Page 16 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 11 SIGNALS AND SYSTEMS 5

signal x(t) (known as a dc signal). For a constant signal x(t) the fundamental period is
undefined since x(t) is periodic for any choice of T (and so there is no smallest positive
value). Any continuous-time signal which is not periodic is called a nonperiodic (or
aperiodic ) signal.
Periodic discrete-time signals are defined analogously. A sequence (discrete-time
signal) x[n] is periodic with period N if there is a positive integer N for which
x[n + N] =x[n] all n (1.9)

An example of such a sequence is given in Fig. 1-3(b). From Eq. (1.9) and Fig. 1-3(b) it
follows that

for all n and any integer m. The fundamental period No of x[n] is the smallest positive
integer N for which Eq. (1.9) holds. Any sequence which is not periodic is called a
nonperiodic (or aperiodic sequence.
Note that a sequence obtained by uniform sampling of a periodic continuous-time
signal may not be periodic (Probs. 1.12 and 1.13). Note also that the sum of two
continuous-time periodic signals may not be periodic but that the sum of two periodic
sequences is always periodic (Probs. 1.14 and 1 .l5).

G. Energy and Power Signals:
Consider v(t) to be the voltage across a resistor R producing a current dt). The
instantaneous power p( t) per ohm is defined as

Total energy E and average power P on a per-ohm basis are
3:
E=[ i2(t)dt joules
-?O
i2(t) dt watts

For an arbitrary continuous-time signal x(t), the normalized energy content E of x(t) is
defined as

The normalized average power P of x(t) is defined as

Similarly, for a discrete-time signal x[n], the normalized energy content E of x[n] is
defined as