Fourier Transform

\begin{align*} \tilde{x}(t) &= \frac{1}{T} \sum_{k=-\infty}^{\infty}X_a \left( k\frac{2\pi}{T} \right) e^{j\frac{2\pi}{T}kt} \\ X_a \left( k\frac{2\pi}{T} \right) &= \int_{-\frac{T}{2}}^{\frac{T}{2}} \tilde{x}(t) e^{-j\frac{2\pi}{T}kt} dt \end{align*}

\begin{align*} \tilde{x}(n) &= \frac{1}{N} \sum_{k=0}^{N-1} \tilde{X}(k)e^{j\frac{2\pi}{N}kn} \\ \tilde{X}(k) &= \sum_{n=0}^{N-1} \tilde{x}(n) e^{-j\frac{2\pi}{N}nk} \end{align*}

\begin{align*} F(j\omega) &= \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt \\ &= |F(j\omega)|e^{j\phi(\omega)} \\ &= R(\omega) + jX(\omega) \\ f(t) &= \frac{1}{2\pi} \int_{-\infty}^{\infty} F(j\omega) e^{j\omega t} d\omega \end{align*}

\begin{align*} a_1 f_1(t) + a_2 f_2(t) &\leftrightarrow a_1 F_1(j\omega) + a_2 F_2(j\omega) \\ f_1(t) * f_2(t) &\leftrightarrow F_1(j\omega) F_2(j\omega) \\ f_1(t) f_2(t) &\leftrightarrow \frac{1}{2\pi} F_1(j\omega) * F_2(j\omega) \\ f(-t) &\leftrightarrow F(-j\omega) \\ f(t+t_0) &\leftrightarrow F(j\omega) e^{j\omega t_0} \\ f(t) e^{j\omega_0 t} &\leftrightarrow F[j(\omega - \omega_0)] \\ F(jt) &\leftrightarrow 2\pi f(-\omega) \\ f(at) &\leftrightarrow \frac{1}{|a|} F \left( j\frac{\omega}{a} \right) \\ \frac{d^nf(t)}{dt^n} &\leftrightarrow (j\omega)^n F(j\omega) \\ (-jt)^n f(t) &\leftrightarrow \frac{d^n F(j\omega)}{d\omega^n} \\ \int_{-\infty}^t f(\tau)d\tau, f(-\infty)=0. &\leftrightarrow \pi F(0) \delta(\omega) + \frac{1}{j\omega} F(j\omega) \\ \pi f(0) \delta(t) - \frac{1}{jt} f(t) &\leftrightarrow \int_{-\infty}^{\omega} F(j \eta) d\eta, F(-j\infty) = 0. \\ f(t) \sum_{n=-\infty}^{\infty} \delta(t-nT) &\leftrightarrow \frac{1}{T} \sum_{n=-\infty}^{\infty} F \left[ j \left( \omega - n\frac{2\pi}{T} \right) \right] \\ \frac{1}{\omega_0} \sum_{n=-\infty}^{\infty} f \left( t - n \frac{2\pi}{\omega_0} \right) &\leftrightarrow F(j\omega) \sum_{n=-\infty}^{\infty} \delta(\omega-n \omega_0) \end{align*}

\begin{align*} \delta(t) &\leftrightarrow 1 \\ \varepsilon(t) &\leftrightarrow \frac{1}{j\omega} + \pi \delta(\omega) \\ t\varepsilon(t) &\leftrightarrow j\pi \delta^{\prime}(\omega) - \frac{1}{\omega^2} \\ \delta^{(k)}(t) &\leftrightarrow (j\omega)^k \\ \delta(t-t_0) &\leftrightarrow e^{-j\omega t_0} \\ \text{sign}(t) &\leftrightarrow \frac{2}{j\omega} \\ \cos(\omega_0 t) &\leftrightarrow \pi [\delta(\omega+\omega_0) + \delta(\omega-\omega_0)] \\ \sin(\omega_0 t) &\leftrightarrow j\pi [\delta(\omega+\omega_0) - \delta(\omega-\omega_0)] \\ \text{rect}_{\tau}(t) &\leftrightarrow \tau \text{Sa} \left( \frac{\omega \tau}{2} \right) \\ \left( 1 - \frac{|t|}{\tau} \right) \text{rect}_{2\tau}(t) &\leftrightarrow \tau \text{Sa}^2 \left( \frac{\omega \tau}{2} \right) \\ e^{-at} \varepsilon(t), a > 0. &\leftrightarrow \frac{1}{a+j\omega} \\ te^{-at} \varepsilon(t), a > 0. &\leftrightarrow \frac{1}{(a+j\omega)^2} \\ \frac{t^{k-1}e^{-at}}{(k-1)!}\varepsilon(t), a > 0. &\leftrightarrow \frac{1}{(a+j\omega)^k} \\ e^{-a|t|}, a > 0. &\leftrightarrow \frac{2a}{\omega^2 + a^2} \\ e^{-a|t|}\text{sign}(t), a > 0. &\leftrightarrow -\frac{2j\omega}{\omega^2+a^2} \\ e^{-at} \cos(\omega_0t)\varepsilon(t), a > 0. &\leftrightarrow \frac{a+j\omega}{(a+j\omega)^2 + \omega_0^2} \\ e^{-at} \sin(\omega_0t)\varepsilon(t), a > 0. &\leftrightarrow \frac{\omega_0}{(a+j\omega)^2 + \omega_0^2} \\ e^{-a|t|}\cos(\omega_0t), a > 0. &\leftrightarrow \frac{2a(\omega^2 + \omega_0^2 + a^2)}{[\omega^2-(a^2+\omega_0^2)]^2 + 4a^2\omega^2} \\ e^{-a|t|}\sin(\omega_0t), a > 0. &\leftrightarrow -\frac{j4a\omega_0 \omega}{[\omega^2-(a^2+\omega_0^2)]^2 + 4a^2\omega^2} \\ e^{-\left(\dfrac{t}{\tau} \right)^{2}} &\leftrightarrow \sqrt{\pi} \tau e^{-\left(\dfrac{\omega\tau}{2}\right)^2} \\ \sum_{n=-\infty}^{\infty} \delta(t-nT) &\leftrightarrow \frac{2\pi}{T} \sum_{n=-\infty}^{\infty} \delta \left( \omega - n\frac{2\pi}{T} \right) \\ \sum_{n=-\infty}^{\infty} F_n e^{jn\omega_0t} &\leftrightarrow 2\pi \sum_{n=-\infty}^{\infty}F_n\delta(\omega - n\omega_0) \end{align*}

