\documentclass{amsart}
\usepackage{graphicx}

\def\R{{\mathchoice{\rm I\hskip-.2em R}{\rm I\hskip-.2em R}
{\rm I\hskip-.15em R}{\rm I\hskip-.1em R}}}

\title{How to Tune a Radio}
\author{Put your name here}

\begin{document}
\maketitle
\section{Introduction}
In the previous lab, we looked in some detail as to when
and how functions could be expressed in a Taylor series.  The big
idea was that very general functions could be built from summing
polynomials together ($x$, $x^2$, $x^3$, etc).

Now we look at another way of building functions- from sines
and cosines- such sums are called Fourier Series.  We'll look
at this in the context of understanding
how a radio tunes in a signal\footnote{This project is taken
from ``How to Tune a Radio'', Applications in Calculus, MAA Notes
Number 29, 1997}.  To stay closer to our central theme of Fourier
Series, we'll focus on {\it amplitude modulation} (or AM) rather
than {\it frequency modulation} (or FM).

To tune a radio, we turn the dial to a specific frequency.  Somehow,
that picks out the signal from a specific radio station, decodes
it, and we can hear music or news.

Every radio station produces a signal, $f(t)$.  We will not go
into the physics of what that signal is; we're just taking it
as some function of $t$.  The antenna picks up a sum of signals
from lots of radio stations;  suppose there are $n$ of them.  Then
the signal picked up is the sum:
$$
F(t)=f_1(t)+f_2(t)+\ldots +f_n(t)
$$
where $f_j(t)$ is the signal produced by radio station $j$.  Let's
be even more specific:  Suppose I want to listen to $f_1(t)$.  The
problem is that I'm picking up a sum of signals- How do I isolate
my music in $f_1$?

\begin{itemize}
\item {\bf QUESTION 0:}  If we know only $f(t)$, where
$f(t)=g(t)+h(t)$, can we recover $g(t)$?
\end{itemize}

In this lab, we'll see that radio signals are NOT general sums,
but are summed in a special way that allows us to isolate
desired signals while deleting undesired signals.  Moreover,
we will be introduced to an amazing discovery of Joseph Fourier,
made in about 1807- that most functions can be written in terms
of sines and cosines.

\section{Trigonometric Identities and Integrals}
To get ready for what we're about to do, use a calc book
(or look on the web), and answer the following questions:
\begin{enumerate}
\item Let $n\neq 0$.  What is $\displaystyle \int_0^{\pi}\cos(nt)\,dt$
(do this by hand).  Plot $\cos(nt)$ for $n=1, 2, 3$.  How does
changing $n$ change $\cos(nt)$?
\item Look up the following identities (either in a calc book or
on the web under ``product to sum formulas''):
\begin{itemize}
\item $\sin(A)\sin(B)=$
\item $\cos(A)\cos(B)=$
\end{itemize}
\item Compute the following, using your previous identities.
You can use Maple to double
check your answer, but also provide all of the details by hand.
\begin{itemize}
\item $\displaystyle \int_0^{\pi} \sin(nt)\sin(mt)\,dt$, where $n\neq m$
and where $n=m$.
\item $\displaystyle \int_0^{\pi} \cos(nt)\cos(mt)\,dt$, where $n\neq m$
and where $n=m$.
\end{itemize}
\item Plot $t\sin(20t)$, $t^2\sin(20t)$, $\sin(t)\sin(20t)$.  Can
you find the functions $t$, $t^2$, and $\sin(t)$ in the graphs?
(Explain).

