Difference between revisions of "Table DT Fourier Transforms" - Rhea

Revision as of 07:57, 3 October 2012

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Discrete-time Fourier Transform Pairs and Properties
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} \$		$\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	$\sum_{n=-\infty}^{\infty}\left\| x[n] \right\|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}\|X(e^{j\omega})\|^2d\omega$

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 | align="right" style="padding-right: 1em;" |
 | align="right" style="padding-right: 1em;" |
-| <math>\cos\left(\omega _0 n\right) \ </math>
+| <math>\sin\left(\omega _0 n\right) \ </math>
 |
 | <math>\frac{\pi}{j} \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)-\delta(\omega+\omega_0-2\pi k))</math>