Signals and systems/GF Fourier

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

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

Identities

$e^{j\theta }=\cos \theta +j\sin \theta$

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

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

$$\begin{gathered} ⟨ \\ Bra\mid Ket \\ ⟩=Ket\cdot Bra \end{gathered}$$

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

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

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