Difference between revisions of "CT DT Fourier transform ECE438F10" - Rhea

Revision as of 06:42, 23 October 2010

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) $