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

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

Identities

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

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

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

$\left\langle n\mid m\right\rangle =T\delta _{n,m}\,$