I would like to replicate the real response of an instrument to some signal. Here's what I have in mind: I generate some ideal signal. I then add Gaussian noise to it to produce a realistic signal s(t). For a given frequency, I know the detector's sensitivity by taking the PSD of the detector's signal over a long time period. Should I Fourier transform s(t), convolve that with the PSD and then inverse FT the result to get the noise+systematics detector response?
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$\begingroup$ I suggest that you attach a block diagram with your question because this way stated it is quite ambiguous at best. Just convolving a PSD with the FT of a time signal is likely to lead to a wrong result, but again I am not sure what you really want. Usually, a time signal is convolved with the impulse response of the receiver filter and probably you also need the detector's nonlinear characteristic following the filter. $\endgroup$– hyportnexCommented Jun 25, 2022 at 0:36
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$\begingroup$ If you don't a good answer here in a day or so, you may want to consider posting this at Cross Validated $\endgroup$– Kyle KanosCommented Jun 25, 2022 at 1:30
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If you know the device transfer function- $T(\omega)$ or an impulse response - $h(t)$, the output signal is
$$s_{out}(t) = h(t)\otimes s_{in}$$ in frequency domain $\hat{s}_{out} (\omega) = \hat{s}_{in} (\omega) \times T(\omega) $
So if you know the input signal which is sum of a deterministic and stochastic component $ s_{in} = A(t) + \underline{\delta A} (t) $.
Know if you are looking to find a transfer function you can do two things that I see: First, you send a white noise with enough power and the output spectral density will be a transfer function $|T(\omega)|^2$. Otherwise, secondly you can send a harmonic signal with enough power and to measure attenuated signal. Which also gives you a transfer function.
