Continuous convolution is graphically introduced in many optics or signal analysis textbooks. I was trying to trace the earliest version of the graphical version of the following integral:
$$ x(t)=\int_{0}^{t} f(\tau) \cdot h(t-\tau) d \tau $$
Currently, it is depicted as, and flip-shift method, where one of the function is reflected across the y-axis. Searching early articles, I found one from the 1950s, Graphical Evaluation of a Convolution Integral, in Mathematical Tables and Other Aids to Computation Vol. 13, No. 67 (Jul., 1959), pp. 202-212 paper on JSTOR.
There the author does the reflection (or folding) but does not change the sign of the $\tau$ axis, instead, he suggests that:
- Plotting $h(\tau),$ curve $H$ in Fig. $1 \mathrm{~A}$
- Folding $f(\tau),$ i.e., plotting it with $+\tau$ axis to the left, curve $F$ in Fig. $1 \mathrm{~B}$
- Translating Fig. $1 \mathrm{~B}$ on Fig. $1 \mathrm{~A}$ such that the $\tau$ axes coincide; for evaluating $x(t)$ at $t=t_{1},$ slide Fig. $1 \mathrm{~B}$ on Fig. $1 \mathrm{~A}$ until the origin of the $\tau$ axis in Fig. $1 \mathrm{~B}$ falls on $\tau=t_{1}$ of the $\tau$ axis in Fig. $1 \mathrm{~A},$ see Fig. $1 \mathrm{C}$.
In modern books, when one of the functions is flipped the positive $\tau$ axis is kept to the right and the function is plotted against the negative $\tau$ values.
(i) My question is about Step 2, why didn't the author change the sign of $\tau$ values after folding $f(\tau)$.
(ii) Is there any earlier version of graphical interpretation of convolution integral, perhaps in non-English texts?
Thanks.
