Applying naive Fourier analysis to a real signal
in a calculus II class, using EXCEL for hands-on integration
The Harvard Consortium Calculus text nicely contrasts
Taylor approximations and Fourier approximations (without
any claims to a deep analysis of Fourier expansions).
Motivated by the textbook problem (#14) which provides the
data (function values) graphically, we tried out a slightly
more realistic signal found on the WWW and digitized it in
Originally we had our eyes set on acoustics -- but the real
data came in a date late -- for a naive analysis of
sound date provided by Eric Brewe using MATLAB try out
Whereas MATLAB clearly is the tool of choice for
advanced calculations --
compare the demos, in the end
EXCEL did a nicer job in the class:
The data set was just large enough to give an inkling that
real world data sets might be larger than what can be
worked with by hand, but they still fit on a few screens.
More important, nothing is as tangible and methodic as
creating one column at a time, filling down the values
of cos(k*t), then of the product f(t)*cos(k*t), creating
a running sum, and finally calculating and overlaying
the Fourier approximation.
In class we started with a
table of function values,
and working together calculated the coeffcients a0 and a1
(don't underestimate the details like time shift, and
time-scaling....). Then each team was assigned a different
coefficient ak or bk -- and we almost reached the final
goal of plotting the jointly calculated Fourier approximation.
The above image is calculated from a
much larger sample solution
which systematically calculates 21 Fourier coefficients.
AFTER this hands-on calculation, students SHOULD by all means
take the next step to refer to each column of data by a simple
name, and work with such variables in MATLAB. For a naive