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Assume a single-input single-output model $y = f(x)$ where time series $x = (x_0, .., x_{n})$ is the input, time series $y = (y_0, .., y_n)$ is the output and $f$ a function mapping $x$ to $y$.

If the system $f$ is linear and time-invariant, it is completely determined by its impulse/frequency response.

However, if this is not the case, how to find $f$?

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In general, if f is nonlinear, you cannot find f.

If f is linear but shift-variant, then f could be a n by n matrix with $n^2$ unknowns, so you still cannot determine f since it's not unique. However, what you can do is to measure the output of each 'point' input $$(1,0\cdots,0), \cdots, (0,\cdots,1,\cdots,0),\cdots,(0,\cdots,0,1)$$ These outputs are the "impulse response" function for linear shift-variant system, called point response function (PRF). After the PRFs are measured, you can use them to compute the output of any input according to the superposition law since the system is still linear.

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Would the situation change if one is looking only an approximation of $f$? Anyway, thank you for introducing the point response function. I will look it up for more details. – user48819 Nov 9 '12 at 19:01

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