**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