Various signal properties translate into specific properties of the Fourier series. If we can identify these properties before hand, we can save ourselves from doing unnecessary calculations. If the periodic signal has a DC offset, then the Fourier Series of the signal will include a zero frequency component, known as the DC component. If the signal does not have a DC offset, the DC component has a magnitude of 0. Due to the linearity of the Fourier series process, if the DC offset is removed, we can analyse the signal further (e.g. for symmetry) and add the DC offset back at the end.

Fouries Series

This is a tutorial made solely for the purpose of education and it was designed for students taking Applied Math 0340. It is primarily for students who have some experience using Mathematica. If you have never used Mathematica before and would like to learn more of the basics for this computer algebra system, it is strongly recommended looking at the APMA 0330 tutorial. As a friendly reminder, don'r forget to clear variables in use and/or the kernel.

Finally, the commands in this tutorial are all written in bold black font, while Mathematica output is in regular fonts. This means that you can copy and paste all commands into Mathematica, change the parameters and run them. You, as the user, are free to use the scripts to your needs for learning how to use the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.

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Motivated Examples

2.5.7. Motivated Examples from Music

As an example, note A above middle C is the note on which most tunings of instruments is based: 440 Hz. In radians, we could describe this note A with

sin(440*(2*\pi t)) = sin (880*pi *t)

Musical tones are restricted to a limited set of frequencies we call
notes. How we judge combinations of notes is partly subjective, but
most people would agree on some simple basics. For example, notes C
and G blend nicely together and produce a stable harmony. Change note
G to F# and now the combination of C and F# is unstable, producing
tension. Adjacent note combinations such as C and C# simply clash and
are unpleasant to the ear. Can these effects be seen mathematically?
That is the purpose for graphing note combinations here.

For purposes of this project musical notes are assigned relative
frequencies with the lowest note, C, having a frequency of 1 cycle per
second, or 1Hz. (Hz is a "Hertz.") While these are not true
frequencies, the math is easier and the comparative results the same
as if true frequencies were used. (Actual middle C has a frequency
close to 262 Hz.) Subsequent notes are tuned by the well-tempered
system of tuning in use since 1720. The ratio of frequencies of
consecutive chromatic notes is the twelfth root of 2, approximately
1.05946. This means that multiplying the frequency of a note by
1.05946 will yield the frequency of the next note above it on the
chromatic (all-inclusive) scale.

For a chart of relative frequencies over two octaves (with lowest C
being 1 Hz) click here: Frequencies. For a more comprehensive
discussion of tuning notes, see notes C and G.

Here are graphs of two note combinations.

 

Fourier Series, Transforms, and Boundary Value Problems: Second Edition By J. Ray Hanna, John H. Rowland Askey, R. and Haimo, D. T. "Similarities between Fourier and Power Series." Amer. Math. Monthly 103, 297-304, 1996. Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959