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Is the system $y(t)= dx(t)/dt$ invertible or not? If yes, please determine the inverse system for it.

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It is not. It suffices to find a counter example. Let me tell you that "I am flat". Could you derive my actual value?

So, any constant function $f(t) =c$ is differentiable, and have the same derivative, $f'(t)=0$. Only from knowing that $f'(t)=0$, you cannot recover the original (constant) function. Hence the system is not invertible.

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The system is not invertible because you can always add an arbitrary constant $c$ to any function $x(t)$ and the system will map it to same differentiated function $y(t)$. So, the mapping is not unique or one-to-one and hence not invertible.

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Delta coding (with no quantization) is quite close to what you are asking. Let:

x=randi([0 10], 8,1); delta = diff([0; x]); x_hat = cumsum(delta); assert(all(x-x_hat)==0);

But then you have a «silent agreement» to preprepend a zero such that you have a known starting point. And differentiation/integration is replaced by its discrete equivalents.

Knut

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