Questions tagged [integration]
The integration tag has no usage guidance.
112
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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:
$$
\int^\infty_{-\infty} \psi(t) dt = 0
$$
$$
\...
3
votes
2
answers
333
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Why is sampling a signal equivalent with multiplying with a Dirac comb?
Given a continuous time signal $f(t)$, we can sample it signal by multiplying with a Dirac comb (impulse train)
$$\bar{f}(t) = \sum_{n=-\infty}^{\infty} f(nT) \delta(t-nT) \tag{1}$$
where each impulse ...
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Does the high-pass filter cause the accelerometer to miscalculate displacement?
I have an accelerometer and I've written a program for it in the ESP32. The program uses a 750sps ADC sampling rate and an accurate 100Hz interrupt for calculations. P.S. The high-pass filter is at 0....
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Fourier Transform of $|t|$
I was going through Papoulis' book (The Fourier Integral and its Applications) when I came across the Fourier Transform for $|t|$. To find it he writes $|t|$ as (I am not sure how):
$$|t| = -\frac{1}{\...
- 1,533
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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 ...
- 3,386
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200
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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 ...
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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 ...
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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:
$$ W(...
- 21
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200
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Complex exponential Fourier series coefficient of periodic convolution
Let the complex exponential Fourier series coefficients of two periodic signals $x_1(t)$ and $x_2(t)$ be $C_{1n}$ and $C_{2n}$, respectively, with $T_0$ being the fundamental time period of both the ...
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What filtering I could use to save my displacement data?
I am double integrating my acceleration to obtain the displacement. However, the displacement looks so strange.
Basically this is the measured acceleration of a system under a impulse movement. and I ...
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126
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Interpretation of the initial value of an integrator
I came across this example the stream processing chapter of a programming book that I'm reading:
Streams as signals
We began our discussion of streams by describing them as computational analogs of ...
- 113
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359
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How to compensate phase delay introduced by the digital integrator?
Let's say I have a digital integrator with transfer function in following form
$$
\frac{Y(z)}{U(z)} = \frac{T}{2}\cdot\frac{z + 1}{z - 1}
$$
I have been looking for a mechanism how to compensate the ...
- 395
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Integral Calculation in Matlab for Filter Optimization
I am trying to calculate the area under my FIR filter kernel using trapz but I'm getting weird results. The reason I want to know the area is eventually I would ...
- 151
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212
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Other Methods for Numerical Integration
I know four common methods for numerical integration of signals such as Midpoint, Trapezoid, Simpson's rule, and FFT integration property. Are there other methods?
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396
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How to double integrate signal in time domain using FFT (python)
I am looking for a python code to double integrate signals in the time domain using Fast Fourier Transformation (FFT).
The idea is to apply FFT to the signal data in the time domain. Then dividing the ...
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883
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Correct way to get velocity and movement spectrum from acceleration signal sample
I have an acceleration signal sample and I need to get velocity and movement spectrum (amplitudes and phases) from it. I try to solve this problem in two ways:
First method
Get FFT of acceleration ...
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Problem with the existence of inverse DTFT
I am having trouble on the following exercise and I can't figure out where I am doing something wrong:
Given an LTI system described by the following difference equation: $$y(n)=x(n)+2x(n-2)+y(n-1)$$
...
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195
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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 ...
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31
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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}}^{-...
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69
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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, ...
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65
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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 ...
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106
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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 ...
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227
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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 ...
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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
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1
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32
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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]...
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84
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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 ...
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838
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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{\...
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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?
- 301
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854
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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 ...
- 395
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130
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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 ...
- 395
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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 ...
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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 ...
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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 $2$...
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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 ...
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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 ...
- 523
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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, $$x(n)= \frac{1}{2\pi j}\oint_c X(z) z^{n-1} \...
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Variance of Integral of a real white Gaussian Noise Process
In this question, is the answer not equal to infinity ? Answer is mentioned as 6. But my doubt is cant we think of it like a linear combination of many independent random variables each having ...
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Why this system is linear?
Hi guys i'm studying signals and systems, and my professor told us that
$$y(t) = \int\limits_{ t+T }^{t-T/2} {x(a+T/2)}\mathrm{d} a$$
is a linear system.
But a primitive of $x$ isn't $ x^2$ ? How it'...
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1
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80
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two dimensional integration of a trigonometric function
I am working on a detection problem which finally, I have to solve the following 2-D integral:
$$\int\limits_{a}^{b} \int\limits_{c}^{d}e^{A\sin(x)\cos(y-B)}\, \mathrm{d}x \, \mathrm{d}y \ ,$$
where $...
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Energy or power signal for continuous signal
I have solved $w(t) = 1dt$ and determined it as power but when $w(t) =\Pi\left(\frac{t}{T_0}\right)$ it is energy.
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Inverse DTFT of phase shifted complex exponential
I have been working on this problem for a few days now and I think this is the closest I have gotten.
I am getting an Answer of zero and I would like to know if that is correct and if someone could ...
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Twisted convolution - simplifying multiple integral via substitution
I need help with understanding a proof from a paper. Following is the description:
Let
$$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$$
...
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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 &...
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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 ...
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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 ...
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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 ...
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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.
...
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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 ...
- 171
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392
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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 ...
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Coherent detection in radar and radio signals
Coherent demodulation\detection is usually done via matched filtering in RF signal processing. In radar signals, coherent integration can be done if the phases of the returns are known. In terms of ...
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