1.1 Continuous-time and discrete-time signals
Continuous-time signal $x (t) $.
Discrete-time signal $x[n]$.
Total energy within $t_{1} \leq t \leq t_{2}$ for a continuous time signal $x (t) $ defined as $\int_{t_{1}}^{t_{2}}\left | X (t) \right |^{2}dt$.
where $\left | X (t) \right |$ is a modulo of $x$ (possibly plural).
Dividing it by the interval length $t_{2}-t_{1}$ the average power within that interval.
The total energy within $n_{1} \leq n \leq n_{2}$ is defined as $x[n]$ for a continuous time signal $\sum_{n=n_{1}}^{n_{2}}\left | X[n] \right |^{2}$.
Dividing it by the number of points in the interval $n_{2}-n_{1}+1$ the average power within that interval.
Power and energy within an infinite range
$e_{\infty}\triangleq \lim_{t\rightarrow \infty}\int_{-t}^{t}\left in continuous-time situations | X (t) \right |^2dt = \int_{-\infty}^{\infty}\left | X (t) \right |^2dt$.
In the case of discrete time $e_{\infty}\triangleq \lim_{n\rightarrow \infty}\sum_{n=-n}^{+n}\left | X[n] \right |^2 = \sum_{n=-\infty}^{+\infty}\left | X[n] \right |^2$.
Average power in an infinite interval
$p_{\infty}\triangleq \lim_{t\rightarrow \infty} \frac{1}{2t}\int_{-t}^{t}\left in the case of continuous time | X (t) \right |^2dt$.
$p_{\infty}\triangleq \lim_{n\rightarrow \infty}\frac{1}{2n+1}\sum_{n=-n}^{+n}\left in discrete-time situations | X[n] \right |^2$.
1) The signal has a finite total energy $e_{\infty}< \infty $, the average power must be zero $p_{\infty}=0$.
2) Average Power limited $p_{\infty}> 0$, there is bound to be $e_{\infty}=\infty $.
3) $P _{\infty}$ and $e_{\infty}$ are not limited.
1.2 Transformation of the Independent variable
Move Time Shift
Time reversal reversal
Time scale Transformation Tmie scaling
$x (\alpha T+\beta) $
Periodic signal
Cycle periodic
$x (t) =x (X+MT) $
$x [n]=x[n+mn]$
Fundamental period: The minimum positive value of the $t_{0}$ fundamental period
Non-periodic aperiodic
Even (even) signal
$x (-T) =x (t) $
$x [-n]=x[n]$
Odd (odd) signal
$x (-T) =-x (t) $
$x [-n]=-x[n]$
Any signal can be decomposed into a sum of an even signal and a singular signal.
$Ev \left \{x (t) \right \}=\frac{1}{2}[x (t) +x (-t)]$
$Od \left \{x (t) \right \}=\frac{1}{2}[x (T)-X (-T)]$
$Ev \left \{x (t) \right \}$ and $od\left \{x (t) \right \}$ are referred to as $x (t) $ of the I (even part) and odd (odd part) respectively.
Signals and Systems Second Edition-alan v. Oppenheim,alan S. willsky,s. Hamid Nawab