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7 votes
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z-Transform Methods: Definition vs. Integration Rule

Just to avoid a misunderstanding: the $\mathcal{Z}$-transform is a transform defined for sequences, comparable to the Laplace transform for continuous functions. What you are talking about is not the $...
Matt L.'s user avatar
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6 votes
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Derivative of delta function

You can see this by using integration by parts: $$\begin{align}\int_{-\infty}^{\infty}x(t)\delta'(t-T)dt&=x(t)\delta(t-T){\Big|}_{-\infty}^{\infty}-\int_{-\infty}^{\infty}x'(t)\delta(t-T)dt\\&...
Matt L.'s user avatar
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6 votes

Numericaly computing an integral in MATLAB

Use integral function. q = integral(g,xmin,xmax) For example $g(t) = e^{-t}$ $$q = \int_0^{\infty}e^{-t}\mathrm{d}t = 1$$ <...
AlexTP's user avatar
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6 votes
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Correct way to get velocity and movement spectrum from acceleration signal sample

That's because you are applying the integration formula for continuous Fourier transform in one case, and in the other you are using the discretized signal. Actually, it is interesting to do this, as ...
Bob's user avatar
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5 votes
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Difference between these two integrators

There are different methods to approximate integration in discrete time. The most straightforward ones are the forward and backward Euler methods, and the trapezoidal method. A discrete-time system ...
Matt L.'s user avatar
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5 votes
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Fourier series of cycloid

The Fourier series of the cycloid can be expressed in terms of the Bessel functions of the first kind: $$J_n(x)=\frac{1}{\pi}\int_0^{\pi}\cos(nt-x\sin t)dt,\qquad n\in\mathbb{Z}\tag{1}$$ Using the ...
Matt L.'s user avatar
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4 votes

Removing drift from integration of accelerometer data

I can say a couple of things about this problem since I have been working on a similar one for a few weeks. First thing to note is that you should subtract the mean of a section of noise from your ...
K Puri's user avatar
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4 votes

Why is $\int^\infty _{0^-}\delta(t-nT)e^{-st}dt = e^{-nsT}$?

$\delta(t)$ is an impulse that's infinitely thin and infinitely high. The area under it is 1 though. $\delta(t - nT)$ places the impulse at time $nT$. Now this is being multiplied by some function, ...
null's user avatar
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4 votes
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Why is $\int^\infty _{0^-}\delta(t-nT)e^{-st}dt = e^{-nsT}$?

Remember the definition of the delta function: it integrates $1$ all over the t-axis but it is zero for any $t\neq 0$. This means that if we multiply any function by $\delta (t)$, then the function is ...
Tendero's user avatar
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4 votes
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integration property of fourier series

Note that the antiderivative of a function is only defined up to a constant. Furthermore, note that if you integrate a periodic function, the result is not necessarily periodic. Let $$x(t)=\sum_{k=-\...
Matt L.'s user avatar
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4 votes
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Why Cramér spectral representation and not DTFT for stochastic process

I will introduce some terminology and intuition that will be helpful when reading other references. It will be neither complete nor completely rigorous. The measures that we first encounter in real ...
Joe Mack's user avatar
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4 votes
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How can I get the function of a curve from a dataset without using a curve fitting tool?

It's not necessary to first fit a curve to the data and then compute the integral. You can directly approximate the integral from the data using numerical integration methods. The most straightforward ...
Matt L.'s user avatar
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4 votes

Fourier series of cycloid

I haven't managed to completely evaluate the integral, but I've made some progress and perhaps someone can pick up where I leave off. The integral you gave is $$ c_n = \int_0^{2\pi}\left(\frac{1-\cos\...
Dan Pollard's user avatar
3 votes
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BIBO stability of $y(t)=\int_{-\infty}^{t}{x(\tau)d\tau}$

Let us try with another hints: could you imagine a bounded input signal which could result in a non bounded output? general suggestion whenever analyzing a system: try a few "simple to compute" ...
Laurent Duval's user avatar
3 votes

BIBO stability of $y(t)=\int_{-\infty}^{t}{x(\tau)d\tau}$

Since this is most likely homework, here is a hint. Write the integral you have displayed in the form $\int_{-\infty}^\infty x(\tau)h(t-\tau) d\tau$ where you get to choose what the function $h(\cdot)...
Dilip Sarwate's user avatar
3 votes
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How to calculate the displacement from the measured acceleration data?

