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

Revision as of 10:18, 30 October 2010

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 x_1(t) + \beta x_2(t)] = \alpha X_1(\omega) + \beta X_2(\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 |} \cdot 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) $

The Discrete Time Fourier Transform (DTFT)

Back to ECE438, Fall 2010, Prof. Boutin