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Suppose I want to measure something with a sensor which in response to $u(t)$ produces an output signal in form of $v_{out}(t) = (1-e^{-t/ \tau})u(t)$, where $u(t)$ is the Heaviside-function and $\tau=40ms$.

How to solve the deconvolution?

How do I determine the optimal Wiener-filter if I have an additional white noise $n(t)=n_0$ with $\omega_{max}=300s^{-1}$?

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The noise which you defined will only shift your signal and only has zero frequency. i think you intended to define the noise's power. Also in addition to the noise power you need to know your signal's power spectrum to obtain Wiener filter.

Wiener filter will be (1/H)*(S/(S+N)). in this equation H is frequency response of your sensor, S is your signal's power spectrum and N is your noise power spectrum.

to obtain H you have to take derivative of your sensor unit step response to obtain unit impulse response, then take Fourier transform of unit impulse response.

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