Continuous-Time and DiscreteContinuous-Time and Discrete-Time Signals

A signal x(t) is a continuous-time signal if t is a continuous variable. If t is a discrete
variable, that is, x(t) is defined at discrete times, then x(t) is a discrete-time signal. Since a
discrete-time signal is defined at discrete times, a discrete-time signal is often identified as
a sequence of numbers, denoted by {x,) or x[n], where n = integer. Illustrations of a
continuous-time signal x(t) and of a discrete-time signal x[n] are shown in Fig.


A discrete-time signal x[n] may represent a phenomenon for which the independent
variable is inherently discrete. For instance, the daily closing stock market average is by its
nature a signal that evolves at discrete points in time (that is, at the close of each day). On
the other hand a discrete-time signal x[n] may be obtained by sampling a continuous-time signal x(t) such as

x(to), +,)' . 7 ~ ( t , )., . *

or in a shorter form as
x[O], x[l], ..., x[n], . ..
or xo, x ~ ,. .. , x,, . . .
where we understand that
x, =x[n] =x(t,)
and x,'s are called samples and the time interval between them is called the sampling
interval. When the sampling intervals are equal (uniform sampling), then
x,, =x[n] =x(nT,)
where the constant T, is the sampling interval.
A discrete-time signal x[n] can be defined in two ways:
1. We can specify a rule for calculating the nth value of the sequence.