# Questions tagged [integration]

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### Need help with deriving a recursive formula for a control system optimization integral

I need some help with a problem that appears in one of the exercises of "Introduction to Stochastic Control Theory" by Karl J. Åström: Chapter 5, page 141, problem 8. It is about deriving a ...
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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 ...
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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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### 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 $x[n]$, y[n] = x[n]...
87 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 ...
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