Fourier Series and Transform

Syllabus: Continuous-time Signals: Fourier series and Fourier transform, sampling theorem and applications.

10 hand-picked visual resources on this topic, 5 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

▶Interactive labFourier series: build a wave from harmonics ▶Interactive labFourier transform pairs and properties

Transform pairs worth memorising

Most questions are a standard pair plus one property. Deriving from the integral is almost always the slow route.

$e^{-at}u(t)\leftrightarrow\frac{1}{a+j\omega}$
$a>0$
$\frac{1}{1+t^{2}}\leftrightarrow\pi e^{-|\omega|}$
$e^{-t^{2}}\leftrightarrow\sqrt{\pi}\,e^{-\omega^{2}/4}$
$\text{rect}(t/T)\leftrightarrow T\,\text{sinc}\!\left(\frac{\omega T}{2\pi}\right)$

See it

Properties that do the work

Differentiation, scaling and modulation turn a known pair into the one you need.

$\frac{dx}{dt}\leftrightarrow j\omega X(j\omega)$
builds odd signals from even ones

Symmetry shortcuts

A real even signal has a real even transform; a real odd signal has a purely imaginary odd transform. This alone eliminates half the options in many questions.

$x(t)\ \text{real even}\Rightarrow X(\omega)\ \text{real even}$
$x(t)\ \text{real odd}\Rightarrow X(\omega)\ \text{imaginary odd}$

Fourier series: the two forms

Any periodic signal meeting the Dirichlet conditions is a sum of harmonics of its fundamental. The exponential form is the one to work in — the coefficients are a single integral, and the trigonometric coefficients drop out of it.

$x(t)=\sum_{n=-\infty}^{\infty}c_ne^{jn\omega_0t},\qquad c_n=\frac1T\int_Tx(t)e^{-jn\omega_0t}dt$

Symmetry kills half the coefficients

Spotting the symmetry of a waveform before integrating is worth more than any integration trick — it tells you which coefficients are zero without computing them.

$\text{even }x(t)\Rightarrow b_n=0\ \text{(cosine only)}$
$\text{odd }x(t)\Rightarrow a_0=a_n=0\ \text{(sine only)}$

Fourier transform of a periodic signal

A periodic signal has a line spectrum: its Fourier transform is a train of impulses at the harmonics, weighted by the Fourier-series coefficients. This is how the two tools meet, and it is what makes sampling analysis work.

$x(t)=\sum c_ne^{jn\omega_0t}\ \leftrightarrow\ X(j\omega)=2\pi\sum_nc_n\,\delta(\omega-n\omega_0)$

Correlation and energy spectral density

Correlation measures how much one signal resembles a shifted copy of another; autocorrelation is the special case that peaks at zero lag. For energy signals it transforms to the energy spectral density, the same way autocorrelation of a random process transforms to its PSD.

$R_{xy}(\tau)=\int x(t)y^*(t-\tau)\,dt=x(\tau)*y^*(-\tau)$

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