Questions tagged [integration]
The integration tag has no usage guidance.
107
questions
9
votes
2
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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 $2$...
- 193
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 $$\mathcal{F} \left( \int_{-\infty}^{t} x(\tau) \mathrm{d} \tau \right) = \frac{1}{j\...
user33568
5
votes
5
answers
3k
views
Weird behaviour converting velocity to displacement
I'm trying to convert accelerometer data from accelerations to displacement,
I assume that the initial velocity is zero and the initial position is also zero.
Here's my code,
...
- 73
5
votes
2
answers
15k
views
Removing drift from integration of accelerometer data
I am trying to get a positional data from the accelerometer data using the following steps:
Re-zero the accelerometer value
Removing mean from accelerometer value
First integration of accelerometer ...
- 185
5
votes
1
answer
358
views
Why Cramér spectral representation and not DTFT for stochastic process
In a lot of time-series analysis references I find (written by mathematicians or statisticians rather than engineers), I find the following signal decomposition for a stochastic process, termed the &...
- 2,202
4
votes
1
answer
172
views
Numericaly computing an integral in MATLAB [closed]
I have the following integral
$$\int_{-\infty}^{\infty} g(t-\tau) \, \mathrm d\tau$$
where $g(\cdot)$ is a known function, e.g., $g(t) = \sin(t)$, and I want to evaluate its value with MATLAB. How ...
- 141
4
votes
2
answers
377
views
Running Integral of sine and cosine functions
In typical signal processing course we were taught that the integral of signal $x(t)$ is given by
$$y(t) = \int_{-\infty}^{t}x(\tau) d\tau$$
How can we use this definition to evaluate the integrals of ...
- 331
4
votes
0
answers
200
views
How to solve Hilbert Transform with empirical discrete data in frequency domain?, from zero to infinity
I have a filter/LTI system frequency response in form of list of values in the frequency domain. I want to get the phase curve/data from magnitude data.
Input data can have either linear spaced points ...
- 163
3
votes
3
answers
2k
views
integration property of fourier series
Please help me sort this issue out.
The integration property in Fourier series is as follows:
So, for proving the above property, i took this approach:
This is where my doubt is. Some books and ...
- 171
3
votes
2
answers
4k
views
How to calculate the displacement from the measured acceleration data?
I have an acceleration data which is collected from a vehicle. I do also have a displacement measurement.
I would like to calcuate the displacement vector from the measured acceleration data, but no ...
- 141
3
votes
1
answer
325
views
Variance of Integral of a real white Gaussian Noise Process
In this question, is the answer not equal to infinity ? Answer is mentioned as 6. But my doubt is cant we think of it like a linear combination of many independent random variables each having ...
- 33
3
votes
2
answers
750
views
Help with finishing this integral, to obtain the power spectral density of a pure cosine wave
I am trying to evaluate the power spectral density $S_{xx}(f)$ of a cosine signal $x(t) = A\cos(2\pi f_0t)$, by starting from its definition for deterministic power signals
$$S_{xx}(f) = \lim_{T\...
- 681
3
votes
1
answer
56
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: $$y(n)=x(n)+2x(n-2)+y(n-1)$$
...
- 145
3
votes
1
answer
399
views
Mutual information of $ \infty $-PAM
I'm trying to compute using matlab the mutual information for an $ \infty $-PAM input (the limit of a very dense PAM constellation) for a range of snr and I got stuck.
I'm working with a real-valued ...
- 171
3
votes
1
answer
141
views
Dimensional analysis of integrated white noise process
This question is somewhat related to this post.
Let us consider we have a white noise current source $i_n(t)$, with a variance $\sigma_i^2$, and mean, $\mu_n=0$.
Assume that this current is passed ...
- 65
2
votes
3
answers
274
views
Derivative of delta function
I've come across the below. Can you please help me understand the proof for this?
$$\int_{-\infty}^{\infty}x(t)\delta'(t-2.5)dt=-\frac{dx(t)}{dt}{\Big |}_{t=2.5}$$
- 335
2
votes
2
answers
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views
z-Transform Methods: Definition vs. Integration Rule
The definition of the z-transform is defined as $z = e^{sT}$ where "s" is complex frequency for continuous-time systems and "T" is the sample period. Why are rules such as the forward rectangular ...
2
votes
2
answers
217
views
Why is $\int^\infty _{0^-}\delta(t-nT)e^{-st}dt = e^{-nsT}$?
I'm currently in the process of going over the $\mathcal Z$-transform and more specifically its derivation. I understand and I am able to follow it up until the final step whereby involving the ...
- 59
2
votes
1
answer
747
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 ...
- 47
2
votes
2
answers
3k
views
Integration of square wave
trying to program an integrator. My input is a square wave and my expected output should be a triangle wave. However, whenever I pass it through my low pass filter algorithm (just a 2nd order ...
- 209
2
votes
1
answer
92
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 ...
- 155
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'...
2
votes
1
answer
226
views
Integral of an upsampled signal, without actually resampling it
Given a signal X which is sampled at a certain frequency. The value we currently compute is given as the integral of the upsampled signal. Thus: Y = X but 100 times upsampled, by means of sinc ...
user7488
2
votes
1
answer
168
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 ...
- 95
2
votes
1
answer
714
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{\...
- 143
2
votes
1
answer
724
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 ...
- 375
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 ...
- 135
2
votes
2
answers
3k
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Why is the Fourier transform valid only for absolutely integrable signals?
Why is the Fourier transform valid only for absolutely integrable signals?
For example, why can't we do the Fourier transform of exponential order functions?
- 41
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:
$$ W(...
