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 $T\neq 0$.

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 $\int _{-T/2}^{T/2}$ 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 $\sum _{m=0}^{\infty }c_{n}\cos \left({\frac {2\pi mt}{T}}+\Theta _{m}\right)$

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

Identities

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

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

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

$\left\langle \ Bra\mid Ket\ \right\rangle =Ket\cdot Bra$

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

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