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Need help with deriving a recursive formula for a control system optimization integral

I need some help with a problem that appears in one of the exercises of "Introduction to Stochastic Control Theory" by Karl J. Åström: Chapter 5, page 141, problem 8. It is about deriving a ...
mattTheMathLearner's user avatar
1 vote
1 answer
27 views

Normalisation of Complex Morlet Wavelet

I am currently writing an essay on Wavelet transforms, and as part of such, I am trying to show that the Morlet wavelet satisfies the standard criteria: $$ \int^\infty_{-\infty} \psi(t) dt = 0 $$ $$ \...
Isaac Mortiboy's user avatar
3 votes
2 answers
348 views

Why is sampling a signal equivalent with multiplying with a Dirac comb?

Given a continuous time signal $f(t)$, we can sample it signal by multiplying with a Dirac comb (impulse train) $$\bar{f}(t) = \sum_{n=-\infty}^{\infty} f(nT) \delta(t-nT) \tag{1}$$ where each impulse ...
Carl's user avatar
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0 answers
67 views

Does the high-pass filter cause the accelerometer to miscalculate displacement?

I have an accelerometer and I've written a program for it in the ESP32. The program uses a 750sps ADC sampling rate and an accurate 100Hz interrupt for calculations. P.S. The high-pass filter is at 0....
leo_0916's user avatar
2 votes
1 answer
114 views

Fourier Transform of $|t|$

I was going through Papoulis' book (The Fourier Integral and its Applications) when I came across the Fourier Transform for $|t|$. To find it he writes $|t|$ as (I am not sure how): $$|t| = -\frac{1}{\...
Ahsan Yousaf's user avatar
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1 vote
0 answers
50 views

Continuous Phase Modulation - Phase trajectory expansion

For continuous phase modulation (CPM), the circular phase trajectory is expressed as follows: $$\phi(t) = 2\pi h\int_0^t\sum_k \xi_k g(\tau - kT_s)d\tau + \Phi_0\tag{1.1}$$ Where: $h$ is the ...
Gilles's user avatar
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2 votes
1 answer
213 views

What is the Fourier Transform of $\operatorname{sgn}(t) \cdot \operatorname{sgn}(t)$?

I am wondering what the Fourier Transform of $\operatorname{sgn}(t) \cdot \operatorname{sgn}(t)$ will be, where $\operatorname{sgn}(t)$ indicates the signum function. It would seem obvious that this ...
Finn Heijink's user avatar
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0 answers
88 views

Are longer integration times equally good or better than averaging over several shorter timespans?

Consider the signal of e.g. a photodiode or a spectrometer, where a dark measurement has been taken to account for baseline noise. The signal we want to measure is quite weak, so we need a decently ...
John Doe's user avatar
  • 101
2 votes
1 answer
80 views

Integrating over the translation on a DWT

I'm really a begginer at Wavelet transform and I'm starting to use the pywt module. I have some difficulties understanding the link between the following integral and the coefficients of the DWT: $$ W(...
user68569's user avatar
1 vote
1 answer
232 views

Complex exponential Fourier series coefficient of periodic convolution

Let the complex exponential Fourier series coefficients of two periodic signals $x_1(t)$ and $x_2(t)$ be $C_{1n}$ and $C_{2n}$, respectively, with $T_0$ being the fundamental time period of both the ...
Kushagr Jaiswal's user avatar
0 votes
1 answer
77 views

What filtering I could use to save my displacement data?

I am double integrating my acceleration to obtain the displacement. However, the displacement looks so strange. Basically this is the measured acceleration of a system under a impulse movement. and I ...
qyu38's user avatar
  • 1
1 vote
2 answers
142 views

Interpretation of the initial value of an integrator

I came across this example the stream processing chapter of a programming book that I'm reading: Streams as signals We began our discussion of streams by describing them as computational analogs of ...
user51462's user avatar
  • 113
0 votes
1 answer
410 views

How to compensate phase delay introduced by the digital integrator?

