## 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 MATLAB
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 this m-file.
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 sample implementation see here.