Questions tagged [integration]

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3 votes
4 answers
2k views

integration property of fourier series

Please help me sort this issue out. The integration property in Fourier series is as follows: So, for proving the above property, i took this approach: This is where my doubt is. Some books and ...
2 votes
1 answer
171 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 ...
0 votes
1 answer
45 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 ...
4 votes
0 answers
201 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 ...
0 votes
0 answers
48 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 ...
2 votes
1 answer
69 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(...
3 votes
2 answers
4k views

How to calculate the displacement from the measured acceleration data?

I have an acceleration data which is collected from a vehicle. I do also have a displacement measurement. I would like to calcuate the displacement vector from the measured acceleration data, but no ...
1 vote
1 answer
138 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 ...
1 vote
1 answer
3k views

Cancel Drift after numerical integration

I am trying to integrate angular acceleration obtained from a set of accelerometers positioned specifically at opposite corners of a cube, based on the paper EcoIMU: A Dual Triaxial-Accelerometer ...
1 vote
2 answers
102 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 ...
0 votes
1 answer
180 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 ...
1 vote
1 answer
31 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 ...
0 votes
1 answer
173 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?
0 votes
1 answer
285 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 ...
2 votes
1 answer
751 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 ...
1 vote
0 answers
182 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 ...
3 votes
1 answer
57 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: $$y(n)=x(n)+2x(n-2)+y(n-1)$$ ...
0 votes
1 answer
671 views

What are and why are sine and cosine modulated integrals used?

I have found the definition of the following formulas in a paper regarding active vibration control, where they are called sine and cosine modulated integrals. $y$ is measurement signal with a strong ...
2 votes
1 answer
3k views

Integral over power spectral density

The Wikipedia entry on Spectral Density (PSD) has one confusing line: Summation or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical ...
0 votes
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}}^{-...
0 votes
1 answer
59 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, ...
1 vote
1 answer
186 views

Verlet integration first step

I'm trying to implement color-to-grayscale method from this paper. And they use Verlet's integration as: $$L^*(t+\Delta t)=\frac{F(t)}{m}\Delta t^2+2L^*(t)-L^*(t-\Delta t),$$ for computing new ...
0 votes
1 answer
59 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 ...
0 votes
0 answers
104 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 ...
0 votes
0 answers
71 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 ...
0 votes
1 answer
31 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]...
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 ...
0 votes
0 answers
73 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 ...
2 votes
1 answer
727 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{\...
1 vote
1 answer
97 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?
1 vote
1 answer
126 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 ...
2 votes
1 answer
730 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 ...
1 vote
0 answers
415 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 ...
0 votes
3 answers
451 views

Causality as applied to capacitors

This question stems from a point of confusion that I still have about the causality, linearity, and time-invariance in LCCDEs. I wanted to use the capacitor as an example. Consider a capacitor with ...
0 votes
0 answers
752 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 ...
1 vote
1 answer
2k views

Differential Equation and Transfer Function of Multiple Summer Integrator Block Diagram

I've been working on this more than 5 hour, I know how to solve feedback, summer, but was not able to solve this one, I've tried to simplify it by choose A(i) to be the first output of first summer ...
6 votes
3 answers
9k views

Deriving the integration property of the Fourier Transform

I want to derive the property of the Fourier Transform that states that if $X(j\omega) = \mathcal{F} (x(t))$ then $$\mathcal{F} \left( \int_{-\infty}^{t} x(\tau) \mathrm{d} \tau \right) = \frac{1}{j\...
9 votes
2 answers
922 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 $2$...
2 votes
1 answer
95 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 ...
0 votes
1 answer
57 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 ...
1 vote
1 answer
89 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} \...
3 votes
1 answer
326 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 ...
2 votes
2 answers
95 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'...
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 $...
0 votes
1 answer
64 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 ...
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.
0 votes
2 answers
163 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 ...
0 votes
1 answer
53 views

Twisted convolution - simplifying multiple integral via substitution

I need help with understanding a proof from a paper. Following is the description: Let $$r_{1}(t) = \int \int h_{1}\left(\tau, \nu\right) e^{j 2\pi \nu (t - \tau)} s\left(t - \tau\right) d\tau d\nu$$ ...
5 votes
1 answer
360 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 &...
1 vote
1 answer
106 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 ...