Definition of modulus for integers
Let \(n\in\mathbb{Z}\) and \(N \in \mathbb{N}\). The mod operator \[n \% N = \left\{ \begin{array}{cc}
\text{remainder}\left(\frac{n}{N}\right) & n \geq 0\\
N - \text{remainder}\left(\frac{|n|}{N}\right) & n < 0
\end{array}
\right.\] where \(\text{remainder}\) is the remainder after dividing \(n\) by \(N\).
Table of Representative Convolution Integrals
CT Convolution Table
| \(u(t)\) |
\(e^{a t}u(t)\) |
\(\frac{1-e^{a t}}{-a}u(t)\) |
| \(u(t)\) |
\(u(t)\) |
\(tu(t)\) |
| \(e^{a_1 t}u(t)\) |
\(e^{a_2 t}u(t)\) |
\(\frac{e^{a_1 t}-e^{a_2 t}}{a_1 - a_2}u(t)\) for \(a_1 \neq a_2\) |
| \(e^{a t}u(t)\) |
\(e^{a t}u(t)\) |
\(te^{a t}u(t)\) |
| \(te^{a_1 t}u(t)\) |
\(e^{a_2 t}u(t)\) |
\(\frac{e^{a_2 t}-e^{a_1 t} + (a_1-a_2)te^{a_1 t}}{(a_1 - a_2)^2}u(t)\) for \(a_1 \neq a_2\) |
| \(e^{a_1 t}\cos(\beta t + \theta)u(t)\) |
\(e^{a_2 t}u(t)\) |
\(\frac{\cos(\theta - \phi)e^{a_2 t} - e^{a_1 t}\cos(\beta t + \theta - \phi)}{\sqrt{(a_1 + a_2)^2 + \beta^2}}u(t)\) |
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\(\phi = \arctan\left( \frac{-\beta}{a_1 + a_2}\right)\) |
Table of Representative Convolution Sums
DT Convolution Table
| \(u[n]\) |
\(u[n]\) |
\((n+1)u[n]\) |
| \(\gamma^{n}u[n]\) |
\(u[n]\) |
\(\frac{1-\gamma^{n+1}}{1-\gamma}u[n]\) |
| \(\gamma_1^{n}u[n]\) |
\(\gamma_2^{n}u[n]\) |
\(\frac{\gamma_1^{n+1}-\gamma_2^{n+1}}{\gamma_1-\gamma_2}u[n]\) for \(\gamma_1 \neq \gamma_2\) |
| \(\gamma^{n}u[n]\) |
\(\gamma^{n}u[n]\) |
\((n+1)\gamma^{n}u[n]\) |
| \(|\gamma_1|^{n}\cos\left(\beta n + \theta \right)u[n]\) |
\(|\gamma_2|^{n}u[n]\) |
\(\frac{1}{R}\left[ |\gamma_1|^{n+1}\cos\left( \beta (n+1) + \theta - \phi\right) - |\gamma_2|^{n+1}\cos\left( \theta - \phi\right)\right]u[n]\) |
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\(R = \left[ |\gamma_1|^2 + |\gamma_2|^2 -2|\gamma_1||\gamma_2|\cos(\beta)\right]^{\frac{1}{2}}\) |
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\(\phi = \arctan\left( \frac{|\gamma_1|\sin(\beta)}{|\gamma_1|\cos(\beta) - |\gamma_2|} \right)\) |
Table of Representative CT Fourier Transform Pairs
CT Fourier Transform Table
| \(1\) |
\(2\pi\delta(\omega)\) |
| \(\delta(t)\) |
\(1\) |
| \(u(t)\) |
\(\pi\delta(\omega) + \frac{1}{j\omega}\) |
| \(e^{-at}u(t)\) |
\(\frac{1}{a + j\omega}\) for \(Re\{a\} > 0\) |
| \(te^{-at}u(t)\) |
\(\frac{1}{\left(a + j\omega\right)^2}\) for \(Re\{a\} > 0\) |
| \(e^{j\omega_0 t}\) |
\(2\pi\delta(\omega-\omega_0)\) |
| \(\cos(\omega_0 t)\) |
\(\pi\left[ \delta(\omega-\omega_0) + \delta(\omega+\omega_0)\right]\) |
| \(\sin(\omega_0 t)\) |
\(j\pi\left[ \delta(\omega+\omega_0) - \delta(\omega-\omega_0)\right]\) |