Let's say I have a digital integrator with transfer function in following form $$ \frac{Y(z)}{U(z)} = \frac{T}{2}\cdot\frac{z + 1}{z - 1} $$ I have been looking for a mechanism how to compensate the ...
Steve's user avatar
  • 397
1 vote
1 answer
32 views

Integral Calculation in Matlab for Filter Optimization

I am trying to calculate the area under my FIR filter kernel using trapz but I'm getting weird results. The reason I want to know the area is eventually I would ...
PrematureCorn's user avatar
0 votes
1 answer
222 views

Other Methods for Numerical Integration

I know four common methods for numerical integration of signals such as Midpoint, Trapezoid, Simpson's rule, and FFT integration property. Are there other methods?
Yazan Alatoom's user avatar
0 votes
1 answer
449 views

How to double integrate signal in time domain using FFT (python)

I am looking for a python code to double integrate signals in the time domain using Fast Fourier Transformation (FFT). The idea is to apply FFT to the signal data in the time domain. Then dividing the ...
Yazan Alatoom's user avatar
2 votes
1 answer
975 views

Correct way to get velocity and movement spectrum from acceleration signal sample

I have an acceleration signal sample and I need to get velocity and movement spectrum (amplitudes and phases) from it. I try to solve this problem in two ways: First method Get FFT of acceleration ...
bvl's user avatar
  • 57
3 votes
1 answer
75 views

Problem with the existence of inverse DTFT

I am having trouble on the following exercise and I can't figure out where I am doing something wrong: Given an LTI system described by the following difference equation: $$y(n)=x(n)+2x(n-2)+y(n-1)$$ ...
Peter Allen's user avatar
1 vote
0 answers
201 views

Fourier transform of a top-hat function in the Faraday Measurement synthesis context

I'm currently trying to calculate the Fourier transform of a top-hat function in the context of Faraday Measurement Synthesis. This is pretty straightforward, however, I cannot understand why I cannot ...
Miguel Cárcamo's user avatar
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0 answers
31 views

n-dimentional integral over Multivariate Gaussian

given the prior distribution of $\mathbf{a}=[a_1,\ldots,a_K]^T$ as \begin{equation} p_{\mathbf{a}}(\mathbf{a})= \frac{1}{\pi^K \det{\mathbf{R_{\mathbf{a}}}}} e^{\mathbf{a}^H \mathbf{R}_{\mathbf{a}}^{-...
zahraesb's user avatar
  • 135
0 votes
1 answer
70 views

Representing changing sample rate in block diagram and frequency response plot

I have an discrete integrator which sums over a block of input samples to produce output samples at a lower rate. The integrator sums a block of 8 input samples, multiplies the sum by a coefficient, ...
Campground's user avatar
0 votes
1 answer
73 views

Evaluating the accuracy of an integrator

I'm designing a digital signal processing chain that includes integration of a periodic input followed by decimation/resampling. The processor is integer/fixed point only, so the primary source of ...
Campground's user avatar
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0 answers
109 views

Numerical Integration Drift

I have collected force platform data to which I calculate acceleration and velocity. The velocity data is presenting what I understand as drift from the numerical integration process that I undertook ...
teli95's user avatar
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4 votes
0 answers
230 views

How to solve Hilbert Transform with empirical discrete data in frequency domain?, from zero to infinity

I have a filter/LTI system frequency response in form of list of values in the frequency domain. I want to get the phase curve/data from magnitude data. Input data can have either linear spaced points ...
Leandro Alsina's user avatar
0 votes
0 answers
73 views

Derivation of equation 3.8 in Fundamentals of Wireless Communication by Tse and Vishwanath

In the book “Fundamentals of Wireless communication”by Tse and Vishwanath, Can someone help with the how the integration helps derive equation 3.8 ? The equation is $$ p_e= \mathbb P\left\{\left\lvert ...
Vageesh Dambal's user avatar
0 votes
1 answer
32 views

Undo finite difference with arbitrary timesteps

I'm wondering if there is a way to undo a finite difference filter with arbitrary timesteps. In the simplest case of a two-sample finite difference of a time-series $x[n]$, \begin{equation} y[n] = x[n]...
Fallobst's user avatar
0 votes
0 answers
87 views

How to prove that the integral of Hilbert transform is not equal to the Hilbert transform of the integral?

