Discrete-time (DT) Fourier Transforms Pairs and Properties

(used in ECE301, ECE438, ECE538)

DT Fourier transform and its Inverse
DT Fourier Transform $\,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\,$
Inverse DT Fourier Transform $\,x[n]=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{0}^{2\pi}\mathcal{X}(\omega)e^{j\omega n} d \omega\,$
DT Fourier Transform Pairs
$x[n] \$ $\longrightarrow$ $\mathcal{X}(\omega) \$
DTFT of a complex exponential $e^{jw_0n} \$ $\ 2\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \$
(info) DTFT of a rectangular window $w[n]= \$ $\text{add formula here} \$
$a^{n} u[n], |a|<1 \$ $\frac{1}{1-ae^{-j\omega}} \$
$(n+1)a^{n} u[n], |a|<1 \$ $\frac{1}{(1-ae^{-j\omega})^2} \$
$\sin\left(\omega _0 n\right) u[n] \$ $\frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right)$
$\cos\left(\omega _0 n\right) \$ $\pi \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)+\delta(\omega+\omega_0-2\pi k))$
$\sin\left(\omega _0 n\right) \$ $\frac{\pi}{j} \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)-\delta(\omega+\omega_0-2\pi k))$
$1 \$ $2\pi\sum^{\infty}_{k=-\infty}\delta(\omega-2\pi k)$
DTFT of a Periodic Square Wave

$\left\{\begin{array}{ll}1, & |n|<N_1,\\ 0, & N_1<|n|\leq\frac{N}{2}\end{array} \right. \text{ and } x[n+N]=x[n]$

$2\pi\sum^{\infty}_{k=-\infty}a_k\delta(\omega-\frac{2\pi k}{N})$
$\sum^{\infty}_{k=-\infty}\delta[n-kN]$ $\frac{2\pi}{N}\sum^{\infty}_{k=-\infty}\delta(\omega -\frac{2\pi k}{N})$
$\delta [n] \$ $1 \$
$u[n] \$ $\frac{1}{1-e^{-j\omega}}+\sum^{\infty}_{k=-\infty}\pi\delta(\omega-2\pi k)$
$\delta[n - n_0] \$ $e^{-j\omega n_0}$
$(n + 1)a^n u[n], \quad |a| < 1$ $\frac{1}{(1-ae^{-j\omega})^{2}}$
DT Fourier Transform Properties
$x[n] \$ $\longrightarrow$ $\mathcal{X}(\omega) \$
multiplication property $x[n]y[n] \$ $\frac{1}{2\pi} \int_{-\pi}^{\pi} X(\theta)Y(\omega-\theta)d\theta$
convolution property $x[n]*y[n] \$ $X(\omega)Y(\omega) \!$
time reversal $\ x[-n]$ $\ X(-\omega)$
Differentiation in frequency $\ nx[n]$ $\ j\frac{d}{d\omega}X(\omega)$
Linearity $ax[n]+by[n] \$ $aX(\omega)+bY(\omega) \$
Time Shifting $x[n - n_0] \$ $e^{-j\omega n_0}X(\omega)$
Frequency Shifting $e^{j\omega_0 n}x[n]$ $X(\omega - \omega_0) \$
Conjugation $x^* [n] \$ $X^* (-\omega) \$
Time Expansion $x_{(k)}[n]=\left\{\begin{array}{ll}x[n/k], & \text{ if n = multiple of k},\\ 0, & \text{else.}\end{array} \right.$ $X(k\omega) \$
Differentiating in Time $x[n] - x[n - 1] \$ $(1 - e^{-j\omega}) X (\omega) \$
Accumulation $\sum^{n}_{k=-\infty} x[k]$ $\frac{1}{1-e^{-j\omega}}X(\omega)$
Symmetry $x[n] \ \text{ real and even} \$ $X(\omega) \ \text{ real and even} \$
$x[n] \ \text{ real and odd} \$ $X(\omega) \ \text{ purely imaginary and odd} \$
Other DT Fourier Transform Properties
Parseval's relation $\sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|X( \omega )|^2d\omega$

ECE462 Survivor

Seraj Dosenbach