\chapter{Excerpt from Undergraduate Thesis by John Piscitello}

The following is a passage from Chapter 3 of John Piscitello's thesis,
describing the screen in his application to show how convolution works.

\paragraph{Excerpt}

PMF A and PMF B
are still in the same place on the screen as they were before, but
now their bars have different colors. A third barchart, labeled
``Convolving PMF A and PMF B'', shows the two barcharts overlapping,
with one of the barcharts mirrored across the y-axis and 
translated to the right.
A slider, labeled C(w), reads how many points to the right that the
mirrored PMF has been translated; adjusting the slider will slide the
PMF left and right. When C(w) = 0, the mirrored PMF's rightmost
value is located at x = 0.

Two toggle buttons are present, exactly one of which is selected
at any given time. The choose whether PMF A or PMF B is to be mirrored and
slid across the x-axis. These buttons exist to show the symmetry of the
convolution operation. Selecting PMF A to be slid
across the axis to compute PMF C corresponds to this form
of the convolution operation:

\begin{center}
${\it p}_c({\it w}_0)=\sum_{{\it x}_0=-{\infty}}^{\infty}{\it p}_a
({\it w}_0-{\it x}_0)*{\it p}_b({\it x}_0)$
\end{center}

Because the parameter ${\it w}_0$ is what is changing in the equation,
we can see that PMF A will be mirrored and translated
across the axis. If the toggle button ``Slide PMF B'' is selected, the
symmetrical case holds,

\begin{center}
${\it p}_c({\it w}_0)=\sum_{{\it x}_0=-{\infty}}^{\infty}{\it p}_a({\it x}_0)*{\it p}_b({\it w}_0-{\it x}_0)$
\end{center}
and now it is PMF B which is slid across the axis. This symmetry is also
displayed in Figure 2-4. The only difference between the two pictures
is that from one to the next, the ``Slide B'' button was selected
instead of the ``Slide A'' button. Notice that the barcharts in the
``Convolving PMF A and PMF B'' graph have exchanged roles; in Figure
2-5, now it is PMF B which has been reversed and translated across the axis,
rather than PMF A as in Figure 2-4.

The product of the two PMFs in the operation is a vector dot 
product. In Figure 2-4,
the vector dot product is displayed by the overlapping barcharts. 
Overlapping
bars are multiplied together when computing the dot product.
%%\begin{figure}[t]

The fourth graph displays the PMF C that in gray.
In addition, the probability of the value of random variable {\it c}
which is currently being computed (as indicated in the slider) is
drawn in black.

Below these graphs is a series of labels. These present exactly the same
information as the barcharts do, but in a different way. The label
to the right of the label marked ``PMF A'' in Figure 2-4
is PMF A, reversed and translated 
two steps to the right. The label next to ``PMF B'' lists
PMF B's values in their correct order, and remains stationary,
as it does on the graph. The 
values on each label line up as they did on the barchart, and form
columns. Below each lined-up column is the product of the
two values; in Figure 2-4,

\begin{itemize}
\item  0.70 and 0.50 are lined up to form the product 0.35.
\item 0.35 = 0.35, the value of PMF C at w = 0.
\end{itemize}

Finally, at the bottom of the screen, we see the partial listing
of the values of PMF C as they are being calculated. All these labels
and barcharts are controlled through the actions of a single slider
and two toggle buttons.
