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I'm reading the paper "Signal Flow Graphs-Computer-Aided System Analysis and Sensitivity Calculations" by Arnold Y Lee and my question in about expression/equality (59) in the paper. The expression looks like this:

$$ \Sigma_{i=1}^r \hat{y}_i(t) * \Delta y_i(t) + \Sigma_{i=1}^q \hat{x}_i(t) * \Delta x_i(t) = \Sigma_{i=1}^m \hat{u}_i(t) * \Delta u_i(t) + \Sigma_{i=1}^q \hat{v}_i(t) * \Delta v_i(t) $$

I do understand the derivation of the foregoing theorem and I also follow the derivation of (40) and (46), which is:

$$ \Sigma_{i=1}^r \hat{y}_i y_i + \Sigma_{i=1}^q \hat{x}_i x_i = \Sigma_{i=1}^m \hat{u}_i u_i + \Sigma_{i=1}^q \hat{v}_i v_i $$

and I also understand the steps leading to (48)

$$ \Sigma_{i=1}^r \hat{y}_i \Delta y_i + \Sigma_{i=1}^q \hat{x}_i \Delta x_i = \Sigma_{i=1}^m \hat{u}_i \Delta u_i + \Sigma_{i=1}^q \hat{v}_i \Delta v_i $$

But I do not understand how to derive or explain (59).

The appearance of the convolution operator is not really explained and it is not obvious to me how to explain it. I hope some smart people here can help (?)

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  • $\begingroup$ I would avoid linking to a copy of the paper that is not accessible from "reasonably open" sources (e.g. CiteSeer, Arxiv,....maybe ResearchGate could be added to this list although in a sort of borderline way). $\endgroup$ – A_A Jan 23 at 15:25

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