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Suppose we have a linear linearly time varying system $S$ such that the output depends on the input + the time at which the system becomes excited so $S(t,t_{0}) \rightarrow g(t_{0})S(t)$ and it is true that if $t_{2} = t_{1}+t_{0}\rightarrow h(t,t_{2}) = h(t,t_{1})+h(t,t_{0}) \rightarrow g(t_{2})h(t) = g(t_{1})h(t)+g(t_{0})h(t)$.Since the above equations hold $g(t)$ must be a linear function.Is that correct?

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  • $\begingroup$ You're saying that $t_{2} = t_{1}+t_{0}\rightarrow h(t,t_{2}) = h(t,t_{1})+h(t,t_{0}) \rightarrow g(t_{2})h(t) = g(t_{1})h(t)+g(t_{0})h(t)$ is a condition on $S$? $\endgroup$
    – TimWescott
    Apr 15, 2023 at 21:26
  • $\begingroup$ @TimWescott Yes S has the property that if $t_{2} = t_{1}+t_{0}$ for arbitary values for $t_{2},t_{1},t_{1}$ then the impulse response at $t_{2}$ will be the sum of the impulse responses at $t_{1}$ and $t_{0}$ $\endgroup$
    – Volpina
    Apr 15, 2023 at 22:04

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