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

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

Identities

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

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

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

$⟨n\mid m⟩=T{\delta }_{n,m}$