Transforms

Transforms are decompositions of a function $f(x)$ into some basis functions $Ø(x, u)$ . $u$ indicates which basis function.

Important transform pairs:

$f(x)=1 \Leftrightarrow F(u)=\delta(u)$
$f(x)=e^{j2\pi f_0x} \Leftrightarrow F(u)=\delta(u- f_0)$
$f(x)=\cos(2\pi f_0x) \Leftrightarrow F(u)=\frac{1}{2}(\delta(u-f_0)+\delta(u+f_0))$
$f(x)=\sin(2\pi f_0x) \Leftrightarrow F(u)=1(\delta(u-f_0)-\delta(u+f_0))$
$f (x) = \begin{cases} 1, x < x_0 \\ 0, \text{otherwise} \end{cases} \Leftrightarrow F(u) = \frac{\sin(2\pi x_0u)}{\pi u} = 2x_0 \mathrm{sinc}(2x_0u)$ , where, $\mathrm{sinc}(t)=\frac{\sin(\pi t)}{\pi t}$

delta function: $\delta(x)=\infty$ , if $x=0$ , else (when $x \neq 0$ ), $\delta(x)=0$ . $\int_{-\infty}^{\infty} \delta(x) \, dx = 1$

Decomposition in vector space:

$f = \alpha_1 \phi_1 + \alpha_2 \phi_2 + \alpha_3 \phi_3$