Operations on discrete signals
Or a discrete signal \(\left\{x(n)\right\}=\left\{x\left(n\right),\ n\in\mathbb{Z}\right\}\) , multiplication by a scalar, time shift, sum, and product of two discrete signals are discrete signals.
\(a\left\{x(n)\right\}=\left\{ax\left(n\right),\ n\in\mathbb{Z}\right\}\)
\(\left\{x(n)\right\}+\left\{y(n)\right\}=z\left(n\right)=\left\{x\left(n\right)+y\left(n\right),\ n\in\mathbb{Z}\right\}\)
\(\left\{y(n)\right\}=\left\{z\left(n-n_0\right),\ n\in\mathbb{Z}\right\}\)
\(\left\{x\left(n\right)\right\}.\left\{y(n)\right\}=z\left(n\right)=\left\{x\left(n\right).y(n),\ n\in\mathbb{Z}\right\}\)
Note :
Noticed :
The sum of two periodic signals is not strongly periodic. When the sum fails to produce a periodic signal, we refer to the signals as having incommensurable periods, indicating a lack of rationality in their periods. We refer to these signals as quasi-periodic or pseudo-periodic signals.