2.9b.2 Continuous-Time Fourier Transform

Continuous-Time Fourier Transform (CTFT)

The continuous-time Fourier transform is a mathematical tool used in signal processing to analyze the frequency content of non-periodic continuous-time signals into continuous functions of frequency, providing a complete representation of the signal’s frequency spectrum, such that
$x(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega ⟺X(\omega )={\int }_{-\infty }^{\infty }x(t)e^{-j\omega t}dt.$
As in discrete time, the frequency response of $x(t)$ is equivalent to its Fourier transform.

Intuition

Properties

(Dirac Input) $x(t)=\delta (t)⟺X(\omega )=1$.

(Unit Input) $x(t)=1⟺X(\omega )=2\pi \delta (\omega )$

(Sine Input) $x(t)=\frac{A}{\pi t}\sin (Bt)⟺X(\omega )=\{\begin{array}{ll}A& \text{if }-B\le \omega \le B\\ 0& \text{otherwise}\end{array}$.

(Time Shift) $x(t-t_0)⟺X(\omega )e^{-j\omega t_0}$.

X(\omega) &= \int^{\infty}_{-\infty}x(t-t_0)e^{-j\omega t}dt \\ &= \int^{\infty}_{-\infty}x(u)e^{-j\omega(u+t_0)}du \\ &= e^{-j\omega t_0} \int^{\infty}_{-\infty}x(u)e^{-j\omega u}du \\ &= e^{-j\omega t_0}X(\omega), \end{align}$$ where $u=t+t_0$ is a change of variables. #### (*Frequency Shift*) $X(\omega - \omega_0) \iff x(t)e^{-j\omega_0 t}$. +the special case $x(t)=e^{j\omega_0 t} \iff X(\omega)=2\pi \delta(\omega-\omega_{0})$ #### (*Convolution*) $y(t) = (x * h)(t) \iff Y(\omega) = X(\omega) H(\omega)$. #### (*Modulation*) $y(t) = x(t) \cdot c(t) \iff Y(\omega) = \frac{1}{2\pi}(X * C)(\omega)$. $$\begin{align} Y(\omega) &= \int^{\infty}_{-\infty}y(t)e^{-j\omega t}dt \\ &=\int^{\infty}_{-\infty} x(t)c(t)e^{-j\omega t}dt \\ &= \int^{\infty}_{-\infty}c(t) \left(\frac{1}{2\pi}\int^{\infty}_{-\infty} X(\lambda)e^{j\lambda t}d\lambda\right) e^{-j\omega t}dt \\ &= \frac{1}{2\pi}\int^{\infty}_{-\infty} X(\lambda)\left(\int^{\infty}_{-\infty}c(t)e^{j(\omega - \lambda) t}dt\right)d\lambda \\ &= \frac{1}{2\pi}\int^{\infty}_{-\infty} X(\lambda)C(\omega - \lambda)d\lambda \\ &= \frac{1}{2\pi} (X * C)(\omega), \end{align}$$ where the two integrals are swapped in order to isolate the coefficient $C(\omega - \lambda)$. #### (*Differentiation*) $\frac{d^k}{dt^k}x(t) \iff (j\omega)^kX(\omega )$. #### (*Integration*) $\int x(t)dt = \frac{1}{j\omega}X(\omega )$ # Examples >[!example] AM Radio > >