| \(e^{-at}\cos(\omega_0 t)u(t)\) |
\(\frac{a+j\omega}{(a+j\omega)^2 + \omega_0^2}\) for \(Re\{a\} > 0\) |
| \(e^{-at}\sin(\omega_0 t)u(t)\) |
\(\frac{\omega_0}{(a+j\omega)^2 + \omega_0^2}\) for \(Re\{a\} > 0\) |
| \(\delta(t - t_0)\) |
\(e^{-j t_0 \omega}\) |
| \(K_0\) |
\(2 K_0 \pi \delta(\omega)\) |
| \(e^{-a|t|}\), \(\text{Re}\{a\} > 0\) |
\(\frac{2a}{a^2 + \omega^2}\) |
| \(u(t + T) - u(t - T)\) |
\(2T \frac{\sin{(\omega T)}}{\omega T}\) |
| \(\frac{\sin{({W}t)}}{W t}\) |
\(\frac{\pi}{W} [u(\omega + W) - u(\omega - W)]\) |
| \(e^{-\frac{t^2}{2 \sigma^2}}\) |
\(\sigma \sqrt{2 \pi} e^{-\frac{\sigma^2 \omega^2}{2}}\) |
| \(\sum\limits_{k=-\infty}^{\infty} a_k e^{j k \omega_0 t}\) |
\(2 \pi \sum\limits_{k=-\infty}^{\infty} a_k \delta{(\omega - k \omega_0)}\) |
| \(\sum\limits_{n=-\infty}^{\infty} \delta(t - nT)\) |
\(\omega_0 \sum\limits_{k=-\infty}^{\infty} \delta{(\omega - k \omega_0)}\), \(\omega_0 = \frac{2 \pi}{T}\) |
Table of Representative DT Fourier Transform Pairs
DT Fourier Transform Table
| \(\delta[n]\) |
\(1\) |
| \(\delta[n - n_0]\) |
\(e^{-j \omega n_0}\) |
| \(u[n]\) |
\(\frac{1}{1 - e^{-j \omega}} + \pi \sum\limits_{k = - \infty}^{\infty} \delta(\omega - 2 k \pi)\) |
| \(K_0\) |
\(2 K_0 \pi \sum\limits_{k = - \infty}^{\infty} \delta(\omega - 2 k \pi)\) |
| \(a^n u[n]\), \(|a| < 1\) |
\(\frac{1}{(1 - ae^{-j \omega)}}\) |
| \(n a^n u[n]\), \(|a| < 1\) |
\(\frac{a e^{-j \omega}}{(1 - a e^{-j \omega})^2}\) |
| \(a^{|n|}\), \(|a| < 1\) |
\(\frac{1-a^2}{1 - 2 a \cos{\omega} + a^2}\) |
| \(e^{j \omega_0 n}\) |
\(2 \pi \sum\limits_{k = - \infty}^{\infty} \delta{(\omega - \omega_0 - 2 k \pi)}\) |
| \(\cos{(\omega_0 n)}\) |
\(\pi \sum\limits_{k = - \infty}^{\infty} [\delta(\omega + \omega_0 - 2 k \pi) + \delta(\omega - \omega_0 - 2 k \pi)]\) |
| \(\sin{(\omega_0 n)}\) |
\(j \pi \sum\limits_{k = - \infty}^{\infty} [\delta(\omega + \omega_0 - 2 k \pi) - \delta(\omega - \omega_0 - 2 k \pi)]\) |
| \(u[n + n_0] - u[n - n_0]\) |
\(\frac{\sin{(\omega (n_0 + 0.5))}}{\sin{0.5 \omega}}\) |
| \(\frac{\sin{({W}n)}}{\pi n}\) |
\(\sum\limits_{k = - \infty}^{\infty} X_1(\omega - 2 k \pi), X_1(\omega) = \begin{array}{cc} 1 & 0 \le |\omega| \le W, 0 < W < \pi\\ 0 & W < |\omega| \le \pi, 0 < W < \pi \end{array}\) |
| \(\sum\limits_{k=n_0}^{n_0+N-1} a_k e^{j k \omega_0 n}\) |
\(2 \pi \sum\limits_{k=-\infty}^{\infty} a_k \delta{(\omega - k \omega_0)}\), \(\omega_0 = \frac{2 \pi}{N}\), |
| \(\sum\limits_{k=-\infty}^{\infty} \delta[n - kN]\) |
\(\omega_0 \sum\limits_{k=-\infty}^{\infty} \delta{(n - k \omega_0)}\), \(\omega_0 = \frac{2 \pi}{N}\), |