Input of RC low pass filter from the output

Question

In general, for an RC low pass filter, the output signal is related to the input signal by $$ Q_{out}(\omega) = H(\omega) Q_{in} (\omega), $$ where the \$Q\$s are the Fourier transforms of the time domain signals (the voltage signals \$v(t)\$) and \$ H(\omega) \$ is the transfer function $$ H(\omega) = \left( \frac{1}{1+j \omega \tau} \right)^{n} $$ where \$ \tau \$ is the time constant and \$ n \$ is the filter order.

This implies if I know the output signal and the transfer function, I can calculate the input signal $$ Q_{in} ( \omega ) = \frac{Q_{out}(\omega)}{H(\omega)} . $$

Is there ever a scenario where the above formula for \$ Q_{in}(\omega) \$ does not hold true?

I.e., where despite knowing the output signal, I cannot find the input signal?

The computed input signal will be noisier than your measurements, given the division operation. But otherwise, generally yes. Have a look in one of the versions of "Numerical Recipes" where they discuss this operation in more detail. I would be a bit more concerned if the input were some incarnation of the Weierstrass function. Might try that one as a concept validation test. ;) — periblepsis
– periblepsis, Commented Sep 22, 2024 at 14:17

Andy aka · Accepted Answer · 2024-09-22 13:09:56Z

Any number of cascaded RC low-pass filters do not behave in a bidirectional way. It's as simple as that and you are abusing the mathematical model of an RC filter by expecting it works in the reverse (if that is what you mean).

Just think about an RC low pass filter. If you apply a signal to the output (say 1 volt RMS) then what is seen at the input is exactly 1 volt RMS but fed through what was the original input resistor i.e. the resistor modifies nothing AND, it is not a bidirectional circuit described by the normal RC transfer function.

I mean, if you cranked-over a lawn mower engine by hand, would you expect it to magically start filling the petrol tank with fuel?

But, if the output signal is produced by a correctly applied (but unknown) input signal then sure, you can work out what the input signal is by inverting the transfer function.

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