# Expression related to Tellegen's theorem and signal flow graphs

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 (?)

• 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). – A_A Jan 23 at 15:25