Fourier Series

Thrm Any waveform in time or space can be uniquely expressed as a sum over different frequencies of the correct phase and amplitude.

Thrm Suppose we have a function that specifies over a domain to and has period . And it satisfies the Dirichlet’s conditions:

• It has a finite integral over the domain.
• It has a finite number of extrema over the domain.
• It has a finite number of non-infinite discontinuities over the domain. Then it is possible to express this function as an infinite sum of harmonic functions:where is the lowest frequency term in the series.

Prop The above parameters are given by:

• The constant is the average of
• If is even, it will be built out of only cosine functions
• If is odd, it will be built out of only sine functions.

Prop Alternative form of Fourier Series Let be the angular frequency, and we will have:

Def The Gibbs Phenomenon For a step discontinuity, the overshoot is 9% on each side of the step. As you increase the number of terms the width of the overshoot tends to zero but the amplitude is constant.