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Suppose you have an LTI system for which the (complex) frequency response $H(j\omega)$ has been measured in some frequency window $[\omega_1,\omega_2]$. Now imagine that you want to provide an input $x(t)$ to this system, such that the output is a specific, known waveform $y(t)$ that has negligible frequency content outside $[\omega_1,\omega_2]$.

My question is:

What are the necessary and sufficient conditions that our measured $H(j\omega)$ values in $[\omega_1,\omega_2]$ need to satisfy such that the required input $X(j\omega) = Y(j\omega)/H(j\omega)$ exists, and makes practical sense? For example, the system can clearly not have any zeros on the $j\omega$ axis in our frequency window.

I realize that a sufficient condition is for the system to be invertible. However, no such knowledge of the system is available, as we only have access to the frequency response in a specific frequency interval. What can we say about the possibility of backing out the frequency response only with the information at hand?

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  • $\begingroup$ This sounds exactly like you're building an equalizer for a channel that you can't know before observing it. $\endgroup$ Dec 3, 2019 at 22:55
  • $\begingroup$ @MarcusMüller Even though having an equalizer would solve the problem, wouldn't it be overkill? Since my question is about equalizing a specific waveform. Cascading the system with an equalizer would compensate the effect of the transfer function for any input signal. I just want to find the input that results in one particular signal to appear at the output of the system (if it exists). $\endgroup$ Dec 4, 2019 at 2:50
  • $\begingroup$ To me, something that flattens the channel wherever necessary is an equalizer (that term imho doesn't require any all-frequencies behaviour, anyway, and that would be impossible to begin with – there can't be an infinitely wide equalizer), and it's a problem that is solved with equalizer in communications theory all the time: Divide observable spectrum into one or multiple bands of interest, and revert the channel on that/these individually, with only the goal of reverting the channel in all respects that are detrimential to your measurement/estimation problem. Exactly modern wideband comms! $\endgroup$ Dec 4, 2019 at 8:30

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