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
- 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
Euler's identity linking rectangular and polar coordinates
The dirac delta has an infinite height and an area of 1