To prove that $\int_{-\infty} ^\infty \mathcal{H}(g(t))(t)\text{d}t\neq\mathcal{H}(\int_{-\infty} ^\infty g(t) \text{d}t)$, where $\mathcal{H}(\cdot)$ is the Hilbert transform operator My approach to ...
UserHuffmann's user avatar
2 votes
1 answer
897 views

Fast computation of a convolution integral with Gaussian kernel

Given a convolution integral with gaussian kernel $$ g(y) =\int_a^b\varphi(y-x)f(x)dx=\int_{-\infty}^{+\infty}\varphi(y-x)f(x)\mathbb{I}_{[a,b]}(x)dx $$ where $\varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\...
NN2's user avatar
  • 143
1 vote
1 answer
113 views

Integral of Kaiser window

I want to sample (evaluate at regular intervals) the integral of the Kaiser window. Is there a nice closed-form solution to this, so I don't have to do it numerically?
cloudfeet's user avatar
  • 301
2 votes
1 answer
943 views

Digital integrator

I have been implementing a control software where I need to calculate a magnetic flux based on the measurement of the phase voltages of a three phase grid (basically three sinewaves) according to the ...
Steve's user avatar
  • 397
1 vote
1 answer
131 views

Phase locked loop for three phase grid

I have been implementing a control software where one of its core functionalities is the ability to synchronize with the three phase grid. Basically I need to implement some form of the phase locked ...
Steve's user avatar
  • 397
1 vote
0 answers
524 views

Eliminating drift generated from double integration of acceleration signal using Envelope Method

I'm trying to remove the drift generated upon the double integration of a noisy acceleration signal. But this question discusses only removing the drift upon single integration to generate velocity ...
MajorMajorMajorMajor's user avatar
0 votes
0 answers
861 views

Integration of FTIR Absorbance Spectrum in Python

Hello I have a FTIR (fourier transform infrared) absorbance spectrum of a sample, Integrating the single lines is correlated to the quantity of substance present in the sample. The spectrum part I'm ...
Alessandro Romancino's user avatar
10 votes
2 answers
1k views

Fourier series of cycloid

What is the Fourier series representation of a cycloid? The parametric representation of the curve is as follows. $$ t=\dfrac{\theta-\sin\theta}{\pi}\\ x=\dfrac{1-\cos\theta}{\pi} $$ The period is $2$...
tfpf's user avatar
  • 203
2 votes
1 answer
112 views

How can I get the function of a curve from a dataset without using a curve fitting tool?

previous relevant question here. I have a dataset of a signal captured from real hardware that produces a curve. Depending on the curve characteristics I can determine whether metal is present or not ...
ChrisD91's user avatar
  • 155
0 votes
1 answer
78 views

How to solve this Laplace integral for an averaging function in an iterative numerical (finite difference) model?

I am trying to implement a mathematical model for vibrational damping described in this article. They provide an equation for damping force ($F$) as a function of: a spring constant ($k$) a damping ...
mike's user avatar
  • 533
1 vote
1 answer
94 views

confusion related to finding inverse Z-Transform using Complex Integral Method

I am facing problems related to evaluation of inverse Z-Transform using Complex Integral Method; Consider $X(z)=\frac{z}{z-2} $ and $ROC: |z|>2$ then, $$x(n)= \frac{1}{2\pi j}\oint_c X(z) z^{n-1} \...
Suresh's user avatar
  • 287
3 votes
1 answer
414 views

Variance of Integral of a real white Gaussian Noise Process

In this question, is the answer not equal to infinity ? Answer is mentioned as 6. But my doubt is cant we think of it like a linear combination of many independent random variables each having ...
Sreejith's user avatar
2 votes
2 answers
98 views

Why this system is linear?