- 21
2
votes
1
answer
218
views
Orthogonal signal generator using integer arithmetic
I have a problem with implementing an orthogonal signal generator (OSG) algorithm on a microcontroller using integer arithmetic. I use this algorithm for a single-phase phase-locked loop (PLL) ...
- 163
2
votes
0
answers
946
views
Designing a practical integrator
One way to describe a practical integrator ("leaky integrator") is
$$
H(s) = \frac{g R}{1 + sRC}
$$
I am trying to understand how to choose the values $g$, $R$ and $C$ because in practice, I will ...
- 473
1
vote
1
answer
193
views
How does integration of a signal change the spectrum?
It's a very basic question, but I havn't found the question on google or dsp.stackexchange.com:
How does integraion of a signal change the spectrum?
- 165
1
vote
1
answer
3k
views
Difference between these two integrators
I am an electrical engineering student but new to digital signal processing. However while searching references for my senior project I came across following two different discrete integrator blocks ...
- 113
1
vote
2
answers
100
views
Interpretation of the initial value of an integrator
I came across this example the stream processing chapter of a programming book that I'm reading:
Streams as signals
We began our discussion of streams by describing them as computational analogs of ...
- 113
1
vote
1
answer
31
views
Integral Calculation in Matlab for Filter Optimization
I am trying to calculate the area under my FIR filter kernel using trapz but I'm getting weird results. The reason I want to know the area is eventually I would ...
- 151
1
vote
1
answer
80
views
two dimensional integration of a trigonometric function
I am working on a detection problem which finally, I have to solve the following 2-D integral:
$$\int\limits_{a}^{b} \int\limits_{c}^{d}e^{A\sin(x)\cos(y-B)}\, \mathrm{d}x \, \mathrm{d}y \ ,$$
where $...
- 13
1
vote
1
answer
1k
views
Convolution Integral of Harmonic Signal (Cosine) with the Sinc Function
I was asked to show that this convolution integral results in the answers also given in the image. Not quite sure how to approach this integral, everything seems to be coupled together.
Does anyone ...
- 11
1
vote
1
answer
371
views
How to convolve an arbitrary signal with a causal decaying exponential?
I need to simplify the following convolution
$$x(t)\star [e^{-2t} u(t)]$$
where $u(t)$ is the unit step function. I'm very confused with this. Using the definition of convolution of continuous-time ...
- 25
1
vote
1
answer
3k
views
Cancel Drift after numerical integration
I am trying to integrate angular acceleration obtained from a set of accelerometers positioned specifically at opposite corners of a cube, based on the paper
EcoIMU: A Dual Triaxial-Accelerometer ...
- 11
1
vote
1
answer
96
views
Integral of Kaiser window
I want to sample (evaluate at regular intervals) the integral of the Kaiser window.
Is there a nice closed-form solution to this, so I don't have to do it numerically?
- 261
1
vote
1
answer
126
views
Phase locked loop for three phase grid
I have been implementing a control software where one of its core functionalities is the ability to synchronize with the three phase grid. Basically I need to implement some form of the phase locked ...
- 375
1
vote
1
answer
106
views
Behaviour of integrator at steady state
I wanted to calculate response of integrator of sinusoidal input at steady state via these two methods as mention in image but these two methods give two different answers at steady state, so where ...
- 269
1
vote
1
answer
2k
views
Absolute and Square Summability & Integration of sinc function
Is the sinc function both absolutely summable (L1 norm for Continuous time signals and l1 norm for Discrete time signals) and square summable (L2 norm for Continuous time signals and l2 norm for ...
- 163
1
vote
1
answer
85
views
How to model a generic low frequency signal?
I'm trying to apply Fourier analysis to a specific problem I have.
I have essentially an integral like the following
$$
\int_{\Omega} f(t) g(t) dt
$$
And I'm trying to assume that $g$ is a narrow ...
- 395
1
vote
1
answer
2k
views
Differential Equation and Transfer Function of Multiple Summer Integrator Block Diagram
I've been working on this more than 5 hour,
I know how to solve feedback, summer, but was not able to solve this one,
I've tried to simplify it by choose A(i) to be the first output of first summer ...
- 177
1
vote
2
answers
208
views
What does the $n$ do in $1/(j \omega)^n$
I'm currently working through a research paper on beam forming. In this paper a magnitude compensation is introduced to compensate for frequency dependency. Due to other calculations low frequencies ...
- 11
1
vote
1
answer
3k
views
The essential bandwidth of a rectangular pulse
The problem is to estimate the essential bandwidth of a rectangular pulse
\begin{equation}
g(t) = \Pi(t/T),
\end{equation}
Where the essential bandwidth must contain at least $90\%$ of the pulse ...
- 13
1
vote
1
answer
220
views
Graph of $[u(\tau)-u(\tau-4)]\cdot[u(t-\tau)-u(t-\tau-4)]$
I'm trying to visualize what would the following signal be like
$$[u(\tau)-u(\tau-4)]\cdot[u(t-\tau)-u(t-\tau-4)]$$
Consider that $\tau$ is the independent variable here, $t$ is the shift variable.
...
- 287
1
vote
1
answer
227
views
Evaluating the inverse $\mathcal Z$-transform on the unit circle
I am trying to understand the math. The inverse $\mathcal Z$-transform is given by:
$$x[n] = \displaystyle\frac{1}{j2\pi} \int_cX(z)z^{n-1}dz$$
where $\displaystyle \int_c$ is a contour integral. ...
- 618
1
vote
1
answer
3k
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Inverse z transform - contour integration
Here is my task:
Find inverse z transform of $X(z)=\frac{1}{2-3z}$, if $|z|>\frac{2}{3}$
I need to find it using definition formula, $x(n)=\frac{1}{2\pi j}\oint_{C}^{ } X(z)z^{n-1}dz$. How can I ...
- 127