Generally, the observed effect occurs when there is an offset and/or linear drift present in your measured data. Double integration then leads to the quadratic (or higher order) effect observed in the ...
user883521's user avatar
3 votes
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How to convolve an arbitrary signal with a causal decaying exponential?

Indeed you have reached what can be reached, may be the following additional line can be obtained by moving the $t$ function $e^{-2t}$ out of the integral and replacing the $u(t-\tau)$ by $1$ as: $$ ...
Fat32's user avatar
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3 votes

Derivative of delta function

The proof comes from the Dirac delta function property: \begin{align} \int\limits_{-\infty}^{\infty} x(t) \delta^{(n)}(t-t_0)dt=(-1)^{n}\frac{d^n}{dt^n}x(t)\bigg\vert_{t=t_0} \end{align} where $x(t)$...
Crypted_39's user avatar
3 votes

Removing drift from integration of accelerometer data

Double integration amplifies any offsets, non-linearities and noise. These can't be removed without the use of some type of external reference point measurements (e.g. not from the accelerometer) or ...
hotpaw2's user avatar
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3 votes

Cancel Drift after numerical integration

Numerical integration may reduce noise under some conditions: the noise is zero-average, ergodic, its properties do not vary too much over time. Since numerical integration seems linear, pre-filtering ...
Laurent Duval's user avatar
3 votes

Convolution Integral of Harmonic Signal (Cosine) with the Sinc Function

As $T\rightarrow \infty$, the integral becomes the convolution integral. You can use the fact that convolution in the time domain is multiplication in the frequency domain and then you have a single ...
Engineer's user avatar
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3 votes
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two dimensional integration of a trigonometric function

Since, the Taylor series expansion for $e^z$ about $z=0$ is $$e^z = \sum_{n=0}^{\infty}\dfrac{z^n}{n!}$$ then, ignoring the question of convergence, you can say $$\begin{align*} \int_a^b \int_c^d e^{A\...
Andy Walls's user avatar
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3 votes
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Variance of Integral of a real white Gaussian Noise Process

Since $W(t)$ is assumed to be zero-mean, also the RV $Y$ is zero-mean. Hence, the variance of $Y$ is given by $$\begin{align}\sigma_Y^2&=E\left\{Y^2\right\}\\&=E\left\{\int_{-\infty}^{\infty}W(...
Matt L.'s user avatar
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3 votes

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

Integrators are tricky. There time continuous transfer function is $$H(\omega) = \frac{1}{j\omega} \tag{1}$$ which unfortunately means that $H(0) \rightarrow \infty$ and that an integrator is not ...
Hilmar's user avatar
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3 votes
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Integral Calculation in Matlab for Filter Optimization

However, if you look at my ideal filter window its area is 2e3. Nope. It's 10000. The spectrum is a metric of spectral density, not frequency. In your case your bandwidth is 2000 Hz and the frequency ...
Hilmar's user avatar
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3 votes
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Fourier Transform of $|t|$

For the derivation of Eq. $(1)$ in your question, Papoulis refers to Eq. ($I$-$32$) in the appendix on distributions. That equality is basically a consequence of how the derivative of a distribution ...
Matt L.'s user avatar
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3 votes

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

Sampling with an impulse train is a mathematical model that is sometimes convenient. Note that the sampled signal $$x_s(t)=\sum_{n=-\infty}^{\infty}x(t)\delta(t-nT)=\sum_{n=-\infty}^{\infty}x(nT)\...
Matt L.'s user avatar
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2 votes
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Graph of $[u(\tau)-u(\tau-4)]\cdot[u(t-\tau)-u(t-\tau-4)]$

As this signal (function) exists inside an integration operator, and also assuming that there are no impulses residing at any discontinuities introduced by $u(t)$ step functions, then we can simply ...
Fat32's user avatar
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2 votes
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Differential Equation and Transfer Function of Multiple Summer Integrator Block Diagram

First, transform the variables into Laplace domain for dealing with algebraic rather than differential equations, which greatly simplifies the labor. And then properly re-route those two feedback ...
Fat32's user avatar
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2 votes

How to calculate the displacement from the measured acceleration data?

Hi I have a similar problem, but I fixed by filtering the RAW acceleration data to remove noise, I use a fifth order butterworth filter and most of the problem was solved.
James88's user avatar
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