# Dragon Notes

UNDER CONSTRUCTION
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# DTFT Pairs

 Eq $$\ds \hspace{10px} x[n] = \frac{1}{2\pi}\int_{\Omega_1}^{\Omega_1 + 2\pi}\hspace{-20px}\bn{X}(e^{j\Omega})e^{j\Omega n}d\Omega$$ $$\ds\bn{X}(e^{j\Omega}) = \iisum{n} x[n] e^{-j\Omega n} = \t{DTFT}\lrbra{x[n]}$$ $$[1]$$ $$\delta[n]$$ ⬌ $$\ds 1$$ $$[2]$$ $$\delta[n-m]$$ ⬌ $$\ds e^{-jm\Omega}$$ $$[3]$$ $$1$$ ⬌ $$\ds 2\pi \iisum{k} \delta(\Omega -2\pi k)$$ $$[4]$$ $$u[n]$$ ⬌ $$\ds \frac{e^{j\Omega}}{e^{j\Omega}-1} + \iisum{k}\pi\delta(\Omega - 2\pi k)$$ $$[5]$$ $$\bn{a}^n u[n]$$ ⬌ $$\ds \frac{e^{j\Omega}}{e^{j\Omega}-\bn{a}}$$ $$[6]$$ $$n\bn{a}^n u[n]$$ ⬌ $$\ds \frac{\bn{a}e^{j\Omega}}{(e^{j\Omega}-\bn{a})^2}$$ $$[7]$$ $$e^{j\Omega_0 n}$$ ⬌ $$\ds 2\pi \iisum{k}[\delta(\Omega - \Omega_0 - 2\pi k) + \delta(\Omega + \Omega_0 - 2\pi k)]$$ $$[8]$$ $$\bn{a}^{-n} u[-n-1]$$ ⬌ $$\ds \frac{\bn{a}e^{j\Omega}}{1-\bn{a}e^{j\Omega}}$$ $$[9]$$ $$\Cos{\Omega_0 n}$$ ⬌ $$\ds \pi \iisum{k}[\delta(\Omega - \Omega_0 - 2\pi k) + \delta(\Omega + \Omega_0 -2\pi k)]$$ $$[10]$$ $$\Sin{\Omega_0 n}$$ ⬌ $$\ds \frac{\pi}{j} \iisum{k}[\delta(\Omega - \Omega_0 - 2\pi k) - \delta(\Omega + \Omega_0 -2\pi k)]$$ $$[11]$$ $$\bn{a}^n \Cos{\Omega_0 n + \theta}u[n]$$ ⬌ $$\ds \frac{e^{j2\Omega}\Cos{\theta} - \bn{a}e^{j\Omega}\Cos{\Omega_0 - \theta}}{e^{j2\Omega} - 2\bn{a}e^{j\Omega}\Cos{\Omega_0}+\bn{a}^2}$$ $$[12]$$ \begin{align}\t{rect}\lrbra{n/N} &\\=u[n+N] - u[n-1-N] & \end{align} ⬌ $$\ds \frac{\t{sin}[\Omega\lrpar{N+\sfrac{1}{2}}]}{\sin(\sfrac{\Omega}{2})}$$ $$[13]$$ $$\ds\frac{\t{sin}[\Omega_0 n]}{\pi n}$$ ⬌ $$\ds \iisum{k}[u(\Omega + \Omega_0 - 2\pi k) - u(\Omega - \Omega_0 -2\pi k)]$$