Page 27 - Schaum's Outline of Theory and Problems of Signals and Systems

SIGNALS AND SYSTEMS [CHAP. 1

Real Exponential Sequences:
If C and a in Eq. (1.57) are both real, then x[n] is a real exponential sequence. Four
distinct cases can be identified: a > 1,0 < a < 1, - 1 < a < 0, and a < - 1. These four real
exponential sequences are shown in Fig. 1-12. Note that if a = 1, x[n] is a constant
sequence, whereas if a = - 1, x[n] alternates in value between +C and -C.

D. Sinusoidal Sequences:
A sinusoidal sequence can be expressed as

If n is dimensionless, then both R, and 0 have units of radians. Two examples of
sinusoidal sequences are shown in Fig. 1-13. As before, the sinusoidal sequence in Eq.
(1.58) can be expressed as

As we observed in the case of the complex exponential sequence in Eq. (1.52), the same G.
observations [Eqs. (1.54) and (1.5611 also hold for sinusoidal sequences. For instance, the
sequence in Fig. 1-13(a) is periodic with fundamental period 12, but the sequence in Fig.
l-13( b) is not periodic.

1.5 SYSTEMS AND CLASSIFICATION OF SYSTEMS
A. System Representation:

A system is a mathematical model of a physical process that relates the input (or
excitation) signal to the output (or response) signal.
Let x and y be the input and output signals, respectively, of a system. Then the system
is viewed as a transformation (or mapping) of x into y. This transformation is represented
by the mathematical notation

where T is the operator representing some well-defined rule by which x is transformed
into y. Relationship (1.60) is depicted as shown in Fig. 1-14(a). Multiple input and/or
output signals are possible as shown in Fig. 1-14(b). We will restrict our attention for the
most part in this text to the single-input, single-output case.

Y - XI b
System -
T Sy stem

(a) (b)
Fig. 1-14 System with single or multiple input and output signals.