# Fourier Series

## Contents

## Fourier series

The Fourier series is used to analyze arbitrary periodic functions by showing them as a composite of sines and cosines.

A function is considered periodic if for .

The exponential form of the Fourier series is defined as

## Determining the coefficient

- The definition of the Fourier series

- Integrating both sides for one period. The range of integration is arbitrary, but using instead of scales nicely when extending the Fourier series to a non-periodic function

- Multiply by the complex conjugate

- Using L'Hopitals to evaluate the case. Note that n & m are integers

## Linear Time Invariant Systems

Must meet the following criteria

- Time independance
- Linearity
- Superposition (additivity)
- Scaling (homogeneity)

## The Dot Product, Complex Conjugates, and Orthogonality

Geometrically, the dot product is a scalar projection of a onto b

Arthimetically, multiply like terms and add

Lets imagine that we are only have one dimension

In order to get the real parts and imaginary parts to multiply as like terms, we need to take the complex conjugate of one of the terms

To test for orthogonality, take the complex conjugate of one of the vectors and multiply.

## Changing Basis Functions

We'd like to change from to

If we assume , then to make the imaginary parts cancel out

Changing variables

## Identities

Euler's identity linking rectangular and polar coordinates

The dirac delta has an infinite height and an area of 1