Lecture 4: Exponential Fourier Series
We have seen how sin and cosine functions can be represented in terms of complex exponentials. It turns out that we can use complex exponentials to represent Fourier series. In many respects, this makes for a simpler representation.
Let's go back to the compact Fourier series representation function, and express
it in terms of complex exponentials:
Note that this is true for all values of
(there are no special cases when
and there is only one formula (not two, as for sins and cosines). This is my
preferred form! In fact, due to its similarity with the Fourier transform to
be discussed soon, it is the most common form of the Fourier series. It is,
of course, possible to convert from one form to another. For example,
Suppose (as is most often the case) that
is a real function. Then
Important observation: To compute the F.S. coefficients, multiply the function by an exponential with negative exponent. There are many transforms that electrical engineers use -- Laplace transforms, Z-transforms, Fourier series, Fourier transforms, etc. In all of these, the exponent is negative. Don't forget it!