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. According to Wikipedia's definition (I couldn't find a better one, unfortunately), the Fourier transform is an operation that transforms one complex-valued function of a real variable into another. In signal processing, the domain of the original function is typically in the time domain, while the domain of the new function is typically called the frequency domain. The new function itself is called the frequency domain representation of the original function and it describes which frequencies are present in the original function.
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) $
