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 $T\ne 0$.

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

Determining the coefficient ${\alpha }_n$

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

• The definition of the Fourier series

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

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

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

• Multiply by the complex conjugate

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

• $\frac{T{e}^{j2\pi (n-m)t/T}}{j2\pi (n-m)}{|}_{-T/2}^{T/2}=T\frac{e^{j\pi (n-m)}-e^{-j\pi (n-m)}}{j2\pi (n-m)}=T\frac{\sin \pi (n-m)}{\pi (n-m)}=\left\{\begin{array}{c}T,n=m\\ 0,n\ne m\end{array}\right\}=T{\delta }_{n,m}$

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

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

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

• $\overrightarrow{a}\cdot \overrightarrow{b}=\left|a\right|\left|b\right|\cos \theta$

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

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

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.

• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\varphi }_n(t){\varphi }_m^{*}(t)dt=0$

Changing Basis Functions

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

$x(t)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_n{e}^{j2\pi nt/T}=\underset{n^{\prime }=-n}{\underbrace{\sum _{n=-\mathrm{\infty }}^{-1}{\alpha }_n{e}^{j2\pi nt/T}}}+{\alpha }_0+\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n{e}^{j2\pi nt/T}=\underset{m=n^{\prime }}{\underbrace{\sum _{n^{\prime }=1}^{\mathrm{\infty }}{\alpha }_n{e}^{j2\pi nt/T}}}+{\alpha }_0+\underset{m=n}{\underbrace{\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n{e}^{j2\pi nt/T}}}$

$={\alpha }_0+\sum _{m=1}^{\mathrm{\infty }}({\alpha }_m{e}^{j2\pi mt/T}+{\alpha }_{-m}{e}^{-j2\pi mt/T})$

If we assume $$\begin{gathered} x(t)\in \mathrm{\Re } \\ \mathrm{\forall } \\ m \end{gathered}$$, then to make the imaginary parts cancel out

• ${\alpha }_{-m}={\alpha }_m^{*}$
• $u+u^{*}=2\mathrm{\Re }[u]$
• ${\alpha }_m=|{\alpha }_m|e^{j\varphi m}$

$={\alpha }_0+\sum _{m=1}^{\mathrm{\infty }}2\mathrm{\Re }\left[{\alpha }_m{e}^{j2\pi mt/T}\right]={\alpha }_0+\sum _{m=1}^{\mathrm{\infty }}2\mathrm{\Re }[|{\alpha }_m|e^{j\varphi m}{e}^{j2\pi mt/T}]={\alpha }_0+\sum _{m=1}^{\mathrm{\infty }}|{\alpha }_m|2\mathrm{\Re }\left[e^{j(2\pi mt/T+\varphi m)}\right]={\alpha }_0+\sum _{m=1}^{\mathrm{\infty }}|{\alpha }_m|2\cos (\frac{2\pi mt}{T}+{\varphi }_m)$

Changing variables

• $c_0={\alpha }_0$
• $c_m=2|{\alpha }_m|$
• ${\mathrm{\Theta }}_m={\varphi }_m$

$=\sum _{m=0}^{\mathrm{\infty }}{c}_m\cos (\frac{2\pi mt}{T}+{\mathrm{\Theta }}_m)$

Identities

$e^{j\theta }=\cos \theta +j\sin \theta$ Euler's identity linking rectangular and polar coordinates

$\sin x=\frac{e^{jx}-e^{-jx}}{2j}$

$\cos x=\frac{e^{jx}+e^{-jx}}{2}$

$<u|v>=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{u}^{*}(x)v(x)dx$

${\alpha }_{-m}={\alpha }^{*}$

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