Discrete-Time Fourier Transforms

Definition

$\large x[n]$ be a discrete-time signal. Then,

D T F T {x [n]} = X (e^{j ω}) = \sum_{n = - \infty}^{\infty} x [n] e^{- j ω n}

$\large DTFT\{x[n]\} = X(e^{j\omega})$ Discrete-Time Fourier Transform $\large x[n]$ .

Conversely,

x [n] = \frac{1}{2 π} \int_{- π}^{π} X (e^{j ω}) e^{j ω n} d ω

Note that the Discrete-Time Fourier Transform is obtained by restricting the $\large \mathcal{Z}$ -Transform to the unit circle on the complex plane.

Properties

Linearity

$\Large DTFT\{c_1x_1[n] + c_2x_2[n]\} = c_1X_1(e^{j\omega}) + c_2X_2(e^{j\omega})$

Time Advance

$\Large DTFT\{x[n + k]\} = e^{jk\omega}X(e^{j\omega})$

Time Delay

$\Large DTFT\{x[n - k]\} = e^{-jk\omega}X(e^{j\omega})$

Frequency Shifting

$\Large DTFT\{e^{j\omega_0 n}x[n]\} = X(e^{j(\omega - \omega_0)})$

Modulation

$\Large DTFT\{x[n]cos(\omega_0 n)\} = \frac{1}{2}X(e^{j(\omega + \omega_0)}) + \frac{1}{2}X(e^{j(\omega - \omega_0)})$

$n$

$\Large DTFT\{nx[n]\} = -j\frac{dX(e^{j\omega})}{d\omega}$

Folding

$\Large DTFT\{x[-n]\} = X(e^{-j\omega})$

Convolution

$\Large DTFT\{x*y\} = X(e^{j\omega})Y(e^{j\omega})$

Windowing

$\Large DTFT\{x[n] w[n]\} = \frac{1}{2\pi}\int_{2\pi} X(e^{j\theta}) W(e^{j(\omega - \theta)}) d\theta$

Parseval's Theorem

$\Large \sum\limits_{n = -\infty}^\infty x_1[n] x^*_2[n] = \frac{1}{2\pi} \int_{2\pi} X_1(e^{j\omega}) X^*_2(e^{j\omega}) d\omega$
$\Large \sum\limits_{n = -\infty}^\infty \abs{x[n]}^2 = \frac{1}{2\pi} \int_{2\pi} \abs{X(e^{j\omega})}^2 d\omega$