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Questions tagged [integration]

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10 votes
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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 $2$...
tfpf's user avatar
  • 203
6 votes
3 answers
10k 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\...
user avatar
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, ...
SamuelNLP's user avatar
5 votes
2 answers
17k 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 ...
Sharah's user avatar
  • 185
5 votes
1 answer
427 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
4 votes
1 answer
174 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 ...
zahra's user avatar
  • 141
4 votes
2 answers
399 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
4 votes
0 answers
237 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
3 votes
4 answers
3k 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 ...
Jarvis1997's user avatar
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 ...
Yirmidokuz's user avatar
3 votes
2 answers
364 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
  • 416
3 votes
1 answer
424 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
3 votes
2 answers
884 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\...
teeeeee's user avatar
  • 681
3 votes
1 answer
76 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
3 votes
1 answer
415 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 ...
SNR's user avatar
  • 181
3 votes
1 answer
158 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 ...
Saqib Shah's user avatar
2 votes
3 answers
316 views

Derivative of delta function

I've come across the below. Can you please help me understand the proof for this? $$\int_{-\infty}^{\infty}x(t)\delta'(t-2.5)dt=-\frac{dx(t)}{dt}{\Big |}_{t=2.5}$$
Chandrahas Balleda's user avatar
2 votes
2 answers
5k 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 ...
Help Appreciated's user avatar
2 votes
2 answers
220 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 ...
Tricks's user avatar
  • 59
2 votes
1 answer
1k 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
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 ...
yun's user avatar
  • 209
2 votes
1 answer
115 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
2 votes
2 answers
99 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
2 votes
1 answer
247 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 ...
user avatar
2 votes
1 answer
118 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
  • 1,583
2 votes
1 answer
221 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
2 votes
1 answer
939 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
2 votes
1 answer
988 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
2 votes
1 answer
4k 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 ...
cheersmate's user avatar
2 votes
2 answers
3k views

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?
khaled Osama's user avatar
2 votes
1 answer
82 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
2 votes
1 answer
227 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) ...
Marko Gulin's user avatar
2 votes
0 answers
1k 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 ...
divB's user avatar
  • 473
1 vote
1 answer
218 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?
Semjon Mössinger's user avatar
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 ...
Deep's user avatar
  • 113
1 vote
2 answers
150 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
1 vote
1 answer
33 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
1 vote
1 answer
81 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
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 ...
Alalalala's user avatar
1 vote
1 answer
412 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 ...
anon's user avatar
  • 25
1 vote
1 answer
4k 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 ...
Nithin G A's user avatar
1 vote
1 answer
117 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
1 vote
1 answer
132 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
1 answer
109 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
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 ...
nurabha's user avatar
  • 163
1 vote
1 answer
90 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 ...
user8469759's user avatar
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 ...
Unknown123's user avatar
1 vote
2 answers
215 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 ...
Denny Beulen's user avatar
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 ...
Engine_ear's user avatar
1 vote
1 answer
224 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. ...
Hossam Houssien's user avatar