563489578934

Sec. 2–1 Properties of Signals and Noise 37
It is seen that this operator is a linear operator, since, from Eq. (2–1), the average of the
sum of two quantities is the same as the sum of their averages: †
8a 1 w 1 (t) + a 2 w 2 (t)9 = a 1 8w 1 (t)9 + a 2 8w 2 (t)9
(2–2)
Equation (2–1) can be reduced to a simpler form given by Eq. (2–4) if the operator is operating
on a periodic waveform.
DEFINITION.
A waveform w(t) is periodic with period T 0 if
w(t) = w(t + T 0 ) for all t
where T 0 is the smallest positive number that satisfies this relationship. ‡
(2–3)
For example, a sinusoidal waveform of frequency f 0 = 1/T 0 hertz is periodic, since it
satisfies Eq. (2–3). From this definition, it is clear that a periodic waveform will have significant
values over an infinite time interval (-q, q) . Consequently, physical waveforms
cannot be truly periodic, but they can have periodic values over a finite time interval. That
is, Eq. (2–3) can be satisfied for t over some finite interval, but not for all values of t.
THEOREM.
reduced to
If the waveform involved is periodic, the time average operator can be
8[ #
1 ]9 =
T 0 L
T 0 /2+a
-T 0 /2+a
[ # ] dt
(2–4)
where T 0 is the period of the waveform and a is an arbitrary real constant, which may
be taken to be zero.
Equation (2–4) readily follows from Eq. (2–1) because, referring to Eq. (2–1), integrals
over successive time intervals T 0 seconds wide have identical areas, since the waveshape is
periodic with period T 0 . As these integrals are summed, the total area and T are proportionally
larger, resulting in a value for the time average that is the same as just integrating over one
period and dividing by the width of that interval, T 0 .
In summary, Eq. (2–1) may be used to evaluate the time average of any type of waveform,
whether or not it is periodic. Equation (2–4) is valid only for periodic waveforms.
DC Value
DEFINITION. The DC (direct “current”) value of a waveform w(t) is given by its time
average, 8w(t)9.
Thus,
1
W dc = lim
T: q T L
T/2
-T/2
w(t) dt
(2–5)
† See Eq. (2–130) for the definition of linearity.
‡ Nonperiodic waveforms are called aperiodic waveforms by some authors.