# Difference between revisions of "Fourier transform"

An initially identity that is useful: $x(t)=\int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}\, dt$
Suppose that we have some function, say $\beta (t)$, that is nonperiodic and finite in duration.
This means that $\beta(t)=0$ for some $T_\alpha < \left | t \right |$
Now let's make a periodic function $\gamma(t)$ by repeating $\beta(t)$ with a fundamental period $T_\zeta$.
The Fourier Series representation of $\gamma(t)$ is
$\gamma(t)=\sum_{k=-\infty}^\infty \alpha_k e^{j2\pi fkt}$ where $f={1\over T_\alpha}$
and $\alpha_k={1\over T_\alpha}\int_{-{T_\alpha\over 2}}^{{T_\alpha\over 2}} \gamma(t) e^{-j2\pi kt}\,dt$