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

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

$⟨Bra\mid Ket⟩$ Notation

$$\begin{gathered} ⟨ \\ n\mid m \\ ⟩=m\cdot n \end{gathered}$$

Identities

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

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

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