Hi guys i'm studying signals and systems, and my professor told us that $$y(t) = \int\limits_{ t+T }^{t-T/2} {x(a+T/2)}\mathrm{d} a$$ is a linear system. But a primitive of $x$ isn't $ x^2$ ? How it'...
Giovanni Cerciello's user avatar
1 vote
1 answer
80 views

two dimensional integration of a trigonometric function

I am working on a detection problem which finally, I have to solve the following 2-D integral: $$\int\limits_{a}^{b} \int\limits_{c}^{d}e^{A\sin(x)\cos(y-B)}\, \mathrm{d}x \, \mathrm{d}y \ ,$$ where $...
Mahdi's user avatar
  • 13
0 votes
0 answers
38 views

Energy or power signal for continuous signal

I have solved $w(t) = 1dt$ and determined it as power but when $w(t) =\Pi\left(\frac{t}{T_0}\right)$ it is energy.
sr_m's user avatar
  • 1
0 votes
2 answers
180 views

Inverse DTFT of phase shifted complex exponential

I have been working on this problem for a few days now and I think this is the closest I have gotten. I am getting an Answer of zero and I would like to know if that is correct and if someone could ...
Dom's user avatar
  • 61
0 votes
1 answer
58 views

Twisted convolution - simplifying multiple integral via substitution

I need help with understanding a proof from a paper. Following is the description: Let $$r_{1}(t) = \int \int h_{1}\left(\tau, \nu\right) e^{j 2\pi \nu (t - \tau)} s\left(t - \tau\right) d\tau d\nu$$ ...
MaxFrost's user avatar
  • 383
5 votes
1 answer
419 views

Why Cramér spectral representation and not DTFT for stochastic process

In a lot of time-series analysis references I find (written by mathematicians or statisticians rather than engineers), I find the following signal decomposition for a stochastic process, termed the &...
Robert L.'s user avatar
  • 2,222
1 vote
1 answer
108 views

Behaviour of integrator at steady state

I wanted to calculate response of integrator of sinusoidal input at steady state via these two methods as mention in image but these two methods give two different answers at steady state, so where ...
user215805's user avatar
0 votes
0 answers
50 views

Resampling with factor linearly moving from one value to another: calculate expected target length

I am resampling an audio (1D) signal, using a resampling factor that moves linearly from startFactor to endFactor; say the input ...
0__'s user avatar
  • 113
1 vote
1 answer
1k views

Complex Data type not accepted in Simulink

I am having an issue with the interaction of the Matlab function block in the Simulink with my model block. Below you can also find the picture. I have created a Matlab function with three inputs and ...
Prateek Tiwari's user avatar
0 votes
1 answer
27 views

integral with vectoric varying coeficient

i have a basic function exp(-x.^2).*log(x).^2 which i integrate in a certain interval. i want to multiply my basic function with a vectoric coefficient called coef_vec that varies with the interval. ...
rocko445's user avatar
  • 171
0 votes
1 answer
66 views

representing simple definite integral as convolution in matlab using CONV

A definite integral is a convolution with idial all '1' filter(by the theory) so i tried to test this issue with MATLAB. i have taken a function called fun made an integral over it in 1e-6 - 20*e-6 ...
rocko445's user avatar
  • 171
4 votes
2 answers
395 views

Running Integral of sine and cosine functions

In typical signal processing course we were taught that the integral of signal $x(t)$ is given by $$y(t) = \int_{-\infty}^{t}x(\tau) d\tau$$ How can we use this definition to evaluate the integrals of ...
user5045's user avatar
  • 331