Some General Purpose Formulas and Definitions
| General Purpose Formulas | |
|---|---|
| Series | |
| Finite Geometric Series Formula | $ \sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &, \text{ else}\end{array}\right. $ |
| Infinite Geometric Series Formula | $ \sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1}{1-x}&, \text{ if } |x|\leq 1\\ \text{diverges} &, \text{ else }\end{array}\right. $ |
| Euler's Formula and Related Equalities | |
| Euler's formula | $ e^{jw_0t}=\cos w_0t+j\sin w_0t $ |
| Cosine function in terms of complex exponentials | $ \cos\theta=\frac{e^{j\theta}+e^{-j\theta}}{2} $ |
| Sine function in terms of complex exponentials | $ \sin\theta=\frac{e^{j\theta}-e^{-j\theta}}{2j} $ |
| Definition of some Basic Functions (what engineers call "Signals") | |
| sinc function | $ sinc(\theta)=\frac{sin(\pi\theta)}{\pi\theta} $ |
| CT unit step function | $ u(t)=\left\{ \begin{array}{ll}1, & t\geq 0 \\ 0, & \text{ else}\end{array}\right., \text{ for }t\in {\mathbb R} $ |
| DT unit step function | $ u[n]=\left\{ \begin{array}{ll}1, & n\geq 0 \\ 0, & \text{ else}\end{array}\right., \text{ for }n\in {\mathbb Z} $ |
| Function (or Signal) Metrics | |
| CT signal energy | $ E_\infty=\int_{-\infty}^\infty | x(t) |^2 dt $ |
| DT signal energy | $ E_\infty=\sum_{n=-\infty}^\infty | x[n] |^2 $ |
