CTFT

T

→

\infty

x(t) = \tilde{x}(t)

for any finite time interval

As a polar form

X(j\omega) = |X(j\omega)|e^{j\angle X(j\omega)}

x(t)

is real, magnitude is even function of

\omega

and phase is an odd funtion of

\omega

Fourier transform

X(jt) = \int_{-\infty}^\infty x(t)e^{-jwt}dt

Inverse Fourier transform

x(t) = \frac{1}{2\pi}\int_{-\infty}^\infty X(j\omega)e^{j\omega t}d\omega

f(x) \cdot g(x) \xleftrightarrow{FT} F(jw) * G(jw)

Inner product → orthogonal → below functions → these properties

Mixing linearity and time shifting make it easy to compute complex function’s FT like this

CTFT Properties

\delta(t) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{j\omega t}d\omega \newline \delta(\omega) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{j\omega t}dt

Fourier Transform Examples