Fourier Series

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 $x(t)=x(t+T)\,$ for $T\neq 0$ .

The exponential form of the Fourier series is defined as $x(t)=\sum _{n=-\infty }^{\infty }\alpha _{n}e^{{j2\pi nt}/T}\,$

Determining the coefficient $\alpha _{n}\,$

$x(t)=\sum _{n=-\infty }^{\infty }\alpha _{n}e^{{j2\pi nt}/T}\,$

The definition of the Fourier series

$\int _{-T/2}^{T/2}x(t)\,dt=\sum _{n=-\infty }^{\infty }\alpha _{n}\int _{-T/2}^{T/2}e^{{j2\pi nt}/T}dt$

Integrating both sides for one period. The range of integration is arbitrary, but using $\int _{-T/2}^{T/2}$ instead of $\int _{0}^{T}$ scales nicely when extending the Fourier series to a non-periodic function

$\int _{-T/2}^{T/2}x(t)e^{{-j2\pi mt}/T}dt=\sum _{n=-\infty }^{\infty }\alpha _{n}\int _{-T/2}^{T/2}e^{{j2\pi nt}/T}e^{{-j2\pi mt}/T}dt=\sum _{n=-\infty }^{\infty }\alpha _{n}\int _{-T/2}^{T/2}e^{{j2\pi (n-m)t}/T}dt$

Multiply by the complex conjugate

$\int _{-T/2}^{T/2}x(t)e^{{-j2\pi mt}/T}dt=\sum _{n=-\infty }^{\infty }\alpha _{n}{\frac {Te^{{j2\pi (n-m)t}/T}}{j2\pi (n-m)}}{\bigg |}_{-T/2}^{T/2}=\sum _{n=-\infty }^{\infty }\alpha _{n}T\delta _{n,m}=T\alpha _{m}$

${\frac {Te^{{j2\pi (n-m)t}/T}}{j2\pi (n-m)}}{\bigg |}_{-T/2}^{T/2}=T{\frac {e^{j\pi (n-m)}-e^{-j\pi (n-m)}}{j2\pi (n-m)}}=T{\frac {\sin \pi (n-m)}{\pi (n-m)}}={\begin{Bmatrix}T,n=m\\0,n\neq m\end{Bmatrix}}=T\delta _{n,m}$

- Using L'Hopitals to evaluate the ${\frac {T\cdot 0}{0}}$ case. Note that n & m are integers

$\alpha _{m}={\frac {1}{T}}\int _{-T/2}^{T/2}x(t)e^{{-j2\pi mt}/T}dt$

Linear Time Invariant Systems

Must meet the following criteria

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

The Dot Product, Complex Conjugates, and Orthogonality

File:300px-Scalarproduct.gif

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

${\vec {a}}\cdot {\vec {b}}=\left|a\right\vert \left|b\right\vert \cos \theta$

Arthimetically, multiply like terms and add

$(3,2,1)\cdot (5,6,7)=3\cdot 5^{*}+2\cdot 6^{*}+1\cdot 7^{*}$

Lets imagine that we are only have one dimension

$(a+jb){\hat {i}}\cdot (a+jb){\hat {i}}\neq a^{2}+b^{2}$

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

$(a+jb){\hat {i}}\cdot (a-jb){\hat {i}}=a^{2}+b^{2}$

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

$\int _{-\infty }^{\infty }\phi _{n}(t)\phi _{m}^{*}(t)dt=0$

Changing Basis Functions

We'd like to change from $\sum _{n=-\infty }^{\infty }\alpha _{n}e^{j2\pi nt/T}$ to $\sum _{m=0}^{\infty }c_{m}\cos \left({\frac {2\pi mt}{T}}+\Theta _{m}\right)$

$x(t)=\sum _{n=-\infty }^{\infty }\alpha _{n}e^{j2\pi nt/T}=\underbrace {\sum _{n=-\infty }^{-1}\alpha _{n}e^{j2\pi nt/T}} _{n'=-n}+\;\alpha _{0}\;+\sum _{n=1}^{\infty }\alpha _{n}e^{j2\pi nt/T}=\underbrace {\sum _{n'=1}^{\infty }\alpha _{n}e^{j2\pi nt/T}} _{m=n'}+\;\alpha _{0}\;+\underbrace {\sum _{n=1}^{\infty }\alpha _{n}e^{j2\pi nt/T}} _{m=n}$

$=\alpha _{0}+\sum _{m=1}^{\infty }\left(\alpha _{m}e^{j2\pi mt/T}+\alpha _{-m}e^{-j2\pi mt/T}\right)$

If we assume $x(t)\in \Re \ \forall \ m$ , then to make the imaginary parts cancel out

$\alpha _{-m}=\alpha _{m}^{*}$
$u+u^{*}=2\Re [u]$
$\alpha _{m}=|\alpha _{m}|e^{j\phi m}\,$

$=\alpha _{0}+\sum _{m=1}^{\infty }2\Re \left[\alpha _{m}e^{j2\pi mt/T}\right]=\alpha _{0}+\sum _{m=1}^{\infty }2\Re \left[|\alpha _{m}|e^{j\phi m}e^{j2\pi mt/T}\right]=\alpha _{0}+\sum _{m=1}^{\infty }|\alpha _{m}|2\Re \left[e^{j(2\pi mt/T+\phi m)}\right]=\alpha _{0}+\sum _{m=1}^{\infty }|\alpha _{m}|2\cos \left({\frac {2\pi mt}{T}}+\phi _{m}\right)$