A work in progress.
The Continuous Time Fourier Transform (CTFT)
CTFT:
$ X(\omega) = \int_{-\infty}^{\infty} \! x(t)e^{-j \omega t} dt $
Inverse CTFT:
$ x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \! X(\omega)e^{j \omega t} dw $
Example:
Let $ x(t) = \delta (t) $
$ \begin{align} X(\omega) &= \int_{-\infty}^{\infty} \! x(t)e^{-j \omega t} dt \\ &= \int_{-\infty}^{\infty} \! \delta (t)e^{-j \omega t} dt \\ &= 1\end{align} $
Therefore, CTFT of $ \delta (t) = 1 $
Properties
Linearity:
$ CTFT[x(t)] = X(\omega) $
Then $ CTFT[\alpha x1(t) + \beta x2(t)] = \alpha X1(\omega) + \beta X2(\omega) $
Time Reversal:
$ CTFT[x(t)] = X(\omega) $
Then $ CTFT[x(-t)] = X(-\omega) $
Time/Frequency Shift:
Time: $ CTFT[x(t-T)] = X(\omega)e^{-j \omega T} $
Frequency: $ CTFT[x(t)e^{j W t}] = X(\omega - W) $
Time Scaling:
$ CTFT[x(\alpha t)] = \frac{1}{|\alpha |} = X(\frac{\omega}{\alpha}) $
Conjugate Symmetry:
Assume x(t) is real,
$ CTFT[x(t)] = X(\omega) $
Then $ X(\omega) = X^*(\omega) $
Time Domain Multiplication/Convolution:
$ CTFT[x(t)\cdot y(t)] = \frac{1}{2\pi}X(\omega)*Y(w) $
$ CTFT[x(t)*y(t)] = X(\omega)Y(\omega) $