\begin{align*} X(e^{j\omega}) &= \sum_{n=-\infty}^{\infty} x(n) e^{-j\omega n} \\ x(n) &= \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega})e^{j\omega} d\omega \end{align*}

\begin{align*} x_1(n) * x_2(n) &\leftrightarrow X_1(e^{j\omega}) X_2(e^{j\omega}) \\ x^{*}(n) &\leftrightarrow X^{*}(e^{-j\omega}) \\ x(-n) &\leftrightarrow X(e^{-j\omega}) \\ x_e(n) &\leftrightarrow X_r(e^{j\omega}) \\ x_o(n) &\leftrightarrow jX_i(e^{j\omega}) \\ x_r(n) &\leftrightarrow X_e(e^{j\omega}) \\ jx_i(n) &\leftrightarrow X_o(e^{j\omega}) \end{align*}

where subscript \(r\), \(i\), \(e\), and \(o\) mean the real, imagnary, even, and odd parts of an expression, respectively, i.e,

\begin{align*} x(n) &= x_e(n) + x_o(n) \\ &= x_r(n) + j x_i(n) \\ x_e(n) &= \frac{x(n) + x^{*}(-n)}{2} \\ x_o(n) &= \frac{x(n) - x^{*}(-n)}{2} \\ X(e^{j\omega}) &= X_e(e^{j\omega}) + X_o(e^{j\omega}) \\ &= X_r(e^{j\omega}) + j X_i(e^{j\omega}) \\ X_e(e^{j\omega}) &= \frac{X(e^{j\omega}) + X^{*}(e^{-j\omega})}{2} \\ X_o(e^{j\omega}) &= \frac{X(e^{j\omega}) - X^{*}(e^{-j\omega})}{2}. \end{align*}

\begin{align*} X(k) &= \sum_{n=0}^{N-1} x(n) e^{-j\frac{2\pi}{N}nk} \\ x(n) &= \frac{1}{N} \sum_{k=0}^{N-1} e^{j\frac{2\pi}{N}kn} \end{align*}

\begin{align*} x^{*}(n) &\leftrightarrow X^{*}(N-k) \\ x^{*}(N-n) &\leftrightarrow X^{*}(k) \\ x_e(n) &\leftrightarrow X_r(k) \\ x_o(n) &\leftrightarrow jX_i(k) \\ x_r(n) &\leftrightarrow X_e(k) \\ jx_i(n) &\leftrightarrow X_o(k) \end{align*}

where subscript \(r\), \(i\), \(e\), and \(o\) mean the real, imagnary, conjugate symmetric, and conjugate antisymmetric parts of an expression, respectively, i.e,

\begin{align*} x(n) &= x_e(n) + x_o(n) \\ &= x_r(n) + j x_i(n) \\ x_e(n) &= \frac{x(n) + x^{*}(N-n)}{2} \\ x_o(n) &= \frac{x(n) - x^{*}(N-n)}{2} \\ X(k) &= X_e(k) + X_o(k) \\ &= X_r(k) + j X_i(k) \\ X_e(k) &= \frac{X(k) + X^{*}(N-k)}{2} \\ X_o(k) &= \frac{X(k) - X^{*}(N-k)}{2}. \end{align*}