A Note About Symmetry and Fourier Series

A Note About Symmetry and Fourier Series

          There are functions which are even and functions which are odd.  Some functions are neither even or odd.  However, when you have a function that is even or odd there are implications of that which help in calculation of Fourier Series coefficients.

• For an even function, f(t) = f(-t).

• Even functions are symmetric around t = 0, and look the same when reflected through the t = 0 line.  Here is an example of an even signal.

• For an odd function, f(t) = -f(-t).

• Odd functions are anti-symmetric around t = 0, and seem to be reflected through the origin point.  Here is an example of an odd signal.

Now, we also know that a sine function is an odd function and a cosine function is an even function.  The implications of that are:

• If you want the Fourier Series coefficients for an even function, there will be no sine terms (all of the b's will be zero) since the sine function is odd.  There will only be cosine terms since the cosine function is even.

• If you want the Fourier Series coefficients for an odd function, there will be no cosine terms (all of the a's will be zero) since the cosine function is even.  There will only be sine terms since the sine function is odd.

Example

        Here is a triangle signal that is symmetric around t = 0.  This is an even function, so there will not be any sine terms in the Fourier Series expansion.