A Fourier series is a way to representing a periodic function as a linear combination of (potentially infinite amount of) harmonically related sine and cosine functions. It is analogous to a Taylor series, which expresses functions as a sum of monomial terms.
For functions that are not periodic, we need to use Fourier transform. For functions of two variables that are periodic in both variables, the trigonometric basis in the Fourier series is replaced by the spherical harmonics.
Key Features
The sinusoidal functions in Fourier series are integer multiples of a fundamental frequency related to the period of the original function. Sine waves with this characteristic are referred to as harmonics.
The sine and cosine functions used in the Fourier series are orthogonal to each other, meaning they are linearly independent over the interval of periodicity 1:
Fourier Series Approximation of Functions
See also: Gibbs phenomenon
In a Fourier series representation of a function, lower frequency components primarily shape affect the large-scale behavior of the function. Conversely, higher frequency components contribute to the fine structure.
The accuracy of the Fourier series approximation depends significantly on the nature of a function. smooth functions have amplitude spectra that decay rapidly as frequency increases. As a result, these functions can be well-approximated using just a few terms in their Fourier series.
On the other hand, functions with sharp changes (e.g. jump discontinuity) have their amplitude spectra decay much more slowly. And as a result, they require many high frequency components to approximate well.
Trigonometric Form
The Fourier series can be defined in several different ways. The most common way is as a sum of sine and cosine terms.
Fourier Series, Trigonometric Form
The Fourier series of a periodic function
of period is for some set of Fourier coefficients
and defined by the integrals
We often refer to the cosine terms as the “even part” and the sine terms as the “odd part”. An insight from decomposing decomposing Fourier series into even and odd components is that when
- for even functions,
and - for odd functions,
and
Notation Variations
Different sources may use alternative notations:
- Some use
instead of as the variable. - Lowercase letters
and may be used instead of and . is sometimes defined as to avoid dividing by 2 in the main series. - The integration limits may be from
to rather than from to . - The angular frequency
may be used for simplification. An equivalent formulation incorporating some of these variations is:
where
Alternative Representations
Compact Trigonometric Form
If we perform some phase shift, we can combine the sine and cosine terms
Fourier Series, Compact Trigonometric Form
where
This form direct represent the amplitude (
Compact Exponential Form
Another way to represent the Fourier series is by combining terms via the Euler’s formula:
Fourier Series, Compact Exponential Form
where
Complex Fourier Series
One of the great advantage of the compact exponential form of the Fourier series is that it gives us new insight of the Fourier Series as a linear combination of circular motions on a complex plane. 2
Consider a single term in the Fourier series:
- the Fourier coefficient
is the radius of the circle - the term
is the angular frequency of the circular motion - As
increase, the term causes a rotation in the complex plane. Positive values result in counterclockwise rotation, while negative values lead to clockwise rotation.
Subtopics
- intuition for Fourier coefficients - where do those Fourier coefficients come from
- Gibbs phenomenon - Behavior of the partial sum of a Fourier series on discontinuity
- Parseval’s theorem - Related signal power of a periodic signal to its Fourier coefficients