A signal is a quantity that varies with an independent variable and carries information about a physical or abstract process. In many engineering applications the independent variable is time.

A continuous-time signal is written

$$x(t),$$

while a discrete-time signal is written

$$x[n].$$

Continuous and discrete time

A continuous-time signal is defined for every time in an interval. A discrete-time signal is defined only at separated index values such as

$$\ldots,x[-1],x[0],x[1],\ldots$$

The index $n$ need not itself be a physical time; a sampling period can map it to times $t=nT_s$.

Amplitude is separate from time

A discrete-time signal can still have continuous-valued amplitudes. Discretizing the time axis and quantizing amplitude are different operations.

Common operations

Signals can be shifted, scaled and combined. For example,

$$y(t)=x(t-t_0)$$

shifts a continuous-time signal by $t_0$, while

$$y[n]=x[n-n_0]$$

shifts a discrete-time sequence.

Periodic signals

A continuous-time signal is periodic when some $T>0$ satisfies

$$x(t+T)=x(t)$$

for all relevant $t$. A discrete-time signal is periodic when an integer $N>0$ satisfies

$$x[n+N]=x[n].$$

Signals provide a common language for later topics such as sampling, filtering, communication and control.