Page 13 - Schaum's Outline of Theory and Problems of Signals and Systems

SIGNALS AND SYSTEMS [CHAP. 1

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. For example,

2. We can also explicitly list the values of the sequence. For example, the sequence
shown in Fig. l-l(b) can be written as

(x,) = (..., 0,0,1,2,2,1,0,1,0,2,0,0 ,... )
T

We use the arrow to denote the n = 0 term. We shall use the convention that if no
arrow is indicated, then the first term corresponds to n = 0 and all the values of the
sequence are zero for n < 0.

(c,) = a(a,) + C, = aa, a = constant

B. Analog and Digital Signals:
If a continuous-time signal x(l) can take on any value in the continuous interval (a, b),
where a may be - 03 and b may be + m, then the continuous-time signal x(t) is called an
analog signal. If a discrete-time signal x[n] can take on only a finite number of distinct
values, then we call this signal a digital signal.

C. Real and Complex Signals:
A signal x(t) is a real signal if its value is a real number, and a signal x(t) is a complex
signal if its value is a complex number. A general complex signal ~(t) is a function of the