\bigskip

\noindent {\it NOTE:  Functions of the form $g(t)\sin(nt)$ where
$g$ is slowly varying with respect to the rapidly varying
$\sin(nt)$ are called Amplitude Modulated signals (or AM signals)}
\item Let $a_k$ be some real constant.
\begin{itemize}
\item Rewrite the expression
$$
a_k\sin(kt)\sin(At)
$$
using the trig identities found previously.
\item Using your previous answer as a substitution, determine the
values of $m$ for which the following integral will NOT be zero
(and in those cases, say what the integral will be):
$$
a_k \int_0^{\pi} \sin(kt)\sin(At)\cos(mt)\,dt
$$
\end{itemize}
\end{enumerate}
\section{Radio Signals:  A Specific Example}
Suppose we have three radio stations, $R_1, R_2, R_3$.  Each of
them are assigned a rapidly varying carrier wave, $\sin(nt)$.  For
this example, let's take $\sin(8t), \sin(16t), \sin(24t)$,
respectively.  The three radio stations want to transmit their
programming, and let's call these $S_1, S_2$ and $S_3$, where
$$
S_1(t)=3\sin(t)-2\sin(t)
$$
$$
S_2(t)=5\sin(t)+6\sin(2t)
$$
$$
S_3(t)=4\sin(t)+7\sin(2t)
$$
The radio stations actually send $S_1(t)\sin(8t)$,
$S_2(t)\sin(16t)$ and $S_3(t)\sin(24t)$.  Our radio antenna picks
up the sum,
$$
f(t)=S_1(t)\sin(8t)+S_2(t)\sin(16t)+S_3(t)\sin(24t)
$$
Let's suppose we want to pick out signal $S_2$.  We know that the
incoming signal is of the form $a_1\sin(t)+a_2\sin(2t)$, so we
only need $a_1, a_2$ in order to put together the signal from the
second radio station.

\noindent {\bf Exercise 6:} Find a constant $k$ and a value for
$m$ so that:
$$
k\int_0^{\pi}f(t)\cos(mt)\,dt=5 \quad (a_1 \mbox{ for radio
station 2})
$$
and $k$ and a value for $m$ so that:
$$
k\int_0^{\pi}f(t)\cos(mt)\,dt=6 \quad (a_2 \mbox{ for radio
station 2})
$$
\noindent {\it HINT:  Use the second part of your answer to
question (5) above!}

\bigskip

\noindent Congratulations!  You have recovered the signal from the
second radio station!  That's AM radio.

\bigskip

\noindent {\bf Exercise 7:}  Repeat Exercise 6 to recover the
signal from the third radio station.

\section{The Big Picture}

The procedure we described works pretty well, as long as:
\begin{itemize}
\item All music, speech, etc. can be written as:
$$
a_1\sin(t)+a_2\sin(2t)+\ldots a_k\sin(kt)
$$
for some value of $k$.
\item The carrier signal has a frequency that is much larger than
the $k$ in the previous item (actually, about twice as big).
\item The space between two carrier waves is large enough so that
we have room to compute our $m$'s (like in Exercise 6 and 7).
\item There exists electronics components that can perform the
desired integrations (they do exist, this is how a radio tunes a
signal).
\end{itemize}
We won't discuss the last three items (you might ask a
physicist!).  Rather, we'll focus on the first item.

There is a beautiful theorem by Fourier, where he said that any
continuous function can be written as a sum of sines and cosines.
We'll keep our discussion to the Fourier sine series (versus the
complete Fourier series).

I want to write a function $g(t)$ defined on $(0, \pi)$ as a sum
of sine functions. Therefore,
$$
g(t)=a_1\sin(t)+a_2\sin(2t)+a_3\sin(3t)+\ldots
$$
What should $a_k$ be for $k=1, 2, \ldots$ ?  We can compute it by
observing that integrating both sides of our equation gives:
$$
\int_0^{\pi}g(t)\sin(kt)\,dt=0+0+\ldots +a_k\cdot
\frac{\pi}{2}+0+0+\ldots
$$
so that
$$
a_k=\frac{2}{\pi}\int_0^{\pi}g(t)\sin(kt)\,dt
$$
Now we see what radio stations do- they transmit the sound by
transmitting a truncated Fourier sine series (you would keep
enough terms of the sum to get a good sound).

Let's see how this works by downloading the Fourier sine series
worksheet on our class website.  Read over the worksheet and see
if you can understand what the new Maple commands are doing!

{\bf Exercise:}  Change the worksheet to see the Fourier sine
series for a different $g(t)$- Be creative!
\end{document}
