A work in progress.
The Continuous Time Fourier Transform (CTFT)
The CTFT transforms an infinite length continuous signal in the time domain into an infinite length signal in the frequency domain. The formulae are:-
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) $
