ...
2
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26 views

Reconstructing a signal at twice the bit-rate

I have a discrete-time signal as: $\alpha_l = \sum_k a_k g(lT/2 -k(T+\Delta T))$ where $a_k \in \lbrace\pm 1\rbrace$, $g(t)=\frac {sin(2\pi t/T)}{2\pi t/T}$, $T$ is the bit period, and $\Delta T$ is ...
2
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22 views

Estimation of Linear Projection

Given a linear system: $Y=AX+W$ Where: $X$ is the input signal of size $N \times K$ $Y$ is the output signal of size $M \times K$ $A$ is a projection of size $M\times N$; with $M >> N$ $W$ ...
2
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105 views

Inverse Fast Fourier Transform to find the voltage across a capacitor of a RC circut

Fourier transform of a RC circuit The following example of a RC circuit describes the use of the fourier transform in order to receive the output voltage across the capacitor. My questions ...
2
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142 views

Expectation and convolution question.

I am learning in an image processing course, and the professor did the following: As part of a derivation, has this: What I do not understand, is how he was able to remove $r(i,j)$ to the ...
2
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0answers
20 views

Transforms with $O(N \log N)$ Complexity

Beside the Discrete Fourier and Walsh-Hadamard operators, are there any non-trivial, bijective operators that admit an evaluation algorithm of $O(N \log N)$ time complexity or better, whose inverses ...
2
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0answers
22 views

How do I estimate the derivative of the current position, when I have only values from past to present?

If I have a discrete real-time signal $x[n]$, with its latest value $x[i]$ and all its past values $x[i-t]$, how can I estimate the derivative at $x[i]$?
2
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83 views

Proof of Caratheodory's theorem about the unique determination of a linear combination of sinusoids

Following is a statement of Caratheodory's Theorem about a positivelinear combination of sinusoids :- Any positive linear combination of k sinusoids is uniquely determined by its value at time t ...
2
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64 views

Why process noise model is$ \dfrac{T^4}{4}$ in Kalman filter.

I am using Kalman filter for filtering noise on 2D object movement. I read a lot of examples, but no one has been explained, why noise model is distance powered by 2: $$ s \times s = \dfrac{T^2}{2} ...
2
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0answers
255 views

Checking if (discrete) signal is stationary, BIBO stable and

I have this discrete signal y[n] = sum (x[k+1]h[k-1]), where k goes from -inf to +inf. I need to check if this signal is stable, stationary, and if it's invertible, i need to find it's inverse ...
2
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0answers
305 views

Lloyd-Max Quantizer Problem

Consider the function $z = x^2 + y^2$ , for $0 \leq x \leq 10$ and $0 \leq y \leq 10$. Take a regular discretization with steps $\Delta x = \Delta y = 0.1$ and its histogram as the probability ...
2
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0answers
250 views

MFCC - Why 13 Coefficients

Basically I am trying to computer a MFCC and wondered if you can help. This is the FFT of 1 of the Frames (After I have multiplied the Hamming Window by the Mel Bank Filters) : Here is the DCT of ...
2
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0answers
360 views

Understanding Discrete Cosine Transformation

I'm currently working on some software and a key component is 2D DCT. But my question is more general, as I'm trying to understand the DCT in general, let's say from engineers point of view. For ...
2
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0answers
197 views

How to solve the recursive relation in Kalman filter?

I was wondering how to solve the Kalman filter's recursive equation (also see the appendix at the end of this post) for the estimated state $\hat{\textbf{x}}_{n|n}$ at time $n$, over discrete times ...
2
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0answers
84 views

Scale invariance and $1/f^2$ power spectrum

In the paper Occlusion Models for Natural Images : A Statistical Study of a Scale-Invariant Dead Leaves Model; Lee, A. B. Mumford, D. B. Huang, J.; International Journal of Computer Vision I read ...
2
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0answers
690 views

How can I use the time-frequency uncertainty principle?

I have a signal composed of the summation of a set of sine waves of different frequencies. The amplitude of these sub-signals can change so many times a second. I have been told that, if I want to ...
2
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0answers
168 views

Singular Value Decomposition

I want to decompose an image $A$ using the Discrete Wavelet Transform and then find the singular values, $S$, such that $A=USV$. I will then do the same to another image such that $B=USV$. I will ...
2
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0answers
69 views

Means to classify data streams or compare similarity

Last year I converted some Matlab code into c to run on embedded Linux. I'm an engineer and normally shy away from maths, but this got me thinking about different ways to classify data or compare the ...
2
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0answers
275 views

FFT signal post processing

This is more a "post a suggestion" topic rather than a question. And thank you if you are willing to read this whole. I've been studing the code in the Nvidia Cuda SDK regarding how to operate a ...
2
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0answers
121 views

Deriving Isosensitivity Functions in ROC space from elementary signal detection parameters

In signal detection, an observer is assigned the task of discerning the presence (or absence) of some signal with accompanying noise. There are four possible outcomes: a hit ($H$), a miss, a false ...
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0answers
21 views

How to decorrelate/Whiten a non-white additive random variable?

I have a signal processing problem where I have the Additive Noise Model (assume Gaussian noise). $$ y = x + w $$ where, $y$ is corrupted signal, $x$ is original signal & $w$ is a non-white ...
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0answers
21 views

Haar wavelet transformation of a binary vector data over $GF(2)$

I am trying to perform Haar wavelet transformation on the following vector which is defined over $GF(2)$. $[1, 0, 1, 0, 1, 0, 1, 0]$ I am doing it as follows. $[1, 0, 1, 0, 1, 0, 1, 0]$ $\implies ...
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0answers
15 views

Nonparametric changepoint detection for point process

This is a replication of a question I've recently asked on Cross Validated. It hasn't received an answer or much attention, so I've posted it here. I have a family of point processes representing ...
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0answers
38 views

Recovering Time Shift Using DFT of Translated Square Pulse?

As an exercise, I attempted to manually translate a pulse $n_0$ steps to the right and recover the translation using the time-shift property. The problem I'm encountering is that the phase unwrapping ...
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0answers
39 views

Is there an exact solution for this resampling (synchronization) problem?

I want to know if there is an exact solution for the following problem and how to approach solving it: I have a discrete-time signal where the Nyquist theorem is satisfied: $$ r_k = \sum_i ...
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0answers
52 views

Compare between Short Time Fourier Transform and Wavelets

Fourier transform is localised in only frequency domain but Short time Fourier transform(STFT) is localised both in time and frequency domain same as in wavelets. I want to know How are STFT and ...
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0answers
21 views

What is the range on a fourier transform?

In particular, I want to know the range of the coefficients on the type-IV discrete cosine transform. Assuming no normalization factor or window is applied, what interval can I expect the coefficients ...
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0answers
41 views

Can a DTFT have a period different of $2\pi$?

I think almost everything is in the title. In an exercise, a DTFT is given : $$X(e^{j\Omega}) = \sin(\Omega) + \cos(\Omega/2)$$ The period of this DTFT is $4\pi$. Is that possible? I mean, the ...