Trigonometric Fourier Series

The Trigonometric Fourier Series is an example of Generalized Fourier Series with sines and cosines substituted in as the orthogonal basis set. Thus a function or signal f(t) with period T₀ can be expressed as

[0 < t < T₀]

where is called the fundamental frequency or base frequency (first resonant frequency = 1/T) and all other nw₀ frequencies are called harmonics (every other component of the series used to create f(t)). In relating this to actual experimental data collected and graphed in Matlab, we note that n here is a positive integer, so that all other frequencies are integral multiples of the base frequency. Thus if one finds the base frequency to be, say, 150 Hz, then one should also observe signal components at 300, 450, 600 and so on. Seeing this agreement between theoretically predicted behavior and actual experimental data is an important experience possible here.

To find a₀, a_n, and b_n we will use an equation very similar to the one for calculating c for signals. This equation calls for dividing by the energy in front of the integral. So we need to find the energy before we can set up the equation to find a₀, a_n, and b_n.

Following the development in the Generalized Fourier Series that you just read, we compute the E_n:

energy for a₀ “DC Term”

energy for all cosine and sine terms (involves using a double angle formula for sin² or cos² ):

Now that we have the energies we can directly write the coefficient equations.

constant term (average value, DC)

cosine coefficients

sine coefficients

Next, go back to the Fourier Series Page to continue with exponential Fourier Series.

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