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.

Deterministic and Random Signals

Deterministic and Random Signals:

Deterministic signals are those signals whose values are completely specified for any
given time. Thus, a deterministic signal can be modeled by a known function of time I .

Random signals are also called non deterministic signals are those signals that take random values at any given time and must be characterized statistically.

Even and Odd Signals

signal x ( t ) or x[n] is referred to as an even signal if

x ( - t ) = x ( r )
x [ - n ] = x [ n ]

A signal x ( t ) or x[n] is referred to as an odd signal if

Examples of even x ( - t ) = - x ( t ) , x [ - n ] = - x [ n ] and odd signals are shown in pictures

Periodic and Nonperiodic Signals

A continuous-time signal x ( t ) is said to be periodic with period T if there is a positive
nonzero value of T for which

x(t + T ) = x ( t ) all t

An example of such a signal is given in fig. From Eq it follows
that

x (t+ mT) = x(t)

for all t and any integer m. The fundamental period T, of x ( t ) is the smallest positive
value of T for which Eq holds. Note that this definition does not work for a constant

Feedback Systems

A special class of systems of great importance consists of systems having feedback. In a
feedback system, the output signal is fed back and added to the input to the system

Stable Systems

A system is bounded-input/bounded-output (BIBO) stable if for any bounded input x
defined by

x < k1


the corresponding output y is also bounded defined by

x < k2


where k1 , and k2, are finite real constants. Note that there are many other definitions of
stability.

Linear Time-Invariant Systems

If the system is linear and also time-invariant, then it is called a linear rime-invariant
(LTI) system.